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Notebook[{

Cell[CellGroupData[{
Cell[" FRW Models ", "Title",
  FontSize->24],

Cell[TextData[{
  "This notebook numerically integrates the Friedman equation (18.78) to find \
a homogeneous isotropic cosmological model. The inputs are the four \
parameters that specify a FRW model \[LongDash]the Hubble constant ",
  Cell[BoxData[
      \(TraditionalForm\`H\_\(\(0\)\(\ \)\)\)]],
  "and the three \[CapitalOmega]'s for radiation, matter, and vacuum energy. \
The program assumes these parameters are such that the universe started with \
a big bang [cf Figure 18.10]."
}], "Text"],

Cell[CellGroupData[{

Cell["Clearing the variables used:", "Subsection"],

Cell["\<\
First,  clear all the variables that will be used in the \
calculation:\
\>", "Text"],

Cell[BoxData[
    \(Clear[omr, omm, omv, omc, h, Th, \ x, y, a0, t0, omcrit, yend, 
      ymax]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Parameters of a FRW model:", "Subsection"],

Cell[TextData[{
  "Four parameters specify a FRW cosmological model. First, there are the \
three",
  StyleBox[" \[CapitalOmega] '",
    FontSlant->"Italic"],
  "s which are the ratios of the present density to critical density for \
radiation, matter, and vacuum energy. These dimensionless numbers are denoted \
by ",
  StyleBox["omr",
    FontWeight->"Bold"],
  ", ",
  StyleBox["omm",
    FontWeight->"Bold"],
  ", and ",
  StyleBox["oml ",
    FontWeight->"Bold"],
  "respectively. These are specified in the following three statements:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(omr = 0.00008\)], "Input"],

Cell[BoxData[
    \(0.00008`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(omm =  .3\)], "Input"],

Cell[BoxData[
    \(0.3`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(omv =  .7\)], "Input"],

Cell[BoxData[
    \(0.7`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The last parameter is the Hubble constant ",
  Cell[BoxData[
      \(TraditionalForm\`H\_0\)]],
  ", or equivalently the Hubble time ",
  Cell[BoxData[
      \(TraditionalForm\`T\_0\)]],
  "=1",
  StyleBox["/",
    FontSlant->"Italic"],
  Cell[BoxData[
      \(TraditionalForm\`H\_0\)]],
  " which we denote here by ",
  StyleBox["Th. ",
    FontWeight->"Bold"],
  "  ",
  Cell[BoxData[
      \(TraditionalForm\`T\_0\)]],
  " has the dimensions of time.  A billion years (Gyr)  is a convenient unit \
of time for cosmology and ",
  Cell[BoxData[
      \(TraditionalForm\`T\_0\)]],
  StyleBox["=9.788 ",
    FontSlant->"Italic"],
  Cell[BoxData[
      \(TraditionalForm\`h\^\(-1\)\ Gyr, \ where\ \ h = H\_0\)]],
  "/[100 (km/s)/Mpc]. Thus, for ",
  Cell[BoxData[
      \(TraditionalForm\`H\_0\)]],
  "=72",
  StyleBox["(km/sec)/Mpc",
    FontVariations->{"CompatibilityType"->0}],
  StyleBox[",  ",
    FontSlant->"Italic"]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(h =  .72\)], "Input"],

Cell[BoxData[
    \(0.72`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Th = 9.788/h\)], "Input"],

Cell[BoxData[
    \(13.594444444444445`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "It is also convenient to define an ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_c\)]],
  " for curvature as in the text. Here we denote it by ",
  StyleBox["omc ",
    FontWeight->"Bold"],
  "and its related to the other ",
  StyleBox["\[CapitalOmega]",
    FontSlant->"Italic"],
  "'s by:"
}], "Text"],

Cell[BoxData[
    \(omc := 1 - omr - omm - omv\)], "Input"],

Cell["For the parameters above ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(omc\)], "Input"],

Cell[BoxData[
    \(\(-0.00007999999999985796`\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Integrating the FRW equation in dimensionless variables:", "Subsection"],

Cell[TextData[{
  "Its convenient to rewrite the FRW equation in terms of  dimensionless \
variables as in(18.78). In the text we used  ",
  Cell[BoxData[
      \(TraditionalForm\`t\&~\)]],
  StyleBox["=",
    FontSlant->"Italic"],
  Cell[BoxData[
      \(TraditionalForm\`H\_0\)]],
  StyleBox["t=t/",
    FontSlant->"Italic"],
  Cell[BoxData[
      \(TraditionalForm\`T\_h\)]],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`\(a\&~\)\)]],
  StyleBox["=a(t)/a(",
    FontSlant->"Italic"],
  Cell[BoxData[
      \(TraditionalForm\`t\_0\)]],
  ").  (The value of ",
  Cell[BoxData[
      \(TraditionalForm\`a\&~\)]],
  " at the present time ",
  Cell[BoxData[
      \(TraditionalForm\`t\_0\)]],
  " is therefore always ",
  StyleBox["1.",
    FontSlant->"Italic"],
  ")   Here its's notationally convnient to write ",
  StyleBox["x",
    FontSlant->"Italic"],
  " for ",
  Cell[BoxData[
      \(TraditionalForm\`t\&~\)]],
  ", and",
  StyleBox[" y ",
    FontSlant->"Italic"],
  "for ",
  Cell[BoxData[
      \(TraditionalForm\`a\&~\)]],
  ". The FRW equation is then the same as that of a non-relativistic particle \
moving in a potential U",
  StyleBox["(y) ",
    FontSlant->"Italic"],
  "where ",
  StyleBox["y ",
    FontSlant->"Italic"],
  "is the particle's position  and ",
  StyleBox["x ",
    FontSlant->"Italic"],
  "is the time. "
}], "Text"],

Cell[BoxData[
    \(U[y_] := \((1/2)\) \((\ \(-omr\)/y^2\  - omm/y - omv\ y^2)\)\)], "Input"],

Cell[TextData[{
  "That is, the FRW equation (18.78) becomes:\n\n       ",
  Cell[BoxData[
      FormBox[
        SuperscriptBox[
          StyleBox[\((dy/dx)\),
            FontSlant->"Italic"], "2"], TraditionalForm]]],
  "= ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_c\)]],
  "(",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_r\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(\[CapitalOmega]\_m\)\(,\)\)\(\ \)\)\)]],
  " ",
  Cell[BoxData[
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  ")-2 U",
  StyleBox["(y; ",
    FontSlant->"Italic"],
  Cell[BoxData[
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  ", ",
  Cell[BoxData[
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  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_v\)]],
  ")  . "
}], "Text"],

Cell[TextData[{
  "We plot this ",
  StyleBox["U(y) ",
    FontSlant->"Italic"],
  "out to a value ",
  StyleBox["y=yend, ",
    FontSlant->"Italic"],
  "also showing a horizontal line at the valueof ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_c\)]],
  "/2. (You might have to change ",
  StyleBox["yend ",
    FontWeight->"Bold"],
  StyleBox["and edit ",
    FontVariations->{"CompatibilityType"->0}],
  " ",
  StyleBox["PlotRange ",
    FontWeight->"Bold"],
  StyleBox["in the ",
    FontVariations->{"CompatibilityType"->0}],
  StyleBox["Show  ",
    FontWeight->"Bold",
    FontVariations->{"CompatibilityType"->0}],
  StyleBox["statement below to get aninformative plot.) ",
    FontVariations->{"CompatibilityType"->0}]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(yend = 3\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(plotu := 
      Plot[U[y], {y, 0, yend}, AxesLabel \[Rule] {"\<y\>", "\<U\>"}, 
        DisplayFunction \[Rule] Identity]\)], "Input"],

Cell[BoxData[
    \(plotomc := 
      Plot[omc/2, \ {y, 0. , yend}, DisplayFunction -> Identity]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[plotu, plotomc, PlotRange \[Rule] {{0, yend}, {0, \(-3\)}}, 
      DisplayFunction -> $DisplayFunction]\)], "Input"],

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  ImageRangeCache->{{{91.5625, 320.938}, {585.562, 444.25}} -> {-1.74315, \
7.92215, 0.0123192, 0.019933}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[TextData[{
  "To solve for a big bang FRW model (one that starts at ",
  StyleBox["a=0 ",
    FontSlant->"Italic"],
  "or ",
  StyleBox["y=0)  ",
    FontSlant->"Italic"],
  "is equivalent to carrying out the integral:\n\n       ",
  StyleBox["x(y)=",
    FontSlant->"Italic"],
  Cell[BoxData[
      FormBox[
        RowBox[{\(\[Integral]\_0\%y\), " ", 
          
          RowBox[{\(\([\(\[CapitalOmega]\_c\)(\ \[CapitalOmega]\_r, \ \
\[CapitalOmega]\_m, \[CapitalOmega]\_v) - 
                    2 \( U(w; \ \[CapitalOmega]\_r, \ \[CapitalOmega]\_m, \ \
\[CapitalOmega]\_v)\)]\)\^\((\(-1\)/2)\)\), 
            StyleBox[
              RowBox[{
                StyleBox["d",
                  FontSlant->"Italic"], "w"}]]}]}], TraditionalForm]]],
  "                                           (*)\n  "
}], "Text"],

Cell[BoxData[
    \(x[y_] := 
      NIntegrate[\((omc - 2  U[w])\)^\((\(-1\)/2)\), \ {w, 0, y}]\)], "Input"],

Cell["\<\
That's the answer. We will plot some specific cases in the next \
subsection. \
\>", "Text"],

Cell[TextData[{
  "For values of ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_c\)]],
  " above the maximum of the potential the expansion is unbounded in time;  \
for values that are below the expansion is bounded and the universe \
recollapses.  The value just at the maximum divides these two behaviors. We \
denote that value by ",
  StyleBox["omcrit.",
    FontWeight->"Bold"]
}], "Text"],

Cell[BoxData[
    \(crit := \ FindMinimum[\(-2\) U[yy], {yy, \  .5}]\)], "Input"],

Cell[BoxData[
    \(omcrit := If[omv >  .000001, \(-crit[\([1]\)]\), \ 0]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(omcrit\)], "Input"],

Cell[BoxData[
    \(\(-0.7522180297530352`\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "For bounded cases the integration in (*) can extend only to the first \
turning point where the denominator vanishes.  This represents the expansion \
of a universe up to its maximum scale factor ",
  StyleBox["ymax.  ",
    FontSlant->"Italic"],
  "Since the integral is singular at the turning point it is necessary to \
stop it just a bit before by small fraction \[Epsilon] ."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(eps =  .0001\)], "Input"],

Cell[BoxData[
    \(0.0001`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The following list of statements looks for the turning point which is the \
smallest positive value of ",
  StyleBox["y ",
    FontSlant->"Italic"],
  "where ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_c\)]],
  "/2=U(tp). This is relevant only for closed models where ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_c\)]],
  "/2 < ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(U\_max\)\(.\)\(\ \)\)\)]],
  " (If the program fails to identify the corret turning point, its possible \
that you might have to insert a  statement ",
  StyleBox["ymax=  ",
    FontWeight->"Bold"],
  StyleBox["to give it the correct value and then run the program again.) ",
    FontVariations->{"CompatibilityType"->0}],
  "\n"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(listtprules = NSolve[omc \[Equal] 2  U[tp], \ tp]\)], "Input"],

Cell[BoxData[
    \({{tp \[Rule] \(\(0.3770878687751771`\)\(\[InvisibleSpace]\)\) + 
            0.6528939073182644`\ \[ImaginaryI]}, {tp \[Rule] \
\(\(0.3770878687751771`\)\(\[InvisibleSpace]\)\) - 
            0.6528939073182644`\ \[ImaginaryI]}, {tp \[Rule] \
\(-0.7539090709026381`\)}, {tp \[Rule] \(-0.0002666666477155029`\)}}\)], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(listtp = tp /. listtprules\)], "Input"],

Cell[BoxData[
    \({\(\(0.3770878687751771`\)\(\[InvisibleSpace]\)\) + 
        0.6528939073182644`\ \[ImaginaryI], \(\(0.3770878687751771`\)\(\
\[InvisibleSpace]\)\) - 
        0.6528939073182644`\ \[ImaginaryI], \(-0.7539090709026381`\), \
\(-0.0002666666477155029`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(listpostp = If[omc < omcrit, Select[listtp, # > 0\  &], {}]\)], "Input"],

Cell[BoxData[
    \({}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(listpostp = Sort[listpostp]\)], "Input"],

Cell[BoxData[
    \({}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(If[omc < omcrit, \ ymax := listpostp[\([1]\)] \((1 - eps)\), 
      ymax := \ yend]\)], "Input"],

Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[TextData[{
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  StyleBox["is introduced which runs from "],
  StyleBox["0",
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  StyleBox[" to "],
  StyleBox["1",
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*)




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End of Mathematica Notebook file.
***********************************************************************)