(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 8009179, 168253] NotebookOptionsPosition[ 7777853, 164599] NotebookOutlinePosition[ 7781286, 164683] CellTagsIndexPosition[ 7781206, 164678] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Last modified: 15, July 2021", "Text", CellTags-> "LastModified",ExpressionUUID->"15342bcd-8b75-4681-b35d-f4ff31bd4d8e"], Cell[CellGroupData[{ Cell["Perfect fluid collapse in circular symmetry - Static Fluid", "Title", CellChangeTimes->{{3.760958989235032*^9, 3.760959042200191*^9}, { 3.7616460032155113`*^9, 3.7616460137341127`*^9}, 3.761646158177384*^9, { 3.773735783579138*^9, 3.773735796169636*^9}},ExpressionUUID->"69660c37-bd53-40a9-85da-\ a4b646e85f2a"], Cell[CellGroupData[{ Cell["Configurations", "Subsubsection", CellChangeTimes->{{3.758357588711343*^9, 3.758357589303277*^9}, { 3.760967268887353*^9, 3.7609672705428553`*^9}, {3.76139869026855*^9, 3.761398693588196*^9}, {3.761400098643458*^9, 3.761400103978828*^9}},ExpressionUUID->"00b21309-951e-49fb-b37f-\ a0fcca324a3f"], Cell[CellGroupData[{ Cell["Packages", "Subsubsubsection", CellChangeTimes->{{3.757921188484313*^9, 3.757921194500388*^9}, { 3.758275040571062*^9, 3.7582750444989567`*^9}, 3.7582750873053493`*^9, { 3.761753620208064*^9, 3.7617536231942577`*^9}},ExpressionUUID->"79abc90e-f71e-41cf-8818-\ dbf77cd9f120"], Cell[TextData[{ "\t\t\tTake the notation package to make sure that any subscripted variables \ are treated as a single variable. \n\t\t\tOtherwise, you cannot use Clear on \ it since Clear only works on symbols and subscripted variables are actually \ functions. 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The StandardForm is needed if using the notation \ package." }], "Text", CellChangeTimes->{{3.7601919042878942`*^9, 3.760192013317361*^9}, { 3.761758056639923*^9, 3.761758063903331*^9}, {3.770032519536647*^9, 3.770032545482421*^9}},ExpressionUUID->"b85a16a0-bb4e-4042-801d-\ 8f75ecd3becd"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ RowBox[{"CurrentValue", "[", RowBox[{ RowBox[{"EvaluationNotebook", "[", "]"}], ",", RowBox[{"{", RowBox[{"InputAutoReplacements", ",", "\"\\""}], "}"}]}], "]"}], ":=", RowBox[{"RowBox", "[", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\<[\>\"", ",", RowBox[{"RowBox", "[", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\<\[SelectionPlaceholder]\>\"", ",", "\"\<]\>\""}], "}"}], "]"}], ",", "\"\<]\>\""}], "}"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.760191741926221*^9, 3.76019174192828*^9}, { 3.760191819083969*^9, 3.7601918687946672`*^9}, {3.760192014215056*^9, 3.760192014952837*^9}, {3.7617580591026497`*^9, 3.7617580702793207`*^9}}, CellLabel-> "In[136]:=",ExpressionUUID->"e00e7753-c115-4002-b7f8-91cc25676cd2"] }, Closed]], Cell[CellGroupData[{ Cell["Palettes and docked buttons", "Subsubsubsection", CellChangeTimes->{{3.762002502126274*^9, 3.762002506162354*^9}, { 3.7673339124027567`*^9, 3.767333919106395*^9}},ExpressionUUID->"0fca8d1d-d5be-4fed-8fde-\ 8e022f046e26"], Cell["\<\ \t\t\tBelow is the full docked buttons code. 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Of course, this assumes \[CapitalLambda] = 0. We \ also need the choice R[r] = r. For general R, need transf R -> R/s instead.\n\ \t\tAlso, note that the PDEs are singular at the lightcone V = \[PlusMinus] 1 \ and at the sonic point V \[PlusMinus] ", Cell[BoxData[ FormBox[ RowBox[{ SqrtBox["\[Kappa]"], "."}], TraditionalForm]],ExpressionUUID-> "f9bda41c-e9f2-466d-b5c8-0f35012d03de"], " This is however not relevant in that we will look for static solutions \ next." }], "Text", CellChangeTimes->{{3.777700840759742*^9, 3.777700949056971*^9}, { 3.777701129059104*^9, 3.77770117729011*^9}, {3.777706382036964*^9, 3.77770640102076*^9}, {3.777706464946413*^9, 3.7777064911696177`*^9}, { 3.777898911134407*^9, 3.777898924479423*^9}, {3.777899228387809*^9, 3.777899228387848*^9}, {3.781362414846664*^9, 3.7813624171575727`*^9}, { 3.781362460499943*^9, 3.781362497259079*^9}, {3.781362642424869*^9, 3.78136268273354*^9}, {3.781363169034169*^9, 3.7813631786619864`*^9}, { 3.7821246199519873`*^9, 3.78212466980053*^9}, {3.782127781255434*^9, 3.7821278007836323`*^9}},ExpressionUUID->"406a27b4-f7fc-4f47-821d-\ 3f5efa406dee"] }, Closed]], Cell[CellGroupData[{ Cell["Static ansatz equations", "Section", CellChangeTimes->{{3.761936497385799*^9, 3.761936503109478*^9}, { 3.761936589115622*^9, 3.761936597363204*^9}, {3.773745369442532*^9, 3.773745371098764*^9}, {3.784015195923874*^9, 3.784015196076852*^9}},ExpressionUUID->"e6d28d04-11f6-4d85-a8ce-\ a00ce2482cd1"], Cell[CellGroupData[{ Cell["Static equations", "Subsection", CellChangeTimes->{{3.761936751908092*^9, 3.761936760176235*^9}, { 3.7737453763461523`*^9, 3.773745378057621*^9}},ExpressionUUID->"f0185fa7-2f2f-45a6-b214-\ 09e6ac2e75cd"], Cell[TextData[{ "\tWe now apply the Static Ansatz. 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\[OpenCurlyDoubleQuote]M\[CloseCurlyDoubleQuote] and hence \ \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] this time! Note that the \ algebraic equation simply relates V, \[CurlyRho] and \[Psi] and in particular \ does not depend on the \[OpenCurlyDoubleQuote]mass\[CloseCurlyDoubleQuote] M \ unlike its 3+1 counterpart.\n\t\tWe shall use two methods from here on and \ check consistency of both results at the end. We can then compare which \ method is numerically more efficient.\n\t\t1) We will keep the equations as \ is and use the alg. eqn for consistency checks/requirements. This keeps the \ equations simpler.\n\t\t2) We will use the algebraic equation to get rid of \ one variable. One equation will thus be redundant modulo the algebraic \ equation." }], "Text", CellChangeTimes->{{3.759143007206635*^9, 3.7591430355978737`*^9}, 3.759143603780449*^9, {3.759229563708259*^9, 3.759229591563796*^9}, { 3.759231638953876*^9, 3.759231642409706*^9}, {3.761936643646139*^9, 3.761936672318465*^9}, {3.7619367844057503`*^9, 3.761936792995001*^9}, { 3.773124099212357*^9, 3.773124172769533*^9}, {3.7731242598938217`*^9, 3.7731242599505167`*^9}, {3.781363609664627*^9, 3.781363612009007*^9}, { 3.781363647385418*^9, 3.781363657375039*^9}},ExpressionUUID->"dd8b6334-651b-42a1-af03-\ 5246e5c2362d"] }, Closed]], Cell[CellGroupData[{ Cell["Consistency check and fluid' s boundary", "Subsection", CellChangeTimes->{{3.761936715529932*^9, 3.761936724552167*^9}},ExpressionUUID->"19399ab0-0ec2-4633-8cb4-\ 8100a4b1d07a"], Cell[TextData[{ "\tThe vanishing of the denominator in \[CurlyRho]\[CloseCurlyQuote] and V\ \[CloseCurlyQuote] indicates the sonic (fluid) boundary. 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We also make sure to remove any \ denominator term in the ODEs.\ \>", "Text", CellChangeTimes->{{3.751384913927721*^9, 3.751384939452557*^9}, { 3.751639088228458*^9, 3.751639088724523*^9}, {3.7518084743652897`*^9, 3.751808518668796*^9}, {3.752333575638295*^9, 3.752333575813356*^9}, { 3.752335686776292*^9, 3.7523357158797827`*^9}, 3.752336267490324*^9, { 3.752488521108032*^9, 3.7524885529389343`*^9}, {3.753194231743269*^9, 3.7531942372632008`*^9}, {3.753903237282598*^9, 3.75390324102495*^9}, { 3.7539042521824713`*^9, 3.753904252785947*^9}, {3.761937392499332*^9, 3.7619373958544197`*^9}, {3.773124389921288*^9, 3.7731243944892893`*^9}, { 3.773138837481094*^9, 3.773138840775717*^9}, {3.773413562342326*^9, 3.773413567054257*^9}, {3.7778930179567127`*^9, 3.777893022311718*^9}, 3.777893524260903*^9, {3.781364425350546*^9, 3.781364428446306*^9}},ExpressionUUID->"ddaf04ed-2452-4d31-9145-\ 4fc5bfa2e72a"], Cell[CellGroupData[{ Cell["Redundancy Check", "Subsubsection", Evaluatable->False, CellChangeTimes->{{3.753904274159281*^9, 3.753904285408024*^9}, { 3.761937409145499*^9, 3.761937409529654*^9}},ExpressionUUID->"d536914a-11a5-4f20-8f73-\ 9dcad0191864"], Cell[TextData[{ "\t\t", StyleBox["Unevaluable.", FontColor->RGBColor[1, 0, 0]], "\n\t\tThe question that we may ask is which of the new variables ODE, g\ \[CloseCurlyQuote] or M\[CloseCurlyQuote], are redundant? It is rather clear \ that \[Psi] can be left out.\n\t \tAfter all, M only depends on a. We can \ then view the M\[CloseCurlyQuote] equation as the one for \ a\[CloseCurlyQuote]. g\[CloseCurlyQuote] is consequentially seen as the \ equation for \[Alpha], which is redundant by Bianchi. \n\t \tInterestingly \ though, the above change of variable introduced a new kind of symmetry. \ Indeed, it turns out that we can now drop the M\[CloseCurlyQuote] equation \ instead. Intuitively, it is because the Bianchi tells us that\n\t \twe really \ need the a\[CloseCurlyQuote] equation, but the g\[CloseCurlyQuote] equation \ also encodes information about a\[CloseCurlyQuote]. The information is thus \ not lost! 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Here,I manually added a minus \ sign, because Mathematica possibly can get it wrong because it does not know \ that R < ", Cell[BoxData[ FormBox[ SubscriptBox["R", "\[Star]"], TraditionalForm]],ExpressionUUID-> "3476c23a-a9f1-408a-9ba1-1bc4389291ca"], ", which then gives the \[OpenCurlyDoubleQuote]wrong sign\ \[CloseCurlyDoubleQuote] inside logarithm.\nRecall, that we also have a gauge \ freedom to rescale the time coordinate. (from above, this comes from the \ integration constant, which with exponentiation gives arbitrary factor in \ from of \[Psi]). This residual gauge\n causes \[Alpha] \[Rule]\[Alpha]/k, and \ thus the same for \[Psi]. 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ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )], Scaled[{0.8, 0.25}], ImageScaled[{0.5, 0.5}], BaseStyle->{FontSize -> Larger}, FormatType->StandardForm]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{ FormBox[ "\"x/\\!\\(\\*SubscriptBox[\\(x\\), \\(\[Star]\\)]\\)\"", TraditionalForm], None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultGraphicsInteraction" -> { "Version" -> 1.2, "TrackMousePosition" -> {True, False}, "Effects" -> { "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> { "freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 1}, {-0.9999999999999982, 0.9999999999999996}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, 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0.385626, 0.209179], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #6}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), Editable->True, InterpretationFunction:>(RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.368417, 0.506779, 0.709798], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.24561133333333335`, 0.3378526666666667, 0.4731986666666667], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.368417`", ",", "0.506779`", ",", "0.709798`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.368417, 0.506779, 0.709798], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.880722`", ",", "0.611041`", ",", "0.142051`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], 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FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.772079`", ",", "0.431554`", ",", "0.102387`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", 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Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #6}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { 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DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.368417, 0.506779, 0.709798], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.880722`", ",", "0.611041`", ",", "0.142051`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", 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FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.3523253333333333, 0.3137493333333333, 0.46756733333333333`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.528488`", ",", "0.470624`", ",", "0.701351`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], 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BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", 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Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #3}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #6}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), Editable->True, InterpretationFunction:>(RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.368417, 0.506779, 0.709798], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.24561133333333335`, 0.3378526666666667, 0.4731986666666667], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.368417`", ",", "0.506779`", ",", "0.709798`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.368417, 0.506779, 0.709798], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.880722`", ",", "0.611041`", ",", "0.142051`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.3523253333333333, 0.3137493333333333, 0.46756733333333333`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.528488`", ",", "0.470624`", ",", "0.701351`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.772079`", ",", "0.431554`", ",", "0.102387`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )], Scaled[{1., 0.5}], ImageScaled[{0, 0.5}], BaseStyle->{FontSize -> Larger}, FormatType->StandardForm]}, AspectRatio->NCache[GoldenRatio^(-1), 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TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.368417, 0.506779, 0.709798], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.24561133333333335`, 0.3378526666666667, 0.4731986666666667], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.368417`", ",", "0.506779`", ",", "0.709798`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; 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Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.3523253333333333, 0.3137493333333333, 0.46756733333333333`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.528488`", ",", "0.470624`", ",", "0.701351`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", 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RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )], Scaled[{1., 0.5}], ImageScaled[{0, 0.5}], BaseStyle->{FontSize -> Larger}, FormatType->StandardForm]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->False, AxesLabel->{ 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Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #3}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #6}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), Editable->True, InterpretationFunction:>(RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.368417, 0.506779, 0.709798], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.24561133333333335`, 0.3378526666666667, 0.4731986666666667], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.368417`", ",", "0.506779`", ",", "0.709798`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.368417, 0.506779, 0.709798], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.880722`", ",", "0.611041`", ",", "0.142051`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.3523253333333333, 0.3137493333333333, 0.46756733333333333`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.528488`", ",", "0.470624`", ",", "0.701351`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.772079`", ",", "0.431554`", ",", "0.102387`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )], Scaled[{1., 0.5}], ImageScaled[{0, 0.5}], BaseStyle->{FontSize -> Larger}, FormatType->StandardForm]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->False, AxesLabel->{ FormBox["\"r\"", TraditionalForm], FormBox["\"M(r)\"", TraditionalForm]}, AxesOrigin->{0, 0}, 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RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.880722`", ",", "0.611041`", ",", "0.142051`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.3523253333333333, 0.3137493333333333, 0.46756733333333333`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.528488`", ",", "0.470624`", ",", "0.701351`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.772079`", ",", "0.431554`", ",", "0.102387`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", 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We will change the paradigm/notation a bit though.\nMetric in \ generalised coordinates is: ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["ds", RowBox[{"2", " "}]], "="}], TraditionalForm]],ExpressionUUID-> "49acce28-e7b9-462a-9137-1940c107b824"], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", SuperscriptBox["\[Alpha]", "2"]}], " ", SuperscriptBox["dt", "2"]}], " ", "+", " ", RowBox[{ SuperscriptBox["a", "2"], SuperscriptBox[ RowBox[{"(", RowBox[{"R", "'"}], ")"}], "2"], " ", SuperscriptBox["dr", "2"]}], "+", RowBox[{ SuperscriptBox["R", "2"], SuperscriptBox["d\[Theta]", "2"]}]}], TraditionalForm]],ExpressionUUID-> "602ccc7b-6185-4147-a876-41d9a70bdf0c"], " =", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", SuperscriptBox["\[Alpha]", "2"]}], " ", SuperscriptBox["dt", "2"]}], " ", "+", " ", RowBox[{ SuperscriptBox["a", "2"], " ", SuperscriptBox["dR", "2"]}], "+", RowBox[{ SuperscriptBox["R", "2"], SuperscriptBox["d\[Theta]", "2"]}]}], TraditionalForm]],ExpressionUUID-> "ca50e441-adb3-4bee-aaa8-34b744b34317"], ". We therefore need to express all the coefficients in terms of R.\nWe \ therefore need to find r = r(R) to plug it into expressions for metric \ coefficients.\nIn this case, the function that needs inverting is ", Cell[BoxData[ FormBox[ SuperscriptBox["R", "2"], TraditionalForm]],ExpressionUUID-> "116c9258-70fd-4c7f-8206-6736b0572ec8"], ": = -\[CapitalLambda] ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SuperscriptBox["r", "2"], "-", SuperscriptBox["r0", "2"]}]}], TraditionalForm]],ExpressionUUID-> "87bacb8a-fbda-4cf4-ab20-09571bc0a30f"], ") + D ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["r", RowBox[{"1", "-", RowBox[{"1", "/", "\[Kappa]"}]}]]}], TraditionalForm]],ExpressionUUID-> "c93d5e8b-1567-470a-9e81-bd277796beba"], " - ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", RowBox[{"1", "-", RowBox[{"1", "/", "\[Kappa]"}]}]], TraditionalForm]],ExpressionUUID-> "800d6e23-1b17-4192-9723-934af22d6ac1"], "). \nFor \[CapitalLambda] = 0, can be done analytically and checked to \ agree with previous solution." }], "Text", CellChangeTimes->{{3.777123098609562*^9, 3.777123422730713*^9}, { 3.77712346787115*^9, 3.777123483542753*^9}, {3.777731317149441*^9, 3.777731319436082*^9}, {3.7777335578863487`*^9, 3.777733694401929*^9}, { 3.777733738161071*^9, 3.777733787737813*^9}, {3.777733827038034*^9, 3.7777339759782753`*^9}, {3.777734542216073*^9, 3.7777345442321568`*^9}, { 3.77789500687954*^9, 3.7778950712221527`*^9}, {3.781424819977089*^9, 3.7814248207947607`*^9}, {3.781441899181651*^9, 3.78144192448748*^9}},ExpressionUUID->"281c0a6b-89f2-4d2f-b6ed-\ ff457231b563"], Cell[CellGroupData[{ Cell["General", "Section", CellChangeTimes->{{3.775801198266048*^9, 3.775801204690735*^9}},ExpressionUUID->"a28a63c6-1990-48ec-80bf-\ 699a42ad69dc"], Cell[TextData[{ StyleBox["Note that in the notes, \[Lambda] and \[CapitalOmega] are the \ dimensionless rescales quantities of \[CapitalLambda] and ", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "0"], TraditionalForm]],ExpressionUUID-> "9c4730c7-b8d5-4981-bf24-47a2baebeaad"], ". See notes. ", StyleBox[" Also, r is a dimensionfull quantity of dimension length, defined \ as ", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["r", "_"], "/", "s"}], TraditionalForm]], FontColor->RGBColor[1, 0, 0],ExpressionUUID-> "e40e00fc-f493-4a92-b806-d48f95cf888d"], ", where s = ", Cell[BoxData[ FormBox[ RowBox[{"Sqrt", "[", SubscriptBox[ OverscriptBox["r", "_"], "0"]}], TraditionalForm]],ExpressionUUID-> "c3cd52ba-3150-4218-a7b4-a8253143d305"], "], where bar means Cataldo quantities, see notes.\nWe shall make this \ general for now (i.e. with rotation). 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RGBColor[0, 0, 1]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[1, 0, 0]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[1, 0, 0]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[0.5, 0, 0.5]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[0.5, 0, 0.5]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #6}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], GrayLevel[0]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], GrayLevel[0]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #7}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), Editable->True, InterpretationFunction:>(RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[ Rational[2, 3], Rational[2, 3], 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.4444444444444444, 0.4444444444444444, 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{ FractionBox["2", "3"], ",", FractionBox["2", "3"], ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[ Rational[2, 3], Rational[2, 3], 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[ Rational[2, 3], Rational[2, 3], 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[1, 0.5, 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.6666666666666666, 0.33333333333333337`, 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"1", ",", "0.5`", ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0.5, 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[1, 0.5, 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0, 1, 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0., 0.6666666666666666, 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0", ",", "1", ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0, 1, 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0, 1, 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0, 0, 1], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0., 0., 0.6666666666666666], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0", ",", "0", ",", "1"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0, 0, 1]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0, 0, 1], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[1, 0, 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.6666666666666666, 0., 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0, 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[1, 0, 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.5, 0, 0.5], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.33333333333333337`, 0., 0.33333333333333337`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.5`", ",", "0", ",", "0.5`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.5, 0, 0.5]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.5, 0, 0.5], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> GrayLevel[0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["GrayLevelColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], GrayLevel[0], Editable -> False, Selectable -> False]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6, ",", #7}], "}"}]}], "]"}]& )], Scaled[{0.8, 0.6}], ImageScaled[{0.5, 0.5}], BaseStyle->{FontSize -> Larger}, FormatType->StandardForm]}, AspectRatio->1, Axes->{True, True}, AxesLabel->{ FormBox["\"x\"", TraditionalForm], FormBox["\"a\"", TraditionalForm]}, AxesOrigin->{0, 0}, DisplayFunction->Identity, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, ImageSize->{196.5, Automatic}, Method->{ "DefaultGraphicsInteraction" -> { "Version" -> 1.2, "TrackMousePosition" -> {True, False}, "Effects" -> { "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> { "freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, "ScalingFunctions" -> None}, PlotRange->{{0, 12}, {0, 9}}, PlotRangeClipping->True, PlotRangePadding->{{0, 0}, {0, 0}}, Ticks->{Automatic, Automatic}], 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GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[1, 0, 0]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[0.5, 0, 0.5]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], RGBColor[0.5, 0, 0.5]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], GrayLevel[0]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], AbsoluteThickness[1.6], GrayLevel[0]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #6}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), Editable->True, InterpretationFunction:>(RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0, 0, 1], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0., 0., 0.6666666666666666], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0", ",", "0", ",", "1"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0, 0, 1]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0, 0, 1], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[1, 0.5, 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.6666666666666666, 0.33333333333333337`, 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"1", ",", "0.5`", ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0.5, 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[1, 0.5, 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0, 1, 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0., 0.6666666666666666, 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0", ",", "1", ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0, 1, 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0, 1, 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[1, 0, 0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.6666666666666666, 0., 0.], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0, 0]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[1, 0, 0], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.5, 0, 0.5], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[0.33333333333333337`, 0., 0.33333333333333337`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.5`", ",", "0", ",", "0.5`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.5, 0, 0.5]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.5, 0, 0.5], Editable -> False, Selectable -> False], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> 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Made \ non-evaluable.\ \>", "Text", CellChangeTimes->{{3.77797348889506*^9, 3.77797350662286*^9}, { 3.781447463772642*^9, 3.781447471502831*^9}},ExpressionUUID->"425036a2-12ac-4f7a-83f8-\ 7b06bc128021"], Cell[CellGroupData[{ Cell["alg eqn to polar radial with only parameter C", "Section", CellChangeTimes->{{3.774185187275897*^9, 3.774185202091655*^9}, { 3.777123703974516*^9, 3.7771237063074217`*^9}},ExpressionUUID->"fdac5a20-4e6d-44e6-9448-\ 62cadeccff4c"], Cell[TextData[{ "First, re-write alg eqn for f(r) (see notes) with only parameter C. This \ obtained from ", Cell[BoxData[ FormBox[ RowBox[{"C", "-", SuperscriptBox["Df", RowBox[{ RowBox[{"(", RowBox[{"\[CapitalGamma]", "-", "1"}], ")"}], "/", "\[CapitalGamma]"}]]}], TraditionalForm]],ExpressionUUID-> "fdb46466-a93f-4103-bb84-4c649c24e309"], Cell[BoxData[ FormBox[ RowBox[{"-", SuperscriptBox["\[CapitalLambda]f", "2"]}], TraditionalForm]], ExpressionUUID->"9e4a4228-67a4-4cbd-aa1d-3f612835d6cb"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["r", "2"], TraditionalForm]],ExpressionUUID-> "9e8f8fde-63e0-4ad2-8c51-1cad3dd8c047"], ". From f\[CloseCurlyQuote]/r = 1 at center, we find the following relation \ for C:" }], "Text", CellChangeTimes->{{3.774177096889629*^9, 3.7741771471705437`*^9}, { 3.777726272464439*^9, 3.777726356158531*^9}},ExpressionUUID->"1bdec868-e4ca-45f1-87e2-\ 78cc09283eca"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"cf0expr", " ", "=", " ", RowBox[{"C", " ", "-", " ", RowBox[{ RowBox[{"\[CapitalGamma]", "/", RowBox[{"(", RowBox[{"1", "-", "\[CapitalGamma]"}], ")"}]}], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "f0"}], " ", "+", " ", RowBox[{"\[CapitalLambda]", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[CapitalGamma]"}], ")"}], "/", "\[CapitalGamma]"}], " ", RowBox[{"f0", "^", "2"}]}]}], ")"}]}]}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.7741680000444717`*^9, 3.7741680432702293`*^9}, { 3.774174456541109*^9, 3.77417446159091*^9}, {3.7741870015368834`*^9, 3.774187016381207*^9}}, CellLabel-> "In[747]:=",ExpressionUUID->"f655acd5-e771-4085-a057-b58b1b539100"], Cell[BoxData[ RowBox[{"C", "-", FractionBox[ RowBox[{"\[CapitalGamma]", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "f0"}], "+", FractionBox[ RowBox[{ SuperscriptBox["f0", "2"], " ", RowBox[{"(", RowBox[{"1", "+", "\[CapitalGamma]"}], ")"}], " ", "\[CapitalLambda]"}], "\[CapitalGamma]"]}], ")"}]}], RowBox[{"1", "-", "\[CapitalGamma]"}]]}]], "Output", CellChangeTimes->{ 3.774168043878619*^9, 3.774172019804495*^9, 3.774174462050193*^9, 3.774175921290401*^9, {3.774176183782992*^9, 3.774176203788772*^9}, { 3.774176412698784*^9, 3.774176436267997*^9}, 3.774176492447069*^9, 3.774176645404173*^9, {3.774176699124817*^9, 3.774176739460401*^9}, 3.774185314025024*^9, 3.774186223888115*^9, {3.774186694667149*^9, 3.774186708944906*^9}, 3.774187017722682*^9, 3.77712229917557*^9, 3.777125790809973*^9, 3.777141158296233*^9, 3.777177869674015*^9, 3.7771784289828568`*^9, 3.777726729510179*^9, 3.777894310578723*^9, 3.77789935286569*^9, 3.777899501014763*^9, 3.781447181504648*^9}, CellLabel-> "Out[747]=",ExpressionUUID->"9211e88b-e06c-4e0c-a516-347914a059ab"] }, Closed]], Cell["\<\ Solve for f0. Note that there are two solutions. Both should exhibit positive \ f(r), as they should as \[Alpha] = f(r). This is readily true for 2nd, \ although first needs C > 0. Also, the first has the correct limit as \[CapitalLambda] -> 0 (coincides \ with what it should be had we taken \[CapitalLambda]=0 to start with), while \ the other is not well defined. Roughly, the first\[CloseCurlyQuote]s solution \ makes the \[CapitalLambda] -> 0 and \[CapitalLambda]=0 coincide, the second \ does not. 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The value of \ A = 0.5 below is motivated from the Fortran code as it is with that value \ that the variables look very similar to the corresponding static sol for \ \[CapitalLambda]=0.\nSo, if we want instead the static sol to be \ \[CapitalLambda] < 0, we need it to be close to its \[CapitalLambda]=0 \ counterpart.\n", StyleBox["Note", FontColor->RGBColor[1, 0, 0]], ": the below is achieved for C which is 1/80 its maximal value. In general, \ the closer C is to 0, the closer both static sol will look alike. This might \ be because this is equivalent to fixing C in the solution and taking \ \[CapitalLambda] -> 0 instead?\nHeuristically, as we saw above, this is also \ because the smaller C, the smaller the total mass M. 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"In[282]:=",ExpressionUUID->"857e6a18-477e-40c7-b5f6-ce720afc3c44"] }, Closed]], Cell[CellGroupData[{ Cell["Numerical solution", "Section", CellChangeTimes->{{3.774185236066221*^9, 3.774185244050281*^9}, { 3.777974985088896*^9, 3.777974985711557*^9}, 3.779618629040471*^9, 3.780733047110858*^9},ExpressionUUID->"c590d9c3-2c3f-4e38-8c55-\ 6490330cb012"], Cell[TextData[{ "Numerically solve for f. Note that for ", Cell[BoxData[ FormBox[ SuperscriptBox["c", "2"], TraditionalForm]],ExpressionUUID-> "e7ded3f1-1c1b-4562-9abd-5802ecba4df7"], "=1/2 as in here, get 3rd order poly eqn and could thus find analytical \ solution." }], "Text", CellChangeTimes->{{3.7741771827610407`*^9, 3.774177219120552*^9}, { 3.777128967011868*^9, 3.777128970951404*^9}},ExpressionUUID->"14f92e4c-a778-4516-907f-\ eb308c616ec9"], Cell[BoxData[ RowBox[{ RowBox[{"rToRTransformationEqnInvertibleSolRuleF", "[", RowBox[{"R_", ",", "sscale_", ",", "\[Mu]scale_"}], "]"}], " ", ":=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"NSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"rToRTransformationEqnF", "[", RowBox[{"R", ",", "r"}], "]"}], " ", "\[Equal]", " ", "0"}], " ", "/.", " ", RowBox[{"fullRuleF", "[", RowBox[{"sscale", ",", "\[Mu]scale"}], "]"}]}], ",", "r", ",", " ", "Reals"}], "]"}], ",", " ", RowBox[{"R", " ", ">", " ", "0"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.774170730651545*^9, 3.774170757689899*^9}, { 3.774170814258068*^9, 3.774170815863853*^9}, {3.774170888055372*^9, 3.774170921292441*^9}, {3.774174511679319*^9, 3.7741745123058357`*^9}, { 3.7741872726238947`*^9, 3.774187297196485*^9}, {3.777124342933028*^9, 3.777124361022649*^9}, {3.7771244076074457`*^9, 3.777124411808696*^9}, 3.777124516903782*^9, 3.77712466860316*^9, {3.7771256004704533`*^9, 3.777125606842494*^9}, {3.7771290386470327`*^9, 3.7771290565315943`*^9}, { 3.7771291551131783`*^9, 3.777129163680395*^9}, {3.777894848973422*^9, 3.777894852819272*^9}, {3.777970324304924*^9, 3.777970340244162*^9}, { 3.777975017355064*^9, 3.777975069317926*^9}, {3.777975213299247*^9, 3.777975228344694*^9}, 3.7779925037733803`*^9, {3.777992645536354*^9, 3.77799264947847*^9}, {3.7779947143119507`*^9, 3.777994728056698*^9}, { 3.7779948754884787`*^9, 3.7779948793602324`*^9}, {3.7807246573935633`*^9, 3.780724662486803*^9}, {3.780727690917964*^9, 3.7807276953992767`*^9}, { 3.781439171099032*^9, 3.7814391835502043`*^9}}, CellLabel-> "In[287]:=",ExpressionUUID->"cf0a0a1b-9d66-43e9-a7bf-38e02c13105f"], Cell["\<\ One easily checks that 2nd solution produces negative valued r and should \ thus be discarded. The first solution is discontinuous at the shown value of R. The discontinuity for 1st solution depends of course on the parameters. 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FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle 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EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], 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FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.772079`", ",", "0.431554`", ",", "0.102387`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", 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RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.880722`", ",", "0.611041`", ",", "0.142051`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.560181`", ",", "0.691569`", ",", "0.194885`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.922526, 0.385626, 0.209179]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.922526, 0.385626, 0.209179], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.528488, 0.470624, 0.701351], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.3523253333333333, 0.3137493333333333, 0.46756733333333333`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.528488`", ",", "0.470624`", ",", "0.701351`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.772079`", ",", "0.431554`", ",", "0.102387`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.772079, 0.431554, 0.102387]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.772079, 0.431554, 0.102387], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4, ",", #5, ",", #6}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", 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RGBColor[0.528488, 0.470624, 0.701351]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.528488, 0.470624, 0.701351], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.772079, 0.431554, 0.102387], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.5147193333333333, 0.28770266666666666`, 0.06825800000000001], FrameTicks -> 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We end back to our static solution, since V=0 \ will inevitably cause A(\[Tau])=constant! So, unlike for the Yang-Mills case \ (see Bizon and Ovchinnikov), if \[Chi](r) is a static solution, then \ \[Chi](r/s(t)) is a solution, ", StyleBox["only if", FontColor->RGBColor[1, 0, 0]], " s(t) = constant." }], "Text", CellChangeTimes->{{3.77769827999851*^9, 3.777698308622014*^9}, { 3.777699666297991*^9, 3.777699830621099*^9}, {3.7778961614906816`*^9, 3.7778961908415537`*^9}, {3.777896280760353*^9, 3.777896328453486*^9}, { 3.777898766914818*^9, 3.777898770578583*^9}},ExpressionUUID->"ae926098-ec29-471d-b2f5-\ 33c1d909f72f"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Find contracting static solution", "Chapter", CellChangeTimes->{{3.778231846285042*^9, 3.7782318523969097`*^9}},ExpressionUUID->"c0ca7f41-a733-44a9-9bba-\ e07871b3e683"], Cell[TextData[{ "From above, we needed V = 0. The idea instead is to look for perturbation \ such that v ~ \[Epsilon] and f = ", Cell[BoxData[ FormBox[ SubscriptBox["f", "0"], TraditionalForm]],ExpressionUUID-> "b9ea0e91-f1fb-4d84-9b7b-e19861056abc"], "(R) + ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Epsilon]", "2"], SubscriptBox["f", "2"]}], TraditionalForm]],ExpressionUUID-> "19c0bee4-e73a-45dc-9de1-6f19a6ae6789"], "(t,R) where ", Cell[BoxData[ FormBox[ OverscriptBox["s", "."], TraditionalForm]],ExpressionUUID-> "5d4dfae9-83ed-404b-871d-7ea1b8b53ae3"], "~\[Epsilon] and obviously f stands for all variables other than v.\nThis is \ formalised by introducing slow time: s = s(\[Epsilon] t). Then, for \ \[Epsilon]=0, we retrieve leading order static solution with scale s(0) which \ is some constant.\nFrom above, leading order is satisfied and leading \ correction order gives a relation between perturbed parts.\nWe can find a \ solution like that instead. In other words, compared to above, the static \ part is only leading order and velocity v appears at the next order only, \ whereas above, f has no correction and v is of order 1." }], "Text", CellChangeTimes->{{3.781448894218561*^9, 3.781448962208767*^9}, { 3.781449175769924*^9, 3.781449415867989*^9}, {3.781449565599626*^9, 3.781449581384715*^9}, {3.781449843120779*^9, 3.781449933219984*^9}},ExpressionUUID->"b807e33a-e70a-4b57-a14e-\ d281a70b8af5"], Cell[TextData[{ "Recall that we have a static solution ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "static"], TraditionalForm]],ExpressionUUID-> "096d335a-3d86-47a7-a1aa-f6d7ed728b6a"], "(r) for both \[CapitalLambda]=0 and \[CapitalLambda]\[NotEqual]0. And both \ can be written (for specific \[Epsilon] in latter case) as functions of R. \n\ We now try to find a solution of the form f(t,r) = ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "static"], TraditionalForm]],ExpressionUUID-> "a867bfda-aefd-4baa-9314-25b2f0bb80c1"], "(R/s(t)), where s(t) needs to be found. Here f is g, M or \[CurlyRho].\nOn \ the other hand, V gets updated to V(t,r). So, need to equations. If we plug \ this ansatz in the PDE, get a seemingly overdetermined system, but for now, \ wont be bothered by that.\nWe will choose the ", Cell[BoxData[ FormBox[ SubscriptBox["M", RowBox[{",", "r"}]], TraditionalForm]],ExpressionUUID-> "ce477f03-1868-48d0-83c5-2f02533c207f"], " and ", Cell[BoxData[ FormBox[ SubscriptBox["M", RowBox[{",", "\[Tau]"}]], TraditionalForm]],ExpressionUUID-> "a24a7e28-b652-4d60-b984-30bd4ea6e4c4"], " equations. \nIn the above section, we tried doing that with \ \[CapitalLambda]=0 in the equations, but found that s needs to be a constant. \ Now, assume \[CapitalLambda]\[NotEqual]0 in the PDE system.\nFor the static \ solution, we will first try the one from \[CapitalLambda]=0, as we have a \ closed form result. 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So, next section, we will see if we \ can make the static solution simplier, such as by assuming that ", Cell[BoxData[ FormBox[ SubscriptBox["r", "0"], TraditionalForm]],ExpressionUUID-> "f8c8e820-56ea-441f-bd6f-0e1b779bdf74"], " is small." }], "Text", CellChangeTimes->{{3.779622179745163*^9, 3.779622219687993*^9}},ExpressionUUID->"a02770ff-b9a5-4cc8-a99f-\ 0f41d92f03cb"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"fromgprToprRule", " ", "=", " ", RowBox[{"R", " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", "x"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MxcontractingEqn", " ", "=", " ", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"M", "[", RowBox[{"\[Tau]", ",", "x"}], "]"}], ",", "x"}], "]"}], " ", "-", " ", "metricMatterEqnSolMx"}], " ", "\[Equal]", " ", "0"}], " ", "/.", " ", "\[Psi]M\[CurlyRho]\[CapitalLambda]Not0\[Epsilon]2AnsatzRule"}], "/.", " ", "\[Gamma]TocsRule"}], " ", "/.", " ", "fromgprToprRule"}], " ", "/.", " ", "\[Epsilon]Rule"}], " ", "/.", " ", RowBox[{"\[Epsilon]", " ", "\[Rule]", " ", RowBox[{"1", "/", RowBox[{"Sqrt", "[", "2", "]"}]}]}]}], ")"}], "]"}], " ", "//", " ", "Simplify"}]}], "\[IndentingNewLine]", RowBox[{"M\[Tau]contractingEqn", " ", "=", " ", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"M", "[", RowBox[{"\[Tau]", ",", "x"}], "]"}], ",", "\[Tau]"}], "]"}], " ", "-", " ", "metricMatterEqnSolM\[Tau]"}], " ", "\[Equal]", " ", "0"}], " ", "/.", " ", "\[Psi]M\[CurlyRho]\[CapitalLambda]Not0\[Epsilon]2AnsatzRule"}], "/.", " ", "\[Gamma]TocsRule"}], " ", "/.", " ", "fromgprToprRule"}], ")"}], "]"}], " ", "//", " ", "Simplify"}]}]}], "Input", Evaluatable->False, CellChangeTimes->{{3.778233628469618*^9, 3.778233629052843*^9}, { 3.778233660522078*^9, 3.7782336662916813`*^9}, {3.778234054376129*^9, 3.778234074687043*^9}, {3.77823414362237*^9, 3.7782341746841717`*^9}, { 3.77823452403491*^9, 3.778234525000985*^9}, {3.77824006173354*^9, 3.778240063866558*^9}, 3.778240376214027*^9, {3.778240631337521*^9, 3.778240641682947*^9}, {3.7782412609698772`*^9, 3.7782412836574497`*^9}, { 3.78066329244182*^9, 3.780663299322006*^9}},ExpressionUUID->"86ba7008-26db-4bc4-99e6-\ 855cba8296a8"], Cell[BoxData["$Aborted"], "Output", CellChangeTimes->{ 3.778233597327121*^9, 3.778233674027869*^9, {3.778234033512743*^9, 3.778234075208497*^9}, 3.778234177031624*^9, 3.77823452610992*^9, 3.778240048353457*^9, 3.7782403688011837`*^9}, CellLabel-> "Out[392]=",ExpressionUUID->"d3a1ce9f-e547-423c-9b13-6f396459ce84"], Cell[BoxData["$Aborted"], "Output", CellChangeTimes->{ 3.778233597327121*^9, 3.778233674027869*^9, {3.778234033512743*^9, 3.778234075208497*^9}, 3.778234177031624*^9, 3.77823452610992*^9, 3.778240048353457*^9, 3.778240386096089*^9}, CellLabel-> "Out[393]=",ExpressionUUID->"a8aa0775-b91b-47e6-b541-4b5a11af78f2"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Try to simplify obtained expression for M assuming r0 small", "Section", CellChangeTimes->{{3.77831602012208*^9, 3.778316028562311*^9}, { 3.778322943744885*^9, 3.7783229658167152`*^9}},ExpressionUUID->"3c442c9c-ea6e-4eac-bfdc-\ 6596bc2f0896"], Cell["For now, just try to simplify the expression for M.", "Text", CellChangeTimes->{{3.778318623296195*^9, 3.778318637224393*^9}},ExpressionUUID->"f7aeefc3-a0a7-4644-81a6-\ 34c57f0d0d02"], Cell[TextData[{ "Note: could try to only consider r0 small. Note that the rule for M can be \ written as (- ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["C", RowBox[{"2", "/", "3"}]]}], TraditionalForm]],ExpressionUUID-> "2020cb95-82cf-4eeb-8787-61e98c598410"], " + AC / ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"C", "^", RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}]}], " ", "+", " ", "B"}], ")"}], "^", "2"}], " ", "+", " ", "B"}], ")"}], ")"}], "^", "2"}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"3", "^", RowBox[{"(", RowBox[{"4", "/", "3"}], ")"}]}], " ", RowBox[{"r0", "^", "2"}], " ", RowBox[{"C", "^", RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}]}]}], ")"}]}], " ", "-", " ", RowBox[{ RowBox[{"R", "^", "2"}], RowBox[{"\[CapitalLambda]", "/", SuperscriptBox["s", "2"]}]}]}], TraditionalForm]],ExpressionUUID-> "8741d009-6f41-4a38-8168-c50d4bae0dc9"], ".Now, A ~ ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "5"], TraditionalForm]],ExpressionUUID-> "eab2a995-70da-461a-9afa-734369246952"], ", while B is either ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"~", "r0"}], ")"}], "2"], TraditionalForm]],ExpressionUUID-> "63c4452d-8813-4ee6-bf44-11410d97ebf0"], "(general) or ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"~", "r0"}], ")"}], "3"], TraditionalForm]],ExpressionUUID-> "911e2653-6ab8-436b-b676-7d5ab57a7bfd"], " if R small.\nIn both cases ", Cell[BoxData[ FormBox[ SuperscriptBox["C", RowBox[{"2", "/", "3"}]], TraditionalForm]],ExpressionUUID-> "e791966e-0ea1-4f0a-b6f9-22ee4231c56a"], " ~ -B and in fact ", Cell[BoxData[ FormBox[ SuperscriptBox["C", RowBox[{"2", "/", "3"}]], TraditionalForm]],ExpressionUUID-> "06f1b170-8ab2-43e8-b7b2-b9f9917e9dba"], "+B ~ 0(r0). To find the factor for r0, can re-write (say for the case where \ R=0), ", Cell[BoxData[ FormBox[ SuperscriptBox["C", RowBox[{"2", "/", "3"}]], TraditionalForm]],ExpressionUUID-> "d87c7727-6eb9-46de-bd5e-f7f9126fbe0c"], "+B = ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "3"], TraditionalForm]],ExpressionUUID-> "25c89ae3-cb32-4957-8d17-8edea41639d6"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"sqrt", "(", RowBox[{"r0", " ", OverscriptBox["A", "~"]}]}]}], TraditionalForm]],ExpressionUUID-> "1e1f8981-89d9-4d4c-9912-93bdbb785e57"], "^2 - ", Cell[BoxData[ FormBox[ OverscriptBox["B", "~"], TraditionalForm]],ExpressionUUID-> "478b78a6-c09e-46fa-aad6-fa70ca9d04c6"], "^3) + r0 ", Cell[BoxData[ FormBox[ OverscriptBox["A", "~"], TraditionalForm]],ExpressionUUID-> "d02bfae8-e81f-4919-808a-ee7da8b88850"], ")^(2/3)+ ", Cell[BoxData[ FormBox[ OverscriptBox["B", "~"], TraditionalForm]],ExpressionUUID-> "1f3f9dbb-1543-4a35-992b-76c0693f0310"], "), where the tilde mean that terms are of order 1 and are appropriately \ rescaled versions\nof A and B. In both case, this means that for the \ expression for M, need only consider the \[OpenCurlyDoubleQuote]middle\ \[CloseCurlyDoubleQuote] term as it is the one which, when divided with \ denominator, is of order 1, the others are of some power of r0." }], "Text", CellChangeTimes->{{3.778315543049232*^9, 3.778315995217901*^9}},ExpressionUUID->"f9295932-23b5-4e90-85ac-\ 45c04a9e9f8b"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"AtmpRule", " ", "=", RowBox[{"myA", " ", "\[Rule]", " ", RowBox[{"9", " ", RowBox[{"r0", "^", "5"}], " ", RowBox[{"\[CapitalLambda]", "^", "2"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"r0", " ", "\[CapitalLambda]"}]}], ")"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"BtmpRule", " ", "=", " ", RowBox[{"myB", " ", "\[Rule]", " ", RowBox[{ RowBox[{"3", "^", RowBox[{"(", RowBox[{"1", "/", "3"}], ")"}]}], " ", RowBox[{"r0", "^", "2"}], " ", "\[CapitalLambda]", " ", RowBox[{"(", RowBox[{ RowBox[{"r0", " ", RowBox[{"(", RowBox[{"2", "+", 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same as M\[CapitalLambda]Not0\[Epsilon]2AnsatzRule, \ although Mathematica has trouble seeing that.\ \>", "Text", CellChangeTimes->{{3.778316043825251*^9, 3.7783160681611843`*^9}},ExpressionUUID->"a9ed0331-7fb1-4068-8cab-\ fe42f00d3647"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{ RowBox[{ RowBox[{"myM", " ", "/.", " ", "ABCRule"}], " ", "/.", " ", RowBox[{ RowBox[{"s", "[", "\[Tau]", "]"}], " ", "\[Rule]", " ", "1"}]}], " ", "/.", " ", RowBox[{"fullRuleF", "[", "100000", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"R", ",", "0", ",", "0.01"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0.002"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0.2"}], "}"}]}], "}"}]}]}], "]"}]], "Input",\ CellChangeTimes->{{3.77831892899916*^9, 3.7783189596553173`*^9}, { 3.77831900362469*^9, 3.778319006925975*^9}}, CellLabel-> 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Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.778316084682968*^9, {3.778318929331846*^9, 3.778318960041267*^9}, { 3.7783190040941067`*^9, 3.778319007332375*^9}, 3.785587919757469*^9}, CellLabel-> "Out[312]=",ExpressionUUID->"48d6b255-349b-43fc-9312-3ae4c6d8cf2f"] }, Closed]], Cell["With the above, comment, can take:", "Text", CellChangeTimes->{{3.7783165968878927`*^9, 3.778316605576202*^9}},ExpressionUUID->"98973944-2b27-4d64-a2ea-\ f4e93c5dde85"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r0smallM", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"myA", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"-", "myB"}], ")"}], "^", RowBox[{"(", RowBox[{"3", "/", "2"}], ")"}]}], "/", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"myC", "^", RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}]}], "+", "myB"}], ")"}], "^", "2"}]}]}], ")"}], "^", "2"}], " ", "/", RowBox[{"(", " ", RowBox[{ RowBox[{"3", "^", RowBox[{"(", RowBox[{"4", "/", "3"}], ")"}]}], " ", "myB", " ", RowBox[{"r0", "^", "2"}]}], ")"}]}], "-", RowBox[{"\[CapitalLambda]", " ", RowBox[{"R", "^", "2"}]}]}], " ", "/.", " ", "CtmpRule"}]}]], "Input", CellChangeTimes->{{3.778316114819665*^9, 3.778316165694634*^9}, { 3.778316211918976*^9, 3.7783162326841793`*^9}, 3.778316548053199*^9, { 3.778316587340728*^9, 3.7783166362318153`*^9}}, CellLabel-> "In[313]:=",ExpressionUUID->"559bbed6-b994-41e7-a926-d0667fb1b7cb"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["myA", "2"], " ", SuperscriptBox["myB", "2"]}], RowBox[{"3", " ", SuperscriptBox["3", RowBox[{"1", "/", "3"}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"myB", "+", SuperscriptBox[ RowBox[{"(", RowBox[{"myA", "+", SqrtBox[ RowBox[{ SuperscriptBox["myA", "2"], "-", SuperscriptBox["myB", "3"]}]]}], ")"}], RowBox[{"2", "/", "3"}]]}], ")"}], "4"], " ", SuperscriptBox["r0", "2"]}]]}], "-", RowBox[{ SuperscriptBox["R", "2"], " ", "\[CapitalLambda]"}]}]], "Output", CellChangeTimes->{ 3.7783165482676163`*^9, 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3.780663358210895*^9}}, CellLabel-> "In[314]:=",ExpressionUUID->"7204d025-dc2a-4cc9-84d3-ce614fcf43fa"], Cell[BoxData[ GraphicsBox[{{}, {}, {}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 0.002}, {-1, 0.2}}, PlotRangeClipping->True, PlotRangePadding->{{0, 0}, {0, 0}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.7783161827873783`*^9, {3.778316235391964*^9, 3.778316243173943*^9}, { 3.778316610697822*^9, 3.7783166392387753`*^9}, {3.778316715769305*^9, 3.7783167308471537`*^9}, {3.778319011011003*^9, 3.778319015022551*^9}, 3.785587919804349*^9}, CellLabel-> "Out[314]=",ExpressionUUID->"5182a5f1-e18c-4a61-823c-496aa856e673"] }, Closed]], Cell["\<\ The above is not very good numerically speaking. Mathematica has visible \ trouble with it. One can see that as beyond some R, solution becomes \ seemingly complex and also if the scale factor is not very small, then M is \ not about -1 at center. This is due to the fact that there are lots of r0 in the above expression, \ with \[OpenCurlyDoubleQuote]big\[CloseCurlyDoubleQuote] powers, so numerical \ computation is error prone.\ \>", "Text", CellChangeTimes->{{3.778319022731838*^9, 3.778319134953053*^9}, { 3.778319241933923*^9, 3.7783192926047*^9}},ExpressionUUID->"18918e23-63dc-4c1d-886a-41a453c1773f"], Cell[TextData[{ "Can simplify the expression further by \[OpenCurlyDoubleQuote]extracting\ \[CloseCurlyDoubleQuote] and cancelling out the r0 as much as possible. The \ corresponding variables of order 1 are denoted by tilde. The most tricky part \ is the ", Cell[BoxData[ FormBox[ SuperscriptBox["C", RowBox[{"2", "/", "3"}]], TraditionalForm]],ExpressionUUID-> "8bff1b01-6da7-4104-b5d3-2114e5a81c3b"], "+B where we use that:" }], "Text", CellChangeTimes->{{3.778318266107395*^9, 3.778318326098526*^9}, { 3.7783186900854597`*^9, 3.778318706646801*^9}, {3.778319125986021*^9, 3.778319127993661*^9}},ExpressionUUID->"7058543f-4a9b-427d-adea-\ eb2a0239f79e"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"C2o3pBtmp", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{ RowBox[{"r0", "^", "10"}], " ", RowBox[{"Atilde", "^", "2"}]}], " ", "-", " ", RowBox[{ RowBox[{"r0", "^", "6"}], " ", RowBox[{"Btilde", "^", "3"}]}]}], "]"}], "+", RowBox[{ RowBox[{"r0", "^", "5"}], " ", "Atilde"}]}], ")"}], "^", RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}]}], "+", " ", RowBox[{ RowBox[{"r0", "^", "2"}], " ", "Btilde"}]}]}]], "Input", CellChangeTimes->{{3.778312262522801*^9, 3.778312318879128*^9}, { 3.778312463054242*^9, 3.7783124767622957`*^9}, {3.778313534877255*^9, 3.778313557471973*^9}, {3.7783170207188883`*^9, 3.778317024899632*^9}, { 3.778317095044217*^9, 3.778317102280788*^9}, {3.778317417425976*^9, 3.778317417534924*^9}, {3.778318343005336*^9, 3.7783183497695093`*^9}, { 3.778318384435821*^9, 3.778318414687572*^9}, {3.7783187110018997`*^9, 3.7783187217664948`*^9}}, CellLabel-> "In[315]:=",ExpressionUUID->"aa41242e-194f-4787-ae3f-8ba526bc9089"], Cell[BoxData[ RowBox[{ RowBox[{"Btilde", " ", SuperscriptBox["r0", "2"]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"Atilde", " ", SuperscriptBox["r0", "5"]}], "+", SqrtBox[ RowBox[{ RowBox[{ RowBox[{"-", SuperscriptBox["Btilde", "3"]}], " ", SuperscriptBox["r0", "6"]}], "+", RowBox[{ SuperscriptBox["Atilde", "2"], " ", SuperscriptBox["r0", "10"]}]}]]}], ")"}], RowBox[{"2", "/", "3"}]]}]], "Output", CellChangeTimes->{ 3.778312319798029*^9, 3.778312477095756*^9, 3.778313557794427*^9, 3.778317026074342*^9, {3.7783170978003473`*^9, 3.7783171025127573`*^9}, 3.778317418067421*^9, {3.778318330778433*^9, 3.778318351836687*^9}, { 3.7783183989769983`*^9, 3.778318415179776*^9}, 3.7783187221662493`*^9, 3.785587919817236*^9}, CellLabel-> "Out[315]=",ExpressionUUID->"ee03c66f-884b-49ed-8941-4ed8fc992693"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Series", "[", RowBox[{"C2o3pBtmp", ",", RowBox[{"{", RowBox[{"r0", ",", "0", ",", "4"}], "}"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"Btilde", " ", "<", "0"}], ",", " ", RowBox[{"r0", " ", ">", " ", "0"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7783124850343637`*^9, 3.778312491025691*^9}, { 3.7783125293364143`*^9, 3.778312532872512*^9}, {3.778313561412169*^9, 3.7783135662075157`*^9}, {3.778317031133973*^9, 3.778317031202799*^9}, { 3.778317143465891*^9, 3.778317149743615*^9}, {3.7783183552590027`*^9, 3.778318361721184*^9}, {3.778318401898872*^9, 3.7783184019757977`*^9}, { 3.77831847764041*^9, 3.7783184833497887`*^9}, {3.778318727432795*^9, 3.778318728206361*^9}}, CellLabel-> "In[316]:=",ExpressionUUID->"dd0fc065-6758-49cc-9e59-6bca9e3ce97d"], Cell[BoxData[ InterpretationBox[ RowBox[{ FractionBox[ RowBox[{"2", " ", "Atilde", " ", SuperscriptBox["r0", "4"]}], RowBox[{"3", " ", SqrtBox[ RowBox[{"-", "Btilde"}]]}]], "+", InterpretationBox[ SuperscriptBox[ RowBox[{"O", "[", "r0", "]"}], "5"], SeriesData[$CellContext`r0, 0, {}, 4, 5, 1], Editable->False]}], SeriesData[$CellContext`r0, 0, {(Rational[2, 3] $CellContext`Atilde) (-$CellContext`Btilde)^ Rational[-1, 2]}, 4, 5, 1], Editable->False]], "Output", CellChangeTimes->{ 3.778312533580935*^9, 3.778313566589533*^9, {3.778317028542062*^9, 3.778317031573674*^9}, 3.7783171046205463`*^9, 3.778317150055113*^9, 3.7783174214786*^9, {3.778318356486663*^9, 3.77831836218878*^9}, { 3.77831840366558*^9, 3.778318430963895*^9}, 3.778318483798861*^9, 3.778318735327985*^9, 3.7855879264203577`*^9}, CellLabel-> "Out[316]=",ExpressionUUID->"37b2ebb7-486e-4d77-a216-3ac25bde1f44"] }, Closed]], Cell["\<\ The above makes it look like it is of order 4. The thing to be careful \ though, is that near center R=0, Btilde is actually of order r0!!! Away from \ center, it is of order 1 though, and that is what we took.\ \>", "Text", CellChangeTimes->{{3.778318666855009*^9, 3.778318685663033*^9}, { 3.7783187413967657`*^9, 3.77831878559616*^9}},ExpressionUUID->"8b0a3eb2-7932-4796-987b-\ 467c34d0a681"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"AtildetmpRule", " ", "=", " ", RowBox[{"myAtilde", " ", "\[Rule]", " ", RowBox[{"9", " ", RowBox[{"\[CapitalLambda]", "^", "2"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"r0", " ", "\[CapitalLambda]"}]}], ")"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"BtildetmpRule", " ", "=", " ", RowBox[{"myBtilde", " ", "\[Rule]", " ", RowBox[{ RowBox[{"3", "^", RowBox[{"(", RowBox[{"1", "/", "3"}], ")"}]}], " ", "\[CapitalLambda]", " ", RowBox[{"(", RowBox[{ RowBox[{"r0", " ", RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", " ", "r0", " ", "\[CapitalLambda]"}]}], ")"}]}], "-", RowBox[{ RowBox[{"R", "^", "2"}], "/", RowBox[{ RowBox[{"s", "[", "\[Tau]", "]"}], "^", "2"}]}]}], ")"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"CtildetmpRule", " ", "=", " ", RowBox[{"myCtilde", " ", "\[Rule]", " ", RowBox[{"myAtilde", " ", "+", " ", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"myAtilde", "^", "2"}], "-", RowBox[{"myBtilde", "^", "3"}]}], "]"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"ABCtildeRule", " ", "=", " ", RowBox[{"{", RowBox[{"AtildetmpRule", ",", " ", "BtildetmpRule", ",", " ", RowBox[{"(", RowBox[{ RowBox[{"CtildetmpRule", " ", "/.", " ", "AtildetmpRule"}], " ", "/.", " ", "BtildetmpRule"}], ")"}]}], "}"}]}], " "}], "\[IndentingNewLine]", RowBox[{"r0smallMV2", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"myBtilde", "^", "4"}]}], " ", RowBox[{ RowBox[{"3", "^", RowBox[{"(", RowBox[{"8", "/", "3"}], ")"}]}], " ", "/", " ", RowBox[{"(", RowBox[{ RowBox[{"r0", "^", "4"}], " ", RowBox[{"2", "^", "4"}], " ", RowBox[{"myAtilde", "^", "2"}]}], ")"}]}]}], " ", "-", RowBox[{ RowBox[{"R", "^", "2"}], " ", RowBox[{"\[CapitalLambda]", "/", RowBox[{"s", "[", "\[Tau]", "]"}]}]}]}], " ", "/.", "CtmpRule"}], " ", "//", "Simplify"}]}]}], "Input", CellChangeTimes->{{3.778317998053244*^9, 3.778318150743085*^9}, { 3.7783181818675137`*^9, 3.778318185839768*^9}, {3.780663370705799*^9, 3.78066337398628*^9}}, CellLabel-> "In[317]:=",ExpressionUUID->"9e2d0756-d750-46a9-bef7-fbe59b4ad2f1"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"myAtilde", "\[Rule]", RowBox[{"9", " ", SuperscriptBox["\[CapitalLambda]", "2"], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"r0", " ", "\[CapitalLambda]"}]}], ")"}]}]}], ",", RowBox[{"myBtilde", "\[Rule]", RowBox[{ SuperscriptBox["3", RowBox[{"1", "/", "3"}]], " ", "\[CapitalLambda]", " ", RowBox[{"(", RowBox[{ RowBox[{"r0", " ", RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", " ", "r0", " ", "\[CapitalLambda]"}]}], ")"}]}], "-", FractionBox[ SuperscriptBox["R", "2"], SuperscriptBox[ RowBox[{"s", "[", "\[Tau]", "]"}], "2"]]}], ")"}]}]}], ",", RowBox[{"myCtilde", "\[Rule]", RowBox[{ RowBox[{"9", " ", SuperscriptBox["\[CapitalLambda]", "2"], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"r0", " ", "\[CapitalLambda]"}]}], ")"}]}], "+", SqrtBox[ RowBox[{ RowBox[{"81", " ", SuperscriptBox["\[CapitalLambda]", "4"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", RowBox[{"r0", " ", "\[CapitalLambda]"}]}], ")"}], "2"]}], "-", RowBox[{"3", " ", SuperscriptBox["\[CapitalLambda]", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"r0", " ", RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", " ", "r0", " ", "\[CapitalLambda]"}]}], ")"}]}], "-", FractionBox[ SuperscriptBox["R", "2"], SuperscriptBox[ RowBox[{"s", "[", "\[Tau]", "]"}], "2"]]}], ")"}], "3"]}]}]]}]}]}], "}"}]], "Output", CellChangeTimes->{3.7783181514551353`*^9, 3.7783181863990297`*^9, 3.7855879264776278`*^9}, CellLabel-> "Out[320]=",ExpressionUUID->"d0f4e1c4-691a-4dd5-ade7-e04e347e3e18"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"9", " ", SuperscriptBox["3", RowBox[{"2", "/", "3"}]], " ", SuperscriptBox["myBtilde", "4"]}], RowBox[{"16", " ", SuperscriptBox["myAtilde", "2"], " ", SuperscriptBox["r0", "4"]}]]}], "-", FractionBox[ RowBox[{ SuperscriptBox["R", "2"], " ", "\[CapitalLambda]"}], RowBox[{"s", "[", "\[Tau]", "]"}]]}]], "Output", CellChangeTimes->{3.7783181514551353`*^9, 3.7783181863990297`*^9, 3.785587926479493*^9}, CellLabel-> "Out[321]=",ExpressionUUID->"2b3063a7-b567-4551-9e4b-3aa253a259e4"] }, Closed]], Cell["\<\ As mentioned above, near R=0, Btilde is actually of order r0, which cancels \ with the one from the denominator. The result is what is expected. Note again \ that this only holds in the r0\[Rule]0 limit.\ \>", "Text", CellChangeTimes->{{3.778318834209577*^9, 3.7783188657775784`*^9}, { 3.778319149472249*^9, 3.77831915488056*^9}, {3.778319200230673*^9, 3.778319215358809*^9}},ExpressionUUID->"f39dfc7f-4461-442e-8245-\ 8f240ca1a247"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{"r0smallMV2", " ", "/.", " ", "ABCtildeRule"}], " ", "/.", " ", RowBox[{"R", " ", "\[Rule]", " ", "0"}]}], ",", RowBox[{"r0", " ", "\[Rule]", " ", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.7783188940596933`*^9, 3.778318906456449*^9}}, CellLabel-> "In[322]:=",ExpressionUUID->"7b660c32-b097-4410-a9c9-a612f03ac000"], Cell[BoxData[ RowBox[{"-", "1"}]], "Output", CellChangeTimes->{{3.7783188902536077`*^9, 3.778318906726969*^9}, 3.7855879265160303`*^9}, CellLabel-> "Out[322]=",ExpressionUUID->"a0855062-9ba5-4fcf-aff5-2e2fb101df39"] }, Closed]], Cell[TextData[{ "This approximation should only be up to order ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "2"], TraditionalForm]],ExpressionUUID-> "94feda27-5e6f-4cc0-987d-0ce8bda99947"], "or ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "3"], TraditionalForm]],ExpressionUUID-> "21448610-3027-4f7d-8c75-4faf4228d88a"], ", because the simplification came from ignoring (", Cell[BoxData[ FormBox[ SuperscriptBox["C", RowBox[{"2", "/", "3"}]], TraditionalForm]],ExpressionUUID-> "6e6c4ba5-48c4-4c6d-be9c-a8a1432ba887"], "+B)^2 / ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"r0", "^", "2"}], SuperscriptBox["C", RowBox[{"2", "/", "3"}]]}]}], TraditionalForm]],ExpressionUUID-> "e3518674-c828-4b46-8f26-c109c8cb3cc7"], "). Now, ", Cell[BoxData[ FormBox[ SuperscriptBox["C", RowBox[{"2", "/", "3"}]], TraditionalForm]],ExpressionUUID-> "b0a8e930-06d8-49c9-aaa5-01f40baf4f8c"], "~B and B is of order ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "2"], TraditionalForm]],ExpressionUUID-> "bcf043ea-ad73-4b2c-9e07-1af87ff9da9a"], " far from center, and ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "3"], TraditionalForm]],ExpressionUUID-> "d1bbd13e-79f6-42f5-ae7c-b451d1e749ce"], "close to it.\nSo, the above approximation is a bit better closer to the \ center than away and that means that one expects things to break down \ \[OpenCurlyDoubleQuote]far away\[CloseCurlyDoubleQuote] from the center \ first, as one could naively expect I think." }], "Text", CellChangeTimes->{{3.778319578242455*^9, 3.7783197923337917`*^9}, { 3.779621222386014*^9, 3.779621236761861*^9}},ExpressionUUID->"f0dac9dc-b6c2-49d2-a65d-\ 7459805fe3e3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{ RowBox[{ RowBox[{"r0smallMV2", " ", "/.", " ", "ABCtildeRule"}], " ", "/.", " ", RowBox[{ RowBox[{"s", "[", "\[Tau]", "]"}], " ", "\[Rule]", " ", "1"}]}], " ", "/.", " ", RowBox[{"fullRuleF", "[", "1000", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"R", ",", "0", ",", "0.02"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0.02"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0.2"}], "}"}]}], "}"}]}]}], "]"}]], "Input",\ CellChangeTimes->{{3.7783181893748903`*^9, 3.7783181897513447`*^9}, { 3.778318220128981*^9, 3.7783182207425947`*^9}, {3.778318913618003*^9, 3.778318917128734*^9}, {3.7783191599725027`*^9, 3.7783191915546618`*^9}, { 3.7806633773371058`*^9, 3.780663377610756*^9}}, CellLabel-> "In[323]:=",ExpressionUUID->"d02a124c-b728-47e6-aee1-31ebfcf5dbf4"], Cell[BoxData[ GraphicsBox[{{}, {}, {}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 0.02}, {-1, 0.2}}, PlotRangeClipping->True, PlotRangePadding->{{0, 0}, {0, 0}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.778318191226308*^9, 3.7783182356178226`*^9, {3.7783189141667624`*^9, 3.7783189175141068`*^9}, {3.778319160540739*^9, 3.778319191895632*^9}, 3.785587926563548*^9}, CellLabel-> "Out[323]=",ExpressionUUID->"16fcf31c-03a2-4e33-a605-df253545d187"] }, Closed]], Cell["\<\ This shows what was clear from the get-go: as you take r0\[Rule]0, you \ recover to static \[CapitalLambda]=0 solution, just as you would if you took \ \[CapitalLambda]\[Rule]0. 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We ill neglect ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "\[Epsilon]"], TraditionalForm]],ExpressionUUID-> "b708650f-dc6b-483a-ba0c-60b80a51c091"], " term in the expression for R. 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Closed]], Cell[TextData[{ "As a side note, one ", StyleBox["CANNOT", FontColor->RGBColor[1, 0, 0]], " simplify the above expression for M any further, because if R is small, \ then r0 cannot be neglected in ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r0", "^", "2"}], "-", SuperscriptBox["R", "2"]}], TraditionalForm]],ExpressionUUID-> "26b3948f-5a53-4f2c-94e7-e6754b355f31"], "/\[CapitalLambda]. Also, the power of those terms is negative and thus \ those terms actually get big as R is small and r0 small!" }], "Text", CellChangeTimes->{{3.778321533124954*^9, 3.7783216403789997`*^9}},ExpressionUUID->"0edad64e-8acf-4754-96bc-\ 6c334916fb78"], Cell["Here, we compare this result, with the previous one:", "Text", CellChangeTimes->{{3.778321665288887*^9, 3.7783216827263117`*^9}},ExpressionUUID->"b356ee2d-0b4f-45b7-938c-\ 71158fc9c2ac"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"tmpM", " ", "/.", " ", RowBox[{"fullRuleF", "[", "1000", "]"}]}], ",", RowBox[{"{", RowBox[{"R", ",", "0", ",", "0.2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0.02"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0.1"}], "}"}]}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{ RowBox[{ RowBox[{"r0smallMV2", " ", "/.", " ", "ABCtildeRule"}], " ", "/.", " ", RowBox[{ RowBox[{"s", "[", "\[Tau]", "]"}], " ", "\[Rule]", " ", "1"}]}], " ", "/.", " ", RowBox[{"fullRuleF", "[", "1000", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"R", ",", "0", ",", "0.02"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0.02"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0.2"}], "}"}]}], "}"}]}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.7783216863485622`*^9, 3.778321710281375*^9}, { 3.778321747779195*^9, 3.778321815074582*^9}, {3.77832227040256*^9, 3.778322296311214*^9}, {3.77832250469135*^9, 3.778322506832678*^9}, { 3.7783225960965548`*^9, 3.7783225987651978`*^9}}, CellLabel-> "In[329]:=",ExpressionUUID->"980d02e6-af14-4384-ac65-caa0e757996d"], Cell[BoxData[ GraphicsBox[{{{}, {}, {}}, {{}, {}, {}}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 0.02}, {-1, 0.1}}, PlotRangeClipping->True, PlotRangePadding->{{0, 0}, {0, 0}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.778321710650494*^9, {3.778321749227251*^9, 3.77832181594699*^9}, { 3.778322280297297*^9, 3.778322297397327*^9}, 3.778322507705593*^9, 3.778322600221745*^9, 3.785587927138605*^9}, CellLabel-> "Out[329]=",ExpressionUUID->"6bfb593e-dfac-4367-87fa-b4711a892a5f"] }, Closed]], Cell[TextData[{ StyleBox["So, this does not work at all", FontColor->RGBColor[1, 0, 0]], "! A posteriori, I think that this is because if r is small, but not quite \ r0, then r^(1-\[Epsilon]) is big and so neglecting such terms is not correct. " }], "Text", CellChangeTimes->{{3.778322611283058*^9, 3.778322682945964*^9}},ExpressionUUID->"1ba67959-bf37-4c8d-8e7d-\ 1b02f050c516"] }, Closed]], Cell[CellGroupData[{ Cell["Try ignoring powers of 2 instead", "Subsection", CellChangeTimes->{{3.778321415280669*^9, 3.778321445376534*^9}, { 3.778322852195936*^9, 3.778322860372027*^9}},ExpressionUUID->"6c17f2c8-1147-44cb-82a4-\ 60036d66eb68"], Cell[TextData[{ "This is hopefully a bit more general. We ill neglect ", Cell[BoxData[ FormBox[ SuperscriptBox["r0", "2"], TraditionalForm]],ExpressionUUID-> "c9e2785a-b62c-4fc9-9caa-de7d414f7c10"], " term in the expression for R. 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RowBox[{"(", RowBox[{"\[Mu]", "-", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[CapitalOmega]", "2"]}], ")"}]}]}], ")"}]}]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "+", "\[Kappa]"}]]}]], "Output", CellChangeTimes->{{3.778302103521575*^9, 3.778302113422439*^9}, 3.778302178693829*^9, 3.778320551587517*^9, 3.778322794779255*^9, 3.7783975981878357`*^9, 3.785587927721352*^9}, CellLabel-> "Out[333]=",ExpressionUUID->"066c2b9c-26d5-458b-b3b6-ec6236123911"] }, Closed]], Cell[TextData[{ "As a side note, one ", StyleBox["CANNOT", FontColor->RGBColor[1, 0, 0]], " simplify the above expression for M any further, because if R is small, \ then r0 cannot be neglected in ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r0", "^", "2"}], "-", SuperscriptBox["R", "2"]}], TraditionalForm]],ExpressionUUID-> "b0186f77-611f-4159-af41-02da16514a60"], "/\[CapitalLambda]. Also, the power of those terms is negative and thus \ those terms actually get big as R is small and r0 small!" }], "Text", CellChangeTimes->{{3.778321533124954*^9, 3.7783216403789997`*^9}},ExpressionUUID->"5b1b5651-9317-49e0-b284-\ 97ce0173f7eb"], Cell["Here, we compare this result, with the previous one:", "Text", CellChangeTimes->{{3.778321665288887*^9, 3.7783216827263117`*^9}},ExpressionUUID->"c1a0684e-95e6-402f-a5ca-\ 33c6c4dd84d8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"MOfRr0smallexpr", " ", "/.", " ", RowBox[{"fullRuleF", "[", "1000", "]"}]}], ",", RowBox[{"{", RowBox[{"R", ",", "0", ",", "0.2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0.02"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0.1"}], "}"}]}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{ RowBox[{ RowBox[{"r0smallMV2", " ", "/.", " ", "ABCtildeRule"}], " ", "/.", " ", RowBox[{ RowBox[{"s", "[", "\[Tau]", "]"}], " ", "\[Rule]", " ", "1"}]}], " ", "/.", " ", RowBox[{"fullRuleF", "[", "1000", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"R", ",", "0", ",", "0.02"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0.02"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0.2"}], "}"}]}], "}"}]}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.7783216863485622`*^9, 3.778321710281375*^9}, { 3.778321747779195*^9, 3.778321815074582*^9}, {3.77832227040256*^9, 3.778322296311214*^9}, {3.77832250469135*^9, 3.778322506832678*^9}, { 3.7783225960965548`*^9, 3.7783225987651978`*^9}, {3.7783228068894253`*^9, 3.778322809289432*^9}, 3.778397602773943*^9}, CellLabel-> "In[334]:=",ExpressionUUID->"bd8928f5-3b67-4671-a06c-d5c046ef36d2"], Cell[BoxData[ GraphicsBox[{{{}, {}, {}}, {{}, {}, {}}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, Charting`ScaledFrameTicks[{Identity, Identity}]}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 0.02}, {-1, 0.1}}, PlotRangeClipping->True, PlotRangePadding->{{0, 0}, {0, 0}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.778321710650494*^9, {3.778321749227251*^9, 3.77832181594699*^9}, { 3.778322280297297*^9, 3.778322297397327*^9}, 3.778322507705593*^9, 3.778322600221745*^9, {3.778322797527027*^9, 3.778322809700934*^9}, { 3.778393849631761*^9, 3.778393861202342*^9}, 3.778397603674765*^9, 3.7855879277837267`*^9}, CellLabel-> "Out[334]=",ExpressionUUID->"a25a8545-7da0-4938-8a08-06fa524784f6"] }, Closed]], Cell[TextData[{ StyleBox["This works now", FontColor->RGBColor[1, 0, 0]], "! A posteriori, I think that this is because if r is small, but not quite \ r0, then r^(1-\[Epsilon]) is big and so neglecting such terms is not correct. 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From ", Cell[BoxData[ FormBox[ SubscriptBox["a", RowBox[{",", "t"}]], TraditionalForm]],ExpressionUUID-> "e476746d-e105-4ed3-b496-93baf8d4bb54"], " ~O(\[Epsilon]) equation, we then need \[Alpha] to be corrected by a factor \ ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Epsilon]", "2"], TraditionalForm]],ExpressionUUID-> "9d172e14-bff7-4e0f-9624-ec99e43149db"], " as well. The other equations ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["a", RowBox[{",", "r"}]]}], TraditionalForm]],ExpressionUUID-> "06ba3013-9bbf-4e55-99bb-d9107d719e28"], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", RowBox[{",", "r"}]], TraditionalForm]],ExpressionUUID-> "ef1467d5-5aa7-46e1-87d7-2024bf8870c3"], ") are consistent with this.\nThis can be see also in the Y-balance law, \ where leading correction in the velocity appears only at second order. So, if \ any variable had an order ~ 0(\[Epsilon]), it would not \ \[OpenCurlyDoubleQuote]feel\[CloseCurlyDoubleQuote] the correction in the \ velocity, which is not consistent, \nin the sense that if we switch back off \ the velocity, we should end up with static 0(1) term.\n", StyleBox["Note that it is important that the density \[Rho] is replaced by \ its dimensionless version so that the \[OpenCurlyDoubleQuote]upgrade\ \[CloseCurlyDoubleQuote] s -> s(t) is done correctly.", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.778671370242998*^9, 3.778671387965481*^9}, { 3.778841633657692*^9, 3.778842021272421*^9}, {3.778843384943308*^9, 3.778843416703915*^9}, {3.781506912086034*^9, 3.781507055962738*^9}, { 3.78150946664835*^9, 3.781509536852942*^9}, {3.7816106656318808`*^9, 3.781610704799671*^9}, 3.7875862007807503`*^9, {3.7875871769956503`*^9, 3.787587196499086*^9}, {3.787643602090623*^9, 3.787643602486347*^9}, { 3.816344437194283*^9, 3.816344437397468*^9}, {3.8163445438822737`*^9, 3.816344549297491*^9}},ExpressionUUID->"34e5456a-1a3a-43bb-aef4-\ dcaaf90e634b"], Cell[TextData[{ "Try an ansatz ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"t", ",", "R"}], ")"}], "=", " ", RowBox[{"=", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["f", "\[Star]"], "(", "x", ")"}], " ", "+", " ", RowBox[{ RowBox[{"s", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"], RowBox[{ SubscriptBox["f", "p"], "(", "x", ")"}]}]}], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["f", "\[Star]"], "(", "x", ")"}], " ", "+", " ", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], " ", RowBox[{"S", "'"}], SuperscriptBox[ RowBox[{"(", "T", ")"}], "2"], RowBox[{ SubscriptBox["f", "p"], "(", "x", ")"}]}]}]}]}]}], TraditionalForm]], ExpressionUUID->"745bf6f4-abb5-4a9d-b88b-e3877cab25b5"], ". ", StyleBox["Note that we made the perturbation depend on x up to that factor ", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"~", "s"}], "'"}], RowBox[{"(", "t", ")"}]}], TraditionalForm]], FontColor->RGBColor[1, 0, 0],ExpressionUUID-> "dc787786-1744-4ee5-b653-83556ec7772a"], StyleBox["!", FontColor->RGBColor[1, 0, 0]], "\nWe made the slow time ansatz s(t) = S(\[Epsilon] (t-t*)) = S(T). ", StyleBox["We take t* = 0 here for simplicity.", FontColor->RGBColor[1, 0, 0]], "\nYou can try the more general case of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"t", ",", "R"}], ")"}], "=", " ", RowBox[{"=", " ", RowBox[{ RowBox[{ SubscriptBox["f", "\[Star]"], "(", "x", ")"}], " ", "+", " ", RowBox[{ RowBox[{"s", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"], RowBox[{ SubscriptBox["f", "p"], "(", RowBox[{"t", ",", "R"}], ")"}]}]}]}]}], TraditionalForm]], ExpressionUUID->"f26e6557-19e7-4737-83a8-b15385e3ce77"], " instead, but you will see that the above requirement is actually very \ natural. Note that out of consistency, we will need ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "p"], "(", RowBox[{"t", ",", "R"}], ")"}], " ", "~", " ", RowBox[{ SubscriptBox["f", "p"], "(", "x", ")"}]}], TraditionalForm]], ExpressionUUID->"a8b22ece-8657-4966-97c4-7c9b80a2063d"], " at some point. In the above, we claim\nthat the factor is simply ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"~", "s"}], "'"}], RowBox[{"(", "t", ")"}]}], TraditionalForm]],ExpressionUUID-> "9781c858-e1d1-4c63-a3bf-9f3aaf53330d"], ". This factor is anyway needed for the slow time ansatz for consistency. We \ then don\[CloseCurlyQuote]t need an extra term that is a function of R or \ s(t), but just x=R/s.\nA bit more in details as to why we think this works is \ because, this ansatz is the only one for v that solves the leading equation \ of X-cons law and for the Y-cons law, we see that the equations naturally \ want to be re-arranged into two terms,\none proportional ", Cell[BoxData[ FormBox[ RowBox[{"S", " ", RowBox[{"S", "''"}]}], TraditionalForm]],ExpressionUUID-> "df4f26d5-c640-4212-88a7-51daf3ea2e45"], " and the other to ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"S", "'"}], ")"}], "2"], TraditionalForm]],ExpressionUUID-> "8719ec67-b7bf-499b-a6fd-f997949d6c14"], ". This ansatz is then the only one that makes this happen. So, we are \ attempting to extract some consistent ODE out of those PDEs.\nRecall after \ all that we have somehow to have an ODE for s(t)!! Logically, this should be \ obtained from the Y-cons law, since it is the law that governs how the fluid\ \[CloseCurlyQuote]s velocity moves and should therefore determine s(t).\nOf \ course, to obtain ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Epsilon]", "2"], TraditionalForm]],ExpressionUUID-> "9cf16e62-aadb-43d3-bece-cfb0b33e6b7b"], ", we used ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"s", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"]}], TraditionalForm]],ExpressionUUID-> "ebc92720-45b4-49db-9732-cf7755c21476"], " technically, it could also have been \ s\[CloseCurlyQuote]\[CloseCurlyQuote](t) etc. We here choose the nicest \ option. See later...\nAnyway, the above is really an ", StyleBox["ansatz then!\nNote:", FontColor->RGBColor[1, 0, 0]], " this ansatz is pretty general, in the sense that the function of t should \ be either ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "92afbdeb-79f3-4f93-9067-9beb48342ba5"], " or ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "89533145-0b71-49b0-9e18-f171f2745b13"], ". The reason is that we want the Y law to be an ODE for ", Cell[BoxData[ FormBox[ RowBox[{"s", "(", "t", ")"}], TraditionalForm]],ExpressionUUID-> "4f4d1355-e45f-4b87-b0f0-d32cd1ac3135"], ". Note that Y law only depends on a and \[CurlyRho], so this amounts to \ choosing it for these two.\nIf one of the expansion at least is not of the \ two form given, then we will separately get an ODE for a and \[CurlyRho], and \ I don\[CloseCurlyQuote]t think that the solutions will be consistent with the \ other equations.\nNow, we are then left with three choices: either both are \ ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "fafb7f13-19cb-4ff0-838f-a10632471397"], " or both ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "05a1632e-f3de-4d12-8f11-19f5ad9c9f10"], " or one ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "172e74d1-42c3-49dd-abaf-60c543291f6e"], " while the other is ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "bf6df3b7-947b-4bec-8656-3cf246186c22"], ". 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This comment therefore also holds \ for all other perturbation variables." }], "Text", CellChangeTimes->{{3.781611157676025*^9, 3.781611192292918*^9}},ExpressionUUID->"697ad9b8-e19c-41ea-b84a-\ d8b276882c45"], Cell[TextData[{ StyleBox["Careful:", FontColor->RGBColor[1, 0, 0]], " In the following, I am using this very definition of ", Cell[BoxData[ FormBox[ SubscriptBox["v", "p"], TraditionalForm]],ExpressionUUID-> "f231bb98-530e-4022-8b31-0857dea18c24"], " (whenever I manually give expressions in terms of this function). 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Since we want perturbation to be 0 (in particular finite) at center, we \ require ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "4147066c-6e18-436b-a81b-581144683c8b"], " ~ ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]],ExpressionUUID-> "c6c93a89-6bcb-45c0-85e6-a02092b15a9c"], " at least. This is in fact the case for the rule for ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "1d204283-eea9-4f74-bb5e-386d92e5461a"], " found in the next section. This is also clear in that every other term in \ perturbed Y is ~ x.\nIn fact, it is clear that ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "8373ff14-9b87-4f7c-a129-97bfae8f6ee4"], " ~ ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]],ExpressionUUID-> "96ff69ab-609c-4c2f-a578-db2182b5d36a"], " because the terms in ", Cell[BoxData[ FormBox[ FractionBox[ OverscriptBox["s", ".."], SuperscriptBox["R", "2"]], TraditionalForm]],ExpressionUUID-> "4b2f9bb6-71a6-4010-9fa3-8ff20ae27a2b"], " and ", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox[ OverscriptBox["s", "."], "2"], RowBox[{ SuperscriptBox["R", "2"], "s"}]], TraditionalForm]],ExpressionUUID-> "749441fa-02db-4555-922f-2bbbd62a08dd"], " go like ~ x, so, the perturbed terms had also better be of the same \ magnitude!\nThe fact that ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "p"], TraditionalForm]],ExpressionUUID-> "c08aecda-18a2-458e-9635-55800be1b156"], " does not explicitly appear can be understood in the sense that we can view \ the above as a PDE for ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "f7193315-597b-41f2-a9e8-9ce82e5415a1"], " as well as S\[CloseCurlyQuote]. See later. \nThe later however only \ depends on M (via integration), which only depends on a. Similarly, note that \ the PDE for ", Cell[BoxData[ FormBox[ SubscriptBox["a", "p"], TraditionalForm]],ExpressionUUID-> "286f1309-0904-4a26-9614-99c6bde8f48b"], " is also independent on ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "p"], TraditionalForm]],ExpressionUUID-> "5aefd6b0-04c4-4741-886d-19643eb1a265"], ". Thus, we can say that the ODEs for ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "13ce3c12-4ec3-4fee-8204-bff183db1c8b"], " and ", Cell[BoxData[ FormBox[ SubscriptBox["a", "p"], TraditionalForm]],ExpressionUUID-> "6f22e5a0-20bf-4524-9e39-06cd8aad286a"], " decouple from ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "p"], TraditionalForm]],ExpressionUUID-> "f1aabcaa-d6f3-4a75-a593-6cb126e0360a"], "." }], "Text", CellChangeTimes->{{3.778931592658066*^9, 3.7789316714698553`*^9}, { 3.7789319089541597`*^9, 3.778931930792968*^9}, {3.778931981199916*^9, 3.778932019846526*^9}, {3.778932470231718*^9, 3.7789324970715218`*^9}, { 3.778933983864443*^9, 3.778934084691881*^9}, {3.7791940636954*^9, 3.779194159693342*^9}, {3.779194191924055*^9, 3.779194244258987*^9}, { 3.781604390178389*^9, 3.7816043972826223`*^9}, {3.787650727575358*^9, 3.7876507699264193`*^9}},ExpressionUUID->"3fda5460-7880-4722-86e7-\ bc3439c1dec8"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Re-write Y-law as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"S", " ", OverscriptBox["S", ".."]}], " ", "+", " ", RowBox[{"f", " ", SuperscriptBox[ OverscriptBox["S", "."], "2"]}]}], " ", "=", " ", "0"}], TraditionalForm]],ExpressionUUID->"b5737356-8fdf-4abe-a5be-137d1f3ef160"] }], "Subsection", CellChangeTimes->{{3.787652580699758*^9, 3.78765260418167*^9}},ExpressionUUID->"c6a2270a-1488-422f-b247-\ bed270f20376"], Cell[TextData[{ "Currently, it is written as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"g", " ", OverscriptBox[ RowBox[{"S", " ", "S"}], ".."]}], " ", "+", " ", RowBox[{"h", " ", SuperscriptBox[ OverscriptBox["S", "."], "2"]}]}], " ", "=", " ", "0"}], TraditionalForm]],ExpressionUUID->"fa497797-2945-44d3-8711-d1592239595c"], ". 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mass is still zero, even for the perturbed \ solution (up to ", Cell[BoxData[ FormBox[ RowBox[{"0", RowBox[{"(", SuperscriptBox["\[Delta]", "2"]}]}], TraditionalForm]],ExpressionUUID-> "7ee10a23-45b4-44ea-912c-19b10a4bead9"], ")). Now, since M ~ \[Integral] X dr = \[Integral] r\[Rho] (1+ \ (1+\[Kappa])", Cell[BoxData[ FormBox[ SuperscriptBox["v", "2"], TraditionalForm]],ExpressionUUID-> "7a4da1ab-94b7-4792-b071-3e8275abd77e"], " + ....). So, the next order contribution where the velocity is \ contributing is of order ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"~", "0"}], RowBox[{"(", SuperscriptBox["\[Epsilon]", "2"]}]}], TraditionalForm]],ExpressionUUID-> "be9ee55b-54e3-476b-aee8-222687f583c1"], "). \nIt will be proportional to ", Cell[BoxData[ FormBox[ SuperscriptBox["v", "2"], TraditionalForm]],ExpressionUUID-> "ef7b2942-12de-493a-98fd-927e89a1adba"], " ~ ", Cell[BoxData[ FormBox[ SuperscriptBox[ OverscriptBox["s", "."], "2"], TraditionalForm]],ExpressionUUID-> "b436eeaa-518f-4a42-ab83-9231bd2b39b2"], ". This also means that in the perturbation, the metric variables had also \ better be of that order. \nThis will actually not disappear by itself and we \ note to balance that by a perturbation of the density term: \[Rho] \[Rule] \ \[Rho] + ", Cell[BoxData[ FormBox[ SubscriptBox["\[Rho]", "pert"], TraditionalForm]],ExpressionUUID-> "3e4be505-38ec-4b97-b45b-54d064d4fef7"], ". From the above, we then require to leading order that ", Cell[BoxData[ FormBox[ SubscriptBox["\[Rho]", "pert"], TraditionalForm]],ExpressionUUID-> "ac141747-291d-41da-bf9a-06cfeeb2fe34"], " ~ ", Cell[BoxData[ FormBox[ RowBox[{"O", "(", SuperscriptBox["\[Epsilon]", "2"]}], TraditionalForm]],ExpressionUUID-> "e36fe789-d9b9-4ad0-a2dc-8acc41945c20"], ").\nNow, since M gets modified by a factor of ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Epsilon]", "2"], TraditionalForm]],ExpressionUUID-> "fb51ebf5-8cae-4d51-baac-61f42dcc00ec"], ", we then also need a(t,r) to also be modified by a factor ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Epsilon]", "2"], TraditionalForm]],ExpressionUUID-> "d4a83f3c-c2e8-48cf-88f6-fb40f5ef055c"], ". From ", Cell[BoxData[ FormBox[ SubscriptBox["a", RowBox[{",", "t"}]], TraditionalForm]],ExpressionUUID-> "60bc9b71-3a55-494f-b66d-a9f9951270f2"], " ~O(\[Epsilon]) equation, we then need \[Alpha] to be corrected by a factor \ ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Epsilon]", "2"], TraditionalForm]],ExpressionUUID-> "49f8d333-fabe-4dee-9871-d3d57664b215"], " as well. The other equations ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["a", RowBox[{",", "r"}]]}], TraditionalForm]],ExpressionUUID-> "27179b10-10b5-4d4e-baf1-fb3bffea5071"], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", RowBox[{",", "r"}]], TraditionalForm]],ExpressionUUID-> "780bfce9-78e4-439a-8045-b75864933afa"], ") are consistent with this.\nThis can be see also in the Y-balance law, \ where leading correction in the velocity appears only at second order. So, if \ any variable had an order ~ 0(\[Epsilon]), it would not \ \[OpenCurlyDoubleQuote]feel\[CloseCurlyDoubleQuote] the correction in the \ velocity, which is not consistent, \nin the sense that if we switch back off \ the velocity, we should end up with static 0(1) term.\n", StyleBox["Note that it is important that the density \[Rho] is replaced by \ its dimensionless version so that the \[OpenCurlyDoubleQuote]upgrade\ \[CloseCurlyDoubleQuote] s -> s(t) is done correctly.", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.778671370242998*^9, 3.778671387965481*^9}, { 3.778841633657692*^9, 3.778842021272421*^9}, {3.778843384943308*^9, 3.778843416703915*^9}, {3.781506912086034*^9, 3.781507055962738*^9}, { 3.78150946664835*^9, 3.781509536852942*^9}, {3.7816106656318808`*^9, 3.781610704799671*^9}, 3.7875862007807503`*^9, {3.7875871769956503`*^9, 3.787587196499086*^9}, {3.787643602090623*^9, 3.787643602486347*^9}},ExpressionUUID->"b6486e03-8653-42d5-b906-\ 632f80571e86"], Cell[TextData[{ "Try an ansatz ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"t", ",", "R"}], ")"}], "=", " ", RowBox[{"=", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["f", "\[Star]"], "(", "x", ")"}], " ", "+", " ", RowBox[{ RowBox[{"s", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"], RowBox[{ SubscriptBox["f", "p"], "(", "x", ")"}]}]}], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["f", "\[Star]"], "(", "x", ")"}], " ", "+", " ", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], " ", RowBox[{"S", "'"}], SuperscriptBox[ RowBox[{"(", "T", ")"}], "2"], RowBox[{ SubscriptBox["f", "p"], "(", "x", ")"}]}]}]}]}]}], TraditionalForm]], ExpressionUUID->"beda765d-7e7d-4043-9ef2-b5a22b1dd520"], ". ", StyleBox["Note that we made the perturbation depend on x up to that factor ", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"~", "s"}], "'"}], RowBox[{"(", "t", ")"}]}], TraditionalForm]], FontColor->RGBColor[1, 0, 0],ExpressionUUID-> "81a19a18-5af7-4791-93b2-8feebb79bb3a"], StyleBox["!", FontColor->RGBColor[1, 0, 0]], "\nWe made the slow time ansatz s(t) = S(\[Epsilon] (t-t*)) = S(T). ", StyleBox["We take t* = 0 here for simplicity.", FontColor->RGBColor[1, 0, 0]], "\nYou can try the more general case of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"t", ",", "R"}], ")"}], "=", " ", RowBox[{"=", " ", RowBox[{ RowBox[{ SubscriptBox["f", "\[Star]"], "(", "x", ")"}], " ", "+", " ", RowBox[{ RowBox[{"s", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"], RowBox[{ SubscriptBox["f", "p"], "(", RowBox[{"t", ",", "R"}], ")"}]}]}]}]}], TraditionalForm]], ExpressionUUID->"929c10ac-a209-4728-9a38-a3fa0c92a397"], " instead, but you will see that the above requirement is actually very \ natural. Note that out of consistency, we will need ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "p"], "(", RowBox[{"t", ",", "R"}], ")"}], " ", "~", " ", RowBox[{ SubscriptBox["f", "p"], "(", "x", ")"}]}], TraditionalForm]], ExpressionUUID->"4b02efe9-79fc-4baa-8883-b3045f626a72"], " at some point. In the above, we claim\nthat the factor is simply ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"~", "s"}], "'"}], RowBox[{"(", "t", ")"}]}], TraditionalForm]],ExpressionUUID-> "a96c05c9-7835-4ba1-95ec-4637a21f7126"], ". This factor is anyway needed for the slow time ansatz for consistency. We \ then don\[CloseCurlyQuote]t need an extra term that is a function of R or \ s(t), but just x=R/s.\nA bit more in details as to why we think this works is \ because, this ansatz is the only one for v that solves the leading equation \ of X-cons law and for the Y-cons law, we see that the equations naturally \ want to be re-arranged into two terms,\none proportional ", Cell[BoxData[ FormBox[ RowBox[{"S", " ", RowBox[{"S", "''"}]}], TraditionalForm]],ExpressionUUID-> "d5145c5b-7e52-476d-a2ca-0069fc5208eb"], " and the other to ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"S", "'"}], ")"}], "2"], TraditionalForm]],ExpressionUUID-> "b3781c53-cd63-4a80-91ee-b7542f15fbf6"], ". This ansatz is then the only one that makes this happen. So, we are \ attempting to extract some consistent ODE out of those PDEs.\nRecall after \ all that we have somehow to have an ODE for s(t)!! Logically, this should be \ obtained from the Y-cons law, since it is the law that governs how the fluid\ \[CloseCurlyQuote]s velocity moves and should therefore determine s(t).\nOf \ course, to obtain ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Epsilon]", "2"], TraditionalForm]],ExpressionUUID-> "7d5b796d-b9b1-42d5-8012-5a37ccbd3db5"], ", we used ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"s", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"]}], TraditionalForm]],ExpressionUUID-> "c2bb87b0-7ca5-4db0-9834-7a246796b3ce"], " technically, it could also have been \ s\[CloseCurlyQuote]\[CloseCurlyQuote](t) etc. We here choose the nicest \ option. See later...\nAnyway, the above is really an ", StyleBox["ansatz then!\nNote:", FontColor->RGBColor[1, 0, 0]], " this ansatz is pretty general, in the sense that the function of t should \ be either ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "ef7de9a5-2eda-4e2f-9f9b-ad64bab967d6"], " or ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "c4a83fb8-c345-4419-a665-cfa3a8c40704"], ". The reason is that we want the Y law to be an ODE for ", Cell[BoxData[ FormBox[ RowBox[{"s", "(", "t", ")"}], TraditionalForm]],ExpressionUUID-> "ef77426a-1bb5-4915-8cb3-c095b197459d"], ". Note that Y law only depends on a and \[CurlyRho], so this amounts to \ choosing it for these two.\nIf one of the expansion at least is not of the \ two form given, then we will separately get an ODE for a and \[CurlyRho], and \ I don\[CloseCurlyQuote]t think that the solutions will be consistent with the \ other equations.\nNow, we are then left with three choices: either both are \ ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "03b9fe3d-2d7a-4229-be3b-1b3365f8dc14"], " or both ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "a8459733-dd70-4cd6-b3bd-f2594fa047ba"], " or one ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "c1fbea50-7fed-4a16-b380-a9c818c719e8"], " while the other is ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "ea7b9d44-f169-4a9c-9468-37c10f57978e"], ". This actually makes all that much difference in the end, because the ODE \ for s(t) is: ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"f", SuperscriptBox[ OverscriptBox["s", "."], "2"], "s"}], " ", "+", " ", RowBox[{"g", OverscriptBox["s", ".."]}]}], " ", "=", " ", "0"}]}], TraditionalForm]],ExpressionUUID->"a5f41f18-c49f-45e7-8887-45958d66d850"], ". We then require f = constant * g.\nNow, whether a (and \[CurlyRho]) will \ contribute to f or g depends on whether they have a ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", SuperscriptBox[ OverscriptBox["s", "."], "2"]}], TraditionalForm]],ExpressionUUID-> "cd792800-0091-4dca-9332-4eeef924ab8e"], " or ", Cell[BoxData[ FormBox[ RowBox[{"~", " ", OverscriptBox["s", ".."]}], TraditionalForm]],ExpressionUUID-> "630e1646-299d-4050-8a21-4be8784324e8"], " term. However, from f = constant * g, this then does not really matter, \ since all we need is to redefine the ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "2"], "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "fc3d2567-4757-4f56-8aee-f1391f008d12"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[CurlyRho]", "2"], "(", "x", ")"}], TraditionalForm]], ExpressionUUID->"eb2e4390-da89-417c-8f29-fadad43a2ee9"], ".\n\nThe perturbation is motivated by the fact that the cosmological \ constant only appears in the source term and we have a term that \ schematically looks like: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"~", " ", "\[Rho]"}], " ", "+", " ", RowBox[{"1", "/", RowBox[{ SuperscriptBox["l", "2"], " ", "~", " ", "\[CurlyRho]"}]}], " ", "+", " ", RowBox[{ SuperscriptBox["x", "2"], " ", FractionBox[ SuperscriptBox["S", "2"], SuperscriptBox["l", "2"]]}]}], TraditionalForm]],ExpressionUUID-> "f4a0e4b0-d14a-42f6-8f19-c9d6f0a411a7"], ". 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Since we want perturbation to be 0 (in particular finite) at center, we \ require ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "ebba3527-5eef-445b-b499-d2e438a6f2cf"], " ~ ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]],ExpressionUUID-> "dea6beed-2206-4ba4-a34a-31a21977b4ea"], " at least. This is in fact the case for the rule for ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "71c9f5af-abe6-4430-acbd-f49041f4d500"], " found in the next section. This is also clear in that every other term in \ perturbed Y is ~ x.\nIn fact, it is clear that ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "6fe7bdf7-8572-40f2-a7fa-069f6c7e6fca"], " ~ ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]],ExpressionUUID-> "17d9c126-e167-4026-bcee-20a18e97bb24"], " because the terms in ", Cell[BoxData[ FormBox[ FractionBox[ OverscriptBox["s", ".."], SuperscriptBox["R", "2"]], TraditionalForm]],ExpressionUUID-> "9499e897-eef9-4a18-bc04-e211791e7996"], " and ", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox[ OverscriptBox["s", "."], "2"], RowBox[{ SuperscriptBox["R", "2"], "s"}]], TraditionalForm]],ExpressionUUID-> "bee3d52f-3e3f-48e9-a248-144a4aea0a4e"], " go like ~ x, so, the perturbed terms had also better be of the same \ magnitude!\nThe fact that ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "p"], TraditionalForm]],ExpressionUUID-> "7b0ce005-8c29-4b6f-ab80-2532361d2d6a"], " does not explicitly appear can be understood in the sense that we can view \ the above as a PDE for ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "ce5e5c16-8ff7-494b-80ee-db35a2bb2fb4"], " as well as S\[CloseCurlyQuote]. See later. \nThe later however only \ depends on M (via integration), which only depends on a. Similarly, note that \ the PDE for ", Cell[BoxData[ FormBox[ SubscriptBox["a", "p"], TraditionalForm]],ExpressionUUID-> "1f051f4e-955a-45e8-9eff-568585ca3ec4"], " is also independent on ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "p"], TraditionalForm]],ExpressionUUID-> "288179e5-8be6-4cc8-8fe0-ad3ae2db2295"], ". Thus, we can say that the ODEs for ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "p"], TraditionalForm]],ExpressionUUID-> "b06b5a93-c583-4d78-ba82-b26a5eb2e713"], " and ", Cell[BoxData[ FormBox[ SubscriptBox["a", "p"], TraditionalForm]],ExpressionUUID-> "4e16ad35-eb47-4e8a-80e6-dd68bc4be5c3"], " decouple from ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "p"], TraditionalForm]],ExpressionUUID-> "5a878890-f919-473e-8510-754d69d8c0a1"], "." }], "Text", CellChangeTimes->{{3.778931592658066*^9, 3.7789316714698553`*^9}, { 3.7789319089541597`*^9, 3.778931930792968*^9}, {3.778931981199916*^9, 3.778932019846526*^9}, {3.778932470231718*^9, 3.7789324970715218`*^9}, { 3.778933983864443*^9, 3.778934084691881*^9}, {3.7791940636954*^9, 3.779194159693342*^9}, {3.779194191924055*^9, 3.779194244258987*^9}, { 3.781604390178389*^9, 3.7816043972826223`*^9}, {3.787650727575358*^9, 3.7876507699264193`*^9}},ExpressionUUID->"9f58e7f8-fd3f-4810-b38e-\ ba322a7aebb1"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Re-write Y-law as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"S", " ", OverscriptBox["S", ".."]}], " ", "+", " ", RowBox[{"f", " ", SuperscriptBox[ OverscriptBox["S", "."], "2"]}], " ", "+", " ", RowBox[{"h", " ", SuperscriptBox["S", "2"]}]}], " ", "=", " ", "0"}], TraditionalForm]], ExpressionUUID->"2c8a6901-386a-4399-889d-f2b0fa34865a"] }], "Subsection", CellChangeTimes->{{3.787652580699758*^9, 3.78765260418167*^9}, { 3.789811231986807*^9, 3.78981123605759*^9}, {3.789892639056286*^9, 3.789892642709639*^9}},ExpressionUUID->"b89a0394-1df4-42aa-8482-\ 3345c297074b"], Cell[TextData[{ "Currently, it is written as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"g", OverscriptBox["S", ".."], " ", "S"}], " ", "+", " ", RowBox[{"f", " ", SuperscriptBox[ OverscriptBox["S", "."], "2"]}], " ", "+", " ", RowBox[{"h", " ", SuperscriptBox["S", "2"]}]}], " ", "=", " ", "0"}], TraditionalForm]], ExpressionUUID->"d10ffe56-0b05-430c-b64b-8e0740b69fa9"], ". So, we first extract the g, f, h. 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See notes for more details.\nBy using the rules \ above, the ", Cell[BoxData[ FormBox[ RowBox[{"O", "(", OverscriptBox["s", "."]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "f944e334-ef72-46e7-a7c4-dc0bbd9305e1"], ") term can be written as: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"\[Mu]", " ", RowBox[{"(", RowBox[{ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", SuperscriptBox["x", "2"]}], RowBox[{"\[PartialD]", "y"}]], " ", FractionBox[ RowBox[{"\[PartialD]", SubscriptBox["\[CurlyRho]", "0"]}], RowBox[{"\[PartialD]", "\[Mu]"}]]}], " ", "-", " ", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", SuperscriptBox["x", "2"]}], RowBox[{"\[PartialD]", "\[Mu]"}]], FractionBox[ RowBox[{"\[PartialD]", SubscriptBox["\[CurlyRho]", "0"]}], RowBox[{"\[PartialD]", "y"}]]}]}], ")"}]}], " ", "=", " ", RowBox[{ SuperscriptBox["x", "2"], " ", FractionBox[ RowBox[{"\[PartialD]", "F"}], RowBox[{"\[PartialD]", "y"}]]}]}], ",", " ", RowBox[{"F", " ", ":=", " ", RowBox[{ 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RowBox[{"2", " ", "\[Kappa]", " ", RowBox[{"(", RowBox[{ RowBox[{"\[Kappa]", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[Mu]"}], ")"}]}], "+", "\[Mu]"}], ")"}]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[Kappa]"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[Mu]"}], ")"}], " ", SqrtBox["\[Mu]"]}]]], "Output", CellChangeTimes->{3.816008303554819*^9, 3.821354106653667*^9, 3.821440834079474*^9, 3.821959383108029*^9, 3.8233369501510077`*^9, 3.823337159749761*^9, 3.82567755958252*^9}, CellLabel-> "Out[146]=",ExpressionUUID->"26540bbe-5b2b-47c0-8b10-882b43e8a085"] }, Open ]], Cell[TextData[{ "In particular, since v = ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["s", "."], " ", SubscriptBox["v", "1"]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "e1548537-788c-47e6-a1b8-1fee545a30cd"], ", it follows that for \[Kappa] = 1/2 and small \[Mu], v ~ ", Cell[BoxData[ 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For \ \[Kappa] > 1/2, then v \[Rule] 0, for any sensible s(t) and for \[Kappa] < \ 1/2, v actually just blows up as y \[Rule] \[Infinity], which\nseems to \ suggest that this ansatz is just wrong, at least for large values of y. \ However, if we assume that the critical solution contracts irrespective of \ \[Kappa],\nthen this would mean that our model is actually only good up to \ some point inside the star ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", SubscriptBox["x", "\[Star]"]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "630b23a7-0339-4cba-9e40-5972eb82af61"], ", since in the limit \[Mu] \[Rule]0, y =\[Infinity] gets mapped to ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", SubscriptBox["x", "\[Star]"]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "b6d122a2-d5eb-4dd7-b61c-1219f69eaeaf"], ".\nOf course, it may be that the critical behaviour is just entirely \ different for \[Kappa] < 1/2, which does look suspicious." }], "Text", 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In the below, the condition is generally not satisfied if we consider small \ \[Mu] and the term then goes to zero. 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term vanishes, while the other ones blow up at infinity.\nIf we consider a \ power series ansatz of the form ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["M", "2"], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["m", "0"], " ", "+", " ", FractionBox[ SubscriptBox["m", "1"], "y"], " ", "+"}], " ", "..."}]}], TraditionalForm]],ExpressionUUID->"dc20535a-499b-4c5e-b065-7564ba3fa145"], ", this means that ", Cell[BoxData[ FormBox[ SubscriptBox["m", "0"], TraditionalForm]],ExpressionUUID-> "0adafacc-a904-4cb8-aa1e-5d33c3f262c7"], " is arbitrary, while we have some condition on ", Cell[BoxData[ FormBox[ SubscriptBox["m", "1"], TraditionalForm]],ExpressionUUID-> "4a77353a-75f6-44b5-a141-a7b775ffce83"], " so that LHS vanishes since\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "2"], RowBox[{ RowBox[{ SubscriptBox["M", "2"], "''"}], " ", "~", " ", SubscriptBox["a", "1"]}], RowBox[{ RowBox[{ SubscriptBox["M", "1"], "''"}], " ", "~", " ", "b", " ", "~", " ", "y"}]}], 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With this \ definition, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", "2"], " ", "=", " ", "0"}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "18c73242-3184-40e0-b11c-4b4724a01322"], " by regularity near the centre." }], "Text", CellChangeTimes->{{3.821446715531336*^9, 3.821446717654643*^9}, { 3.823084696135578*^9, 3.8230848095400133`*^9}},ExpressionUUID->"fa765019-42e7-4b10-8d4e-\ d44116641427"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"M2Sol", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"y", "^", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", RowBox[{"1", "/", "\[Kappa]"}]}], ")"}]}], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"y", "^", RowBox[{"(", RowBox[{"1", "-", RowBox[{"1", "/", "\[Kappa]"}]}], ")"}]}]}], ")"}], " ", TemplateBox[{"1"}, "C"]}], "+", RowBox[{ RowBox[{"y", "^", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "/", "\[Kappa]"}], ")"}]}], " ", TemplateBox[{"2"}, "C"]}], "+", FractionBox[ RowBox[{"6", " ", "\[Kappa]", " ", RowBox[{"y", "^", RowBox[{"(", RowBox[{"-", "2"}], ")"}]}], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"y", "^", RowBox[{"(", RowBox[{"2", "-", RowBox[{"2", "/", "\[Kappa]"}]}], ")"}]}]}], ")"}]}], RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Kappa]"}], ")"}], "^", "2"}], " ", RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}]}]], "-", RowBox[{ RowBox[{"y", "^", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", RowBox[{"1", "/", "\[Kappa]"}]}], ")"}]}], " ", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ SubscriptBox["G", "0"], "+", "\[Kappa]", "+", RowBox[{ SubscriptBox["G", "0"], " ", "\[Kappa]"}]}], ")"}], " ", RowBox[{"Log", "[", "y", "]"}]}], RowBox[{"\[Kappa]", " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Kappa]", "^", "2"}]}], ")"}]}]]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"M2SolRuletmp", " ", "=", " ", RowBox[{ SubscriptBox["M", "2"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"y", ",", RowBox[{"Evaluate", "[", "M2Sol", "]"}]}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"M2ODE\[Mu]0", " ", "\[Equal]", " ", "0"}], " ", "/.", " ", "M2SolRuletmp"}], " ", "//", " ", "Simplify"}]}], "Input", CellChangeTimes->{{3.821959991347979*^9, 3.8219600704558153`*^9}, { 3.821960272474681*^9, 3.8219602804012012`*^9}, {3.8219612460657167`*^9, 3.8219612467046337`*^9}, 3.823084691508327*^9, {3.823084836868042*^9, 3.823084847475011*^9}, {3.823084998235065*^9, 3.823085003602528*^9}}, CellLabel-> "In[1983]:=",ExpressionUUID->"53f29e10-ad4d-4487-9acc-fbc8ca2a9df8"], Cell[BoxData["True"], "Output", CellChangeTimes->{ 3.821959977390416*^9, 3.821960072124056*^9, {3.821960273263274*^9, 3.821960280700445*^9}, 3.8219612473162622`*^9, 3.823081394143923*^9, 3.823084692344515*^9, {3.82308484440629*^9, 3.823084848020056*^9}, 3.82308500448342*^9, {3.823092081202922*^9, 3.8230920930878077`*^9}}, CellLabel-> "Out[1985]=",ExpressionUUID->"2f5ebb86-a0fc-43d3-8ac5-434e959d8224"] }, Open ]], Cell["\<\ The perturbation is 0 at the surface. 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This is also the only choice possible when considering ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"eigenmodes", " ", RowBox[{ SubscriptBox["M", "1"], "(", RowBox[{"T", ",", "x"}], ")"}]}], " ", "=", " ", RowBox[{ SuperscriptBox["e", "\[Lambda]t"], SubscriptBox["M", RowBox[{"1", ",", "\[Lambda]"}]], " ", RowBox[{ RowBox[{"(", "x", ")"}], "."}]}]}], TraditionalForm]],ExpressionUUID-> "79e3d572-6bc0-4af6-a5d7-ae36431005f4"], "\nThis is also understood from the point of view that a, \[Alpha] are \ scalars and since in Cartesian coordinates \[OpenCurlyDoubleQuote]", Cell[BoxData[ FormBox[ SuperscriptBox["R", "2"], TraditionalForm]],ExpressionUUID-> "5e63b613-beab-4c6b-b7c0-3075905a1743"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{ SuperscriptBox["y", "2"], "\"\<\>"}]}], TraditionalForm]],ExpressionUUID-> "59e0ba92-8cac-4d24-9dea-bd0ff7cd4280"], " is regular, the expansion of a and \[Alpha] are functions of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["R", "2"], " ", "or", " ", "equivalently", " ", SuperscriptBox["x", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"i", ".", "e", ".", " ", "even"}], " ", "expansion", " ", "in", " ", "R"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "886a2c6a-c7cf-4298-a476-e2339e8c6689"], ".\nV is odd and thus as an odd expansion in R, or \[ScriptCapitalV] has an \ even expansion in R.\nFinally, the leading order power of 4) and 5) induce a \ condition on the leading coefficients of the variables. 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We then find ", Cell[BoxData[ FormBox[ SubscriptBox["M", "1"], TraditionalForm]],ExpressionUUID-> "86b21459-5a6b-4afa-9495-d51161878fe9"], " via algebraic equation and ", Cell[BoxData[ FormBox[ SubscriptBox["g", "1"], TraditionalForm]],ExpressionUUID-> "956cadc8-3559-4e9f-89f6-cd602b8d5a33"], " by integration." }], "Text", CellChangeTimes->{{3.784286334226987*^9, 3.784286404577552*^9}},ExpressionUUID->"e46038ce-7890-43e4-b0d7-\ 15d14b5aa33f"] }, Closed]], Cell[CellGroupData[{ Cell["Analysis", "Subsection", CellChangeTimes->{{3.784287181624728*^9, 3.784287183208315*^9}},ExpressionUUID->"b3512967-a0c6-405d-a5f7-\ f200b62eee00"], Cell[TextData[{ "As described above, we can only focus on the matter equations. Note that \ the ODE can be written as ", Cell[BoxData[ FormBox[ SubscriptBox["f", "1"], TraditionalForm]],ExpressionUUID-> "c101693b-6414-4460-8b90-9b086a8b00f8"], "\[CloseCurlyQuote] = ", Cell[BoxData[ FormBox[ SubscriptBox["Af", "1"], TraditionalForm]],ExpressionUUID-> "83322d50-5257-4c0f-bc97-db8ce0e88fad"], ", since it is linear and A = ", Cell[BoxData[ FormBox[ RowBox[{"A", "(", SubscriptBox["f", "0"]}], TraditionalForm]],ExpressionUUID-> "60d94a40-93f0-4d90-8141-b4a9cd20eba8"], ";\[Kappa],\[Lambda]). Evidently, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "0"], " ", "and", " ", SubscriptBox["f", "1"]}], TraditionalForm]],ExpressionUUID-> "e4de22da-3e81-4a57-b27a-fbde8ae37acc"], " are the set of static and perturbed equations.\nNow, to find a unique \ solution, the matrix A had better not be degenerate, i.e. det A \[NotEqual] \ 0. If we write A = ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["a", "ij"]}], TraditionalForm]],ExpressionUUID-> "12370f36-1b23-4f39-acf7-a5b685601319"], "), we find:" }], "Text", CellChangeTimes->{{3.7842871872404203`*^9, 3.7842873051657743`*^9}, { 3.7842873432364473`*^9, 3.784287438666218*^9}},ExpressionUUID->"bd1295c0-dd4e-45ff-b857-\ bc6bbdf85257"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"tmp", " ", "=", " ", RowBox[{ RowBox[{"Flatten", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{ "ODEReducedg1EMM1EM\[CurlyRho]1EM\[Mu]0InTermsOfxShooting", " ", "\[Equal]", " ", "0"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["g", "1"], "'"}], "[", "x", "]"}], ",", RowBox[{ RowBox[{ SubscriptBox["M", "1"], "'"}], "[", "x", "]"}], ",", RowBox[{ RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "'"}], "[", "x", "]"}]}], "}"}]}], "]"}], "]"}], " ", "//", " ", "Simplify"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a11", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["M", "1"], "'"}], "[", "x", "]"}], " ", "/.", " ", "tmp"}], ",", RowBox[{ SubscriptBox["M", "1"], "[", "x", "]"}]}], "]"}], " ", "//", " ", "Simplify"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a12", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["M", "1"], "'"}], "[", "x", "]"}], " ", "/.", " ", "tmp"}], ",", RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "[", "x", "]"}]}], "]"}], " ", "//", " ", "Simplify"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a21", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "'"}], "[", "x", "]"}], " ", "/.", " ", "tmp"}], ",", RowBox[{ SubscriptBox["M", "1"], "[", "x", "]"}]}], "]"}], " ", "//", " ", "Simplify"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a22", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "'"}], "[", "x", "]"}], " ", "/.", " ", "tmp"}], ",", RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "[", "x", "]"}]}], "]"}], " ", "//", " ", "Simplify"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"a11", " ", "a22"}], "-", RowBox[{"a12", " ", "a21"}]}], "]"}], " ", "//", " ", "Simplify"}], "\[IndentingNewLine]", RowBox[{"Clear", "[", RowBox[{"tmp", ",", "a11", ",", "a12", ",", "a21", ",", "a22"}], "]"}]}], "Input", CellChangeTimes->{{3.784379108073749*^9, 3.784379123789864*^9}, 3.784876346235588*^9}, CellLabel-> "In[502]:=",ExpressionUUID->"a0400657-5fc7-4540-8142-3535de449610"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox["\[Lambda]", "2"], RowBox[{"\[Kappa]", " ", SuperscriptBox[ RowBox[{ SubscriptBox["g", "0"], "[", "x", "]"}], "2"]}]]}], "+", FractionBox[ RowBox[{"8", " ", RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}], " ", SuperscriptBox[ RowBox[{ SubscriptBox["\[CurlyRho]", "0"], "[", "x", "]"}], "2"]}], RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{ SubscriptBox["M", "0"], "[", "x", "]"}], "2"]}]]}]], "Output", CellChangeTimes->{3.784379124494627*^9, 3.784379211540691*^9, 3.7843846237249937`*^9, 3.7844477014336367`*^9, 3.784448375403207*^9, 3.7844699730955133`*^9, 3.784632550141168*^9, 3.784639282745373*^9, 3.784876348421032*^9, 3.785062797171555*^9}, CellLabel-> "Out[507]=",ExpressionUUID->"f3be7485-e8fe-46c7-9102-037a324f3a2a"] }, Open ]], Cell[TextData[{ "The condition then is: ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Lambda]", "2"], " ", "\[NotEqual]", " ", RowBox[{ FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}]}], "\[Kappa]"], SuperscriptBox["x", "2"]}]}], TraditionalForm]], FontColor->GrayLevel[0],ExpressionUUID-> "007a929e-77f8-415d-97f6-694ab323d017"], ". Thus, ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Lambda]", "2"], " ", "\[NotElement]", " ", RowBox[{ RowBox[{"[", RowBox[{"0", ",", " ", RowBox[{"4", FractionBox[ RowBox[{"1", "+", "\[Kappa]"}], RowBox[{"1", "-", "\[Kappa]"}]]}]}], "]"}], "."}]}], TraditionalForm]],ExpressionUUID->"50550e04-f128-43db-8b2b-24d9576d5615"] }], "Text", CellChangeTimes->{{3.784287481519349*^9, 3.784287521670621*^9}, { 3.784287647380719*^9, 3.784287651100525*^9}, {3.784287877141223*^9, 3.784287915592564*^9}, {3.784287947401486*^9, 3.784288000504265*^9}, { 3.784379292747757*^9, 3.7843793518721724`*^9}, {3.7848766739439793`*^9, 3.784876742854162*^9}},ExpressionUUID->"5d9e684d-43dd-4aaf-8b2e-\ 085856af5b8a"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Boundary asymptotics for eigenmodes, \[Mu] = 0", "Subchapter", CellChangeTimes->{{3.782211776361195*^9, 3.782211788681737*^9}, { 3.782211865982449*^9, 3.782211868847074*^9}, {3.787472941431333*^9, 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SubscriptBox["M", "1"], "-", SubscriptBox["\[CurlyRho]", "1"]}], ")"}]}], RowBox[{ SubscriptBox["M", "0"], " ", "x"}]]}]}]}], "}"}]], "Output", CellChangeTimes->{3.784381594162518*^9, 3.78438462374754*^9, 3.784447701461735*^9, 3.784448375430938*^9, 3.7844699731115723`*^9, 3.784632550162848*^9, 3.784639282768373*^9, 3.785062797195393*^9}, CellLabel-> "Out[509]=",ExpressionUUID->"260310e7-4586-41aa-b85f-c7a3278c4bb9"] }, Open ]], Cell[TextData[{ "We easily find that:\n1) ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["dM", "1"], "dx"], TraditionalForm]],ExpressionUUID-> "8bc2ff10-3fe0-4c15-a000-7323286a559a"], " ~ ", Cell[BoxData[ FormBox[ RowBox[{"O", RowBox[{"{", RowBox[{"x", " ", RowBox[{"(", RowBox[{ SubscriptBox["M", "1"], " ", "-", " ", SubscriptBox["\[CurlyRho]", "1"]}], ")"}]}], "}"}]}], TraditionalForm]],ExpressionUUID->"f3dcb944-6540-4493-ab0a-2dd4c2c1e9d5"], "\n2) ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["dg", "1"], "dx"], TraditionalForm]],ExpressionUUID-> "23bf1698-ce68-46ed-a148-1bc73a650e7e"], " ~ ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["dM", "1"], "dx"], TraditionalForm]],ExpressionUUID-> "8cc2b2d8-a89c-4f1e-b536-dadc0c7e3017"], "\n3) ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["d\[CurlyRho]", "1"], "dx"], TraditionalForm]], ExpressionUUID->"24f2c550-6efa-4272-bf12-7b49d5748273"], " ~ ", Cell[BoxData[ FormBox[ RowBox[{"O", RowBox[{"{", RowBox[{ FractionBox[ SubscriptBox["M", "1"], "x"], " ", "+", " ", RowBox[{"x", " ", RowBox[{"(", RowBox[{ SubscriptBox["M", "1"], " ", "-", " ", SubscriptBox["\[CurlyRho]", "1"]}], ")"}]}]}], "}"}]}], TraditionalForm]],ExpressionUUID->"cf174b62-de72-4db2-bc0c-be1dce6ce265"], "\nFrom 3), doing a regular expansion wee see that the leading term will be \ ~ ", Cell[BoxData[ FormBox[ FractionBox["1", "x"], TraditionalForm]],ExpressionUUID-> "71e4f934-2c79-4653-b150-979f11da7776"], ". This term needs to vanish, inducing a regularity condition ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["M", "1"], "(", "0", ")"}], "=", "0"}], TraditionalForm]], ExpressionUUID->"ab000f35-18db-4dea-8bf4-5ca2953d12d2"], ", which is good as this means no conical singularity at the center.\nAs for \ which expansion to consider, since ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]],ExpressionUUID-> "b20d1951-6c29-4019-b5ea-0922e616821b"], " is regular in y, we expect each of these variables to have an expansion as \ a function ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]],ExpressionUUID-> "68f28098-754f-48a1-8d7b-80cfc1a76097"], " (not just x as it is not regular in y).\nThis is also understood from the \ point of view that a,\[Alpha] are scalars and since in Cartesian coordinates \ \[OpenCurlyDoubleQuote]", Cell[BoxData[ FormBox[ SuperscriptBox["R", "2"], TraditionalForm]],ExpressionUUID-> "cc4d7118-a760-42b8-a768-71af4544ae37"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{ SuperscriptBox["y", "2"], "\"\<\>"}]}], TraditionalForm]],ExpressionUUID-> "eab6fb8f-1b4c-4f59-862e-a07606182924"], " is regular, the expansion of a and \[Alpha] are functions of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["R", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"i", ".", "e", ".", " ", "even"}], " ", "expansion", " ", "in", " ", "R"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "15fbb10d-b273-4293-b2f4-447b07db53d8"], ".\nV is odd and thus as an odd expansion in R, or \[ScriptCapitalV] has an \ even expansion in R and thus x. This is also consistent with the above \ asymptotics. Since from 2) and 3), we need ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["M", "1"], " ", "~", " ", SubscriptBox["\[CurlyRho]", "1"]}], TraditionalForm]],ExpressionUUID-> "cc43c0a6-97b7-440c-98cd-0a9331183be5"], " and from 2), we need ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["g", "1"], "~", SubscriptBox["M", "1"], "~", SubscriptBox["\[CurlyRho]", "1"]}], "."}], TraditionalForm]], ExpressionUUID->"43382f3a-4880-4191-8882-396d5f009b08"], "\nAnyway, we are left with 2 d.o.f: ", Cell[BoxData[ FormBox[ SubscriptBox["g", "1"], TraditionalForm]],ExpressionUUID-> "3fd06b74-bd24-4b4a-b568-3868577b53d6"], "(0) (gauge since from above, can shift ", Cell[BoxData[ FormBox[ SubscriptBox["g", "1"], TraditionalForm]],ExpressionUUID-> "b1adbb99-dbbd-4e9f-a2a7-2348536e9802"], " arbitrarily) and ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "1"], TraditionalForm]],ExpressionUUID-> "833482c0-4512-4bed-b481-a46fc54db12f"], "(0).\nIMPORTANT: as stated from above, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubsuperscriptBox["M", "1", "c"], "[", "0", "]"}], "=", "0"}], TraditionalForm]],ExpressionUUID->"4b30625b-1497-44db-97ce-a5303dc0eb02"], " is needed from 4), however this is true only for ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", " ", "\[NotEqual]", " ", RowBox[{ RowBox[{"0", "!!"}], "!"}]}], TraditionalForm]],ExpressionUUID-> "b0b9e0b0-ff01-47c1-b7ef-7b9038b3799f"], " Indeed, if ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0"}], TraditionalForm]],ExpressionUUID-> "275e8fab-6d91-45a7-8cff-d94e76a20404"], " then the term ", Cell[BoxData[ FormBox[ RowBox[{"~", FractionBox[ RowBox[{ SubscriptBox["M", "1"], " "}], "x"]}], TraditionalForm]],ExpressionUUID-> "538da335-9154-4955-b98e-a456a7940791"], " in 3) actually vanishes!! This can also be made clear from the the \ shooting later, where we can\nshoot from the surface all the way to the \ center without problem and in fact we find that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["M", "1"], "(", "0", ")"}], " ", ">", " ", "0"}], TraditionalForm]],ExpressionUUID->"9227a7b7-1610-4a31-9b27-e259cc928050"], ". This is clear from gauge analysis below as well, where if ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0"}], TraditionalForm]],ExpressionUUID-> "38508919-dfc9-44ce-ae12-00437c2b786c"], " then ", Cell[BoxData[ FormBox[ RowBox[{"p", "=", "0"}], TraditionalForm]],ExpressionUUID-> "5c1e7596-6d86-403a-8ad8-211f0124e48a"], " and then the full perturbation term is like ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"~", " ", RowBox[{"-", " ", "x"}]}], " ", RowBox[{ SubscriptBox["\[PartialD]", "x"], SubscriptBox["M", "0"]}]}], TraditionalForm]],ExpressionUUID-> "850c20fe-5be3-485a-97d0-23cf4aa579dc"], " which is non-zero at the center." }], "Text", CellChangeTimes->CompressedData[" 1:eJwdyE0og3EAx/HHyhgHcmBh9tIKYdnycsCy1EQ4PNhh7SDMYaUhl1kJmxQH L2lshlqj4aI1D2tltRO1RqTIaKyJek6UtYPy/z2Hb5/6SofNtJFHUVQ5Caaq auooBau5dKy0wd/4TS/UHp72w5jzWM/9Z5EBThrnc7OJV7OtBbC2a0MMC/O8 UliR0Clg8M2qht3eUOMs8Uj1yRl1i4KQZtVhOMS7eJgjfvzkPcL+l/pXmHLp 03DhLMifJ65XegRwNL1dBBODO0KYbG6/g9qSDk73UzJgIzbl6L8gsys1HBAj zh5O+1S+GUqWTqegPOayQEemdA0K5ccTMaLJn5mBrmWf6FfJasbCLRUwflcm gwn7txzuWazV0BdilPB10bElU7Ea8ULxPjRZT7zQ39ng48y69UPtwDUDo4KB EOT/bd7DCC15h8yILgWnz1lbH3HcSC9Dz0NgFcp0H0fwHzW11Do= "],ExpressionUUID->"1b4d92ad-86ea-465d-8968-f6b9333fc1b5"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "At the surface x = ", Cell[BoxData[ FormBox[ SubscriptBox["x", "\[Star]"], TraditionalForm]],ExpressionUUID-> "7f73e840-03b6-452c-ae10-b69e82b06715"] }], "Section", CellChangeTimes->{{3.782212735563519*^9, 3.7822127423901167`*^9}},ExpressionUUID->"ecf12632-1339-4aa9-a10e-\ d7300fda14d2"], Cell[TextData[{ "We find that (see analysis of the static \[CapitalLambda]=0 solution \ section. We mostly use the fact that ", Cell[BoxData[ FormBox[ SubscriptBox["M", "0"], TraditionalForm]],ExpressionUUID-> "7c81c973-ff50-4e91-84fe-99ef80b26c57"], " ~ ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[CurlyRho]", "0"], " ", SubscriptBox["g", "0"]}], TraditionalForm]],ExpressionUUID-> "9ef1ff10-6ec6-4b69-b8ac-ae95c0d3ca5c"], "):\n\t1) ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox["dM", "1"], "dx"], " "}], TraditionalForm]], ExpressionUUID->"973e6516-0b24-48f0-a636-b503752675c3"], "~ ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ SubscriptBox["M", "1"], "-", SubscriptBox["\[CurlyRho]", "1"]}], RowBox[{ SubscriptBox["x", "\[Star]"], "-", "x"}]], TraditionalForm]], ExpressionUUID->"c1ff76d8-9ee0-443f-bb20-eb1abd2d99c2"], "\n\t2) ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox["dg", "1"], "dx"], " "}], TraditionalForm]], ExpressionUUID->"704ae493-b51b-4208-91da-54a774058025"], "~ ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["dM", "1"], "dx"], TraditionalForm]],ExpressionUUID-> "3f08d63e-cd3b-482c-a4c9-9992bc1405ec"], "\n\t3) ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox["d\[CurlyRho]", "1"], "dx"], " "}], TraditionalForm]], ExpressionUUID->"f77ceb0a-2dc2-46a6-ab84-93d11d7a67d2"], "~ ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["M", "1"], "-", SubscriptBox["\[CurlyRho]", "1"]}], RowBox[{ SubscriptBox["x", "\[Star]"], "-", "x"}]], "."}], TraditionalForm]], ExpressionUUID->"333ce4bb-0b50-49b6-bcd1-87a6e21db7bc"] }], "Text", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQAWIQzR2krvlf942jaFhmIog+yGDeCaI/SM/tBtEdu/e/ b9Z74zhN+BhzC5Bec2WncxuQnnOazw9Ez5PIzgLRb3dW54HoztrDi5YC6YAt VxeD6AvT/y4H0RYnVTaAaC7DoH0gWmq9+n4Q/atg500Q/enBs0cgWmB6x3cQ HcSz9CeInh8ucGw5kL5b43gcRO+Sn8y1Akhz+EXzguhbgg83bQDSJS6vwfSz 9fMZ1PXfOPItnMUEorvKz5l+M3zjWJX01RxEb7u2xgZEL8krCwDRJcf/Xlcy euPoWBh3F0Qvz/ryHERv//vgHYg2MmxnVAbSYs+usYBovhd17B7xbxyZwvvA NACZaqD3 "],ExpressionUUID->"fd956769-f769-44e9-8e69-d363c0f4f9f4"], Cell["\<\ This is of course in-line with the more general analysis done before looking \ for exponential growing mode solutions.\ \>", "Text", CellChangeTimes->{{3.782212382750495*^9, 3.782212416207458*^9}, { 3.782212460029972*^9, 3.782212612737151*^9}, {3.782214391849279*^9, 3.782214405985951*^9}, {3.7822144440938587`*^9, 3.7822144632876883`*^9}, 3.782481947675356*^9, {3.782481984569438*^9, 3.782482029088666*^9}, { 3.782482095822711*^9, 3.782482112630234*^9}, {3.782482160252396*^9, 3.782482342200128*^9}, {3.78248244538386*^9, 3.782482472293269*^9}, { 3.78248382862475*^9, 3.7824838347769938`*^9}, {3.7824839795973797`*^9, 3.782484034154419*^9}, {3.782484089121767*^9, 3.782484145983482*^9}, { 3.782484260308597*^9, 3.782484379145955*^9}, {3.782484421896357*^9, 3.782484454606991*^9}, {3.782553592113492*^9, 3.78255378521806*^9}, { 3.782558855461124*^9, 3.7825591657796183`*^9}, {3.782560970258966*^9, 3.782561055991737*^9}, {3.7842909707631474`*^9, 3.784291001314391*^9}},ExpressionUUID->"af8e3fc4-7dab-40aa-9426-\ d2220d3f05f9"], Cell[CellGroupData[{ Cell["Power ansatz and regularity condition", "Subsection", CellChangeTimes->{{3.782817273724472*^9, 3.7828172912759113`*^9}},ExpressionUUID->"4efcb021-97c8-40af-9cd2-\ 63d009f4f914"], Cell[TextData[{ "From the above, and using the fact that since we are describing a star, we \ do not expect the perturbations to be analytic at the surface. This was tried \ explicitly before and yielding the trivial solution only.\nSo, we will \ instead consider an expansion of the type ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyRho]", "1"], TraditionalForm]],ExpressionUUID-> "556014d1-a439-47e4-bc1a-7679979cc218"], " = \[Sum] ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "[", "n", "]"}], TraditionalForm]], ExpressionUUID->"ffc31a60-bb6f-43ae-9df5-36ad8d3c00d5"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}]}], TraditionalForm]],ExpressionUUID-> "1c7001c2-61a4-4200-9863-fcf6594e7151"], Cell[BoxData[ FormBox[ SuperscriptBox[")", RowBox[{"p", "+", "n"}]], TraditionalForm]],ExpressionUUID-> "ded5ca98-24dd-4e43-9afc-ee5a6c12d006"], " , ", Cell[BoxData[ FormBox[ SubscriptBox["M", "1"], TraditionalForm]],ExpressionUUID-> "9c4aaf5f-a84b-43d2-bcd9-a78c4a149975"], " = \[Sum] ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["M", "1"], "[", "n", "]"}], TraditionalForm]],ExpressionUUID-> "68299db0-f417-4572-b3ee-6078e78d88df"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}]}], TraditionalForm]],ExpressionUUID-> "aaffc08e-4c82-4fa8-a351-153c9a88c7c4"], Cell[BoxData[ FormBox[ SuperscriptBox[")", RowBox[{"p", "+", "n"}]], TraditionalForm]],ExpressionUUID-> "b02e2104-8c55-43f4-853b-891184cfbc81"], ". Then 1) suggests that we take ", Cell[BoxData[ FormBox[ SubscriptBox["g", "1"], TraditionalForm]],ExpressionUUID-> "de1de474-0c01-47a5-886a-f2cd3af10322"], " = \[Sum] ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["g", "1"], "[", "n", "]"}], TraditionalForm]],ExpressionUUID-> "60179488-6363-4124-927d-dae463444cfe"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}]}], TraditionalForm]],ExpressionUUID-> "6efbfc46-2d1e-4d7a-96fd-11e940fb2678"], Cell[BoxData[ FormBox[ SuperscriptBox[")", RowBox[{"p", "+", "n"}]], TraditionalForm]],ExpressionUUID-> "04e7f918-3195-4736-84ca-6668520f9234"], ". p needs to be found.\nSince the to matter equations decouple from metric ", Cell[BoxData[ FormBox[ SubscriptBox["g", "1"], TraditionalForm]],ExpressionUUID-> "e6c4d8bd-70a2-48cf-a2b1-64a25a720fe8"], ", we can look at first order term. From 2) and 3) this then fixes p and one \ of the d.o.f..\nTo leading order, we have that,\n", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["d\[CurlyRho]", "1"], "dx"], TraditionalForm]], ExpressionUUID->"daa31038-ca15-4e11-ab7f-07fa6cc3f751"], " = ", Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox[ RowBox[{"1", "+", "\[Kappa]"}], RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Kappa]"}], ")"}], " ", "\[Lambda]"}]]}], TraditionalForm]],ExpressionUUID->"038479a1-b862-4d7a-a006-a8d7d0f82714"], " ", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}]], TraditionalForm]],ExpressionUUID-> "3c532750-95cd-4022-9f73-9a0240af05c5"], " ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["\[Lambda]", "2"], " ", "-", RowBox[{"4", " ", FractionBox[ RowBox[{"1", "+", "\[Kappa]"}], RowBox[{"1", "-", "\[Kappa]"}]]}]}], ")"}], " ", SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "+", " ", RowBox[{"\[Lambda]", " ", SubscriptBox["\[CurlyRho]", "1"]}]}], "}"}], TraditionalForm]], ExpressionUUID->"6ff0951f-c5b3-4662-b137-c9b0ecf7fb24"], " = ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["B", "1"], " ", SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "+", RowBox[{ SubscriptBox["B", "2"], " ", SubscriptBox["\[CurlyRho]", "1"]}]}], ")"}]}], TraditionalForm]], ExpressionUUID->"6db82068-2ae3-4b2d-b683-ad8fcad43993"], "\n", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["d\[ScriptCapitalV]", "1"], "dx"], TraditionalForm]], ExpressionUUID->"227fc138-ef8f-42a7-abfb-e5d0e43a0fb2"], " = ", Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox["1", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}], SubscriptBox["x", "\[Star]"]}]]}], TraditionalForm]],ExpressionUUID-> "471adc74-6e7c-4c9e-803d-13ddfeb03b64"], " ", Cell[BoxData[ FormBox[ RowBox[{"{", " ", RowBox[{ RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}], " ", SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "-", " ", RowBox[{ FractionBox[ SubscriptBox["x", "\[Star]"], RowBox[{"2", " ", RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}], ")"}]}]], " ", RowBox[{"(", " ", RowBox[{ RowBox[{ RowBox[{"-", "4"}], " ", FractionBox[ RowBox[{"1", "+", "\[Kappa]"}], RowBox[{"1", "-", "\[Kappa]"}]], SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "+", " ", RowBox[{"\[Lambda]", " ", SubscriptBox["\[CurlyRho]", "1"]}]}], ")"}]}]}], "}"}], TraditionalForm]],ExpressionUUID->"9635a56b-c272-42ab-90fb-ed48bd3b80df"], " = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["A", "1"], " ", SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "+", " ", RowBox[{ FractionBox["1", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["A", "2"], " ", SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "+", RowBox[{ SubscriptBox["A", "3"], " ", SubscriptBox["\[CurlyRho]", "1"]}]}], ")"}]}]}], TraditionalForm]], ExpressionUUID->"38c5510b-8063-47d0-b76b-c79c2708a49a"], ". This one is old because now, we use ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["dM", "1"], "dx"], TraditionalForm]],ExpressionUUID-> "db997250-f8a8-4155-9d5d-db5a96424817"], " instead, but I am not doing the analysis again....\n", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["dg", "1"], "dx"], TraditionalForm]],ExpressionUUID-> "40c45a5a-8a85-4b4b-8d2d-77965b5d6902"], " = ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", "\[Kappa]"}], ")"}]}], RowBox[{"\[Lambda]", " ", SubscriptBox["x", "\[Star]"]}]], " ", RowBox[{"(", " ", RowBox[{ RowBox[{ FractionBox[ RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "\[Kappa]"}], ")"}]}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", "\[Kappa]"}], ")"}], "2"]], " ", SubscriptBox["\[ScriptCapitalV]", "1"]}], " ", "-", RowBox[{ FractionBox["\[Lambda]", RowBox[{"1", "-", "\[Kappa]"}]], " ", SubscriptBox["\[CurlyRho]", "1"]}], " ", "+", RowBox[{ FractionBox[ RowBox[{"\[Lambda]", " ", SubscriptBox["x", "\[Star]"]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", "\[Kappa]"}], ")"}], " ", RowBox[{"(", RowBox[{"x", "-", SubscriptBox["x", "\[Star]"]}], ")"}]}]], " ", SubscriptBox["g", "1"]}]}], ")"}]}], TraditionalForm]],ExpressionUUID-> "53b5079a-86ea-47a7-9e30-e5332f96d7ce"], "\nTo leading order, we then find for the matter equations,\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"p", "-", SubscriptBox["A", "2"]}], ")"}], " ", RowBox[{ SubscriptBox["\[ScriptCapitalV]", "1"], "[", "0", "]"}]}], " ", "=", " ", RowBox[{ SubscriptBox["A", "3"], " ", RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "[", "0", "]"}]}]}], TraditionalForm]],ExpressionUUID->"c480d27e-049a-4e42-a9bf-faee4f7e21e4"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"p", "-", SubscriptBox["B", "2"]}], ")"}], " ", RowBox[{ SubscriptBox["\[CurlyRho]", "1"], "[", "0", "]"}]}], " ", "=", " ", RowBox[{ SubscriptBox["B", "3"], " ", RowBox[{ SubscriptBox["\[ScriptCapitalV]", "1"], "[", "0", "]"}]}]}], TraditionalForm]],ExpressionUUID->"63e05808-79ff-451e-9895-50593809478c"], "\nLeading to a 2nd order algebraic equation for p: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"p", "-", SubscriptBox["B", "2"]}], ")"}], " ", RowBox[{"(", RowBox[{"p", "-", SubscriptBox["A", "2"]}], ")"}]}], " ", "=", " ", RowBox[{ SubscriptBox["A", "3"], " ", SubscriptBox["B", "1"]}]}], TraditionalForm]],ExpressionUUID-> "b561cd90-e900-449e-af4a-683dba529c5e"], ". Can solve for p. Note that only positive root p = ", Cell[BoxData[ FormBox[ SubscriptBox["p", "+"], TraditionalForm]],ExpressionUUID-> "6e37bf52-e73c-40b0-b35f-be676620548b"], " is acceptable since we want p > 0 (perturbation does not blow up at \ surface). 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expansion for ", Cell[BoxData[ FormBox[ SubscriptBox["g", "0"], TraditionalForm]],ExpressionUUID-> "f2ee6aaa-425c-41f6-b01f-cefb56b163d5"], " can be done immediately as the expansion terminates at 2nd order.\nSide \ note: if we picked instead to expand under ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "\[Star]"], "-", "x"}], ")"}], RowBox[{"p", "+", "n"}]], TraditionalForm]],ExpressionUUID-> "190e742b-f928-41f2-a467-380120c9c58f"], " instead, we expect to get a factor ~ ", Cell[BoxData[ FormBox[ SubsuperscriptBox["x", "\[Star]", RowBox[{"-", "p"}]], TraditionalForm]],ExpressionUUID-> "f69eb479-c06b-4bdf-9749-06a3a5da6b3a"], " lurking around. If you do this, you will see that this is not the case and \ the reason is that the free parameter is rescaled to \ \[OpenCurlyDoubleQuote]hide\[CloseCurlyDoubleQuote] this factor.\nFor \ example, considering the ODE ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox["x", SubscriptBox["x", "\[Star]"]]}], ")"}], " ", RowBox[{"f", "'"}]}], " ", "=", " ", "f"}], TraditionalForm]], ExpressionUUID->"8403af61-d698-44e2-8010-98352dd5bd9b"], ". The solution is ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", " ", RowBox[{ SubscriptBox["f", "0"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", FractionBox["x", SubscriptBox["x", "\[Star]"]]}], ")"}], RowBox[{"-", SubscriptBox["x", "\[Star]"]}]]}]}], TraditionalForm]],ExpressionUUID-> "bfa0637a-dbf4-4087-a44f-0ce0798c2569"], ". Now, if we re-consider the ODE and make two ans\[ADoubleDot]tze: ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", " ", RowBox[{"c", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", FractionBox["x", SubscriptBox["x", "\[Star]"]]}], ")"}], "p"]}]}], TraditionalForm]], ExpressionUUID->"8dab21a4-4661-4b55-a79b-0a3f37b028ed"], " and ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", " ", RowBox[{ OverscriptBox["c", "_"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "\[Star]"], "-", "x"}], ")"}], "p"]}]}], TraditionalForm]],ExpressionUUID->"a6a10397-f3cb-47d2-be66-55483ea81886"], ". We see that both ans\[ADoubleDot]tze require ", Cell[BoxData[ FormBox[ RowBox[{"p", " ", "=", " ", RowBox[{"-", SubscriptBox["x", "\[Star]"]}]}], TraditionalForm]],ExpressionUUID-> "29ed3aad-510f-47be-adcd-7f4919a23f17"], ". Of course, c and ", Cell[BoxData[ FormBox[ OverscriptBox["c", "_"], TraditionalForm]],ExpressionUUID-> "2212cb7c-1134-4329-9999-2bc5d3df77b2"], " are related by ~ ", Cell[BoxData[ FormBox[ SubsuperscriptBox["x", "\[Star]", "p"], TraditionalForm]],ExpressionUUID-> "64d93bef-034f-42d4-b494-f9668f995a71"], ".\nHowever, if we were to be a bit hand wavy and write ", Cell[BoxData[ FormBox["c", TraditionalForm]],ExpressionUUID-> "ab870bbd-c2b0-4f90-8d3a-da018021f7bc"], " instead of ", Cell[BoxData[ FormBox[ OverscriptBox["c", "_"], TraditionalForm]],ExpressionUUID-> "90567328-4d22-45ac-ac89-f6016b90441e"], ", one would get the impressions that the constant factor in both ans\ \[ADoubleDot]tze is the same, although it isn\[CloseCurlyQuote]t." }], "Text", CellChangeTimes->{{3.7825622421285677`*^9, 3.7825622710568733`*^9}, { 3.7844505011942663`*^9, 3.7844506207356377`*^9}, {3.784450773194859*^9, 3.784451083554532*^9}},ExpressionUUID->"fb6b9fea-249c-45ae-807f-\ 02b75c63916c"], Cell[BoxData[{ 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We solve for p, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubsuperscriptBox["\[ScriptCapitalV]", "1", "s"], "[", "0", "]"}], " ", "and", " ", RowBox[{ SubsuperscriptBox["\[CurlyRho]", "1", "s"], "[", "0", "]"}]}], TraditionalForm]],ExpressionUUID->"b555e192-f69f-4a42-b3af-f65140d01e34"], " explicitly so we don\[CloseCurlyQuote]t put them here." }], "Text", CellChangeTimes->{{3.757076585354084*^9, 3.757076604287318*^9}, 3.757077435340796*^9, {3.761925220536894*^9, 3.76192522092136*^9}, { 3.782049559427824*^9, 3.782049564563072*^9}, 3.7821238599596033`*^9, { 3.7825625956705627`*^9, 3.78256262725541*^9}, {3.782823747784773*^9, 3.78282377782731*^9}, {3.782825491453891*^9, 3.782825552676073*^9}},ExpressionUUID->"22a69ad3-9333-4087-9566-\ 086a63cf50f1"], Cell[BoxData[ RowBox[{ RowBox[{"PS1UnknownSurface", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ SubsuperscriptBox["g", "1", "s"], "[", "n", "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", 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Actual Apply uses two: @@ instead.\ \>", "Text", CellChangeTimes->{{3.758206181789382*^9, 3.758206197023718*^9}, { 3.7582102629436493`*^9, 3.7582103153342237`*^9}, 3.7614000541327057`*^9, { 3.761758279361308*^9, 3.7617582804235163`*^9}, {3.761758312829186*^9, 3.761758338079104*^9}},ExpressionUUID->"fa24428a-fe36-410f-ab72-\ 735a616f9231"], Cell[BoxData[ RowBox[{"\t\t\t\t", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", RowBox[{"Evaluate", "@", RowBox[{"{", RowBox[{ RowBox[{"Context", "[", "]"}], ",", "NotationMakeBoxes", ",", "NotationMakeExpression"}], "}"}]}]}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.7582060437307568`*^9, 3.7582060607461643`*^9}, { 3.758206146343397*^9, 3.758206146911203*^9}, {3.758211219152483*^9, 3.7582112196482677`*^9}, {3.758211324660634*^9, 3.758211325236521*^9}, 3.761400055156755*^9, {3.7617583438548822`*^9, 3.761758344786606*^9}},ExpressionUUID->"d3336325-d33a-4713-b384-\ 2393710523f4"], Cell["\<\ \t\t\t\tNote on the last two, the need to put Evaluate at the very front. \ This is because DumpSave has attribute HoldRest, making it not read/evaluate \ the arguments at level 2 or lower. \t\t\t\tIn this case, the 1st level argument of interest would be {...}. 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"Section", CellChangeTimes->{{3.773747032559412*^9, 3.77374705268666*^9}, { 3.78195891376851*^9, 3.781958921176094*^9}, 3.782201160664762*^9, { 3.7846393418434677`*^9, 3.784639343587453*^9}, 3.790949321149581*^9, 3.7909494609938173`*^9},ExpressionUUID->"1029594e-a0f2-4243-95ac-\ feecff71228a"], Cell[TextData[{ "In general, we would consider a perturbation of the type ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", " ", RowBox[{ SubscriptBox["f", "0"], " ", "+", " ", RowBox[{"\[Epsilon]", " ", SubscriptBox["f", "0"], " ", RowBox[{ SubscriptBox["f", "1"], "[", RowBox[{"T", ",", "y"}], "]"}]}], " ", "+", " ", RowBox[{"\[Mu]", " ", SubscriptBox["f", "0"], " ", RowBox[{ SubscriptBox[ OverscriptBox["f", "_"], "1"], "[", RowBox[{"T", ",", "y"}], "]"}]}]}]}], TraditionalForm]],ExpressionUUID-> "019c0faa-9df2-4841-94c6-be300d9fd330"], ". But, first \[Mu] ~ \[Epsilon], so that, ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", " ", RowBox[{ SubscriptBox["f", "0"], " ", "+", " ", RowBox[{"\[Epsilon]", " ", RowBox[{ SubscriptBox["f", "0"], "(", RowBox[{ RowBox[{ SubscriptBox["f", "1"], "[", RowBox[{"T", ",", "y"}], "]"}], " ", "+", " ", RowBox[{ SubscriptBox[ OverscriptBox["f", "_"], "1"], "[", RowBox[{"T", ",", "y"}], "]"}]}], ")"}]}]}]}], TraditionalForm]], ExpressionUUID->"ea997743-5df5-48f5-8366-92e913948da0"], ". Secondly, when we consider the decomposition of ", Cell[BoxData[ FormBox[ SubscriptBox["f", "1"], TraditionalForm]],ExpressionUUID-> "a9518e1f-92a3-4d4c-9a79-67e9d7bea790"], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ OverscriptBox["f", "_"], "1"], TraditionalForm]],ExpressionUUID-> "7b444d42-c1e7-4f67-ba8c-8880941ea0b0"], " into frequencies:\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["f", "1"], "(", RowBox[{"T", ",", "y"}], ")"}], " ", "=", " ", RowBox[{ SuperscriptBox["e", "\[Lambda]T"], RowBox[{ SubscriptBox["f", "1"], "(", "y", ")"}]}]}], ",", " ", RowBox[{ RowBox[{ SubscriptBox[ OverscriptBox["f", "_"], "1"], "(", RowBox[{"T", ",", "y"}], ")"}], " ", "=", " ", RowBox[{ SuperscriptBox["e", RowBox[{ OverscriptBox["\[Lambda]", "_"], "T"}]], RowBox[{ SubscriptBox[ OverscriptBox["f", "_"], "1"], "(", "y", ")"}]}]}]}], TraditionalForm]],ExpressionUUID->"6d92a959-bbfd-469c-9a68-948247fb787e"], ", we note that for consistency, we better have that ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", " ", "=", " ", OverscriptBox["\[Lambda]", "_"]}], TraditionalForm]],ExpressionUUID-> "5a418aa1-d985-47d7-ab02-00dbf5c9edd0"], ". So, the part that only depends on y can be combined again.\nThus, it \ suffices to look for perturbation of the type: ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", " ", RowBox[{ SubscriptBox["f", "0"], " ", "+", " ", RowBox[{"\[Epsilon]", " ", SubscriptBox["f", "0"], " ", RowBox[{ SubscriptBox["f", "1"], "[", RowBox[{"T", ",", "y"}], "]"}]}]}]}], TraditionalForm]],ExpressionUUID-> "06189477-f2d8-49d9-9ea7-06e5d0887c0a"], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "1"], "[", RowBox[{"T", ",", "y"}], "]"}], " ", "=", " ", RowBox[{ SuperscriptBox["e", "\[Lambda]T"], RowBox[{ SubscriptBox["f", "1"], "(", "y", ")"}]}]}], TraditionalForm]], ExpressionUUID->"b29cbe15-bdd7-4972-ad26-a0e9219ef2c5"], ". Compare this with the contracting solution and the reason this is nicer \ is because in the contracting solution, we could have done the same if S(T) \ was an exponential.\nThis was however 2nd order perturbation and therefore it \ was not required. 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NOte that since we \ consider \[Mu] as a perturbation of the \[CapitalLambda] = 0 case, we replace \ derivative of the background variables with \[Mu] = 0.\nThis is consistent \ since we are considering \[Epsilon] ~ \[Mu] term here and so each background \ variable ", Cell[BoxData[ FormBox[ SubscriptBox["f", "0"], TraditionalForm]],ExpressionUUID-> "f94a2f1b-44da-4ef1-b75e-70ca7e53169f"], " is reattached to a ", Cell[BoxData[ FormBox[ SubscriptBox["f", "1"], TraditionalForm]],ExpressionUUID-> "9688248e-35f2-432f-8a74-ed47cb3a0b05"], ". 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Note that t has dimension length, so we \ scale it so that \[Lambda] is dimensionless.\nThe ODEs are then functions of \ dim-less qtes and the scale factor s does not explicitly appear as it shouldn\ \[CloseCurlyQuote]t.\nNote: this \[Lambda] has of course nothing to do with \ the \[Lambda] from the general static solution!\n", StyleBox["Note:", FontColor->RGBColor[1, 0, 0]], " the matter equations for the linear perturbations do not depend on the \ linear perturbation of g and therefore do not depend on \[Alpha]!!\n", StyleBox["Note:", FontColor->RGBColor[1, 0, 0]], " for the perturbation of \[Mu], the perturbation should only depend on x. \ The reason is that the \[Mu] (being of order \[Epsilon]) will always be \ re-attached to background variables. \nSo, the inhomogeneous part will only \ depend on x and so the correction of the perturbation should also." }], "Text", CellChangeTimes->{{3.7819462191196537`*^9, 3.7819462943185043`*^9}, { 3.7819597719909277`*^9, 3.78195977666776*^9}, 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