(* 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[ 5993481, 131434] NotebookOptionsPosition[ 5837553, 128797] NotebookOutlinePosition[ 5840994, 128881] CellTagsIndexPosition[ 5840914, 128876] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Last modified: 14, August 2021", "Text", CellTags-> "LastModified",ExpressionUUID->"cbbf0e65-2a95-4b6d-aeb5-6c53d59343ac"], Cell["Perfect fluid collapse in spherical symmetry with \[CapitalLambda] = \ 0", "Title", CellChangeTimes->{{3.7609591767684307`*^9, 3.760959199249362*^9}, { 3.761492880256297*^9, 3.761492886815938*^9}},ExpressionUUID->"6029902a-1c6b-4117-8d54-\ dca7ad90cb54"], 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->"cb967a07-00a8-44a3-991e-\ 89c0bcee56b4"], 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->"02fd4322-d16b-424d-97cd-\ 5ddefaef0c66"], 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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Try with and without the above. Check with FullForm to see the difference." }], "Text", CellChangeTimes->{{3.755878536512877*^9, 3.755878588984085*^9}, 3.756818811030196*^9, {3.761758044294409*^9, 3.761758044759638*^9}},ExpressionUUID->"9b9d16aa-87d1-4fdf-a44f-\ e858fe560173"], Cell["\<\ \t\t\tThe below is for pretty print purposes. Allows to use the function \ pdConv.\ \>", "Text", CellChangeTimes->{{3.770033965435131*^9, 3.770033991338285*^9}},ExpressionUUID->"7697ad29-1b12-44d4-87d6-\ 874de4838065"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ "<<", " ", "\"\\"", RowBox[{"(*", RowBox[{"Allows", " ", "for", " ", "nice", " ", "output", " ", "form"}], "*)"}]}]}]], "Input", CellChangeTimes->{{3.770033958091524*^9, 3.7700339588275137`*^9}}, CellLabel->"In[5]:=",ExpressionUUID->"239249fa-5d4d-4185-a508-8877feff3916"] }, Closed]], Cell[CellGroupData[{ Cell["Keyboard shortcuts", "Subsubsubsection", CellChangeTimes->{{3.760191889651951*^9, 3.7601918966393347`*^9}},ExpressionUUID->"5d767d25-fe8b-4574-bde8-\ 63a031c24183"], Cell[TextData[{ "\t\t\tThe following allows to auto-replace stuff when you write \ \[OpenCurlyDoubleQuote]tex[\[CloseCurlyDoubleQuote] into \ \[OpenCurlyDoubleQuote]TeXForm[StandardForm[]]\[CloseCurlyDoubleQuote], which \ is what is needed to output ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox["A", BoxBaselineShift->-0.4, BoxMargins->{{-0.5, -0.3}, {0, 0}}], FontSize->Smaller], "T", AdjustmentBox["E", BoxBaselineShift->0.5, BoxMargins->{{-0.3, 0}, {0, 0}}], "X"}], SingleLetterItalics->False], TraditionalForm]],ExpressionUUID-> "b204f0e8-9b40-460d-98cb-dfac326c3233"], " code of an equation. 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->"681ef5b4-bb6d-46e9-9b00-\ 1cb6708e1621"], 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[6]:=",ExpressionUUID->"33585150-c746-412b-9b60-343e6ea5aae5"] }, Closed]], Cell[CellGroupData[{ Cell["Palettes and docked buttons", "Subsubsubsection", CellChangeTimes->{{3.762002502126274*^9, 3.762002506162354*^9}, { 3.7673339124027567`*^9, 3.767333919106395*^9}},ExpressionUUID->"67cee9f0-fc5f-4c47-997d-\ c61ad1d9f1fb"], Cell["\<\ \t\t\tBelow is the full docked buttons code. 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Cell[CellGroupData[{ Cell["My choice (in progress)", "Subsection", CellChangeTimes->{{3.767503095205633*^9, 3.7675031071747313`*^9}, { 3.767503849923181*^9, 3.767503854951664*^9}, {3.767504512151003*^9, 3.7675045147321997`*^9}, {3.767510201744485*^9, 3.7675102167677603`*^9}, { 3.767510274763578*^9, 3.7675102796465473`*^9}},ExpressionUUID->"16c683b2-5841-4138-a232-\ a10a984ca67c"], Cell[TextData[{ "\tHere we take ", Cell[BoxData[ FormBox[ SubsuperscriptBox["V", "\[Mu]", "X"], TraditionalForm]],ExpressionUUID-> "946f8352-a061-4f91-8b33-8cbf228df5ae"], " = (\[Alpha]/(a sin \[Theta]),0,0,0) and ", Cell[BoxData[ FormBox[ SubsuperscriptBox["V", "\[Mu]", "Y"], TraditionalForm]],ExpressionUUID-> "7bdb86da-cbcf-42b1-94b8-9f462a12f173"], " = ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", RowBox[{ RowBox[{"R", "'"}], "/", RowBox[{"(", SuperscriptBox["R", "2"]}]}]}]}], TraditionalForm]],ExpressionUUID-> "a1111e6b-dd43-4d72-bc49-5c949dda13c1"], "sin \[Theta]),0,0) to extract to two 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On the other hand, we have only 4 variables! 3 \ equations need to be redundant. \tFrom Bianchi, we wish to find equations that are linear combinations of \ other. Intuitively, it is clear that g\[Theta]\[Theta] is related to g\[Phi]\ \[Phi] by just a sin^2(x). Also, g\[Theta]\[Theta] contains 2nd derivative, \ and is therefore probably just \t \tsome linear combinations (include derivatives) of the other terms, modulo \ the Bianchi identity. To see more precisely, we will take a tensor (could be \ Einstein,Energy or even EFE) and write it in the general form \twe have above. (i.e. we write for the first term just gtt,etc for all \ non-trivial terms). 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The speed of the \ fluid will be less than the speed of light. We can see that the boundary of \ causality for the fluid occurs whenever the denominator in the above vanishes. \tV^2=1 cannot happen as this corresponds to the speed of light. In \ Evernote,under EvansColeman_consistency_check,we show this boundary agrees \ with the theoretical prediction.\ \>", "Text", CellGroupingRules->{"GroupTogetherGrouping", 10006.}, CellChangeTimes->{{3.7500722138431377`*^9, 3.7500722413397408`*^9}, 3.750076858372908*^9, 3.750415998456653*^9, 3.750416422718185*^9, { 3.751109104611277*^9, 3.7511091074563437`*^9}, {3.7513841946174717`*^9, 3.75138420744771*^9}, {3.75138433471736*^9, 3.751384494661426*^9}, { 3.753193059851952*^9, 3.753193072154109*^9}, {3.753624241461729*^9, 3.753624271789392*^9}, {3.7536250937976*^9, 3.753625133587139*^9}, { 3.753625272201021*^9, 3.7536252771680613`*^9}, {3.761903363685417*^9, 3.761903385643358*^9}},ExpressionUUID->"3cd38f52-5cc6-406d-890e-\ a4830f7d5986"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"2", " ", "x"}], ")"}]}], "*", RowBox[{"Denominator", "[", "metricMatterEqnSol\[Rho]xCSSOld", "]"}]}], "]"}], " ", "/.", 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Indeed, we need to use the algebraic equation. 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We can then compare which method is numerically more efficient. 1) We will keep the equations as is and use the alg. eqn for consistency \ checks/requirements. This keeps the equations simpler. 2) We will use the algebraic equation to get rid of one variable. 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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}, {3.7619038963639097`*^9, 3.7619039005640993`*^9}},ExpressionUUID->"203f050a-c3c5-47a2-ae6b-\ aa421c256e3e"], Cell[CellGroupData[{ Cell[" Redundancy Check", "Subsubsection", CellChangeTimes->{{3.753904274159281*^9, 3.753904285408024*^9}, { 3.7619039026432447`*^9, 3.761903902908063*^9}},ExpressionUUID->"9060ba2a-4eee-4e46-993c-\ 123a3004e026"], Cell["\<\ \t\tUnevaluable. \t\tThe question that we may ask is which of the new variables ODE, Ng\ \[CloseCurlyQuote] or A\[CloseCurlyQuote], are redundant? 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In what follows, we will denote X = (Ng, \[Omega], V).\n\ ", StyleBox["Made the Full ODEs and the Check below not evaluable", FontColor->RGBColor[1, 0, 0]], ". The results have been checked to be identical to the reduced system below." }], "Text", CellChangeTimes->{{3.751383593015214*^9, 3.751383674533968*^9}, { 3.7525002242023773`*^9, 3.7525002495914783`*^9}, {3.7534553449834757`*^9, 3.753455366319138*^9}, {3.761905739506466*^9, 3.761905739714754*^9}, { 3.770034527786044*^9, 3.770034530121526*^9}},ExpressionUUID->"4d3410bf-7763-4fa0-a63f-\ ddbdea00b102"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Power series ansatz near x = 0", "Section"]], \ "Section", CellChangeTimes->{{3.750576547548811*^9, 3.750576569048753*^9}, { 3.750576831917136*^9, 3.750576845901026*^9}, {3.751633877418722*^9, 3.751633880451054*^9}, {3.751633921977611*^9, 3.7516339265855722`*^9}, { 3.753191519080305*^9, 3.753191572614909*^9}, {3.753191676851458*^9, 3.753191684475602*^9}, {3.753191747979604*^9, 3.753191757537088*^9}, 3.75396724197537*^9, {3.761905552775548*^9, 3.761905555135439*^9}, { 3.761905673139873*^9, 3.761905677027369*^9}},ExpressionUUID->"00fb5091-6da9-4a8a-81d7-\ cd53c6b0896b"], Cell[TextData[{ "We do not gauge fix \[Alpha] (and thus Ng), because we use the gauge \ freedom later to fix sonic point ", Cell[BoxData[ FormBox[ SubscriptBox["x", "s"], TraditionalForm]],ExpressionUUID-> "38a31c21-d0d3-4841-a8a0-e19404ca631d"], " = 1. Also, regularity of the data implies that the power series must only \ contain even powers of r for\na, \[Alpha] and \[Rho] (scalars) and odd \ powers for V (vector). To avoid conical singularity, we also require a(0) = \ 1. A posteriori, we see that Ng0[-1] and \[Omega]0[2] will remain arbitrary \ and we thus solve all other coefficients\nas functions of them. KEEP the \ powertermOrigin even!! (See notes). Otherwise, the psUnkownAtOrigin is not \ correct. At each odd term, we can solve only for an additional new variable \ (a coefficient of V) \ncompared to the previous even order. At each new even \ power, we can solve for a new \[OpenCurlyDoubleQuote]set of\ \[CloseCurlyDoubleQuote] coefficients, which is nicer and simpler." }], "Text", CellChangeTimes->{{3.7525009355659447`*^9, 3.752501034022213*^9}, { 3.7525010974762993`*^9, 3.752501112203932*^9}, {3.7525055114631577`*^9, 3.7525056059086*^9}, {3.7528240986369247`*^9, 3.752824134148138*^9}, { 3.752824246992353*^9, 3.752824252592046*^9}, {3.752824465899531*^9, 3.752824567391101*^9}, {3.753449422512581*^9, 3.753449443407971*^9}, { 3.7539672884895153`*^9, 3.753967371263363*^9}, {3.761905561165205*^9, 3.761905578351048*^9}, {3.761905691285829*^9, 3.761905693593145*^9}, { 3.76190597701252*^9, 3.761905980154962*^9}},ExpressionUUID->"cd8f6fcc-c202-4926-8da8-\ 2cf628f194e5"], Cell["\<\ For defining the below properly, we first need to clear all the symbolized \ structures and we re - instore the notation by evaluating the Initialization \ cells afterwards:\ \>", "Text", CellChangeTimes->{{3.760023463872653*^9, 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We expect to remain two free \ variables. The singularity at x = 0, will be used to parametrize ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["V", "c"], "[", "1", "]"}], TraditionalForm]], ExpressionUUID->"b53a4d5b-2000-4aff-8354-533e7d547aff"], " in terms of ", Cell[BoxData[ FormBox[ SuperscriptBox["Ng", "c"], TraditionalForm]],ExpressionUUID-> "e5853fda-d27e-4962-9689-676ce4d45633"], "[-1] \n\t(note that we could have done it the other way around). 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The above does not show it, \ because it stops at 0th order, but we also have a -1th order equation in V. \ We get two 1st order eqn (coming from Ng\[CloseCurlyQuote] and V\ \[CloseCurlyQuote] respectively) and \na two 2nd order equation coming from A\ \[CloseCurlyQuote] and \[Omega]\[CloseCurlyQuote]. For n=2, then, we have 5 \ equations! Whereas we only want 4 (the leading orders in the expansion of \ the variables)! By comparing with the n=3 case, we see that the 1st order \ equation in V\[CloseCurlyQuote] is actually incomplete at n=2. \nIndeed, it \ misses contribution from ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "c"], TraditionalForm]],ExpressionUUID-> "814b1905-4e36-4c6c-af11-61e47734bfdb"], "[3]. We therefore need to rescale this properly. The pattern continues like \ that at higher orders. We added an extra line to make sure we do not get any \ extra equations. \nThis is normal as the n+1th line should be ignored in our \ PS expansion." }], "Text", CellChangeTimes->{{3.752501326307472*^9, 3.7525013614524937`*^9}, { 3.7525051884509487`*^9, 3.752505204817548*^9}, {3.752505632059435*^9, 3.7525056326352787`*^9}, {3.752505997119279*^9, 3.7525061400054283`*^9}, { 3.752559701884975*^9, 3.752559720708479*^9}, {3.752831575293808*^9, 3.752831637277501*^9}, {3.752831698208342*^9, 3.7528317026803427`*^9}, { 3.7528317746155663`*^9, 3.75283179363125*^9}, {3.753445402757533*^9, 3.753445426134675*^9}, {3.753967605369322*^9, 3.753967645929418*^9}, { 3.757077151543598*^9, 3.7570771647592077`*^9}, {3.761387078786558*^9, 3.761387099858328*^9}, {3.761387167846876*^9, 3.761387168481619*^9}, { 3.761906124838778*^9, 3.761906145269644*^9}},ExpressionUUID->"69b6517f-b8ee-40e0-a73c-\ 17ef734cf4c3"], Cell[TextData[{ "We start by rescaling the V\[CloseCurlyQuote] equations by x. We can then \ solve for the leading order in V (we can already do that at n=1), which will \ now be a 0th order equation. We can solve the leading order equations of the \ other 3 variables, with the exception for \[Omega], \nwhich needs to wait \ until n=4. In a sense, the resolution of the coefficients in \[Omega] are \ always \[OpenCurlyDoubleQuote]one step behind\[CloseCurlyDoubleQuote] the \ others. We will therefore need scale it one point down. So, schematically: at \ 0th order in equations, we now solve for leading term in V\[CloseCurlyQuote], \ \nat 1st order, solve leading terms in N and A, then at 2nd order, solve for \ 2nd coeff. of V. At 3rd order, can finally solve for leading order in \ \[Omega]. The pattern afterwards is clear. (See notes for more details). \n\ In summary: at n=1 (0th order eqn), solve for ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "c"], TraditionalForm]],ExpressionUUID-> "556759ea-17bc-46b0-9803-e2539a0b7c2f"], "[-1], at n=2 (1st order eqn), solve for ", Cell[BoxData[ FormBox[ SuperscriptBox["N", "c"], TraditionalForm]],ExpressionUUID-> "40318cd0-c469-43a4-888f-05fe58ce2fa9"], "[1] and ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "c"], TraditionalForm]],ExpressionUUID-> "dc290322-9169-487b-8341-0363f242ad1b"], "[2]. For n=3 (2nd order eqn), solve for ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "c"], TraditionalForm]],ExpressionUUID-> "1a60401a-9c6a-42d4-a8fb-83e948fdd562"], "[3], n=4 (3rd order eqn), for ", Cell[BoxData[ FormBox[ SuperscriptBox["N", "c"], TraditionalForm]],ExpressionUUID-> "8e1c4501-bc3f-4dee-a012-0a896655f97f"], "[3], ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "c"], TraditionalForm]],ExpressionUUID-> "eab1d663-0eb8-42a0-ac26-7542b6cd975a"], "[4], ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Omega]", "c"], TraditionalForm]],ExpressionUUID-> "3c05a955-9881-4ec0-9afb-8edab7d7fb66"], "[4], etc...\n Also, we will now suppress the explicit variables name in the \ naming of the EOM as this is self-understood." }], "Text", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQB2IQ/fxx5fzTn187ZsRfWAWizYSvRp8B0lJtdYkgWu31 swwQbRURlQOiMxhrW0C0gjv/JBDNECY1HUSzGveuB9F/pH7vBtGThC0Og+hp NfmnQXT73eozINpG5fs3EL3vvsVvEP2j/mxx2NfXjsuWa5SB6MU2t3oLgPQJ Q71+EO1RdSPjK5AOCI/uAtENDx+dV/7+2vH3ooTbIFrpvOkjEH063eY1mP/a 9AOINmpvY1YB0hf8OzaB6G1PTuwH0cveyjIo/HjtyJRwgAtEb5uzVhBEW/k7 iYPoHZMaFEB0XPB7VRDN1T6nCETfdHxfDuaX/Ojd9fu1I8ui/AkgetorwTC+ f68dvygVRYLoh+o3T7uxvXF0X5d3GUSn8avYurC/cexbH+YAogGVQtBR "],ExpressionUUID->"5be77b04-821c-46a7-938a-4dee3ce16e3f"], Cell["\<\ Because it is a 1st order ODE, the nth term in one variable can affect the \ n-1th order equations due to the derivative. 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Cell[CellGroupData[{ Cell["Power series solution at sonic point", "Chapter", CellChangeTimes->{{3.750571111969203*^9, 3.7505711568185463`*^9}, 3.750572766281311*^9, 3.750572818958763*^9, {3.7517153490998*^9, 3.751715361883449*^9}, {3.752318208771442*^9, 3.7523182168754168`*^9}, 3.752499952574058*^9, {3.75283214828432*^9, 3.7528321517320337`*^9}, { 3.753528839580628*^9, 3.753528839876255*^9}, {3.761387354640627*^9, 3.761387354705515*^9}, {3.769957163402075*^9, 3.76995716349942*^9}, { 3.829113760310672*^9, 3.829113761171769*^9}},ExpressionUUID->"b9932e49-2fb9-4eb4-abbc-\ 8807674986a9"], Cell[TextData[{ "Now, we solve near the sonic boundary. \nWe solve the Power Series \ expansion first for the Full (4 variables) ODEs, then the Reduced (3 \ variables) ODEs and see if the results match.\n", StyleBox["Made everything below the Reduced ODEs section non-evaluable this \ time.", FontColor->RGBColor[1, 0, 0]], " Again, results are the same but full system is a bit faster and neater." }], 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We don\[CloseCurlyQuote]t \ need to expand, say in even powers (which was due to regularity ar origin) \ because we are \[OpenCurlyDoubleQuote]faraway\[CloseCurlyDoubleQuote] from \ it. The only free parameter should be \[Rho]S[0], so it is the only one we \ wont/cant solve for and all other coefficients will be solved as functions of \ it. The other 0th order coefficient will be fixed via analyticity condition \ at x = 1. For convenience, we have arranged the unknowns in a matrix. 1st column are \ the Ng\[CloseCurlyQuote]s, 2nd column the \[Omega]\[CloseCurlyQuote]s and 3rd \ column the V\[CloseCurlyQuote]s.\ \>", "Text", CellChangeTimes->{{3.752489900270915*^9, 3.752489950637496*^9}, { 3.753989169154962*^9, 3.7539892380854273`*^9}, 3.755857977621173*^9, { 3.761907943030121*^9, 3.761907951550375*^9}},ExpressionUUID->"248f0f1a-73a6-49fd-abf0-\ fd30f7c5474c"] }, Closed]], Cell[CellGroupData[{ Cell["Full ODEs", "Section", CellChangeTimes->{{3.7524906596299*^9, 3.752490685149213*^9}, { 3.752495922245822*^9, 3.7524959242377234`*^9}, {3.75318526943401*^9, 3.75318527613909*^9}, {3.75398897469769*^9, 3.753989005764893*^9}},ExpressionUUID->"1ed8dbe2-86dd-4794-9376-\ f724e9349273"], Cell[TextData[{ "We reinforce the sonic boundary to be at x = 1. This corresponds to the \ \[OpenCurlyDoubleQuote]left\[CloseCurlyDoubleQuote] (for positive t) (past) \ \[OpenCurlyDoubleQuote]sonic cone\[CloseCurlyDoubleQuote]. We then want our \ fluid to stay to the right of it, i.e. t > -r => x < 1. This is good as we \ want later to match with the solution at r = 0 => x = 0. \nThis forces the \ denominator of \[Omega]\[CloseCurlyQuote] and V\[CloseCurlyQuote] to vanish. \ Regularity implies that we also need the respective numerators to vanish. We \ shall view this as conditions on 0th order terms ", Cell[BoxData[ FormBox[ SuperscriptBox["Ng", "s"], TraditionalForm]],ExpressionUUID-> "e4ccd1ce-8627-4d43-8324-d207d3b05bf2"], "[0], ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Omega]", "s"], TraditionalForm]],ExpressionUUID-> "017311bc-aa9e-4bb9-9b9d-1e3038f95d7b"], "[0], ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "s"], TraditionalForm]],ExpressionUUID-> "08219708-5c70-41bc-9bc7-f6c96801bd41"], "[0]. \nWe have 3 equations (both denominators vanish simultaneously), but \ we shall see that 1 is redundant, leaves us with, say, ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "s"], TraditionalForm]],ExpressionUUID-> "b47e495c-58b6-4d82-8189-f2182f469ebd"], "[0] arbitrary. Afterwards, we manually solve the 1st order equations (in \ terms of ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "s"], TraditionalForm]],ExpressionUUID-> "ed206d1f-ca5e-410b-89a0-24129242f55a"], "[0])." }], "Text", CellChangeTimes->{{3.752497511594502*^9, 3.752497576548102*^9}, { 3.753190934450451*^9, 3.753190940682167*^9}, {3.75398925708788*^9, 3.753989261881328*^9}, {3.753989312909066*^9, 3.753989333582686*^9}, { 3.755857222869049*^9, 3.755857244596541*^9}, {3.7570801209423237`*^9, 3.757080140591591*^9}, {3.760031623260597*^9, 3.760031672206676*^9}, { 3.762005648369154*^9, 3.76200570032365*^9}},ExpressionUUID->"7db552c8-0c71-4aaf-9efa-\ 6b32312ca1ef"], Cell[CellGroupData[{ Cell["Sonic regularity constraint", "Subsection", CellChangeTimes->{{3.751633776621109*^9, 3.7516337925647917`*^9}, { 3.7516339045779247`*^9, 3.7516339072821503`*^9}, {3.751633946480803*^9, 3.751633946768655*^9}, {3.75170140483239*^9, 3.751701410216175*^9}, { 3.7517154090581093`*^9, 3.751715414242201*^9}, {3.752490756419004*^9, 3.7524907589466553`*^9}, {3.753517237628868*^9, 3.753517242900262*^9}, { 3.753989355516618*^9, 3.753989360528226*^9}, {3.753991899071797*^9, 3.7539919045533667`*^9}, 3.761385394804317*^9, {3.761907981732567*^9, 3.7619079820520372`*^9}},ExpressionUUID->"e14b26aa-b21d-4468-b436-\ 8275e52cfe1c"], Cell[CellGroupData[{ Cell["Set up", "Subsubsection", CellChangeTimes->{{3.761380542581998*^9, 3.761380558912282*^9}, 3.7619087773388853`*^9},ExpressionUUID->"2f2c9426-2c0d-4d6a-a574-\ 512b94cb1b1c"], Cell[TextData[{ "\t\tWe impose the sonic boundary at x = ", Cell[BoxData[ FormBox[ SubscriptBox["x", "s"], TraditionalForm]],ExpressionUUID-> "adbc6602-fe7f-425f-b454-973097dc9d00"], ". This implies that the denominator in \[Omega]\[CloseCurlyQuote] and V\ \[CloseCurlyQuote] needs to vanish at that point. Note that in these \ variables, the sonic point conditions actually do not depend on ", Cell[BoxData[ FormBox[ SubscriptBox["x", "s"], TraditionalForm]],ExpressionUUID-> "16e670b6-670e-4581-8296-46fbcaf7bfa0"], "! We first extract the relevant parts,\t\t" }], "Text", CellChangeTimes->{{3.751630969261623*^9, 3.75163110998632*^9}, { 3.751631363235888*^9, 3.7516313838681087`*^9}, {3.751633829788105*^9, 3.751633837931753*^9}, {3.751634181662211*^9, 3.7516341843842783`*^9}, { 3.751793999042799*^9, 3.7517940295624943`*^9}, {3.752323574904912*^9, 3.752323617232377*^9}, {3.753190947802554*^9, 3.7531909555640917`*^9}, { 3.7539893730268497`*^9, 3.753989380403895*^9}, {3.753989425512183*^9, 3.753989435055068*^9}, {3.75399012949205*^9, 3.753990129920924*^9}, { 3.754048117943527*^9, 3.754048119176136*^9}, {3.7558572810683517`*^9, 3.755857290930365*^9}, {3.757079989043118*^9, 3.757079990867262*^9}, { 3.760031679197109*^9, 3.76003169065277*^9}, {3.76138056012901*^9, 3.761380561197027*^9}, {3.761908779635324*^9, 3.761908787022829*^9}},ExpressionUUID->"b088db97-e10e-4cb2-99be-\ 7c5daeaef1be"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{ RowBox[{ RowBox[{ "metricMatterEqnFullCSSSol\[Omega]xNumerator", " ", "=", " ", 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"fc3b5be9-0b78-4ed8-ad35-52881d1e2fa0"], "[0]. The above form indeed 3 independent equations. We FIRST solve for the \ denominator, in terms of Ng:" }], "Text", CellChangeTimes->{{3.7504986376172953`*^9, 3.750498665639257*^9}, 3.750570986582698*^9, 3.750571099472274*^9, 3.750572766282735*^9, 3.750572818960101*^9, {3.750577480444645*^9, 3.750577485428295*^9}, { 3.751631727914044*^9, 3.751631744410145*^9}, {3.751631818263823*^9, 3.751631874422924*^9}, {3.751632303251562*^9, 3.751632320435667*^9}, { 3.7516342047383947`*^9, 3.7516342052746897`*^9}, {3.752323473092393*^9, 3.752323489323636*^9}, {3.7531910090159683`*^9, 3.753191012896144*^9}, { 3.7531911269884033`*^9, 3.7531911314442*^9}, {3.7539907408195667`*^9, 3.75399074288745*^9}, {3.755857314161788*^9, 3.7558573494009113`*^9}, { 3.7600317519310427`*^9, 3.7600317732578993`*^9}, 3.761380578395031*^9, { 3.761908837161977*^9, 3.7619088373716583`*^9}},ExpressionUUID->"68cc9a36-48ff-4fa9-8fda-\ 9c5bcde7cdbd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t\t", RowBox[{ RowBox[{ RowBox[{"assumptionscsVxsRule", " ", "=", " ", 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We want positive (ingoing one) since \[Alpha] should be positive from \ the metric and hence Ng should. \n\t\tWe easily see that at fluid rest ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["V", "s"]}], TraditionalForm]],ExpressionUUID-> "5e60d426-06a8-4efb-8842-dec2db4b0f99"], "[0] = 0), only the 2nd solution is indeed positive. On a different way, the \ first solution would make ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["A", "s"], "[", "0", "]"}], " ", "<", " ", "1"}], ",", " ", RowBox[{ "which", " ", "would", " ", "make", " ", "the", " ", "evolution", " ", "of", " ", "it", " ", "decreasing"}], ",", " ", RowBox[{"which", " ", "is", " ", "not", " ", "good"}], ",", " ", RowBox[{ RowBox[{"since", " ", "we", " ", "want", " ", "A"}], " ", "\[Rule]", " ", RowBox[{ RowBox[{"1", " ", "as", " ", "x"}], " ", "\[Rule]", " ", "0."}]}]}]}], TraditionalForm]],ExpressionUUID->"7b6bb405-56fa-4564-aa05-92ab46c2be71"], " \n\t\t(See Koike,Hara,Adachi: Renorm. crit. collapse). As a side note, we \ could have tried solving for V, but this makes the equations much messier \ later one." }], "Text", CellChangeTimes->{{3.751632382344653*^9, 3.7516324872871733`*^9}, { 3.751634210730715*^9, 3.751634220985821*^9}, {3.751701520093741*^9, 3.7517015512765493`*^9}, {3.7517016268841543`*^9, 3.7517016304588223`*^9}, {3.751701970937688*^9, 3.751701979169952*^9}, { 3.751794929034843*^9, 3.751794931564363*^9}, {3.751795099487308*^9, 3.7517952794790487`*^9}, {3.751795328944036*^9, 3.7517953338334427`*^9}, { 3.751795373032508*^9, 3.751795476517556*^9}, {3.751795885801647*^9, 3.751795899906461*^9}, 3.752323463702092*^9, {3.752323499486107*^9, 3.7523235606978703`*^9}, {3.7531910290632772`*^9, 3.753191045742922*^9}, { 3.7531911433321953`*^9, 3.753191149556164*^9}, {3.753990842311647*^9, 3.753990912120009*^9}, {3.7558580395710163`*^9, 3.7558580575702333`*^9}, { 3.7566466843687067`*^9, 3.75664683117279*^9}, {3.760031874976253*^9, 3.760031918312676*^9}, {3.761380582953517*^9, 3.761380585628957*^9}, { 3.761380678709147*^9, 3.761380702468356*^9}, {3.761908865489538*^9, 3.761908884656135*^9}},ExpressionUUID->"9ae0bddd-e8b5-4ed8-8cad-\ 1c45cd981f3e"] }, Closed]], Cell[CellGroupData[{ Cell[" Update numerators", "Subsubsection", CellChangeTimes->{{3.7613807186630898`*^9, 3.761380728370981*^9}, { 3.761908891719585*^9, 3.7619088918972063`*^9}},ExpressionUUID->"d24cd59f-70a8-4ab0-a67f-\ 6a9686a544f8"], Cell["\<\ \t\tWe now plug this back into the Numerators, before solving for them. Note \ that both expressions agree, up to some none zero factors!\ \>", "Text", CellChangeTimes->{{3.7613806677212343`*^9, 3.761380674586423*^9}, { 3.761908894335579*^9, 3.7619088945240107`*^9}},ExpressionUUID->"a98940c0-c14e-473c-ad53-\ 7f8e2c55babf"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{ RowBox[{ RowBox[{ "metricMatterEqnFullCSSSol\[Omega]xNumeratorSonic2", " ", "=", " ", "\[IndentingNewLine]", "\t\t", RowBox[{"Simplify", "[", RowBox[{ "metricMatterEqnFullCSSSol\[Omega]xNumeratorSonic", " ", "/.", " ", "RegularitySolRuleNgSonicFull"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t\t", RowBox[{ RowBox[{ "metricMatterEqnFullCSSSolVxNumeratorSonic2", " ", "=", "\[IndentingNewLine]", "\t\t", RowBox[{"Simplify", "[", RowBox[{ "metricMatterEqnFullCSSSolVxNumeratorSonic", " ", "/.", " ", "RegularitySolRuleNgSonicFull"}], "]"}]}], ";"}]}]}]], "Input", CellChangeTimes->{{3.751632638859167*^9, 3.751632726296672*^9}, { 3.7516327654573507`*^9, 3.75163277325883*^9}, {3.7516328064334497`*^9, 3.751632806934677*^9}, {3.751634228137005*^9, 3.751634230380248*^9}, { 3.751701809188848*^9, 3.751701829301333*^9}, {3.751795832365608*^9, 3.751795833010848*^9}, {3.751795913339673*^9, 3.751795914944742*^9}, { 3.7517966139610567`*^9, 3.7517966150620937`*^9}, {3.752318224964692*^9, 3.752318247010881*^9}, 3.752318293444228*^9, 3.7523187750785713`*^9, { 3.7523241809592113`*^9, 3.752324182023737*^9}, {3.752491311255464*^9, 3.752491371311639*^9}, {3.7531910485847483`*^9, 3.753191053391582*^9}, { 3.753191153792495*^9, 3.753191156819992*^9}, {3.753434365949514*^9, 3.7534343670700483`*^9}, {3.7534344788861933`*^9, 3.753434481074493*^9}, { 3.7539909173394613`*^9, 3.753990974292053*^9}, {3.753991052929474*^9, 3.753991053951503*^9}, {3.753991129236251*^9, 3.753991157091551*^9}, { 3.754047566012898*^9, 3.7540475801814423`*^9}, {3.7570801650625343`*^9, 3.757080173477091*^9}, {3.761380586461546*^9, 3.761380587951816*^9}, { 3.761908896439795*^9, 3.761908903471182*^9}}, CellLabel-> "In[177]:=",ExpressionUUID->"297e257f-3da5-499c-8236-3301aa72d544"], Cell["\<\ \t\t So, it turns out that we have only 1 constraint left. We solve \ \[Omega]. \ \>", "Text", CellChangeTimes->{{3.7516327485356407`*^9, 3.751632760608041*^9}, { 3.751632811566071*^9, 3.751632855781659*^9}, {3.7516334232705297`*^9, 3.7516334236865273`*^9}, {3.751634231985557*^9, 3.751634232569899*^9}, { 3.75170189280976*^9, 3.7517019254839277`*^9}, {3.751722641064404*^9, 3.751722793492221*^9}, 3.751795942736215*^9, {3.7523241981993713`*^9, 3.7523242187422647`*^9}, {3.753191055662442*^9, 3.753191060422621*^9}, { 3.7531911600881557`*^9, 3.75319116043546*^9}, {3.753991550398368*^9, 3.753991551524585*^9}, 3.7613805896270933`*^9, {3.7619089060950193`*^9, 3.761908906256282*^9}},ExpressionUUID->"c972446d-3657-41f4-9ef2-\ 7c0a6002ebaf"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Set numerator to 0 and define temporary sonic regularity rule\ \>", "Subsubsection", CellChangeTimes->{{3.761380777525202*^9, 3.76138082717581*^9}, { 3.761908914222425*^9, 3.7619089143913918`*^9}},ExpressionUUID->"b8cd83a7-b163-4402-a982-\ 5f381af72c1a"], Cell[BoxData[ 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3.7600319505408916`*^9, 3.7600319555472097`*^9}, {3.761380590458037*^9, 3.761380592471874*^9}, {3.7619089190355663`*^9, 3.761908930406561*^9}}, CellLabel-> "In[179]:=",ExpressionUUID->"13cd6586-fbbf-4d12-b015-e8fdc063c1ca"] }, Closed]], Cell[CellGroupData[{ Cell[" Sanity check", "Subsubsection", CellChangeTimes->{{3.761380839323369*^9, 3.7613808456470003`*^9}, { 3.761908937686885*^9, 3.7619089378619843`*^9}},ExpressionUUID->"e239ab3f-6612-401a-86d9-\ 51ae948b47cf"], Cell["\<\ \t\tAs a sanity check, let us see if the numerators and Denominators indeed \ vanish:\ \>", "Text", CellChangeTimes->{{3.7517018869113092`*^9, 3.7517019182134647`*^9}, { 3.753191071357951*^9, 3.7531910734440107`*^9}, {3.753191170283062*^9, 3.7531911710832787`*^9}, {3.753991553269432*^9, 3.753991554847187*^9}, 3.761380594651163*^9, 3.761908939650197*^9},ExpressionUUID->"f954272c-126b-4040-9214-\ eb6238ecd6aa"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t\t", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ 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(Alternatively, we can fix ", Cell[BoxData[ FormBox[ SuperscriptBox["Ng", "s"], TraditionalForm]],ExpressionUUID-> "b0c1d84c-412d-425d-855a-90f5b53bc45c"], "[0] = 1 (equivalent to \[Alpha][0] = 1) and keep ", Cell[BoxData[ FormBox[ SubscriptBox["x", "s"], TraditionalForm]],ExpressionUUID-> "42dc3e43-c014-49ea-ad7a-f12dc7bdbe8a"], " arbitrary). \n\t\tNote that the above implies that -1 < ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "s"], TraditionalForm]],ExpressionUUID-> "7a8a988f-9144-4eee-b88d-db060d43afa7"], "[0] < ", Cell[BoxData[ FormBox[ SubscriptBox["c", "s"], TraditionalForm]],ExpressionUUID-> "f880d82d-3884-49c2-9a7c-5f86d61e2d53"], "!" }], "Text", CellChangeTimes->{{3.751796914537217*^9, 3.751796935005559*^9}, { 3.751797461135613*^9, 3.7517974616800127`*^9}, {3.752324355018371*^9, 3.752324373514011*^9}, {3.7523244138247547`*^9, 3.752324453632958*^9}, { 3.752494985049614*^9, 3.752495016840777*^9}, {3.75319121834591*^9, 3.7531912290258217`*^9}, {3.753991879437326*^9, 3.7539918865113573`*^9}, { 3.755858159518751*^9, 3.755858167046603*^9}, {3.757080205524097*^9, 3.7570802063881903`*^9}, {3.75975877878259*^9, 3.75975881091751*^9}, { 3.760032016242977*^9, 3.7600320420742064`*^9}, {3.761380603583701*^9, 3.761380605479175*^9}, {3.761909014988003*^9, 3.76190902087586*^9}},ExpressionUUID->"2578070c-4111-4cef-84e6-\ 1256f5699c47"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Solution of 1st order coefficients", "Subsection", CellChangeTimes->{{3.7524958455754766`*^9, 3.7524958774703913`*^9}, { 3.7539893628943663`*^9, 3.753989364654965*^9}, {3.7539919197433147`*^9, 3.75399192208421*^9}, {3.754046751195315*^9, 3.7540467526297007`*^9}, { 3.761907999459414*^9, 3.761908025602763*^9}},ExpressionUUID->"cab7057f-751a-4a26-b38b-\ 74d02e5ba8f8"], Cell[CellGroupData[{ Cell[" Check structure of equations", "Subsubsection", CellChangeTimes->{{3.761383574660701*^9, 3.761383588575479*^9}, { 3.761909030555045*^9, 3.761909030739476*^9}},ExpressionUUID->"13970737-779f-4a6d-a189-\ fa8e45f41814"], Cell["\<\ \t\tThe sonic condition causes the 0th order coefficient for \[Omega]\ \[CloseCurlyQuote] and V\[CloseCurlyQuote] to vanish. 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This is to be expected since the regularity \ condition precisely make the 0th order terms of the denominator and numerator \ to vanish. \t\tSo, we therefore \[OpenCurlyDoubleQuote]start\[CloseCurlyDoubleQuote] at \ 1st order straight away. Now, we need to bring the above structure into the \ correct form. Intuitively, the Ng\[CloseCurlyQuote] and A\[CloseCurlyQuote] \ equations are \[OpenCurlyDoubleQuote]1 step behind\[CloseCurlyDoubleQuote] \ compared to the other equations (or the other 2 are one step ahead). \t\tSo, we shall take, say for n=1, the two 0th order equations (Ng\ \[CloseCurlyQuote] and A\[CloseCurlyQuote]) and the two \[Omega]\ \[CloseCurlyQuote] and V\[CloseCurlyQuote] equations at 1st order, with the \ same pattern at higher orders. \t\tThis is anyway obvious, since you should always try to solve the \ \[OpenCurlyDoubleQuote]first\[CloseCurlyDoubleQuote] lowest orders of each \ equation before going to next order. Here, we shall choose to bring the \ \[Omega]\[CloseCurlyQuote] and V\[CloseCurlyQuote] equations down, instead of \ bringing Ng\[CloseCurlyQuote] equation up. See notes for more details.\ \>", "Text", CellChangeTimes->{{3.751708080841434*^9, 3.751708307781413*^9}, { 3.751708367819371*^9, 3.7517084043549747`*^9}, {3.752494260058723*^9, 3.752494341078596*^9}, {3.752498408224419*^9, 3.752498409792859*^9}, { 3.752498444759883*^9, 3.752498484443273*^9}, {3.7524985257394657`*^9, 3.752498540587482*^9}, {3.752498622470182*^9, 3.752498659318143*^9}, { 3.752501477808817*^9, 3.752501479401825*^9}, {3.752504420931697*^9, 3.752504476832473*^9}, {3.753191256314522*^9, 3.753191268594078*^9}, { 3.7534590401385107`*^9, 3.7534590475790453`*^9}, {3.754041908562928*^9, 3.7540419546612253`*^9}, 3.755858197487213*^9, {3.756103574354474*^9, 3.756103597135755*^9}, {3.7619091745810537`*^9, 3.761909186384363*^9}},ExpressionUUID->"6bb76752-b1e3-4cfe-a385-\ 66bac34e9458"] }, Closed]], Cell[CellGroupData[{ Cell[" Rescale and separate equations", "Subsubsection", CellChangeTimes->{{3.761383654961781*^9, 3.761383681668605*^9}, { 3.761909050722743*^9, 3.761909050901842*^9}},ExpressionUUID->"6ae4ca20-a88b-4c4a-9115-\ 26a55d5d0419"], Cell["\<\ \t\tWe make a matrix of type (order,variables). 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The above form indeed 2 independent equations. We FIRST solve for the \ denominator, in terms of Ng:" }], "Text", Evaluatable->False, CellChangeTimes->{{3.7504986376172953`*^9, 3.750498665639257*^9}, 3.750570986582698*^9, 3.750571099472274*^9, 3.750572766282735*^9, 3.750572818960101*^9, {3.750577480444645*^9, 3.750577485428295*^9}, { 3.751631727914044*^9, 3.751631744410145*^9}, {3.751631818263823*^9, 3.751631874422924*^9}, {3.751632303251562*^9, 3.751632320435667*^9}, { 3.7516342047383947`*^9, 3.7516342052746897`*^9}, {3.752323473092393*^9, 3.752323489323636*^9}, 3.753191350394313*^9, {3.7535172992598963`*^9, 3.7535172996777143`*^9}, {3.754048845492395*^9, 3.754048846653701*^9}, { 3.7558606279354687`*^9, 3.755860636719392*^9}, {3.7595773725407963`*^9, 3.7595773725973463`*^9}, {3.760032372991543*^9, 3.7600323874631157`*^9}, 3.761383297905262*^9, {3.761384819233964*^9, 3.7613848196563377`*^9}, { 3.761385509980126*^9, 3.761385510028817*^9}, {3.7619202627073383`*^9, 3.761920263195665*^9}, 3.761921423098771*^9},ExpressionUUID->"d0924750-3e96-4068-8172-\ 5f50d49b37d3"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{ RowBox[{ RowBox[{"assumptionscsVxsRule", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["c", "s"], ">", "0"}], ",", RowBox[{ RowBox[{"-", "1"}], "<", RowBox[{ SuperscriptBox["V", "s"], "[", "0", "]"}], "<", "1"}], ",", " ", RowBox[{ SubscriptBox["x", "s"], " ", ">", " ", "0"}]}], "}"}]}], ";"}], "\n", "\t\t", RowBox[{ RowBox[{"RegularitySolRuleNgSonicReducedtmp", " ", "=", " ", RowBox[{"Flatten", "[", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Solve", "[", " ", RowBox[{ RowBox[{ "metricMatterEqnReducedCSSSol\[Omega]xDenominatorSonic", "\[Equal]", " ", "0"}], ",", RowBox[{ SuperscriptBox["Ng", "s"], "[", "0", "]"}]}], "]"}], ",", "assumptionscsVxsRule"}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t\t", RowBox[{ RowBox[{"RegularitySolRuleNgSonicReduced", " ", "=", " ", RowBox[{"RegularitySolRuleNgSonicReducedtmp", "[", RowBox[{"[", "2", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t\t", RowBox[{ RowBox[{"Clear", "[", "RegularitySolRuleNgSonicReducedtmp", "]"}], ";"}], " "}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.750404112934824*^9, 3.750404124905686*^9}, { 3.750404815951026*^9, 3.750404817042903*^9}, {3.750404993988186*^9, 3.750404999998363*^9}, {3.750408849836544*^9, 3.750408853364132*^9}, 3.750420078174383*^9, 3.750571099471613*^9, 3.750572766281939*^9, 3.7505728189593477`*^9, {3.750577262971881*^9, 3.750577268050606*^9}, 3.7516326307114477`*^9, {3.751634206442685*^9, 3.751634208722937*^9}, { 3.751701586134488*^9, 3.751701609867333*^9}, {3.751723279704723*^9, 3.751723280487804*^9}, {3.751723663560145*^9, 3.751723684205306*^9}, 3.751723728798403*^9, {3.751724053150432*^9, 3.751724055819357*^9}, { 3.7517935774165277`*^9, 3.751793577516655*^9}, {3.751794291261701*^9, 3.7517943047712383`*^9}, {3.751794363196021*^9, 3.751794406456609*^9}, { 3.7517947809441643`*^9, 3.7517948064790993`*^9}, {3.751794840758872*^9, 3.751794841917797*^9}, 3.751794877294304*^9, {3.751794925413569*^9, 3.751794967579645*^9}, 3.751795086064232*^9, {3.751795299009449*^9, 3.7517953106089077`*^9}, {3.751795507601611*^9, 3.7517955144842787`*^9}, { 3.751795698535685*^9, 3.751795713383498*^9}, {3.751795809453845*^9, 3.751795813156496*^9}, {3.7517974166927*^9, 3.751797427801705*^9}, { 3.751798492042906*^9, 3.7517984942394*^9}, {3.7518087465287657`*^9, 3.751808779853878*^9}, {3.751808907174011*^9, 3.7518089086533537`*^9}, { 3.7523179631470013`*^9, 3.752318015453751*^9}, {3.752318070099513*^9, 3.752318091231449*^9}, {3.752323431873869*^9, 3.752323432470282*^9}, { 3.752323634415495*^9, 3.7523236410157013`*^9}, {3.752491290847983*^9, 3.7524913002046337`*^9}, {3.7524913510856247`*^9, 3.75249135264071*^9}, { 3.752494749076261*^9, 3.752494750210211*^9}, {3.7524979883381987`*^9, 3.752498003149981*^9}, {3.75319135131225*^9, 3.753191352949912*^9}, { 3.753434577810357*^9, 3.753434588626893*^9}, {3.7535173001820917`*^9, 3.753517305635524*^9}, {3.754048623512866*^9, 3.75404885514629*^9}, { 3.755860642872917*^9, 3.755860675094027*^9}, {3.75708003025139*^9, 3.757080034546*^9}, {3.760032391670918*^9, 3.760032401839917*^9}, { 3.761383298458013*^9, 3.761383300946897*^9}, {3.761385512444525*^9, 3.761385512530559*^9}, {3.761920265355753*^9, 3.76192027823503*^9}, { 3.76192142379171*^9, 3.761921428607596*^9}}, CellLabel-> "In[275]:=",ExpressionUUID->"830768e3-a7ea-40e6-992c-639f021d28ed"], Cell[TextData[{ "\t\tWe picked the second solution for ", Cell[BoxData[ FormBox[ SuperscriptBox["Ng", "s"], TraditionalForm]],ExpressionUUID-> "612010d8-77b8-42ce-9690-2278a1be1ee6"], "[0]. We want positive (ingoing one) since \[Alpha] should be positive from \ the metric and hence Ng should. \n\t\tWe easily see that at fluid rest ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["V", "s"]}], TraditionalForm]],ExpressionUUID-> "8d994b72-7904-4f4e-930d-5a3cea0fdb82"], "[0] = 0), only the 2nd solution is indeed positive. As a side note, we \ could have tried solving for V, but this makes the equations much messier \ later one." }], "Text", Evaluatable->False, CellChangeTimes->{{3.751632382344653*^9, 3.7516324872871733`*^9}, { 3.751634210730715*^9, 3.751634220985821*^9}, {3.751701520093741*^9, 3.7517015512765493`*^9}, {3.7517016268841543`*^9, 3.7517016304588223`*^9}, {3.751701970937688*^9, 3.751701979169952*^9}, { 3.751794929034843*^9, 3.751794931564363*^9}, {3.751795099487308*^9, 3.7517952794790487`*^9}, {3.751795328944036*^9, 3.7517953338334427`*^9}, { 3.751795373032508*^9, 3.751795476517556*^9}, {3.751795885801647*^9, 3.751795899906461*^9}, 3.752323463702092*^9, {3.752323499486107*^9, 3.7523235606978703`*^9}, {3.753191354069824*^9, 3.753191355630114*^9}, { 3.7531913880381804`*^9, 3.7531913969009943`*^9}, {3.7535173086671343`*^9, 3.753517311588192*^9}, {3.7540488644142323`*^9, 3.754048888369265*^9}, { 3.7558606786301126`*^9, 3.755860679157854*^9}, {3.7570800396500187`*^9, 3.7570800495536413`*^9}, {3.760032410270453*^9, 3.760032415528373*^9}, { 3.761383302225624*^9, 3.7613833047861423`*^9}, {3.7613834132709312`*^9, 3.761383415365999*^9}, {3.761384837767576*^9, 3.7613848378801394`*^9}, { 3.761385514883883*^9, 3.761385514933728*^9}, {3.76192027931994*^9, 3.761920283779388*^9}, {3.761921430159387*^9, 3.7619214308322573`*^9}},ExpressionUUID->"a9ae330b-9374-476e-91c5-\ f63707b89304"] }, Closed]], Cell[CellGroupData[{ Cell["Update numerators", "Subsubsection", Evaluatable->False, CellChangeTimes->{{3.761383438592333*^9, 3.761383448443883*^9}, { 3.7613848399370203`*^9, 3.761384840035325*^9}, {3.761385517899497*^9, 3.76138551799233*^9}, {3.7619202067646513`*^9, 3.7619202069410553`*^9}, 3.761921386632256*^9},ExpressionUUID->"8f1f440e-c28a-41c7-a394-\ e8035877472a"], Cell["\<\ \t\tWe now plug this back into the Numerators, before solving for them. Note \ that both expressions agree, up to some none zero factors!\ \>", "Text", Evaluatable->False, CellChangeTimes->{{3.76138341756986*^9, 3.761383422069153*^9}, { 3.761384846272893*^9, 3.761384846335104*^9}, {3.761385524243329*^9, 3.7613855242970057`*^9}, {3.7619202944100924`*^9, 3.76192029486633*^9}, 3.7619214361267567`*^9},ExpressionUUID->"525cb217-7f0d-486e-bb27-\ b409f2f9b757"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{ RowBox[{ RowBox[{ "metricMatterEqnReducedCSSSol\[Omega]xNumeratorSonic2", " ", "=", " ", RowBox[{"Simplify", "[", " ", RowBox[{ "metricMatterEqnReducedCSSSol\[Omega]xNumeratorSonic", " ", "/.", " ", "RegularitySolRuleNgSonicReduced"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t\t", RowBox[{ RowBox[{"metricMatterEqnReducedCSSSolVxNumeratorSonic2", " ", "=", " ", RowBox[{"Simplify", "[", " ", RowBox[{ "metricMatterEqnReducedCSSSolVxNumeratorSonic", " ", "/.", " ", "RegularitySolRuleNgSonicReduced"}], "]"}]}], ";"}], " "}]}]], "Input",\ Evaluatable->False, CellChangeTimes->{{3.751632638859167*^9, 3.751632726296672*^9}, { 3.7516327654573507`*^9, 3.75163277325883*^9}, {3.7516328064334497`*^9, 3.751632806934677*^9}, {3.751634228137005*^9, 3.751634230380248*^9}, { 3.751701809188848*^9, 3.751701829301333*^9}, {3.751795832365608*^9, 3.751795833010848*^9}, {3.751795913339673*^9, 3.751795914944742*^9}, { 3.7517966139610567`*^9, 3.7517966150620937`*^9}, {3.752318224964692*^9, 3.752318247010881*^9}, 3.752318293444228*^9, 3.7523187750785713`*^9, { 3.7523241809592113`*^9, 3.752324182023737*^9}, {3.752491311255464*^9, 3.752491371311639*^9}, {3.752498011913493*^9, 3.752498038360862*^9}, { 3.753191400630712*^9, 3.753191406588729*^9}, {3.753517311955295*^9, 3.753517314810788*^9}, {3.7540489046165733`*^9, 3.7540489349402943`*^9}, { 3.754049168615984*^9, 3.754049174268333*^9}, {3.7558633800377483`*^9, 3.755863380574836*^9}, {3.757080058745829*^9, 3.7570800671686153`*^9}, { 3.761383306253007*^9, 3.761383307729895*^9}, {3.761385042187134*^9, 3.761385072024251*^9}, {3.7619202972638397`*^9, 3.761920306978138*^9}, { 3.7619214368786507`*^9, 3.761921437599535*^9}}, CellLabel-> "In[279]:=",ExpressionUUID->"cd14c063-f535-4a4f-a0b3-243c69a667db"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t\t", RowBox[{"Simplify", "[", RowBox[{ "metricMatterEqnReducedCSSSol\[Omega]xNumeratorSonic2", "/", "metricMatterEqnReducedCSSSolVxNumeratorSonic2"}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.7613850759998198`*^9, 3.7613850784558764`*^9}, { 3.7619203121958113`*^9, 3.761920314703621*^9}, 3.7619214383264*^9}, CellLabel-> "In[281]:=",ExpressionUUID->"d04849d9-f758-416d-a794-631816e2308f"], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["\[Omega]", "s"], "[", "0", "]"}], SubscriptBox["c", "s"]]], "Output", Evaluatable->False, CellChangeTimes->{3.761385211516705*^9, 3.761385481044297*^9, 3.827063409937451*^9, 3.827063447228957*^9}, CellLabel-> "Out[281]=",ExpressionUUID->"bc4c0a61-938f-4a5b-b88f-17323b8e6e17"] }, Open ]], Cell["\tSo, it turns out that we have only 1 constraint left.", "Text", Evaluatable->False, CellChangeTimes->{{3.761385090283956*^9, 3.7613850946232023`*^9}, { 3.761385529334221*^9, 3.761385529409397*^9}, {3.761920317679038*^9, 3.761920318176381*^9}},ExpressionUUID->"cc1968fa-d8d5-4167-bbcf-\ 2f2efc1fd81a"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Set numerator to 0 and define temporary sonic regularity rule\ \>", "Subsubsection", Evaluatable->False, CellChangeTimes->{{3.761383468062817*^9, 3.761383469707265*^9}, { 3.761384848736917*^9, 3.761384848814237*^9}, {3.761385534698935*^9, 3.761385534769643*^9}, {3.7619202081967373`*^9, 3.761920208380892*^9}, 3.761921388800338*^9},ExpressionUUID->"5b62fbac-83c2-42af-9426-\ 124265a610c7"], Cell["\t\tWe solve \[Omega]. 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This is to be expected since the regularity \ condition precisely make the 0th order terms of the denominator and numerator \ to vanish. So, we therefore \[OpenCurlyDoubleQuote]start\ \[CloseCurlyDoubleQuote] at 1st order straight away. \t\tNow, we need to bring the above structure into the correct form. \ Intuitively, the Ng\[CloseCurlyQuote] equation is \[OpenCurlyDoubleQuote]1 \ step behind\[CloseCurlyDoubleQuote] compared to the other equations (or \ rather the other 2 are one step ahead). So, we shall take, say for n=1, \t\tthe 0th order equation (Ng\[CloseCurlyQuote]) and the two \[Omega]\ \[CloseCurlyQuote] and V\[CloseCurlyQuote] equations at 1st order, with the \ same pattern at higher orders. Here, we shall choose to bring the \[Omega]\ \[CloseCurlyQuote] and V\[CloseCurlyQuote] equations down, instead of \ bringing Ng\[CloseCurlyQuote] equation up. 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Concerning ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Omega]", "s"], TraditionalForm]],ExpressionUUID-> "200ebc08-09b9-4653-bc8b-8c6a04b2bfa1"], "[1] and ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "s"], TraditionalForm]],ExpressionUUID-> "15f7d5ba-1f30-485f-b1be-cda20ac5b5be"], "[1], they only agree for the last two values of ", Cell[BoxData[ FormBox[ SuperscriptBox["V", "s"], TraditionalForm]],ExpressionUUID-> "531f3e60-bd0f-43aa-8ae2-a9b293057478"], "[1]. Note that this means that ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "1"], TraditionalForm]], ExpressionUUID->"b0b01c3d-7b4c-42ae-b600-9854575a01f5"], " is then not admissible.\nIndeed, the fact that the mass term ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["A", "s"]}], TraditionalForm]],ExpressionUUID-> "2c62240a-21b1-4122-9290-a7f86682ceda"], "[1]) does not agree, \nmeans that at least one of them (in fact the one \ coming from the Full eqn) will not be consistent with the alg eqn. Indeed \ note that the Full eqns only provide a single possible value for ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "s"], TraditionalForm]],ExpressionUUID-> "67fdfe5f-fe26-482a-8124-19749b35b11e"], "[1]. \nThis value (see above) coincide will the value we get for ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "s"], TraditionalForm]],ExpressionUUID-> "83bbb82b-62b0-4236-a141-9f3b885c95f9"], "[1] from the Reduced system, except for ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "1"], TraditionalForm]], ExpressionUUID->"4bb11fb0-0694-4aab-9499-122be5e8052e"], ". Since by construction, the ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "s"], TraditionalForm]],ExpressionUUID-> "06283957-757d-48ac-bf1b-242f02838cfd"], "[1] in the Reduced system is obtained by making the algebraic eqn to\n\ explicitly vanish at 1st order, this means that the expression for ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "s"], TraditionalForm]],ExpressionUUID-> "f4e68b7a-e5f0-4902-b111-f199c92059fa"], "[1] from the Full system will NOT be consistent with the alg eqn in the \ case of ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "1"], TraditionalForm]], ExpressionUUID->"2f32ac99-e171-4160-8e36-9ce32122bda3"], ". All this points towards concluding that ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "1"], TraditionalForm]], ExpressionUUID->"c610b150-8c22-488d-8858-867b97e9a76e"], " is not an\nadmissible solution. The ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "1"], TraditionalForm]], ExpressionUUID->"54cf08a8-ef75-44d2-b019-97d690027639"], " solution is also special in that it SEPARATELY satisfies the LHS and RHS \ of the V\[CloseCurlyQuote] equation. See notes for more details." }], "Text", Evaluatable->False, CellChangeTimes->{{3.753186901738398*^9, 3.7531871061980267`*^9}, { 3.753189976903243*^9, 3.753190057257998*^9}, {3.753190177513302*^9, 3.753190492611515*^9}, {3.753191450707202*^9, 3.753191453626914*^9}, { 3.753433626919795*^9, 3.753433636968464*^9}, {3.753433667356173*^9, 3.753433668238179*^9}, {3.7535175069275084`*^9, 3.753517527839368*^9}, { 3.754054395434456*^9, 3.754054474565196*^9}, {3.7558617772213173`*^9, 3.755861864938383*^9}, {3.760032841305871*^9, 3.760032916201009*^9}, { 3.761921882922226*^9, 3.7619219127309027`*^9}, {3.827063485285668*^9, 3.827063508729949*^9}, {3.827064061705308*^9, 3.827064069771036*^9}},ExpressionUUID->"af2e25e7-052f-48fa-ad4a-\ 7969485ac38c"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Analysis of the ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "2"], TraditionalForm]], ExpressionUUID->"fd320ef6-0a5a-4afa-9cb8-3227b0028703"], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "3"], TraditionalForm]], ExpressionUUID->"c2c07e8e-07e2-4321-9480-66f1e487100a"], " solutions" }], "Section", CellChangeTimes->{{3.7558661000399647`*^9, 3.755866195725531*^9}, { 3.760032948238413*^9, 3.760032953237344*^9}},ExpressionUUID->"e44d1dbd-5211-42f7-b24e-\ e02a563fdc03"], Cell[TextData[{ "Unevaluable. This will also need the full system solution.\nTo find which \ of the remaining two ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{"VS", "[", "1", "]"}], "i"], TraditionalForm]],ExpressionUUID-> "438ecd77-32e2-4915-976e-dd9e27558a07"], " is the most relevant, we shall investigate the causal structure of a point \ near x=1. If we denote by ", Cell[BoxData[ FormBox[ SuperscriptBox["S", "a"], TraditionalForm]],ExpressionUUID-> "3e95a1df-60b5-4907-a020-1b64c3c4bb84"], " the tangent vector at the sonic point, \none can calculate the ratio ", Cell[BoxData[ FormBox[ SubscriptBox["S", "R"], TraditionalForm]],ExpressionUUID-> "cf72e475-ba5d-43de-aa5f-103967096700"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["S", "x"], "/", SuperscriptBox["S", "\[Tau]"], " "}], TraditionalForm]],ExpressionUUID-> "e9bc61e0-9afd-4333-bda7-767548478a8c"], "as a function of x and X, A. 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Below, we worked out (see notes) the equation for ", Cell[BoxData[ FormBox[ SubscriptBox["S", "R"], TraditionalForm]],ExpressionUUID-> "2c114f61-7266-4a52-b8ea-a39353f731ca"], "." }], "Text", CellChangeTimes->{{3.753602500713159*^9, 3.753602540648884*^9}, { 3.7536026971162*^9, 3.753602861319728*^9}, {3.754066204116906*^9, 3.75406622693637*^9}, {3.754066257217752*^9, 3.754066257827744*^9}, { 3.754066304678314*^9, 3.754066332495263*^9}, 3.755862064652424*^9, { 3.755865929007018*^9, 3.755865959717629*^9}, {3.755880757930688*^9, 3.755880758538714*^9}, {3.761386722661539*^9, 3.761386730541181*^9}, { 3.761922004129739*^9, 3.761922007589579*^9}, {3.8270642056063843`*^9, 3.8270642160701227`*^9}},ExpressionUUID->"191c0162-1f76-45a2-956c-\ 98ec2ff194da"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"SonicConeEqntmp", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Alpha]", "[", RowBox[{"\[Tau]", ",", "x"}], "]"}], "^", "2"}], "/", RowBox[{ RowBox[{"(", 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We now solve for ", Cell[BoxData[ FormBox[ SubscriptBox["S", "R"], TraditionalForm]],ExpressionUUID-> "fdca2445-d7a0-44bc-864f-a6c2fdda7066"], "." }], "Text", CellChangeTimes->{{3.7540666680588408`*^9, 3.754066731203116*^9}, 3.7540671800848293`*^9, 3.7619220206925087`*^9},ExpressionUUID->"ae6ddc17-c94c-419e-8070-\ 833e9b5afd58"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SonicConeEqnCSSSolRuleStmp", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"SonicConeEqnCSS", " ", "==", " ", "0"}], ",", SubscriptBox["S", "R"]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.753539806129757*^9, 3.7535398456464157`*^9}, { 3.753539945423623*^9, 3.753539948867365*^9}, {3.75354257358517*^9, 3.753542575838758*^9}, {3.753545312125894*^9, 3.753545312452627*^9}, { 3.753602994199839*^9, 3.75360300996319*^9}, {3.753603083443469*^9, 3.753603084512601*^9}, {3.754066738424027*^9, 3.75406678681144*^9}, { 3.754067186061426*^9, 3.754067187161105*^9}, {3.7540673875470133`*^9, 3.754067412602597*^9}, 3.7619220222794123`*^9}, CellLabel-> "In[214]:=",ExpressionUUID->"9100b228-28d4-4baf-98a8-f44d38269840"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["S", "R"], "\[Rule]", FractionBox[ RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ RowBox[{"Ng", "[", "x", "]"}], " ", RowBox[{"(", RowBox[{ SubscriptBox["c", "s"], "-", RowBox[{"V", "[", "x", "]"}]}], ")"}]}], "+", RowBox[{ SubscriptBox["c", "s"], " ", RowBox[{"V", "[", "x", "]"}]}]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SubscriptBox["c", "s"], " ", RowBox[{"V", "[", "x", "]"}]}]}]]}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["S", "R"], "\[Rule]", FractionBox[ RowBox[{"x", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{ SubscriptBox["c", "s"], " ", RowBox[{"V", "[", "x", "]"}]}], "+", RowBox[{ RowBox[{"Ng", "[", "x", "]"}], " ", RowBox[{"(", RowBox[{ SubscriptBox["c", "s"], "+", RowBox[{"V", "[", "x", "]"}]}], ")"}]}]}], ")"}]}], RowBox[{"1", "+", RowBox[{ SubscriptBox["c", "s"], " ", RowBox[{"V", "[", "x", "]"}]}]}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.761571285223886*^9, 3.8290413308688097`*^9, 3.829112672845639*^9, 3.829124903657786*^9, 3.8291995583482647`*^9}, CellLabel-> "Out[214]=",ExpressionUUID->"9bf3813a-7dac-4db6-9924-b643248f4b8f"] }, Open ]], Cell[TextData[{ "The 2 solutions corresponds to the 2 (future) directed sonic-cones. For our \ purposes, we need to pick the\[CloseCurlyDoubleQuote]left\ \[CloseCurlyDoubleQuote] light cone. This is because we wish to find which of \ the VS[0] is the correct solution, and for the left sonic-cone, we need ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "R"], "'"}], TraditionalForm]],ExpressionUUID-> "3394bbd8-d015-4922-b556-f15e51f7ea5b"], " > 0 at x = 1 (see notes). \nIndeed, we have that for the \ \[OpenCurlyDoubleQuote]left\[CloseCurlyDoubleQuote] cone: ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["S", "-"], "x"], TraditionalForm]],ExpressionUUID-> "7f74aad8-6455-474c-9eb9-e14af77eb63d"], " = 0 at the sonic point, negative before and positive after. The right \ sonic-cone always has ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["S", "+"], "x"], TraditionalForm]],ExpressionUUID-> "067adc70-d8ab-4325-ad50-44f8ff3fa934"], " > 0 for all x. For V = 0, the world-line of the fluid is just a straight \ vertical line.This is because V=0 corresponds \nin the initial variables to W \ = 1 (\[Proportional] ", Cell[BoxData[ FormBox[ SuperscriptBox["U", "t"], TraditionalForm]],ExpressionUUID-> "ac04645d-6f01-45a5-84a5-ff39686f0341"], ") and U = 0 (\[Proportional] ", Cell[BoxData[ FormBox[ SuperscriptBox["U", "r"], TraditionalForm]],ExpressionUUID-> "80354f86-7984-4b72-8bb6-5455acb3e104"], "). I.e. corresponds to moving along time direction. Anyway, the fact that V \ = 0 is a straight line means that we restrict the allowed movements to this \ line. \nThis line is good though to check ", Cell[BoxData[ FormBox[ SubscriptBox["S", "R"], TraditionalForm]],ExpressionUUID-> "b77259bb-9cf2-406b-bfde-69ed99486ce9"], " as there are points on that line that have x < 1 and x > 1." }], "Text", CellChangeTimes->{{3.753542473249645*^9, 3.753542561725857*^9}, { 3.75380033411903*^9, 3.753800334279139*^9}, {3.7538007702343607`*^9, 3.753800832200941*^9}, {3.753800863168022*^9, 3.7538009025431337`*^9}, { 3.7538009906685257`*^9, 3.753801210887342*^9}, {3.7538012624770184`*^9, 3.7538012971805964`*^9}, {3.753801558676646*^9, 3.753801606898773*^9}, { 3.754066805504271*^9, 3.754066895472007*^9}, {3.754066929021207*^9, 3.754066974076736*^9}, {3.754067194647152*^9, 3.754067220700453*^9}, { 3.754067253842491*^9, 3.754067313171856*^9}, {3.7540673477088327`*^9, 3.7540673551965923`*^9}, {3.7540675102014427`*^9, 3.7540675538328867`*^9}, { 3.754067599504994*^9, 3.754067685346611*^9}, {3.755870962619528*^9, 3.755871085154006*^9}, {3.758294376391213*^9, 3.758294376662751*^9}, { 3.761922024023211*^9, 3.761922055149508*^9}},ExpressionUUID->"23e99538-2ead-45a2-9fcc-\ 7f367d3342dd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"SonicConeEqnCSSSolRuleStmp", " ", "/.", " ", RowBox[{ RowBox[{"V", "[", "x", "]"}], " ", "\[Rule]", " ", "0"}]}], " ", "/.", " ", "pp"}]], "Input", CellChangeTimes->{{3.753542279120487*^9, 3.753542299205826*^9}, { 3.7535424544443274`*^9, 3.7535424558176527`*^9}, {3.753542531409968*^9, 3.753542538702681*^9}, {3.7536030241026497`*^9, 3.7536030358219547`*^9}, { 3.7536030906764727`*^9, 3.7536030934177723`*^9}, 3.753800338270933*^9, { 3.754067438910407*^9, 3.7540674439523077`*^9}, 3.7619220587013693`*^9}, CellLabel-> "In[215]:=",ExpressionUUID->"ed14e67c-04d5-4cee-87cf-1ece1642d1ac"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["S", "R"], "\[Rule]", RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SubscriptBox["c", "s"], " ", "Ng"}]}], ")"}]}], " ", "x"}]}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["S", "R"], "\[Rule]", RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{ SubscriptBox["c", "s"], " ", "Ng"}]}], ")"}], " ", "x"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.753542293671575*^9, 3.753542299474249*^9}, 3.753542456472653*^9, 3.753542539479394*^9, 3.753545472253368*^9, 3.753603036933791*^9, 3.753800339418112*^9, 3.753801246570944*^9, 3.753802374062644*^9, 3.753805205764016*^9, 3.7540674474881477`*^9, 3.754135281125567*^9, 3.75587676333776*^9, 3.755878462378303*^9, 3.755935874266893*^9, 3.756020589172597*^9, 3.756038340850996*^9, 3.7560426349585733`*^9, 3.756187987759172*^9, 3.756191157264555*^9, 3.7565567792950478`*^9, 3.756565852925199*^9, 3.756805837551759*^9, 3.75681893048251*^9, 3.7569026958774853`*^9, 3.756912440078079*^9, 3.756987070524756*^9, 3.756987418146245*^9, 3.757055027238078*^9, 3.7570645815387278`*^9, 3.757077889474555*^9, 3.7571653179614773`*^9, 3.757166098693283*^9, 3.757168841772999*^9, 3.7571698763703203`*^9, 3.757177334203245*^9, 3.757177496792729*^9, 3.8290413308871307`*^9, 3.8291126728670692`*^9, 3.829124903679305*^9, 3.8291995583687277`*^9}, CellLabel-> "Out[215]=",ExpressionUUID->"5c8e9286-56f8-4a7d-b828-3092b322f139"] }, Open ]], Cell[TextData[{ "The 2nd solution is always positive, characterising ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["S", "+"], "x"], TraditionalForm]],ExpressionUUID-> "e85fc3c0-6713-4837-8c9f-f3a531627f13"], ", while the first can be + or -. In particular, one can solve for ", Cell[BoxData[ FormBox[ SubscriptBox["S", "R"], TraditionalForm]],ExpressionUUID-> "08aa6193-3f21-4e85-afca-ca442164546b"], " = 0, to get the sonic point condition." }], "Text", CellChangeTimes->{ 3.7540669895916567`*^9, {3.754067026444344*^9, 3.754067074844185*^9}, 3.7540677175269403`*^9, 3.761922060227829*^9},ExpressionUUID->"71ab49ff-1c3c-4753-972e-\ e2190194dc6d"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"SonicConeEqnCSSSolRuleS", "=", " ", RowBox[{"SonicConeEqnCSSSolRuleStmp", "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"SonicConeEqnCSSSolS", " ", "=", " ", RowBox[{ SubscriptBox["S", "R"], " ", "/.", " ", "SonicConeEqnCSSSolRuleS"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Clear", "[", "SonicConeEqnCSSSolRuleStmp", "]"}], ";"}], " "}]}], "Input", CellChangeTimes->{{3.753545306990553*^9, 3.753545328851479*^9}, { 3.753545365732441*^9, 3.753545366362941*^9}, {3.753545413513154*^9, 3.7535454361877613`*^9}, {3.7535455641476097`*^9, 3.753545663685976*^9}, { 3.7536030795520983`*^9, 3.753603124135249*^9}, {3.7536194206867332`*^9, 3.753619420762836*^9}, {3.753619478596342*^9, 3.7536194786811523`*^9}, { 3.753801638454071*^9, 3.753801638529443*^9}, {3.753802064703281*^9, 3.7538020768199987`*^9}, {3.753802120157838*^9, 3.753802120290872*^9}, { 3.753805256800791*^9, 3.753805256886001*^9}, {3.754067710273519*^9, 3.75406777919944*^9}, {3.754067810907139*^9, 3.7540678519427423`*^9}, { 3.761922062031807*^9, 3.7619220646624737`*^9}}, CellLabel-> "In[216]:=",ExpressionUUID->"7b434c5b-18af-4a62-bd8e-7d0e7d2480e8"], Cell[BoxData[ FractionBox[ RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ RowBox[{"Ng", "[", "x", "]"}], " ", RowBox[{"(", RowBox[{ SubscriptBox["c", "s"], "-", RowBox[{"V", "[", "x", "]"}]}], ")"}]}], "+", RowBox[{ SubscriptBox["c", "s"], " ", RowBox[{"V", "[", "x", "]"}]}]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SubscriptBox["c", "s"], " ", RowBox[{"V", "[", "x", "]"}]}]}]]], "Output", CellChangeTimes->{3.761571292799364*^9, 3.8290413309087877`*^9, 3.829112672887714*^9, 3.829124903701234*^9, 3.829199558390746*^9}, CellLabel-> "Out[217]=",ExpressionUUID->"4c33aeb6-c8ce-489b-af71-15cc9d21c557"] }, Open ]], Cell[TextData[{ "We now wish to find which values of ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "1", "]"}], "i"], TraditionalForm]], ExpressionUUID->"51154afd-3565-4d34-862c-8bf844c3a68e"], " does ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["S", "R"], "'"}], ")"}], "-"], TraditionalForm]], ExpressionUUID->"f9890078-dceb-4b46-9eb3-5aafe7ff14f7"], " > 0. We therefore differentiate the above. 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1}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.02]}}, RotateLabel->False, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.8290508647710752`*^9, {3.829112675337483*^9, 3.8291127320096827`*^9}, { 3.829124906309074*^9, 3.829124920828487*^9}, 3.829199560896796*^9}, CellLabel-> "Out[226]=",ExpressionUUID->"9408cf6b-ffec-41a6-af8c-1d07a8e632e5"] }, Open ]], Cell[TextData[{ "This is not conclusive on which solutions are allowed. We might have a \ preference for ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "0", "]"}], "3"], TraditionalForm]], ExpressionUUID->"fdc54165-063e-4c40-b502-d00b6dea2590"], " since it is always positive (except for unphysical case or when its value \ becomes complex).\n", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "0", "]"}], "2"], TraditionalForm]], ExpressionUUID->"6fe09a64-8286-46d7-8612-bd5ee8d17f02"], " on the other hand seems to have an odd \[OpenCurlyDoubleQuote]additional \ constraint\[CloseCurlyDoubleQuote] attached to it. Aka, it is NOT always \ positive and so is probably unphysical. Nonetheless, a formal reasoning is \ missing." }], "Text", CellChangeTimes->{{3.7562082239102583`*^9, 3.75620842075863*^9}, { 3.760033229940851*^9, 3.760033236228704*^9}, {3.7619221085905437`*^9, 3.7619221111460533`*^9}},ExpressionUUID->"bdd581f8-e05a-48bf-b814-\ fce091ad4dff"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Shooting for background fields", "Chapter", CellChangeTimes->{{3.75620853969777*^9, 3.7562085412823153`*^9}, { 3.758024198174676*^9, 3.758024205606633*^9}, {3.7613873805674*^9, 3.761387380687524*^9}},ExpressionUUID->"0ebc4fd9-88e1-45f5-9216-\ ebc8454e6da3"], Cell[TextData[{ "Let us proceed to shooting. We consider the reduced system, with the \ initial condition expansion up to 2nd order: X(\[Epsilon]) = X[0]+\[Epsilon] \ X[1]+\[Epsilon]^2 X[2] (except for Ng, since it starts at ", Cell[BoxData[ FormBox[ SuperscriptBox["x", RowBox[{"-", "1"}]], TraditionalForm]],ExpressionUUID-> "c3aa2a1c-f024-4cff-8bae-b17a8c590b44"], "). We then solve the system with NDSolve up to x = 0.5 with those IC \ (initial conditions). \nThis allows us \[OpenCurlyDoubleQuote]to start\ \[CloseCurlyDoubleQuote] away from the degeneracy. The solution formally \ depends on our arbitrary variables ", Cell[BoxData[ FormBox[ SuperscriptBox["Ng", "c"], TraditionalForm]],ExpressionUUID-> "0f6c75a6-d9ed-4b78-a968-1b36db0084ef"], "[-1] and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Omega]", "c"], "[", "2", "]"}], TraditionalForm]], ExpressionUUID->"93c78dc3-5935-40fe-aecb-85ca8cde5550"], ". We will fix it later via FindRoot.\nThe reason as to why we go up to 2nd \ order and not first is because up to 1st order and in radiation fluid, the \ IC, do not depend on ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Omega]", "c"], "[", "2", "]"}], TraditionalForm]], ExpressionUUID->"60faa86e-1f78-4fec-979e-867a4643f1c1"], " (see below)! We will therefore not be able to solve for it later. \nEven \ for non-radiation fluid, it only occurs in the IC of N(x). However, its \ contribution is negligible compared to the first term, as it has factor \ difference of x^2. 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It would \ anyway not make any difference compared to the ~ 0(1/x) term.\ \>", "Text", CellChangeTimes->{{3.757230433227437*^9, 3.7572304790818787`*^9}, { 3.7619223684099693`*^9, 3.761922368705791*^9}},ExpressionUUID->"e3db6f83-7f1f-4730-9454-\ 770db1657741"] }, Closed]], Cell[CellGroupData[{ Cell["IC at sonic point", "Section", CellChangeTimes->{{3.7562087057323647`*^9, 3.756208716227984*^9}, { 3.761922366754848*^9, 3.761922394424192*^9}, 3.7619225296361103`*^9},ExpressionUUID->"27b626e5-7a4d-41c5-8455-\ 3f151d863756"], Cell[TextData[{ "Now doing the same thing for the x = 1 expansion for both of the ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SubscriptBox["V", "S"], "[", "1", "]"}], "2"], TraditionalForm]], ExpressionUUID->"f05093df-da11-4a6d-ae34-04aff4d4f370"], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SubscriptBox["V", "S"], "[", "1", "]"}], "3"], TraditionalForm]], ExpressionUUID->"c9a2d003-f81c-4d11-99cb-bca842774bd6"], " solution. We will check after the shooting which solution is most \ appropriate. WE DO NOT go to second order here, but only up to 1st order. \n\ The reason is that the result changes very little between 1st and 2nd order \ here (unlike above), although computational times increases noticeably. The \ values of our variables after FindRoot only differ by less than ~ 10^(-10). \n\ We wrote the code for it below for reference." }], "Text", CellChangeTimes->{{3.755964665791424*^9, 3.755964760891275*^9}, { 3.75612832284219*^9, 3.756128428391531*^9}, {3.756128470669856*^9, 3.756128492060915*^9}, {3.75620874418843*^9, 3.756208780955592*^9}, { 3.757226717597755*^9, 3.757226733597147*^9}, {3.761922399373554*^9, 3.76192241376814*^9}, {3.761922598023725*^9, 3.761922599906654*^9}},ExpressionUUID->"d8a999f9-5a4a-45f3-bc1f-\ 0c48369a1a97"], Cell[BoxData[{ RowBox[{ RowBox[{"ICUnitOrder1V2", "[", RowBox[{"VS_", ",", "\[Epsilon]_", ",", "cs_"}], "]"}], " ", ":=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["Ng", "s"], "[", "0", "]"}], "+", RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], " ", RowBox[{ SuperscriptBox["Ng", "s"], "[", "1", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["\[Omega]", "s"], "[", "0", "]"}], 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\[OpenCurlyDoubleQuote]midpoint\ \[CloseCurlyDoubleQuote] and solve for the variables ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["Ng", "c"], "[", RowBox[{"-", "1"}], "]"}], TraditionalForm]],ExpressionUUID-> "9e67279b-01d7-4f83-a4a2-0976e241d93c"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Omega]", "c"], "[", "2", "]"}], TraditionalForm]], ExpressionUUID->"58497b75-1df7-4176-9039-0f025d22c063"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "0", "]"}], TraditionalForm]], ExpressionUUID->"eeac605a-780d-42a1-88bb-47bf22161a63"], " so that the mismatch is negligible.\nNote that we have 3 variables to \ solve for and we will get a non-linear system of 3 equation. 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This reinforce the idea that this \ solution is indeed fallacious. The above has been made unevaluatable as a \ consequence.\ \>", "Text", CellChangeTimes->{{3.756210453742305*^9, 3.756210479822237*^9}, { 3.758199251505666*^9, 3.7581992719905024`*^9}, 3.761923014938521*^9},ExpressionUUID->"b4f79348-735b-42ab-8350-\ 0472c3d64e83"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Ngm1\[Omega]2V0Rule", " ", "=", " ", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"metricMatterODEReducedCSSShootingMismatchV3", "[", RowBox[{"myEvaluation", ",", RowBox[{ SuperscriptBox["Ng", "c"], "[", RowBox[{"-", "1"}], "]"}], ",", RowBox[{ SuperscriptBox["\[Omega]", "c"], "[", "2", "]"}], ",", RowBox[{ SuperscriptBox["V", "s"], "[", "0", "]"}], ",", "my\[Epsilon]", ",", "myc"}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["Ng", "c"], "[", RowBox[{"-", "1"}], "]"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[Omega]", "c"], "[", "2", "]"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ 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" solution is likely wrong and the 3rd solution is the right one.\nPlotting \ the whole solution (both left and right) now, to see how they match up at x = \ 0.5. We consider x Ng[x], since Ng[x] blows up at x = 0, making the plot a \ bit nicer. 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to write the the css \ solutions of our fields, which we will from now on denote as: ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["Ng", "css"], ",", SubscriptBox["A", "css"], ",", SubscriptBox["\[Omega]", "css"], ",", SubscriptBox["V", "css"]}]}], TraditionalForm]],ExpressionUUID-> "30664529-c679-4b2c-9bda-3063a7a57dea"], ")." }], "Text", CellChangeTimes->{{3.7699259394336557`*^9, 3.7699259453800364`*^9}, 3.769926012330496*^9},ExpressionUUID->"c288b6d2-d53a-42bd-b31d-\ 57318f94f460"], Cell["\<\ We plug in the expansion for the unperturbed terms. 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Note that left-hand piece is defined \ only up to the evaluation point, typically it should be 0.5, while the \ right-hand piece starts from 0.5 up to 1000 typically.\ \>", "Text", CellChangeTimes->{{3.757678360698846*^9, 3.757678378154587*^9}, { 3.7576790836044827`*^9, 3.75767916310669*^9}, {3.761923579387989*^9, 3.761923582380706*^9}, 3.7619239031699963`*^9, 3.769925869668808*^9, 3.769926014010066*^9},ExpressionUUID->"c14cecaf-d253-4db5-8d86-\ e57b1ca633a8"], Cell[BoxData[{ RowBox[{ RowBox[{"upCoefficientsLeftRule", " ", "=", " ", RowBox[{ RowBox[{"TrueMetricMatterODEReducedCSSShootingLeftExpansionRule", "[", RowBox[{"my\[Epsilon]", ",", "myc"}], "]"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"Ng", " ", "\[Rule]", " ", SubscriptBox["Ng", "css"]}], ",", RowBox[{"\[Omega]", " ", "\[Rule]", " ", SubscriptBox["\[Omega]", "css"]}], ",", RowBox[{"V", " ", "\[Rule]", " ", SubscriptBox["V", "css"]}]}], "}"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"upCoefficientsRightRule", " ", "=", " ", RowBox[{ RowBox[{"TrueMetricMatterODEReducedCSSShootingRightExpansionRule", "[", RowBox[{"my\[Epsilon]", ",", "myc"}], "]"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"Ng", " ", "\[Rule]", " ", SubscriptBox["Ng", "css"]}], ",", RowBox[{"\[Omega]", " ", "\[Rule]", " ", SubscriptBox["\[Omega]", "css"]}], ",", RowBox[{"V", " ", "\[Rule]", " ", SubscriptBox["V", "css"]}]}], "}"}]}]}], ";"}], " "}]}], "Input", CellChangeTimes->{{3.757679078052918*^9, 3.757679080628892*^9}, { 3.757679193161541*^9, 3.7576792372164097`*^9}, 3.757679373891881*^9, { 3.7576794071241207`*^9, 3.7576794480664*^9}, {3.757679501402855*^9, 3.757679506392725*^9}, {3.760090246992351*^9, 3.760090261238442*^9}, { 3.761923585502727*^9, 3.761923591580546*^9}, {3.761923904225457*^9, 3.761923905025908*^9}, {3.769925871512206*^9, 3.769925872876142*^9}, { 3.76992601503918*^9, 3.769926015783251*^9}}, CellLabel-> "In[384]:=",ExpressionUUID->"85b3254e-8c8a-4a44-bd1f-40816e0de14d"], Cell["\<\ We can use the above to define the unperturbed variables over the full range \ in one go, as piecewise functions:\ \>", "Text", CellChangeTimes->{{3.757946911138104*^9, 3.757946960923272*^9}, 3.761923596219204*^9, 3.761923906618148*^9, 3.769925874564166*^9, 3.769926016970294*^9},ExpressionUUID->"b749b57d-d37d-46cf-8fe9-\ 8085b74e5f7e"], Cell[TextData[{ StyleBox["IMPORTANT NOTE", FontColor->RGBColor[1, 0, 0]], " on the below: the input for the rules DO NOT evaluate the the \ Interpolating function at said point. It really it just to tell which \ functions to pick (left or right).\nIt would have been best if we could \ define myNgupRule etc to take string input like \[OpenCurlyDoubleQuote]left\ \[CloseCurlyDoubleQuote], \ \[OpenCurlyDoubleQuote]right\[CloseCurlyDoubleQuote] and output the obvious \ from there; but I could not make this work. \nSo, to evaluate the lhs, put as \ input evaluation/2 and for rhs, just plug in 3/2 evaluation. So, we will \ define those evaluation points below.\n", StyleBox["NOTE THAT", FontColor->RGBColor[1, 0, 0]], " the \[LessEqual] actually matters or else evaluation at that point gives \ 0, which is not true of course! 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IT DOES NOT evaluate at that specific point!! So, \ to evaluate it for the left, just use as argument evaluation/2 (cannot really \ use 0 unfortunately).\nOtherwise, use, say, 3/4 > 1/2. The former makes \ everything directly depend on the evaluation point. Normally, this just \ stays at 0.5. 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3.7698359153391333`*^9}},ExpressionUUID->"2507765c-3b4f-4843-82a7-\ 864f6c1c493f"], Cell[TextData[{ "Recall the gauge freedom t \[Rule] ct that we still have. This allows us to \ arbitrarily rescale \[Alpha] \[Rule] \[Alpha]/c so that \[Alpha]dt is \ functionally the same, say at the origin. Such a change would also rescale \ \[Rho] ", Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{ RowBox[{"-", "2"}], "\[Tau]"}]], TraditionalForm]],ExpressionUUID-> "286b2249-023e-416a-ae46-a41e40f67fd1"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["t", "2"], TraditionalForm]],ExpressionUUID-> "eaa1dde2-ffae-4755-b6c7-90ef16a3cdb7"], "\[Rho] ", Cell[BoxData[ FormBox[ RowBox[{"->", " ", RowBox[{ SuperscriptBox["t", "2"], "\[Rho]"}]}], TraditionalForm]],ExpressionUUID-> "4a1effd4-dc77-4016-a177-3714fb2a0a89"], " ", Cell[BoxData[ FormBox[ SuperscriptBox["c", "2"], TraditionalForm]],ExpressionUUID-> "4659e6f1-ed68-4c5a-9829-1e691d01d526"], ".\nOn the other hand, note that by definition also that V is invariant. \ Indeed, the denominator is \[Alpha] ", Cell[BoxData[ FormBox[ FractionBox["dt", "d\[Tau]"], TraditionalForm]],ExpressionUUID-> "d69c0967-817c-437d-84d4-8e4bb57eac8c"], " and the combination \[Alpha] dt is invariant under a t transformation.\n\ Overall, the relation N = ", Cell[BoxData[ FormBox[ FractionBox["a", "\[Alpha]x"], TraditionalForm]],ExpressionUUID-> "bda8284e-8f5f-43a0-b0ef-a15ac0b163c1"], " stays functional invariant and so does the formula for \[Omega] = ", Cell[BoxData[ FormBox[ RowBox[{"4", " ", "\[Pi]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["x", "2"], " ", SuperscriptBox["e", RowBox[{ RowBox[{"-", "2"}], "\[Tau]"}]]}], TraditionalForm]],ExpressionUUID-> "a894ccdb-c4a7-4733-8650-bf0cd5740123"], "\[Rho]. On the other hand, note that this also causes for all the variables \ to be re-evaluated at f(x) \[Rule] f(x/c).\nThis is the point of view of \ active transformation. For example, consider N(x) = ", Cell[BoxData[ FormBox[ FractionBox["a", "\[Alpha]x"], TraditionalForm]],ExpressionUUID-> "e6ec1618-da8b-452c-911d-cb17b9b7eda4"], " \[Rule] ", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["ac", "2"], "\[Alpha]x"], TraditionalForm]],ExpressionUUID-> "2f79281e-9bbf-415f-a643-12127dd85b2c"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["c", "2"], RowBox[{"N", "(", RowBox[{"x", "/", "c"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "e33498ca-c2b7-4b81-a31d-aa4816eee98b"], ". Then, from inverting, \[Alpha] (x)= xa(x)N(x). Apply transformation, now \ that we know how each\npart transforms, \[Rule] \[Alpha](x/c) = x a(x/c) \ N(x/c). And now, if we do a change of variable this just reads: \[Alpha](x) = \ c x a(x) N(x). This \[Alpha] indeed satisfies \[Alpha](0)=1. 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As for the unperturbed case, the \ expansion near the origin works to \[OpenCurlyDoubleQuote]high\ \[CloseCurlyDoubleQuote] orders, but at Unit, Mathematica has already some \ trouble for the linear order calculation. The end result should not-depend on its higher \ order terms (as for up case), and so we will focus on the 1st order.\ \>", "Text", CellChangeTimes->{{3.757228717725967*^9, 3.757228828061035*^9}, 3.7572425348760433`*^9},ExpressionUUID->"455a3c68-41a3-43b0-832b-\ d0b0464f5327"], Cell[CellGroupData[{ Cell["Power series solution at Origin", "Section", CellChangeTimes->{{3.757077316343977*^9, 3.757077326848049*^9}},ExpressionUUID->"e314f05e-4c57-490c-adb9-\ b824320f4a3d"], Cell[CellGroupData[{ Cell[" \tPower Series Ansatz near x = 0", "Subsection", CellChangeTimes->{{3.750576547548811*^9, 3.750576569048753*^9}, { 3.750576831917136*^9, 3.750576845901026*^9}, {3.751633877418722*^9, 3.751633880451054*^9}, {3.751633921977611*^9, 3.7516339265855722`*^9}, { 3.753191519080305*^9, 3.753191572614909*^9}, {3.753191676851458*^9, 3.753191684475602*^9}, {3.753191747979604*^9, 3.753191757537088*^9}, 3.75396724197537*^9, {3.756991411590618*^9, 3.756991435269307*^9}, 3.7570759830671463`*^9, 3.7570773479722977`*^9},ExpressionUUID->"19922caa-d5ad-41e0-a190-\ 4f6fe72cb3e3"], Cell[TextData[{ "\t\tRecall that most variables are logs! In order to avoid conical \ singularity, we impose regularity at the center: ", Cell[BoxData[ FormBox[ SubscriptBox[ OverscriptBox["A", "_"], "p"], TraditionalForm]],ExpressionUUID-> "e737273b-9df8-4450-9125-c251fd3e5ff1"], "(\[Tau],0) = 0. \n\t\tOne also imposes the condition that ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "e96d7ca7-8b7b-405d-a09b-af7e43c0d9a8"], "(\[Tau],0) = 0, since the velocity should be an odd function of x. This \ fixes the leading order terms in the variables\[CloseCurlyQuote] respective \n\ \t\texpansions. 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3.757079163614317*^9}, {3.757079242663577*^9, 3.7570793949288597`*^9}, {3.757080966452602*^9, 3.757080986819748*^9}, { 3.75708106693504*^9, 3.7570810695645657`*^9}, {3.757164653743819*^9, 3.757164653838283*^9}, {3.757164756371114*^9, 3.757164818240946*^9}}, CellLabel-> "In[247]:=",ExpressionUUID->"86ef611a-1ba3-40cf-87ba-e545024b5054"], Cell["\<\ \t\tAlso, we write down at each order, which variables we will solve for:\ \>", "Text", CellChangeTimes->{{3.757076585354084*^9, 3.757076604287318*^9}, 3.757077435340796*^9},ExpressionUUID->"d4f58567-3e3e-443a-996c-\ f671ca900ab3"], Cell[BoxData[ RowBox[{"\t\t ", RowBox[{ RowBox[{"PSpUnknownUnit", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ SuperscriptBox[ SubscriptBox["Ngbar", "p1"], "EM"], "[", "n", "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "PSpOrderUnit"}], "}"}]}], "]"}], ",", RowBox[{"Table", "[", RowBox[{ RowBox[{ SuperscriptBox[ SubscriptBox["Abar", "p1"], "EM"], "[", "n", "]"}], ",", 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CellChangeTimes->{{3.757401431323309*^9, 3.757401439931026*^9}},ExpressionUUID->"4420a2fe-4968-4acb-9751-\ ccfd381cb933"], Cell["\<\ Old version. THIS IS NOT CORRECT. I keep it though for exercise purpose and a \ reminder on what can go wrong when dealing with singular behaviour. All 3 methods below agree. \ \>", "Text", CellChangeTimes->{{3.757401451762467*^9, 3.757401579803711*^9}},ExpressionUUID->"c4369cd2-c16b-4466-b087-\ 29ac5e5504c3"], Cell[CellGroupData[{ Cell[TextData[{ " \t", StyleBox["Sonic boundary constraint (Old)", "Section"] }], "Section", CellChangeTimes->{{3.751633776621109*^9, 3.7516337925647917`*^9}, { 3.7516339045779247`*^9, 3.7516339072821503`*^9}, {3.751633946480803*^9, 3.751633946768655*^9}, {3.75170140483239*^9, 3.751701410216175*^9}, { 3.7517154090581093`*^9, 3.751715414242201*^9}, {3.752490756419004*^9, 3.7524907589466553`*^9}, {3.753517237628868*^9, 3.753517242900262*^9}, { 3.753989355516618*^9, 3.753989360528226*^9}, {3.753991899071797*^9, 3.7539919045533667`*^9}, {3.7573299291600246`*^9, 3.757329934128089*^9}, { 3.757332651408251*^9, 3.7573326533121634`*^9}, {3.75733366859134*^9, 3.757333668935115*^9}}, CellLabel-> "In[341]:=",ExpressionUUID->"4441084a-f337-46b1-818b-227384ab581c"], Cell["\<\ \t\t\tThis was the old attempt that calculating the constraint. The logic is \ technically the same as the two section2 below, \t\t\tbut a bit more obtuse... but it makes an interesting exercise, so I \ keep it anyway.\ \>", "Text", CellChangeTimes->{{3.757332662543704*^9, 3.757332718958242*^9}, { 3.757333582930153*^9, 3.7573336285691013`*^9}, {3.75733385657892*^9, 3.757333865537324*^9}, {3.757401563719219*^9, 3.7574015719242697`*^9}},ExpressionUUID->"e839c2e1-0937-44a6-8b72-\ ec682377d2c8"], Cell[BoxData[" "], "Input", Evaluatable->False, CellChangeTimes->{3.7573327241202497`*^9}, CellLabel-> "In[343]:=",ExpressionUUID->"26545c50-1d9d-47f4-84d7-f3e947860eda"], Cell[TextData[{ "\t\t\tNote that this time around, the denominator does not depend on the \ perturbed variables! And we know that it vanished at x = ", Cell[BoxData[ FormBox[ SubscriptBox["x", "S"], TraditionalForm]],ExpressionUUID-> "06635d56-28db-46b0-a377-432d2115af23"], " = 1.\n\t\t\tSo, we need to adjust both numerators so that it also vanishes \ at that point. This allows us to leave ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox[ OverscriptBox["A", "_"], "p1"], "EM"], TraditionalForm]],ExpressionUUID-> "36cda5e8-beb0-48c7-b965-b1ae081293e0"], "[0] as the only free parameter (N is gauge fixed).\n\t\t\tNOTE: it will not \ suffice to take the numerator, since the denominator has the same singular \ structure as for unperturbed case, but WITH A SQUARE!\n\t\t\tSo, the \ numerator, as it is, will have a factor of the denominator with it (BUT ONLY \ AT X=1 !!), and thus is zero when evaluating it. \n\t\t\tIn general, it has \ not the same structure as the denominator. In particular, it will have A and \ \[Omega] factors, which can be related to V via the regularity condition, but\ \n\t\t\tonly when we are at the x=1, otherwise not." }], "Text", CellChangeTimes->{{3.751630969261623*^9, 3.75163110998632*^9}, { 3.751631363235888*^9, 3.7516313838681087`*^9}, {3.751633829788105*^9, 3.751633837931753*^9}, {3.751634181662211*^9, 3.7516341843842783`*^9}, { 3.751793999042799*^9, 3.7517940295624943`*^9}, {3.752323574904912*^9, 3.752323617232377*^9}, {3.753190947802554*^9, 3.7531909555640917`*^9}, { 3.7539893730268497`*^9, 3.753989380403895*^9}, {3.753989425512183*^9, 3.753989435055068*^9}, {3.75399012949205*^9, 3.753990129920924*^9}, { 3.754048117943527*^9, 3.754048119176136*^9}, {3.7558572810683517`*^9, 3.755857290930365*^9}, {3.757079884030293*^9, 3.757079974307239*^9}, { 3.757080249287842*^9, 3.757080250378694*^9}, {3.757080664581623*^9, 3.757080754572687*^9}, {3.757081089872242*^9, 3.757081092311843*^9}, { 3.757150585599409*^9, 3.7571507163242826`*^9}, {3.7572549511172047`*^9, 3.7572550495624447`*^9}, {3.757329914532289*^9, 3.757329914907309*^9}, 3.7573327412387*^9, {3.75733281226753*^9, 3.757332813165228*^9}, { 3.757333886903462*^9, 3.757333887266518*^9}},ExpressionUUID->"b8c40f95-7bcd-458c-8cca-\ 1d323303ef81"], Cell["\<\ \t\t\tFirst, let us check out again the expression that causes the \ irregularity:\ \>", "Text", CellChangeTimes->{{3.757151512020505*^9, 3.757151536880537*^9}, 3.757154850419971*^9, {3.7571565388020163`*^9, 3.7571565626576567`*^9}}, CellLabel-> "In[347]:=",ExpressionUUID->"cb1bbe89-9356-49b7-aec8-168ced250e8c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{"Simplify", "[", RowBox[{"Factor", "[", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubsuperscriptBox["c", "S", "2"]}], ")"}], " ", RowBox[{ SubscriptBox["Ng", "CSS"], "[", "x", "]"}], " ", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}], "+", RowBox[{ SubsuperscriptBox["c", "S", "2"], " ", SuperscriptBox[ RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SubscriptBox["Ng", "CSS"], "[", "x", "]"}], "2"], " ", RowBox[{"(", RowBox[{ SubsuperscriptBox["c", "S", "2"], "-", SuperscriptBox[ RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}], "2"]}], ")"}]}]}], "]"}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.757151123521572*^9, 3.7571511399902477`*^9}, { 3.757156634985817*^9, 3.7571566363044987`*^9}, {3.7573299169746923`*^9, 3.7573299173460083`*^9}, {3.7573327330960207`*^9, 3.757332733471299*^9}, { 3.7573338898440647`*^9, 3.7573338902097473`*^9}},ExpressionUUID->"d710c5bc-e66d-4181-9bcb-\ 643cc8096b78"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ RowBox[{ SubscriptBox["Ng", "CSS"], "[", "x", "]"}], " ", RowBox[{"(", RowBox[{ SubscriptBox["c", "S"], "-", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}], ")"}]}], "+", RowBox[{ SubscriptBox["c", "S"], " ", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{ SubscriptBox["c", "S"], " ", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}], "+", RowBox[{ RowBox[{ SubscriptBox["Ng", "CSS"], "[", "x", "]"}], " ", RowBox[{"(", RowBox[{ SubscriptBox["c", "S"], "+", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}], ")"}]}]}], ")"}]}]], "Output", CellChangeTimes->{{3.7571511260842943`*^9, 3.757151140276463*^9}, 3.757165352055003*^9, 3.757166100092539*^9, 3.757166256098274*^9, 3.757168860945973*^9, 3.75716989635842*^9, 3.757177353568802*^9, 3.757177516386002*^9, 3.757227618798678*^9, 3.757244931416913*^9, 3.7572499159161453`*^9, 3.7572552290498734`*^9, 3.757333708169385*^9}, CellLabel-> "Out[347]=",ExpressionUUID->"c1773728-357c-4c54-b79f-0cea3ed0d492"] }, Closed]], Cell["\<\ \t\t\tIt is the first term that is 0 once we apply the regularity condition:\ \>", "Text", CellChangeTimes->{{3.757156567812686*^9, 3.757156585904283*^9}, { 3.757156637871191*^9, 3.757156638488461*^9}}, CellLabel-> "In[349]:=",ExpressionUUID->"338b40c7-355f-49cb-a024-ff27c18eb117"], Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{ RowBox[{"singularityexpressiontmp", " ", "=", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ RowBox[{ SubscriptBox["Ng", "CSS"], "[", "x", "]"}], " ", RowBox[{"(", RowBox[{ SubscriptBox["c", "S"], "-", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}], ")"}]}], "+", RowBox[{ SubscriptBox["c", "S"], " ", RowBox[{ SubscriptBox["V", "CSS"], "[", "x", "]"}]}]}], ")"}]}], ";"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.757156594755953*^9, 3.757156642695706*^9}, { 3.757156748068253*^9, 3.7571567514361343`*^9}}, CellLabel-> "In[349]:=",ExpressionUUID->"4ff1a16b-18b8-48c6-9d8a-718a5702837f"], Cell["\<\ \t\t\tWe evaluate the Numerators/Denominators at x = 1, which by definition \ is really just a change of labelling of the up and p variables \t\t\t(only the 0th order coefficients count). We also rewrite the needed \ regular conditions rules with the correct namings. \t\t\tAgain, note the special treatment of N due to gauge fixing. \ \>", "Text", CellChangeTimes->{{3.757156670085856*^9, 3.757156777818534*^9}, { 3.757162415834313*^9, 3.757162416630414*^9}, {3.757162447625548*^9, 3.757162480160224*^9}}, CellLabel-> "In[350]:=",ExpressionUUID->"2856a790-d0b1-4b1e-b534-374257596196"], Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{ RowBox[{ RowBox[{"PSupUnitRule", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["Ng", "CSS"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SubscriptBox["Ng", "CSS1"], "[", "0", "]"}]}], "]"}]}], ",", RowBox[{ SubscriptBox["A", "CSS"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SubscriptBox["A", "CSS1"], "[", "0", "]"}]}], "]"}]}], " ", ",", " ", RowBox[{ SubscriptBox["\[Omega]", "CSS"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SubscriptBox["\[Omega]", "CSS1"], "[", "0", "]"}]}], "]"}]}], ",", " ", "\[IndentingNewLine]", "\t\t\t\t\t\t\t\t", RowBox[{ SubscriptBox["V", "CSS"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SubscriptBox["V", "CSS1"], "[", "0", "]"}]}], "]"}]}]}], "}"}]}], ";"}], "\n", "\t\t\t ", RowBox[{ RowBox[{"PSpUnitRule", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ SubscriptBox["Ngbar", "p"], "EM"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", "0"}], "]"}]}], ",", RowBox[{ SuperscriptBox[ SubscriptBox["Abar", "p"], "EM"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SuperscriptBox[ SubscriptBox["Abar", "p1"], "EM"], "[", "0", "]"}]}], "]"}]}], " ", ",", " ", "\[IndentingNewLine]", "\t\t\t\t\t\t\t\t", RowBox[{ SuperscriptBox[ SubscriptBox["\[Omega]bar", "p"], "EM"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SuperscriptBox[ SubscriptBox["\[Omega]bar", "p1"], "EM"], "[", "0", "]"}]}], "]"}]}], ",", " ", RowBox[{ SuperscriptBox[ SubscriptBox["V", "p"], "EM"], " ", "\[Rule]", " ", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{ SuperscriptBox[ SubscriptBox["V", "p1"], "EM"], "[", "0", "]"}]}], "]"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\t\t\t ", RowBox[{ RowBox[{"singularityexpression", " ", "=", " ", RowBox[{"singularityexpressiontmp", " ", "/.", " ", "PSupUnitRule"}]}], ";"}], "\n", "\t\t\t ", RowBox[{ RowBox[{"ruletmp", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["Ng", "S"], "[", "0", "]"}], " ", "\[Rule]", " ", RowBox[{ SubscriptBox["Ng", "CSS1"], "[", "0", "]"}]}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["A", "S"], "[", "0", "]"}], " ", "\[Rule]", " ", RowBox[{ SubscriptBox["A", "CSS1"], "[", "0", "]"}]}], ",", RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "S"], "[", "0", "]"}], " ", "\[Rule]", " ", RowBox[{ SubscriptBox["\[Omega]", "CSS1"], "[", "0", "]"}]}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["V", "S"], "[", "0", "]"}], " ", "\[Rule]", " ", RowBox[{ SubscriptBox["V", "CSS1"], "[", "0", "]"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\t\t\t ", RowBox[{ RowBox[{"RegularitySonicRuleFullup", " ", "=", " ", RowBox[{"RegularitySonicRuleFull", " ", "/.", " ", "ruletmp"}]}], ";"}], "\[IndentingNewLine]", "\t\t\t ", RowBox[{ RowBox[{"Clear", "[", RowBox[{"singularityexpressiontmp", ",", "ruletmp"}], "]"}], ";"}]}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.757155178862041*^9, 3.757155184030118*^9}, { 3.7571567854292183`*^9, 3.757156829569268*^9}, {3.757165870991296*^9, 3.75716588367603*^9}, {3.757168396486146*^9, 3.7571684057469397`*^9}}, CellLabel-> "In[351]:=",ExpressionUUID->"bbc7d84e-7ce9-4b25-9250-21ce11ea7ffd"], Cell["\<\ \t\t\tFirst, let us plug-in the regularity condition from the unperturbed \ variables, except for N. Otherwise, we will get 0. \t\t\tWe then factor the result to extract the singular part of the numerator \ which is meant to be simplified with one of the two copies present in the \ denominator. \t\t\tHere, we don\[CloseCurlyQuote]t do the simplification, we just extract \ the relevant factor which needs to be set to zero.\ \>", "Text", CellChangeTimes->{{3.7571563416082983`*^9, 3.757156443293816*^9}, { 3.7571568550721416`*^9, 3.7571568756558113`*^9}, 3.757157572022525*^9, 3.757162487152388*^9}, CellLabel-> "In[357]:=",ExpressionUUID->"0d4e4b32-bab9-4099-932b-951a7917f3a3"], Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{ RowBox[{ RowBox[{ "metricMatterEqnSol\[Omega]barpEMxNumeratorUnit", " ", "=", " ", "\[IndentingNewLine]", "\t\t\t ", RowBox[{"Simplify", "[", RowBox[{"Numerator", "[", RowBox[{ RowBox[{ RowBox[{ "metricMatterEqnSol\[Omega]barpEMx", " ", "/.", " ", "PSupUnitRule"}], " ", "/.", "\[IndentingNewLine]", "\t\t\t ", "PSpUnitRule"}], " ", "/.", " ", RowBox[{"RegularitySonicRuleFullup", "[", RowBox[{"[", RowBox[{"1", ";;", "2"}], "]"}], "]"}]}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t\t\t ", RowBox[{ RowBox[{ "metricMatterEqnSolVpEMxNumeratorUnit", " ", "=", "\[IndentingNewLine]", "\t\t\t ", RowBox[{"Simplify", "[", RowBox[{"Numerator", "[", RowBox[{ RowBox[{ RowBox[{"metricMatterEqnSolVpEMx", " ", "/.", " ", "PSupUnitRule"}], " ", "/.", "\[IndentingNewLine]", "\t\t\t ", "PSpUnitRule"}], " ", "/.", " ", RowBox[{"RegularitySonicRuleFullup", "[", RowBox[{"[", RowBox[{"1", ";;", "2"}], "]"}], "]"}]}], "]"}], "]"}]}], ";"}]}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.7570797895370207`*^9, 3.757079864943029*^9}, { 3.757141780309106*^9, 3.757141783682675*^9}, {3.757141891231*^9, 3.7571418923908777`*^9}, {3.757141963708433*^9, 3.7571419642796164`*^9}, { 3.757155016929253*^9, 3.757155022721965*^9}, {3.757155054545231*^9, 3.75715513507869*^9}, {3.757155309265375*^9, 3.7571553187212152`*^9}, { 3.75715539220199*^9, 3.7571553929027452`*^9}, {3.757155480215884*^9, 3.757155574584944*^9}, {3.757165894102448*^9, 3.757165899003339*^9}, { 3.7571684109280233`*^9, 3.757168415241989*^9}, {3.757169793593701*^9, 3.757169797886505*^9}, {3.7572536526794252`*^9, 3.75725365701015*^9}, 3.757256292910646*^9}, CellLabel-> "In[358]:=",ExpressionUUID->"1c159a3f-bfbc-4b9f-b8cd-3b34c15f9198"], Cell["\<\ \t\t\tRemove the singular part. Note the simplification applied BEFORE we \ apply the full regularity condition and simplify again.\ \>", "Text", CellChangeTimes->{{3.757248942842906*^9, 3.757248980561932*^9}, 3.7572562831288147`*^9}, CellLabel-> "In[360]:=",ExpressionUUID->"32d04810-b418-4c4b-982a-87fcf772c18c"], Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{ RowBox[{ RowBox[{ "metricMatterEqnSol\[Omega]barpEMxNumeratorUnitNormalized", " ", "=", " ", "\[IndentingNewLine]", "\t\t\t ", RowBox[{ RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ "Factor", "[", "metricMatterEqnSol\[Omega]barpEMxNumeratorUnit", "]"}], "/", "singularityexpression"}], "]"}], " ", "/.", " ", "\[IndentingNewLine]", "\t\t\t ", "RegularitySonicRuleFullup"}], " ", "//", " ", "Simplify"}]}], ";"}], "\[IndentingNewLine]", "\t\t \t", RowBox[{ RowBox[{ "metricMatterEqnSolVpEMxNumeratorUnitNormalized", " ", "=", " ", "\[IndentingNewLine]", "\t\t\t ", RowBox[{ RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Factor", "[", "metricMatterEqnSolVpEMxNumeratorUnit", "]"}], "/", "singularityexpression"}], "]"}], " ", "/.", " ", "\[IndentingNewLine]", "\t\t\t ", "RegularitySonicRuleFullup"}], " ", "//", " ", "Simplify"}]}], ";"}]}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.757156476892275*^9, 3.75715652078734*^9}, { 3.757157223930715*^9, 3.757157240282525*^9}, {3.757157288802491*^9, 3.757157301480279*^9}, 3.757157739115489*^9, {3.7571579845233517`*^9, 3.757158045434928*^9}, {3.757162502057652*^9, 3.757162530574752*^9}, { 3.757165903525818*^9, 3.757165907907243*^9}, {3.7572495647459707`*^9, 3.757249570021104*^9}, {3.7572496034003887`*^9, 3.757249610110676*^9}, 3.7572531813711777`*^9}, CellLabel-> "In[360]:=",ExpressionUUID->"70e3b1be-231d-4574-8cf0-9c572cd87a5e"], Cell["\<\ \t\t\tOne check that the numerators are in fact proportional to each other. \ There is therefore only 1 condition, like in up case:\ \>", "Text", CellChangeTimes->{{3.7571601450348454`*^9, 3.757160172474284*^9}, { 3.7571615926204453`*^9, 3.757161593001813*^9}, {3.7571625390855417`*^9, 3.757162539422039*^9}}, CellLabel-> "In[362]:=",ExpressionUUID->"193d43f0-650c-435f-b734-8ffdbed74546"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{"Simplify", "[", RowBox[{ "metricMatterEqnSol\[Omega]barpEMxNumeratorUnitNormalized", "/", "metricMatterEqnSolVpEMxNumeratorUnitNormalized"}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.7571580965243073`*^9, 3.757158124989018*^9}, { 3.757162540135161*^9, 3.757162541277913*^9}}, CellLabel-> "In[362]:=",ExpressionUUID->"60afdf1a-abbb-4d2e-ba08-52df6d256a49"], Cell[BoxData[ FractionBox["1", RowBox[{ SubscriptBox["c", "S"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox[ RowBox[{ SubscriptBox["V", "CSS1"], "[", "0", "]"}], "2"]}], ")"}]}]]], "Output",\ CellChangeTimes->{ 3.757158064347308*^9, {3.757158096925644*^9, 3.757158125299075*^9}, 3.757165357312756*^9, 3.757166100169086*^9, 3.757166260671871*^9, 3.7571688631046953`*^9, 3.757169808813903*^9, 3.757169901027067*^9, 3.7571773582035913`*^9, 3.7571775211946173`*^9, 3.757227622911751*^9, 3.757244936024605*^9, 3.757249920648988*^9, 3.75725523309741*^9, 3.757333713091463*^9}, CellLabel-> "Out[362]=",ExpressionUUID->"b00bed52-4306-4512-8c5e-089e28097771"] }, Closed]], Cell["\t\t\tWe therefore proceed to solving one of them, say \[Omega] :", \ "Text", CellChangeTimes->{{3.757161601003623*^9, 3.7571616332750473`*^9}, { 3.757162542517501*^9, 3.757162542861878*^9}, {3.7572490101049833`*^9, 3.757249011448845*^9}, 3.757253267205778*^9}, CellLabel-> "In[364]:=",ExpressionUUID->"eb544c32-b557-4d4a-86eb-2693ce67d273"], Cell[BoxData[ RowBox[{"\t\t\t ", RowBox[{ RowBox[{ "RegularitySonicRuleFullPerturbation\[Omega]tmp", " ", "=", " ", "\[IndentingNewLine]", "\t\t\t ", RowBox[{"Simplify", "[", RowBox[{"Flatten", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{ "metricMatterEqnSol\[Omega]barpEMxNumeratorUnitNormalized", " ", "\[Equal]", " ", "0"}], ",", " ", RowBox[{ SuperscriptBox[ SubscriptBox["\[Omega]bar", "p1"], "EM"], "[", "0", "]"}]}], "]"}], "]"}], "]"}]}], ";"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.757157317066317*^9, 3.757157365246386*^9}, { 3.75715743082264*^9, 3.757157434893003*^9}, {3.757160184055306*^9, 3.7571602169783487`*^9}, {3.7571625468313847`*^9, 3.757162550685994*^9}, { 3.757163963250758*^9, 3.757163973096601*^9}, 3.757164164148882*^9, 3.7572532909668922`*^9}, CellLabel-> "In[364]:=",ExpressionUUID->"7d55f149-4645-43e2-b574-9ab3bebc4471"], Cell[TextData[{ "\t\t\tNOTE: BE CAREFUL when handling singular stuff. As an example, try the \ following: do not work with the normalised numerators and try to solve for \ both \[Omega] and V.\n\t\t\tAfter that, update the results (rules) by the \ remaining regularity condition (for N). This WILL WORK and you will get a \ pair of solutions for \[Omega] and N as functions of\n\t\t\tunperturbed \ variable only!!! This makes little sense since we know that, first the \ numerators are actually zero after the full regular update. Now, it could be \ that \n\t\t\tduring \[OpenCurlyDoubleQuote]Solve\[CloseCurlyDoubleQuote], \ Mathematica noticed the factorization and did the cancel itself. Still, the \ \[OpenCurlyDoubleQuote]leftover\[CloseCurlyDoubleQuote] part should coincide \ with the normalised numerators above, \t\t\twhich have been shown to be \ \[Proportional]. So, why do we get nonetheless 2 well-define solutions (which \ can be checked to indeed work!). The reason is that the \ \[OpenCurlyDoubleQuote]real\[CloseCurlyDoubleQuote] \t\n\t\t\tsolution, say \ \[Omega] which depends on ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "46f136f0-bf09-41f5-b79b-56e442a5da2f"], " is a solution to the system FOR ANY ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "4ddb8d69-5603-4af7-a703-197b86a63f59"], ". In particular, one could take ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "9b6bc658-9139-4b95-9b24-750c6c9591ce"], " to be some function of the unperturbed variable; in \n\t\t\twhich case \ \[Omega] will also only depend on it. This is precisely the solution given if \ you do as suggested. \n\t\t\tIn other words, the solution is only for a \ particular choice of ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "d95c6f7b-35a4-4e92-a01c-3b7c064d2c11"], " which is presented as \[OpenCurlyDoubleQuote]the solution\ \[CloseCurlyDoubleQuote] of the linear system. To convince yourself, do the \ above, \n\t\t\tthen update the \[OpenCurlyDoubleQuote]true\ \[CloseCurlyDoubleQuote] solution \[Omega] (depending on ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "5743d5f2-e6b6-43f9-a132-8175f3b585d4"], ") with the ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "c140b58f-22e0-4376-b9d7-78b8c2068d34"], " \[OpenCurlyDoubleQuote]solution\[CloseCurlyDoubleQuote] obtained and you \ will see that this \[Omega] (now depending only on the unperturbed \n\t\t\t\ variable) will coincide with the other \[Omega] obtained. I am not sure how \ this is possible, as I would have anticipated the above to NOT work due to \ some division by \n\t\t\tzero.... Anyway, the moral in the end is to be VERY \ careful when handling stuff like that as you might get seemingly \ \[OpenCurlyDoubleQuote]correct\[CloseCurlyDoubleQuote] results." }], "Text", CellChangeTimes->{{3.757161640002557*^9, 3.757162284359027*^9}, { 3.7571625532589483`*^9, 3.7571626027521667`*^9}, {3.7571653888390417`*^9, 3.757165416422543*^9}}, CellLabel-> "In[365]:=",ExpressionUUID->"11e25349-a380-4709-bb26-45a3638db89c"], Cell[TextData[{ "\t\t\tAlright, we now make use of the algebraic equation (the perturbed \ one). 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Cell[CellGroupData[{ Cell[" \t\tGeneral Set-up", "Subsection", CellChangeTimes->{{3.757242909955887*^9, 3.7572429190035133`*^9}},ExpressionUUID->"25a92939-3865-4020-85e2-\ aa03c5e8e9f3"], Cell["\<\ \t\t\t\tHere, we set up the solutions of the p coefficients. 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We could do the same as the \ above, however, the equations are of the form: f/s, where s is 0 at x = 1. \ If we linearize it like that, we will get a 1/s^2 term by the end of the day. \ \nThis will make the sonic constraint calculation difficult as the numerator \ will then already be 0 at x =1, without imposing anything on the perturbed \ fields. This will require a bit of trickery to \[OpenCurlyDoubleQuote]extract\ \[CloseCurlyDoubleQuote] the correct part. \nSo, here will will instead keep \ stuff linear by writing the linear term in terms of unperturbed \[Omega]\ \[CloseCurlyQuote] (i.e., we DO NOT use the CSS equation to expand this!). \ After all, we know its (linear) PS expansion of \[Omega]! 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As we said, this makes things complicated." }], "Text", CellChangeTimes->{{3.757406670488793*^9, 3.757407037175571*^9}, { 3.75751888459968*^9, 3.7575189302116756`*^9}, {3.757753030747437*^9, 3.757753032114004*^9}, {3.760038562543807*^9, 3.7600385638928547`*^9}, { 3.7619243651799*^9, 3.761924388595017*^9}},ExpressionUUID->"2763fdbb-b5cb-479c-8a03-\ ed5cfac0a6e0"], Cell[BoxData[{ RowBox[{ RowBox[{"\[Omega]barx", " ", "=", " ", RowBox[{"f\[Omega]", "/", "s\[Omega]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Vx", " ", "=", " ", RowBox[{"fV", "/", "sV"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Omega]barxPerturbationtmp", " ", "=", " ", RowBox[{"Series", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"f\[Omega]", "+", RowBox[{"\[Epsilon]", " ", "\[Delta]f\[Omega]"}]}], ")"}], "/", RowBox[{"(", RowBox[{"s\[Omega]", "+", RowBox[{"\[Epsilon]", " ", "\[Delta]s\[Omega]"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"\[Epsilon]", ",", "0", ",", "1"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"VxPerturbationtmp", " ", "=", " ", RowBox[{"Series", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"fV", "+", RowBox[{"\[Epsilon]", " ", "\[Delta]fV"}]}], ")"}], "/", RowBox[{"(", RowBox[{"sV", "+", RowBox[{"\[Epsilon]", " ", "\[Delta]sV"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"\[Epsilon]", ",", "0", ",", "1"}], "}"}]}], "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.757407168745789*^9, 3.7574072889736347`*^9}, { 3.75740739433776*^9, 3.757407415057406*^9}, {3.757407529597557*^9, 3.757407615171103*^9}, {3.757407745393759*^9, 3.757407750247183*^9}, { 3.7574080738518333`*^9, 3.757408081788159*^9}, {3.757408278240038*^9, 3.7574082836262493`*^9}, {3.7574091232431173`*^9, 3.757409133913208*^9}, { 3.757409367877695*^9, 3.7574093699947968`*^9}, {3.75742633210859*^9, 3.7574263333694887`*^9}, {3.757438177633306*^9, 3.757438177702114*^9}, { 3.761924396581318*^9, 3.7619244209217367`*^9}}, CellLabel-> "In[238]:=",ExpressionUUID->"52c66429-8b6c-490a-b9c4-bab0ec10a74e"], Cell[TextData[{ "We see that a ", Cell[BoxData[ FormBox[ RowBox[{"1", "/", SuperscriptBox["s", "2"]}], TraditionalForm]],ExpressionUUID-> "9aa5c9a8-a07e-4552-99b0-e46342641c38"], " emerges. We can remove it by writing this term as a function of \[Omega]\ \[CloseCurlyQuote]. We take the linear term." }], "Text", CellChangeTimes->{{3.7574076362736692`*^9, 3.75740764165763*^9}, { 3.757407693632114*^9, 3.757407741718285*^9}, {3.75740817990452*^9, 3.757408260638728*^9}, {3.75740842012848*^9, 3.7574084638395233`*^9}, { 3.7574320974922533`*^9, 3.7574321013403254`*^9}, 3.761924422517218*^9},ExpressionUUID->"558481aa-27d3-4781-ab36-\ 3c15139735fb"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Omega]barpx", " ", "=", " ", RowBox[{ RowBox[{"Coefficient", "[", RowBox[{ RowBox[{"\[Omega]barxPerturbationtmp", " ", "/.", " ", RowBox[{"f\[Omega]", " ", "\[Rule]", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "css"], "'"}], "[", "x", "]"}], " ", RowBox[{"s\[Omega]", "/", RowBox[{ SubscriptBox["\[Omega]", "css"], "[", "x", "]"}]}]}]}]}], ",", "\[Epsilon]", ",", "1"}], "]"}], " ", "//", " ", "Simplify"}]}], ";"}], " ", RowBox[{"(*", RowBox[{ RowBox[{"Using", " ", "that", " ", RowBox[{ SubscriptBox[ OverscriptBox["\[Omega]", "_"], "css"], "'"}]}], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "css"], "'"}], "/", SubscriptBox["\[Omega]", "css"]}]}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Vpx", " ", "=", " ", RowBox[{ RowBox[{"Coefficient", "[", RowBox[{ RowBox[{"VxPerturbationtmp", " ", "/.", " ", RowBox[{"fV", " ", "\[Rule]", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["V", "css"], "'"}], "[", "x", "]"}], " ", "sV"}]}]}], " ", ",", "\[Epsilon]", ",", "1"}], "]"}], "//", " ", "Simplify"}]}], ";"}]}], "Input", CellChangeTimes->{{3.757407417740788*^9, 3.757407419185183*^9}, { 3.757407525045537*^9, 3.757407526277423*^9}, {3.7574077571246758`*^9, 3.757407849011714*^9}, {3.757407926884335*^9, 3.7574079403204813`*^9}, { 3.757407979019189*^9, 3.757408027957642*^9}, {3.757408085558496*^9, 3.757408091139639*^9}, {3.757408124405534*^9, 3.757408176017289*^9}, { 3.757408284925582*^9, 3.757408286745935*^9}, {3.757408414137643*^9, 3.757408415521412*^9}, {3.757408505499864*^9, 3.757408618770834*^9}, { 3.7574086489576063`*^9, 3.757408649457757*^9}, {3.757432114935244*^9, 3.7574321192759457`*^9}, {3.757438350368884*^9, 3.757438351169095*^9}, { 3.757439863505658*^9, 3.757439903515429*^9}, {3.760038573320489*^9, 3.76003859178642*^9}, {3.760038750884015*^9, 3.760038752782523*^9}, { 3.761924424236074*^9, 3.761924425898193*^9}}, CellLabel-> "In[242]:=",ExpressionUUID->"a1d5cca7-5193-4d7e-b639-6a1662c257d1"], Cell["\<\ The power expansion of these CSS values are known. Here of course, we only \ need to go to 1st order, so they are in fact just constants! We will assume \ this later. 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In order to avoid conical singularity, \ we impose regularity at the center: ", Cell[BoxData[ FormBox[ SubscriptBox[ OverscriptBox["A", "_"], "p"], TraditionalForm]],ExpressionUUID-> "745a0ae7-a0d2-4970-bcb2-fe84924df622"], "(\[Tau],0) = 0. One also imposes the condition that ", Cell[BoxData[ FormBox[ SubscriptBox["V", "p"], TraditionalForm]],ExpressionUUID-> "e5e9923f-bd07-4127-8e4b-a90807f83480"], "(\[Tau],0) = 0, since the velocity should be an odd function of x. \nThis \ fixes the leading order terms in the variables\[CloseCurlyQuote] respective \ expansions. Both of these condition can be seen to be reinforced at the \ leading order of the equations anyway." }], "Text", CellChangeTimes->{{3.7569979487534637`*^9, 3.756997957678698*^9}, { 3.7569985440041*^9, 3.756998563042046*^9}, {3.756998725767076*^9, 3.756998734311111*^9}, {3.75699916180062*^9, 3.756999171960651*^9}, { 3.756999284126835*^9, 3.756999286229828*^9}, {3.7569993421250257`*^9, 3.756999350515615*^9}, {3.757067148293586*^9, 3.757067342271796*^9}, { 3.757072868519288*^9, 3.7570729254619293`*^9}, {3.757076005107607*^9, 3.75707602527481*^9}, {3.757076138431273*^9, 3.7570761618698683`*^9}, { 3.757077356968582*^9, 3.757077379831731*^9}, 3.757511372028911*^9, { 3.761925144139748*^9, 3.761925150468246*^9}},ExpressionUUID->"3ed03703-eaff-41e0-861a-\ ff7a0700c9ab"], Cell["\<\ We define the expansion for both unperturbed (up) and perturbed (p) \ variables. The unperturbed expansion are of of course known (see: Power \ series solution at Origin), and in particular their expansion is readily \ known. The perturbed ones\[CloseCurlyQuote] are really ans\[ADoubleDot]tze. A \ posteriori, one can improve on it as the first 3 turn out to be even and the \ last one odd. See the end of next section for more details.\ \>", "Text", CellChangeTimes->{{3.75707621525948*^9, 3.757076435675205*^9}, { 3.757076527426024*^9, 3.75707653004275*^9}, {3.757077382758913*^9, 3.75707742390101*^9}, {3.76192515274797*^9, 3.761925163307619*^9}},ExpressionUUID->"a20007e4-c1b1-47d6-945a-\ ea193d051d8e"], Cell[TextData[{ "Note: be very careful on how you decide how many coefficients are enough. \ For example n=4 is needed in order to solve for ", Cell[BoxData[ FormBox[ OverscriptBox["\[Omega]", "_"], TraditionalForm]],ExpressionUUID-> "09bcac6b-6b6e-4d23-b9e0-5b8c5a3d8fe5"], "[2] correctly. 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By inspection, one sees that the \ matter equation of V has a term up to ~ 1/x. \tWe therefore multiply it by x and bring the equation down by one, to make \ everything on the same level. This means that now, we solve the leading order \ equations ~ 0(1) for the linear coefficients, the ~0(x) equations will be \ solved \tfor the 2nd order coefficients,etc. So, for a given expansion order n, we \ will therefore solve the first n-1 order equations. This makes also the \ calculations much faster, since before, going to say 2nd order means that the \ matter equations will have 6 terms, \tranging from ~0(1/x) to ~0(x^2), which is overkill, if we want only to know \ the first few terms in the expansion. This is why the below is much quicker. \ The 1/x term is in KHA what is referred (in the x -> - \[Infinity] analysis) \ the expanding mode. \tThe solution to make this vanish corresponds to the V[1] solution. 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Note that the denominator is \ only dependent on the up variables, so there is nothing to change here.\nNote \ that had we explicitly expanded ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "css"], TraditionalForm]],ExpressionUUID-> "c092f8f9-2dc6-4427-b5d8-16845aa34b0a"], "\[CloseCurlyQuote] in terms of the CSS variables, we would have gotten and \ additional singular factor in the denominator! This was what was done in the \ Old Attempt. \nThis makes things complicated since after all, \nthis causes a \ degeneracy in the data, which we already solved for, so it is best to leave \ it like that and only expand it as a power series where coefficients are \ known as there were previously calculated. 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This makes things faster, to first \ simplify as much as possible before plugging in the up sonic linear \ coefficients.\ \>", "Text", CellChangeTimes->{{3.7576982061726427`*^9, 3.757698262795166*^9}, { 3.761926078158534*^9, 3.76192607892686*^9}},ExpressionUUID->"192d0da3-57d0-4e79-8d9f-\ f462f3fdf561"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Use algebraic equation as additional constraint to find regularity rule for \ perturbed fields\ \>", "Subsection", CellChangeTimes->{{3.761926086595851*^9, 3.761926098611968*^9}, { 3.7619262038902903`*^9, 3.761926222146296*^9}},ExpressionUUID->"a53b653d-6850-4eec-b2cd-\ b0d3c3b13af5"], Cell[TextData[{ "\tWe have a second algebraic relation at our disposal, via the leading \ order term in the algebraic equation. 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Here, this is a system of two equations. \ We DO NOT use the p regularity condition, as this is too much work. We saw \ that leading order vanishes (assuming we plug everything in) as so, we solve \ the linear order.\ \>", "Text", CellChangeTimes->{{3.757176875170925*^9, 3.757176908978333*^9}, { 3.7571771827294407`*^9, 3.757177205384656*^9}, {3.757231234485441*^9, 3.7572313302742167`*^9}, {3.757231459790965*^9, 3.757231506020542*^9}, { 3.7572439507228003`*^9, 3.7572439541547813`*^9}, {3.7572445905175667`*^9, 3.7572445979164667`*^9}, {3.757244641872061*^9, 3.757244832726761*^9}, 3.757247583351976*^9, 3.757525746026821*^9, {3.757604310438545*^9, 3.757604319799877*^9}, {3.757605833674597*^9, 3.7576058428945513`*^9}, { 3.75766969959781*^9, 3.757669770387615*^9}, {3.761926386844922*^9, 3.761926389023313*^9}},ExpressionUUID->"86d03736-821e-4b95-9ae5-\ ec5114be2453"], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{ RowBox[{"\[Omega]PSpEquation", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"SeriesCoefficient", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"metricMatterODEFullNpAp\[Omega]pVp", "[", 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3.757605283255652*^9}, { 3.757605322169207*^9, 3.757605351839674*^9}, {3.757605663141693*^9, 3.757605714366749*^9}, {3.757606060967361*^9, 3.7576060945444107`*^9}, { 3.7600952453822308`*^9, 3.760095258574032*^9}, {3.7619264114908533`*^9, 3.761926416017271*^9}}, CellLabel-> "In[315]:=",ExpressionUUID->"b2d1d88b-edf6-4899-8f3e-6037dae2783c"], Cell["\<\ \tNote that the solution depends on the values of the linear order \ coefficients of N and A.\ \>", "Text", CellChangeTimes->{{3.757610919114975*^9, 3.7576109478024797`*^9}, 3.7576693692849617`*^9, 3.761926417435813*^9},ExpressionUUID->"21c8f3ae-db17-4284-8ddd-\ a45cadce1790"], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{ RowBox[{"\[Omega]barEMLinearTermRule", " ", "=", " ", RowBox[{"\[Omega]barVEMLinearTermRuletmp", "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t", RowBox[{ RowBox[{"\[Omega]barEMLinearTerm", " ", "=", " ", RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", 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3.761926464041051*^9}},ExpressionUUID->"2ff497ff-8d12-4794-af1f-\ d7c9a70251be"], Cell["\tAgain, let us compile all the rules above:", "Text", CellChangeTimes->{{3.757243630747809*^9, 3.757243643203741*^9}, 3.761926466594242*^9},ExpressionUUID->"5ce3c5c6-1bba-46ac-b62d-\ 9dc294bb373e"], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{ "metricMatterODEFullPSpUnitSolRuleNpAp\[Omega]pVp", " ", "=", " ", RowBox[{"Flatten", "[", RowBox[{"{", RowBox[{ "NgbarEMLinearTermRule", ",", "AbarEMLinearTermRule", ",", "\[Omega]barEMLinearTermRule", ",", "VEMLinearTermRule"}], "}"}], "]"}]}], ";"}]}]], "Input", CellChangeTimes->{{3.757243732761023*^9, 3.757243785671002*^9}, { 3.757243857608354*^9, 3.757243882204137*^9}, {3.757605874596119*^9, 3.757605875775794*^9}, {3.761926495298935*^9, 3.7619264987699947`*^9}}, CellLabel-> "In[320]:=",ExpressionUUID->"c011fad7-ec66-4bcb-86b2-3a7544151348"] }, Closed]] }, Closed]] }, Closed]], Cell["\<\ We see that we are somewhat reaching the limits of what Mathematica can do \ here. We shall therefore proceed to use the known numerical values for the \ unperturbed coefficients. We shall only retain \[Kappa] and the unknown p coefficients as symbolic of \ course. We will plug in the value of \[Kappa] for the shooting.\ \>", "Text", CellChangeTimes->{{3.757605920723751*^9, 3.757606001983989*^9}, { 3.757606124360745*^9, 3.75760614469302*^9}, {3.757678489287199*^9, 3.75767849580735*^9}, {3.761926501129615*^9, 3.761926533357357*^9}},ExpressionUUID->"a98b186d-75b5-4ecd-b808-\ b751b025f20e"], Cell[CellGroupData[{ Cell["Numerics of unperturbed and perturbed coefficients", "Subchapter", CellChangeTimes->{{3.757678509677888*^9, 3.7576785276617126`*^9}},ExpressionUUID->"36e82faa-5a0f-46fc-8b14-\ 38a37db7e2f0"], Cell[CellGroupData[{ Cell["Numerical approximation of the unperturbed solutions", "Section", CellChangeTimes->{{3.757228564897201*^9, 3.7572285842666197`*^9}, { 3.757405748783104*^9, 3.757405749007049*^9}, 3.757678497270618*^9, { 3.757678530125493*^9, 3.757678532413591*^9}, {3.7619265374160137`*^9, 3.761926556223579*^9}},ExpressionUUID->"7d531d7f-f535-4281-afdc-\ 1e11a560baaa"], Cell["\<\ Since Mathematica has trouble above to keep working in closed form, we shall \ plug-in the calculated values for the unperturbed variables. This is of \ course also needed for the shooting.\ \>", "Text", CellChangeTimes->{{3.757228592762446*^9, 3.75722870448707*^9}, { 3.757230842186761*^9, 3.757230842574102*^9}, {3.7576061551597013`*^9, 3.757606167220096*^9}, {3.761926545256297*^9, 3.761926548207945*^9}},ExpressionUUID->"b25ba31b-83af-48db-bb25-\ 587207228ddd"], Cell[CellGroupData[{ Cell["Numerical set-up of unperturbed coefficients", "Subsection", CellChangeTimes->{{3.7572308106196136`*^9, 3.757230832927177*^9}, { 3.761926551695627*^9, 3.761926552775716*^9}},ExpressionUUID->"7e53fe2e-20dc-45a0-9979-\ 4cb91ae1c342"], Cell["\<\ \tWe write up all the solutions we have for the unperturbed coefficients, by \ plugging in the values obtained from the Shooting. mycRule should have been defined when shooting for the background variables!!\ \ \>", "Text", CellChangeTimes->{{3.757230847442219*^9, 3.757230891121099*^9}, { 3.761926562856518*^9, 3.761926563257085*^9}, {3.829199857846979*^9, 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Recall that the regularity condition also fixes 3 out \ of 4 leading order coefficients. \tFor completeness, we shall compile all the values together:\ \>", "Text", CellChangeTimes->{{3.75722960169209*^9, 3.7572296876667433`*^9}, { 3.757230958070939*^9, 3.7572309605111647`*^9}, 3.757605602531889*^9, { 3.7619266227976017`*^9, 3.761926628573859*^9}},ExpressionUUID->"f51afe49-dbdd-4d8a-90bf-\ bfc9fe6b0136"], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{ RowBox[{ "metricMatterODEReducedCSSPSOriginSolRuleN\[Omega]VUnperturbedNumeric", " ", "=", " ", RowBox[{"Flatten", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["A", "css", "c"], "[", "0", "]"}], " ", "\[Rule]", " ", "1"}], ",", " ", RowBox[{"Ngm1\[Omega]2V0RuleUnperturbed", "[", RowBox[{"[", RowBox[{"1", ";;", "2"}], "]"}], "]"}], ",", " ", "metricMatterODEReducedCSSPSOriginSolRuleN\[Omega]\ VUnperturbedNumerictmp"}], "}"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t", RowBox[{ 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Recall that we \ still need to find \[Omega] and V. 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We considered the full system up till now. For \ the shooting, we wish to fix the three free coefficients (two from Origin \ expansion and one from Unit). This means that it is easiest to have\n3 ODEs \ instead of 4. \nAgain, the algebraic equation allows us to remove, say, the A\ \[CloseCurlyQuote] ODE. We are thus left with a system of 3 ODEs in N, \ \[Omega] and V. For the IC (initial conditions) at the origin, we go up to \ 1st order.\nThey are all of the form: X(\[Epsilon]) = X[0]+\[Epsilon] X[1]. \ We then solve the system with NDSolve up to x = 0.5 with those IC. This \ allows us \[OpenCurlyDoubleQuote]to start\[CloseCurlyDoubleQuote] away from \ the degeneracy. The solution formally depends on our arbitrary\nvariables ", Cell[BoxData[ FormBox[ SubsuperscriptBox[ OverscriptBox["Ng", "_"], "p", RowBox[{"em", ",", "s"}]], TraditionalForm]],ExpressionUUID-> "418c04b7-6fc2-4bfd-a950-93ccd07bde5a"], "[0] and ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", ",", "s"}]], "[", "0", "]"}], TraditionalForm]], ExpressionUUID->"c1aea456-6fd9-4d90-b510-ff03874b9c57"], " for this expansion. We will also do and expansion at x = 1, depending on \ ", Cell[BoxData[ FormBox[ SubsuperscriptBox[ OverscriptBox["A", "_"], "p", RowBox[{"em", ",", "s"}]], TraditionalForm]],ExpressionUUID-> "3e725f78-f50c-41b7-9981-495b72eaed5e"], "[0]. The mismatch between those two expansions will be checked at x = 0.5, \ fixing the coefficients\n(we will use FindRoot). The reason as to why in the \ perturbed case, we do not need to go up to 2nd order, unlike the unperturbed \ case, is because we see that each of the variables are non-trivial to linear \ order.\nBefore, recall that \[Omega] needed to go to 2nd order to be \ non-trivial and the IC would not depend on ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Omega]", "c"], TraditionalForm]],ExpressionUUID-> "a9899e05-9434-4400-a4ac-61b8dfa0714c"], "[2] making the system unsolvable later. This is not the case here." }], "Text", CellChangeTimes->{{3.755964453430434*^9, 3.755964651040887*^9}, { 3.756038365501349*^9, 3.75603843907548*^9}, {3.756121037963759*^9, 3.756121051331352*^9}, {3.756121119368618*^9, 3.75612126397336*^9}, { 3.756209700135277*^9, 3.75620976267068*^9}, {3.757674490556622*^9, 3.757674690813409*^9}, {3.757674749073225*^9, 3.757674956789152*^9}, { 3.757675087776957*^9, 3.757675120568536*^9}, {3.757675164902123*^9, 3.757675194234071*^9}, {3.757675271314817*^9, 3.7576754059817047`*^9}, { 3.757675444219111*^9, 3.75767546643647*^9}, {3.757675523225772*^9, 3.757675572288045*^9}, {3.759227117016233*^9, 3.7592271278068953`*^9}, { 3.760095622789071*^9, 3.760095657560725*^9}, {3.7619270217050734`*^9, 3.761927041977783*^9}, {3.826369940244185*^9, 3.826369969782289*^9}},ExpressionUUID->"408af7ad-9087-4b30-a9f0-\ 65439d1b7ed1"], Cell[CellGroupData[{ Cell["Initial conditions and parameters set-up", "Subchapter", CellChangeTimes->{{3.756208573816579*^9, 3.7562085823683863`*^9}, { 3.756208979779703*^9, 3.756208982284487*^9}, {3.756210603577859*^9, 3.756210609241817*^9}, {3.761927121829956*^9, 3.761927122046081*^9}},ExpressionUUID->"113850f6-d6fb-468a-9893-\ 04cd17a042b7"], Cell[CellGroupData[{ Cell["Set-up", "Section", CellChangeTimes->{{3.7619270794793987`*^9, 3.76192709171087*^9}, { 3.761927228961956*^9, 3.761927229375532*^9}},ExpressionUUID->"15556f09-c2fa-42d1-a5d0-\ 9bab84565a4e"], Cell[TextData[{ "The value ", Cell[BoxData[ FormBox[ SubscriptBox["c", "S"], TraditionalForm]],ExpressionUUID-> "388ffa0e-b0f3-40c4-a0bd-3d4c52c2fb65"], "has been defined in the background shooting and of course has to be the \ same here." }], "Text", CellChangeTimes->{{3.7559647760919456`*^9, 3.755964785603692*^9}, { 3.756208953233074*^9, 3.756208960996436*^9}, {3.757694707435781*^9, 3.757694718651593*^9}, {3.758275441173767*^9, 3.758275467365171*^9}, 3.76192709481491*^9, 3.761927234370246*^9, 3.761927295736227*^9},ExpressionUUID->"2eb9ad46-5b74-4dae-b2cd-\ fefd684db96d"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Kappa]KHA", " ", "=", " ", "2.8105525488"}], ";"}], " ", RowBox[{"(*", RowBox[{"Found", " ", "in", " ", RowBox[{"KHA", ".", " ", "This"}], " ", "is", " ", "really", " ", "for", " ", "double", " ", "check", " ", "with", " ", "the", " ", "\[Kappa]", " ", "we", " ", "will", " ", "find", " ", "below"}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Kappa]KHARule", " ", "=", " ", RowBox[{"\[Kappa]", " ", "\[Rule]", " ", "\[Kappa]KHA"}]}], ";"}], "\t\t "}]}], "Input", CellChangeTimes->{{3.755944667938341*^9, 3.7559446753525887`*^9}, 3.755947648460829*^9, {3.7559508343131933`*^9, 3.755950837903113*^9}, 3.755950872920286*^9, {3.755957794459301*^9, 3.755957800577355*^9}, { 3.755958856442112*^9, 3.7559588684796867`*^9}, {3.7559592173988657`*^9, 3.7559592648910303`*^9}, {3.756022349087524*^9, 3.756022352499062*^9}, { 3.756022435385374*^9, 3.756022438267373*^9}, {3.756023285411981*^9, 3.756023285529108*^9}, {3.756023562131857*^9, 3.756023562224275*^9}, { 3.756027880272278*^9, 3.7560278806745853`*^9}, {3.756046219791054*^9, 3.75604622962105*^9}, {3.756046268613481*^9, 3.756046269426812*^9}, { 3.756046527593486*^9, 3.756046527682774*^9}, {3.756121099410001*^9, 3.756121099464566*^9}, {3.756208962006012*^9, 3.756208964604699*^9}, { 3.7562099121357203`*^9, 3.7562099686782303`*^9}, {3.7562100176192007`*^9, 3.756210076619322*^9}, {3.757242512052145*^9, 3.757242525042303*^9}, { 3.7576755904786997`*^9, 3.757675590900558*^9}, {3.7576762971737556`*^9, 3.757676338268343*^9}, {3.757676471168195*^9, 3.7576764712166653`*^9}, { 3.757677911762343*^9, 3.757677912850089*^9}, {3.757774681588874*^9, 3.757774681658544*^9}, {3.757774754466632*^9, 3.7577747547920303`*^9}, { 3.757774931827436*^9, 3.757774933026533*^9}, {3.757776301231409*^9, 3.7577763016541033`*^9}, {3.7577763716691847`*^9, 3.757776372220026*^9}, { 3.757950319552071*^9, 3.7579503205418453`*^9}, {3.7582754284362297`*^9, 3.7582754325505257`*^9}, {3.758275494796468*^9, 3.7582755242113523`*^9}, { 3.760095690212472*^9, 3.760095704338152*^9}, {3.761927096689033*^9, 3.7619271012070417`*^9}, {3.761927296564106*^9, 3.761927298124206*^9}, { 3.776502787399396*^9, 3.776502795582065*^9}}, CellLabel-> "In[475]:=",ExpressionUUID->"1dc4a7c2-7af7-4495-aca0-ece0fee27613"] }, Closed]], Cell["\<\ Note that the point at which we will evaluate the mismatch between the two \ shootings, i.e. myEvaluation, NEEDS to be the same as the one used for \ background case. It can be made to be independent, but not worth the \ additional trouble....\ \>", "Text", CellChangeTimes->{{3.7765027989330397`*^9, 3.7765028573002033`*^9}},ExpressionUUID->"0eb21d7b-3e07-4a76-81e9-\ 29efc6a0465d"], Cell[CellGroupData[{ Cell["IC at origin", "Section", CellChangeTimes->{{3.7562085913760357`*^9, 3.756208601775774*^9}, { 3.761927111549884*^9, 3.761927111886923*^9}, {3.761927159028623*^9, 3.761927159700539*^9}, 3.761927317831353*^9},ExpressionUUID->"2e4c4f5c-0ec1-452d-8e83-\ 855f17c24390"], Cell[BoxData[ RowBox[{ RowBox[{"ICPerturbedOriginOrder1", "[", RowBox[{ "Nbar0_", ",", "\[Omega]bar0_", ",", "\[Epsilon]para_", ",", "\[Kappa]para_"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["Ng", "_"], "p", RowBox[{"em", ",", "c"}]], "[", "0", "]"}], "+", RowBox[{"x", " ", RowBox[{ SubsuperscriptBox[ OverscriptBox["Ng", "_"], "p", RowBox[{"em", ",", "c"}]], "[", "1", "]"}]}]}], " ", ",", RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", ",", "c"}]], "[", "0", "]"}], "+", RowBox[{"x", " ", RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", ",", "c"}]], "[", "1", "]"}]}]}], ",", RowBox[{"x", " ", RowBox[{ SubsuperscriptBox["V", "p", RowBox[{"em", ",", "c"}]], "[", "1", "]"}]}]}], "}"}], " ", "/.", " ", "\[IndentingNewLine]", "metricMatterODEFullPSpOriginSolRuleNpAp\[Omega]pVpNumeric"}], "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"\[Kappa]", " ", "\[Rule]", " ", "\[Kappa]para"}], ",", RowBox[{"x", " ", "\[Rule]", " ", "\[Epsilon]para"}], " ", ",", " ", RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["Ng", "_"], "p", RowBox[{"em", ",", "c"}]], "[", "0", "]"}], " ", "\[Rule]", " ", "Nbar0"}], ",", " ", RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", ",", "c"}]], "[", "0", "]"}], " ", "\[Rule]", " ", "\[Omega]bar0"}]}], "}"}]}]}]], "Input", CellChangeTimes->{{3.756208620464652*^9, 3.756208628552731*^9}, { 3.757675719674056*^9, 3.757675723120756*^9}, {3.757675760865456*^9, 3.7576758356624603`*^9}, {3.757675891075561*^9, 3.757675894428279*^9}, { 3.757675942154173*^9, 3.75767594737798*^9}, {3.7576760104555817`*^9, 3.7576760658381643`*^9}, 3.7576762083317204`*^9, {3.7576762546822367`*^9, 3.75767626759976*^9}, {3.757676344932988*^9, 3.757676386139771*^9}, { 3.760095720052252*^9, 3.760095793160693*^9}, {3.761927131361953*^9, 3.761927145135993*^9}, {3.761927303418401*^9, 3.761927304547822*^9}}, CellLabel-> "In[477]:=",ExpressionUUID->"03278d5c-614e-4e7f-aca8-7a3ff69aec3c"], Cell[TextData[{ "Note that for radiation fluid, the linear term for N is actually 0. It \ would anyway not make any difference compared to the ~ 0(1/x) term. Also, \ note that we do not specify ", Cell[BoxData[ FormBox[ SubscriptBox["c", "S"], TraditionalForm]],ExpressionUUID-> "46724991-893d-4345-a344-f3c004bbc65e"], " as it is already assumed to be radiation fluid." }], "Text", CellChangeTimes->{{3.757230433227437*^9, 3.7572304790818787`*^9}, { 3.7576759631198387`*^9, 3.757675986719479*^9}, {3.761927150189447*^9, 3.761927151437003*^9}, {3.761927306820245*^9, 3.761927308868105*^9}},ExpressionUUID->"d2827af0-fffd-46ed-81c1-\ cc051979df0a"] }, Closed]], Cell[CellGroupData[{ Cell["IC at sonic point", "Section", CellChangeTimes->{{3.7562087057323647`*^9, 3.756208716227984*^9}, { 3.761927156341634*^9, 3.761927161498246*^9}, 3.76192732054464*^9},ExpressionUUID->"7efec88e-74b6-45a5-a440-\ a241907136e3"], Cell["\<\ Now doing the same thing for the x = 1 expansion.We also go to 1st order. As \ for unperturbed case, for small enough \[Epsilon], we do not expect 2nd \ order to make a significant difference.\ \>", "Text", CellChangeTimes->{{3.755964665791424*^9, 3.755964760891275*^9}, { 3.75612832284219*^9, 3.756128428391531*^9}, {3.756128470669856*^9, 3.756128492060915*^9}, {3.75620874418843*^9, 3.756208780955592*^9}, { 3.757226717597755*^9, 3.757226733597147*^9}, {3.7576764164824963`*^9, 3.757676477567315*^9}, {3.761927164292692*^9, 3.761927167420425*^9}, 3.7619273231918097`*^9},ExpressionUUID->"566c39be-f414-4084-980f-\ c2e6e5df3409"], Cell[BoxData[ RowBox[{ RowBox[{"ICPerturbedUnitOrder1", "[", RowBox[{"Abars_", ",", "\[Epsilon]para_", ",", "\[Kappa]para_"}], "]"}], " ", ":=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", RowBox[{ SubsuperscriptBox[ OverscriptBox["Ng", "_"], "p", RowBox[{"em", ",", "s"}]], "[", "1", "]"}]}], ",", RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", ",", "s"}]], "[", "0", "]"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", RowBox[{ SubsuperscriptBox[ OverscriptBox["\[Omega]", "_"], "p", RowBox[{"em", ",", "s"}]], "[", "1", "]"}]}]}], ",", RowBox[{ RowBox[{ SubsuperscriptBox["V", "p", RowBox[{"em", ",", "s"}]], "[", "0", "]"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", RowBox[{ SubsuperscriptBox["V", "p", RowBox[{"em", ",", "s"}]], "[", "1", "]"}]}]}]}], "}"}], " ", "/.", " ", "\[IndentingNewLine]", "metricMatterODEFullPSpUnitSolRuleNpAp\[Omega]pVpNumeric"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"\[Kappa]", " ", "\[Rule]", " ", "\[Kappa]para"}], ",", RowBox[{"x", " ", "\[Rule]", RowBox[{"1", "-", "\[Epsilon]para"}]}], ",", " ", RowBox[{ RowBox[{ SubsuperscriptBox[ OverscriptBox["A", "_"], "p", RowBox[{"em", ",", "s"}]], "[", "0", "]"}], " ", "\[Rule]", " ", "Abars"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.755962993662628*^9, 3.755963070731983*^9}, { 3.75596330054738*^9, 3.755963370154332*^9}, {3.755963403032791*^9, 3.7559635340491447`*^9}, {3.755963608743926*^9, 3.755963615327037*^9}, { 3.7559636810367813`*^9, 3.755963717839611*^9}, {3.755964276513184*^9, 3.755964285038281*^9}, {3.7559643267048187`*^9, 3.755964327508576*^9}, { 3.7559648829107037`*^9, 3.755964888166277*^9}, {3.75596521365269*^9, 3.755965244273182*^9}, 3.7559652914671593`*^9, {3.755965379653305*^9, 3.7559654208993883`*^9}, {3.756027727238014*^9, 3.7560277373772497`*^9}, { 3.756121336973785*^9, 3.756121397092955*^9}, {3.756121451017083*^9, 3.756121481335794*^9}, {3.7561215876303463`*^9, 3.756121589795973*^9}, 3.756121646668242*^9, {3.756121683747176*^9, 3.7561216839053793`*^9}, { 3.7561217389208517`*^9, 3.756121740632749*^9}, {3.756121772583373*^9, 3.756121784445969*^9}, {3.756123085470911*^9, 3.756123107045437*^9}, 3.7561253444297237`*^9, {3.756128845895018*^9, 3.7561288786508083`*^9}, { 3.7562088639646473`*^9, 3.756208924807582*^9}, {3.75767648259525*^9, 3.7576765264313173`*^9}, {3.75767658385395*^9, 3.7576766554593*^9}, { 3.7576768347415247`*^9, 3.757676879315898*^9}, {3.757693882545989*^9, 3.757693882703575*^9}, {3.757694159217514*^9, 3.7576941597351913`*^9}, { 3.757848941104518*^9, 3.757848944910173*^9}, {3.7600958236327953`*^9, 3.760095899556131*^9}, {3.7600959484610453`*^9, 3.7600959584570513`*^9}, { 3.761927170105624*^9, 3.761927178623515*^9}, {3.761927324841633*^9, 3.761927326073951*^9}}, CellLabel-> "In[478]:=",ExpressionUUID->"aa96d1f9-e1a9-407c-ba49-4d907eff6fa5"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Left and right shootings and mismatch resolution", "Subchapter", CellChangeTimes->{{3.7619273358372107`*^9, 3.761927355220564*^9}},ExpressionUUID->"496bf112-ef37-490a-b45b-\ e98b10f46f4a"], Cell[CellGroupData[{ Cell["Left expansion shooting, x = 0", "Section", CellChangeTimes->{{3.756208995075102*^9, 3.756209023882128*^9}, { 3.7619272407019367`*^9, 3.761927264673131*^9}, {3.761927476786089*^9, 3.7619274790078278`*^9}},ExpressionUUID->"4b592849-d232-4b89-8cd1-\ 9e02c323c78d"], Cell["\<\ Solve the ODE system. The 2nd function has as additional input the evaluation \ of the solution at a given point. We define both the rule and the \ corresponding evaluations. 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define the mismatch at the \[OpenCurlyDoubleQuote]midpoint\ \[CloseCurlyDoubleQuote] and solve for the variables ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["Ng", "c"], "[", RowBox[{"-", "1"}], "]"}], TraditionalForm]],ExpressionUUID-> "c709592f-732a-4ce0-84b9-614c9307a6eb"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Omega]", "c"], "[", "2", "]"}], TraditionalForm]], ExpressionUUID->"9eab39a7-09f7-41a5-b11f-a79853a008df"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["V", "s"], "[", "0", "]"}], TraditionalForm]], ExpressionUUID->"8a5f159f-885e-47ab-b1af-8803bd9b726f"], " so that the mismatch is negligible.\nNote that we have 3 variables to \ solve for and we will get a non-linear system of 3 equation. We denote the \ above 3 variables as ", Cell[BoxData[ FormBox["\"\", TraditionalForm]],ExpressionUUID-> "27975cdc-982b-4ddc-9b78-2e87df195c18"], ", \[Omega]0, VS\[CloseCurlyDoubleQuote] respectively." }], "Text", CellChangeTimes->{{3.756209648506431*^9, 3.756209683279525*^9}, { 3.756209779227541*^9, 3.756209873825629*^9}, 3.75621037074594*^9, { 3.756210672785552*^9, 3.756210704159678*^9}, {3.756210938929702*^9, 3.756210939265564*^9}, {3.760096839255679*^9, 3.760096848799114*^9}, { 3.761927563325344*^9, 3.76192759652932*^9}},ExpressionUUID->"841464f0-bea5-4c45-99c3-\ 1fc17fa96bab"], Cell[BoxData[ RowBox[{ RowBox[{"metricMatterODEReducedPerturbationShootingMismatch", "[", RowBox[{ RowBox[{"evaluation_", "?", "NumericQ"}], ",", RowBox[{"Nbar0_", "?", "NumericQ"}], ",", RowBox[{"\[Omega]bar0_", "?", "NumericQ"}], ",", " ", RowBox[{"Abars_", "?", "NumericQ"}], ",", RowBox[{"\[Epsilon]para_", "?", "NumericQ"}], ",", RowBox[{"\[Kappa]para_", "?", "NumericQ"}]}], "]"}], " ", ":=", "\[IndentingNewLine]", "\t", RowBox[{ RowBox[{"metricMatterODEPerturbationReducedShootingLeftExpansion", "[", RowBox[{ "evaluation", ",", "Nbar0", ",", "\[Omega]bar0", ",", "\[Epsilon]para", ",", "\[Kappa]para"}], "]"}], " ", "-", " ", RowBox[{"metricMatterODEPerturbationReducedShootingRightExpansion", "[", RowBox[{ "evaluation", ",", "Abars", ",", "\[Epsilon]para", ",", "\[Kappa]para"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.756021546596631*^9, 3.75602155539357*^9}, { 3.756021627955875*^9, 3.756021759617654*^9}, {3.756209876651957*^9, 3.756209892369196*^9}, {3.75621064214808*^9, 3.756210662865324*^9}, { 3.757696239037424*^9, 3.757696262595703*^9}, {3.757696316955369*^9, 3.757696366942379*^9}, {3.757756049770063*^9, 3.757756055087792*^9}, { 3.75777159283111*^9, 3.757771606136866*^9}, {3.7581996595265007`*^9, 3.758199696827797*^9}, {3.758215458005972*^9, 3.758215459875856*^9}, { 3.760096786321594*^9, 3.760096789647559*^9}, {3.761927601228672*^9, 3.761927612846925*^9}}, CellLabel-> "In[493]:=",ExpressionUUID->"d7a8ec34-299f-4063-8927-618f56b1a4fc"], Cell["\<\ For the Shooting, we need starting values. 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This is not great \ because finding \[Kappa] should be part of the problem and the system is \ anyway not well defined, because the system is homogenous and thus has a \ scaling invariance! The answer happens to be correct however. We keep it for archiving purposes:\ \>", "Text", CellChangeTimes->{{3.758199412156262*^9, 3.758199509193941*^9}, { 3.761927619446742*^9, 3.7619276267975607`*^9}},ExpressionUUID->"c290861c-e1d7-425b-9b20-\ 6795a2ebcf14"] }, Closed]], Cell[CellGroupData[{ Cell["Solve the mismatch", "Section", CellChangeTimes->{{3.760097399521811*^9, 3.760097406135193*^9}, { 3.760097437769991*^9, 3.760097437866605*^9}, {3.761927251713912*^9, 3.7619272718817472`*^9}},ExpressionUUID->"9b4306cb-693e-4f4c-b0e9-\ 440619125064"], Cell[CellGroupData[{ Cell["Solving for the variables given \[Kappa] (old)", "Subsection", CellChangeTimes->{{3.7581995156888027`*^9, 3.758199531984494*^9}, { 3.758282597237022*^9, 3.758282597399164*^9}, 3.76009743988347*^9, { 3.761927647404791*^9, 3.76192766271681*^9}},ExpressionUUID->"0be0b224-e4b3-4c7d-8f51-\ c81d2e4eb7b3"], Cell["\<\ \tThis has been made unevaluatable. It does not work really well and see next \ section as to why. 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The last one, is the \ same as for Evans and Coleman. The middle one (~ can be seen to make function \ only ", Cell[BoxData[ FormBox[ SuperscriptBox["C", "0"], TraditionalForm]],ExpressionUUID-> "8e7c534e-1b56-4596-9ca0-b2965875d0dd"], " at the midpoint 0.5). ", StyleBox["As a side note, this is the drawback of this method.", FontColor->RGBColor[1, 0, 0]], "\n\t\t\tWe only require the mismatch to be very small at the evaluation (by \ default, it is 0.5), but not regular. \n\t\t\tThe middle point is wrong also \ for a different reason. By changing the evaluation point, myEvaluation, the \ middle root actually changes as well! This of course should not be the case. \ The smallest and largest value do not change though.\n\t\t\tThe largest value \ is the Evans and Coleman eigenmode. The first one is a pure gauge mode, see \ Koike Hara Adachi \[OpenCurlyDoubleQuote]Renormalization group and critical \ behaviour in gravitational collapse\[CloseCurlyDoubleQuote] section F. and \ G.2). 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We will therefore move one with the only remaining physical mode." }], "Text", CellChangeTimes->{{3.7764978590494423`*^9, 3.7764980375174093`*^9}, { 3.7765031652106037`*^9, 3.7765032882005167`*^9}, {3.7765067053000174`*^9, 3.776506750909914*^9}, {3.776506885558628*^9, 3.7765069729643373`*^9}, { 3.776507006385545*^9, 3.776507010075729*^9}, {3.776507059249384*^9, 3.776507079569152*^9}, {3.77650760549482*^9, 3.776507681910656*^9}},ExpressionUUID->"2a8db3c8-6b76-4c65-b88c-\ 1e5b1b57691f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ RowBox[{ RowBox[{"my\[Kappa]Rule", " ", "=", " ", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"MisMatchMatrixdet", "[", "\[Kappa]", "]"}], " ", "\[Equal]", " ", "0"}], ",", RowBox[{"{", RowBox[{"\[Kappa]", ",", "2.8"}], "}"}], ",", RowBox[{"WorkingPrecision", "\[Rule]", " ", "6"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", "\t\t\t", RowBox[{"my\[Kappa]", " ", "=", " ", RowBox[{"\[Kappa]", " ", "/.", " ", "my\[Kappa]Rule"}]}]}]}]], 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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$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> 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"SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.922526, 0.385626, 0.209179], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.6150173333333333, 0.25708400000000003`, 0.13945266666666667`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], StyleBox[ RowBox[{"RGBColor", "[", RowBox[{"0.922526`", ",", "0.385626`", ",", "0.209179`"}], "]"}], NumberMarks -> False]], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = 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[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", 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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, 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(Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotLabel->None, PlotRange->{All, All}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}], Scaled[{0.5, 0.5}], Center, Scaled[{1, 1}]], InsetBox[ TemplateBox[{ "\"\\!\\(\\*SubscriptBox[\\(\[Alpha]\\), \\(1\\)]\\)\"", "\"\\!\\(\\*SubscriptBox[\\(a\\), \\(1\\)]\\)\"", "\"4\\!\\(\\*SuperscriptBox[\\(\[Pi]R\\), \ \\(2\\)]\\)\\!\\(\\*SubscriptBox[\\(\[Rho]\\), \\(1\\)]\\)\"", "\"\\!\\(\\*SubscriptBox[\\(v\\), \\(1\\)]\\)\""}, "LineLegend", DisplayFunction->(FormBox[ StyleBox[ StyleBox[ PaneBox[ TagBox[ GridBox[{{ TagBox[ GridBox[{{ GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ 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}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> {"Columns" -> {{0.8, 0.5}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxDividers -> {"Columns" -> {{None}}, "Rows" -> {{None}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> {"Columns" -> {{0}}, "Rows" -> {{1}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontWeight -> Bold, 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[{"{", RowBox[{#, ",", #2, ",", #3, ",", #4}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "Bold", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Row\""}]}], "]"}]& )], Scaled[{0.5, 1}], ImageScaled[{0.5, 0}], BaseStyle->{FontSize -> Larger}, FormatType->StandardForm]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->False, AxesLabel->{ FormBox["\"x\"", TraditionalForm], None}, AxesOrigin->{0., -0.8}, DisplayFunction->Identity, Frame->False, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, ImagePadding->All, ImageSize->{293.9999999999967, Automatic}, LabelStyle->{Bold}, 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]]}& )}}, PlotLabel->None, PlotRange->Automatic, PlotRangeClipping->False, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}], InterpretTemplate[Legended[ Graphics[{{{{{}, {}, Annotation[{ Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], Line[CompressedData[" 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Made it unevaluable of course.\ \>", "Text", CellChangeTimes->{{3.7582061083679457`*^9, 3.7582061311996183`*^9}, { 3.75820630722574*^9, 3.758206323145143*^9}, {3.75821118176105*^9, 3.7582111988886223`*^9}, {3.758211358363481*^9, 3.758211371872242*^9}, 3.761400042277134*^9, {3.761758218494741*^9, 3.7617582191272182`*^9}, { 3.770033052095716*^9, 3.770033068031384*^9}, {3.770033159084667*^9, 3.770033167124207*^9}},ExpressionUUID->"17e1364e-76b6-4a4b-b1fd-\ f489accb3115"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"context", " ", "=", " ", RowBox[{"Context", "[", "]"}]}], "}"}], ",", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", " ", RowBox[{"{", RowBox[{ "context", ",", "NotationMakeBoxes", ",", "NotationMakeExpression"}], "}"}]}], "]"}]}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.75821115096309*^9, 3.758211154826736*^9}, { 3.75821120345455*^9, 3.758211203905016*^9}, {3.758211310677207*^9, 3.7582113469642353`*^9}, {3.761400045509206*^9, 3.761400046589329*^9}, { 3.761758221539482*^9, 3.7617582363722553`*^9}, {3.77003305422393*^9, 3.770033056120089*^9}, {3.770033177980672*^9, 3.770033178307878*^9}},ExpressionUUID->"411365db-0399-4ca0-bb52-\ ebdfe74cc0ac"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", "x"}], "]"}]}], "]"}], "@", "$Context"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.758211162812993*^9, 3.7582111628140287`*^9}, { 3.758211309397113*^9, 3.758211309800643*^9}, 3.761400047405044*^9, { 3.761758244334284*^9, 3.7617582471299963`*^9}, 3.770033058271699*^9, 3.770033183652508*^9},ExpressionUUID->"56222099-2513-47d5-aaa0-\ 871ed99213fa"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", RowBox[{"{", RowBox[{ "\"\\"", ",", "NotationMakeBoxes", ",", "NotationMakeExpression"}], "}"}]}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.758206338430524*^9, 3.758206341793087*^9}, { 3.7582112095448017`*^9, 3.758211209976789*^9}, {3.758211302957556*^9, 3.758211314687251*^9}, 3.761400048117167*^9, {3.761758250761093*^9, 3.761758252341844*^9}, 3.7700330604396152`*^9, 3.770033185356071*^9},ExpressionUUID->"bb42ed05-0067-466c-8f68-\ 490b4076e3d8"], Cell["\<\ \t\t\tThe above assumes that the current context is in fact Global`!\ \>", "Text", CellChangeTimes->{{3.758206216748415*^9, 3.758206234952991*^9}, { 3.758211290109406*^9, 3.7582112914613953`*^9}, 3.7614000488847*^9, { 3.7617582564572077`*^9, 3.7617582569664507`*^9}, 3.770033065039239*^9},ExpressionUUID->"d4ff89d9-d244-4070-9ffd-\ ab783732cbd0"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ RowBox[{ RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", "#"}], "]"}], "&"}], "@", "$Context"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.758211264017559*^9, 3.758211264018753*^9}, { 3.7582113170367804`*^9, 3.758211317455592*^9}, {3.758211384114606*^9, 3.7582113891461897`*^9}, 3.761400050148987*^9, {3.7617582610507507`*^9, 3.761758262647715*^9}, 3.7700331217819633`*^9, 3.770033187412513*^9},ExpressionUUID->"94b35b94-1a7d-42ba-85fd-\ e304b3d20050"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", RowBox[{"Evaluate", "@", "$Context"}]}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.758211270208699*^9, 3.75821127020964*^9}, { 3.7582113178608713`*^9, 3.758211318288128*^9}, 3.761400051100971*^9, { 3.76175826462099*^9, 3.7617582676331367`*^9}, 3.770033123598151*^9, 3.770033188252555*^9},ExpressionUUID->"59496b66-ae4c-425a-a8bf-\ 74f293936e8b"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", "x"}], "]"}]}], "]"}], "@", "$Context"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.758211240146318*^9, 3.758211240147313*^9}, 3.761400051900865*^9, {3.76175827136557*^9, 3.7617582726872797`*^9}, 3.770033125133524*^9, 3.770033189756363*^9},ExpressionUUID->"183ff828-0cc4-4b82-b66e-\ 06ca102216cb"], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{"DumpSave", "[", RowBox[{"\"\\"", ",", RowBox[{"Evaluate", "[", RowBox[{"{", RowBox[{ RowBox[{"Context", "[", "]"}], ",", "NotationMakeBoxes", ",", "NotationMakeExpression"}], "}"}], "]"}]}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{{3.758206158319065*^9, 3.758206167182637*^9}, { 3.758206343274136*^9, 3.758206343753145*^9}, {3.758211214296632*^9, 3.758211214784382*^9}, 3.75821124873524*^9, {3.758211320804739*^9, 3.758211322017871*^9}, 3.761400053215851*^9, {3.7617582760062523`*^9, 3.761758277239504*^9}, 3.7700331263573*^9, 3.770033190828609*^9},ExpressionUUID->"efa1142a-164d-4706-8278-\ b00185b221b2"], Cell["\<\ \t\t\tOr succinctly using the @ shortcut to mean that RHS is the argument of \ LHS. Or that LHS is \[OpenCurlyDoubleQuote]applied\[CloseCurlyDoubleQuote] to \ RHS. 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}, 3.770033127686103*^9},ExpressionUUID->"31174e18-e5c3-42e7-b799-\ 826c787a8906"], Cell[BoxData[ RowBox[{"\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}, 3.77003312920553*^9, 3.770033191948394*^9},ExpressionUUID->"0bc8c30c-1d50-4493-9c56-\ 063bb8a89a41"], Cell["\<\ \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\tIn this case, the 1st level argument of interest would be {...}. The \ stuff inside is at \[OpenCurlyDoubleQuote]2nd level\[CloseCurlyDoubleQuote]. \ I.e., DumbSave sees its 2nd argument: {...} but because of HoldRest does not \ \[OpenCurlyDoubleQuote]dig deeper\[CloseCurlyDoubleQuote]. \t\t\tThis is overridden by putting Evaluate on the 1st level (and therefore \ {...} is now 2nd level and the stuff inside 3rd). \ \>", "Text", CellChangeTimes->{{3.758210913537651*^9, 3.758211127065795*^9}, { 3.758211404562688*^9, 3.758211417858315*^9}, {3.761400055877595*^9, 3.761400064509173*^9}, {3.761758302101101*^9, 3.761758307913857*^9}, { 3.770033130461433*^9, 3.770033133349373*^9}},ExpressionUUID->"4914b044-7102-439d-a65b-\ 4160622dd1a3"] }, Closed]] }, Closed]] }, NotebookEventActions:>{{"MenuCommand", "Save"} :> Module[{$CellContext`dy, $CellContext`mn, $CellContext`yr}, \ {$CellContext`dy, $CellContext`mn, $CellContext`yr} = Map[(LinkWrite[ First[$FrontEnd], FrontEnd`Value[#]]; LinkRead[ First[$FrontEnd]])& , {"Day", "MonthName", "Year"}]; 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