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Big Rip & Big Crunch

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Big Rip & Big Crunch

Verfasst: Mi 4. Dez 2024, 06:01
von Yukterez
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Bild This is the english version.   Bild Für die deutschsprachige Version geht es hier entlang. Bild
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Big Rip

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Evolution of a Big Rip universe with H=H0·√[Ωr/a⁴+Ωm/a³+Ωk/a²+ΩΛ·a] (Equation of state for Λ: w=-4/3); initial conditions at a=1: H0=67.15, Ωm=0.315, ΩΛ=0.685, Ωr=0, Ωk=0 → the Big Rip happens at t=44.8 billion years after the big bang. Spactime diagrams in in proper, comoving & conformal coordinates, the light cone originates at a=1, t=14.4169 Gyr:

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↑ proper, r(t) ◉ ↓ comoving, R(t)

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↑ comoving, R(t) ◉ ↓ conformal, R(η)

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For a comparison with our universe where H=H0·√[Ωr/a⁴+Ωm/a³+Ωk/a²+ΩΛ] see here. Code:

Code: Alles auswählen

   (* | Evolution of a Big Rip Universe | Simon Tyran, Vienna | www.yukterez.net | *)
   
   set = {"GlobalAdaptive", "MaxErrorIncreases"->100,
   Method->"GaussKronrodRule"};                                (* Integration Rule *)
   if[F_] := If[ΩΛ<=0, Nothing, If[ΩK>0, Nothing, F]]       (* Event Horizon Check *)
   n = 100;                                                     (* Recursion Depth *)
   int[f_, {x_, xmin_, xmax_}] :=                                      (* Integral *)
   NIntegrate[f, {x, xmin, xmax},
   Method->set, MaxRecursion->n, WorkingPrecision->wp];
   wp = MachinePrecision;                                     (* Working Precision *)
   im = 320;                                                         (* Image Size *)
   tmax = 488/10 Gyr;                                         (* Integration Limit *)
   ηmax = 59.28619955687554;                               (* Conformal Plot Range *)
   prmax = 70; ptmax = tmax/Gyr;                             (* Regular Plot Range *)
   
   c = 299792458 m/sek;                                              (* Lightspeed *)
   G = 667384*^-16 m^3 kg^-1 sek^-2;                            (* Newton Constant *)
   Gyr = 10^7*36525*24*3600 sek;                                  (* Billion Years *)
   Glyr = Gyr*c;                                             (* Billion Lightyears *)
   Mpc = 30856775777948584200000 m;                                  (* Megaparsec *)
   kB = 13806488*^-30 kg m^2/sek^2/K;                        (* Boltzmann Constant *)
   h = 662606957*^-42 kg m^2/sek;                               (* Planck Constant *)
   ρc[H_] := 3H^2/8/π/G;                                       (* Critical Density *)
   ρR = 8π^5 kB^4 T^4/15/c^5/h^3;                             (* Radiation Density *)
   ρΛ = ρc[H0] ΩΛ;                                          (* Dark Energy Density *)
   T = 2725/1000 K;                                             (* CMB Temperature *)
   kg = m = sek = 1;                                                   (* SI Units *)
   
   ΩR = 0;                             (* Radiation Proportion including Neutrinos *)
   ΩM = 315/1000;                       (* Matter Proportion including Dark Matter *)
   ΩΛ = 1-ΩM-ΩR;                                         (* Dark Energy Proportion *)
   ΩT = ΩR+ΩM+ΩΛ;                           (* Total Density over Critical Density *)
   ΩK = 1-ΩT;                                                 (* Curvature Density *)
   
   H0 = 67150 m/Mpc/sek;                                        (* Hubble Constant *)
   H[a_] := H0 Sqrt[ΩR/a^4+ΩM/a^3+ΩK/a^2+ΩΛ*a]                 (* Hubble Parameter *)
   
   sol = Quiet[NDSolve[{A'[t]/A[t] == H[A[t]], A[0] == 1*^-15},
   A, {t, 0, tmax},
   MaxSteps->∞, WorkingPrecision->wp]];
   
   a[t_] := Evaluate[(A[t]/.sol)[[1]]];                (* Scale Factor a by Time t *)
   т[a_] := int[1/A/H[A], {A, 0, a}];                  (* Time t by Scale Factor a *)
   rP[t_] := a[t] int[c/a[т], {т, 0, t}];          (* Proper Particle Horizon by t *)
   rp[a_] := a int[c/A^2/H[A], {A, 0, a}];         (* Proper Particle Horizon by a *)
   RP[t_] := int[c/a[т], {т, 0, t}];             (* Comoving Particle Horizon by t *)
   Rp[a_] := int[c/A^2/H[A], {A, 0, a}];         (* Comoving Particle Horizon by a *)
   rE[t_] := if[a[t] int[c/a[т], {т, t, tmax}]];      (* Proper Event Horizon by t *)
   re[a_] := if[a int[c/A^2/H[A], {A, a, tmax}]];     (* Proper Event Horizon by a *)
   RE[t_] := if[int[c/a[т], {т, t, tmax}]];         (* Comoving Event Horizon by t *)
   Rε[a_] := if[int[c/A^2/H[A], {A, a, tmax}]];     (* Comoving Event Horizon by a *)
   rL[t0_, t_] := a[t] int[c/a[т], {т, t, t0}];          (* Proper Light Cone by t *)
   rl[a0_, a_] := a int[c/A^2/H[A], {A, a, a0}];         (* Proper Light Cone by a *)
   RL[t0_, t_] := int[c/a[т], {т, t, t0}];             (* Comoving Light Cone by t *)
   Rl[a0_, a_] := int[c/A^2/H[A], {A, a, a0}];         (* Comoving Light Cone by a *)
   rH[t_] := c/H[a[t]];                               (* Proper Hubble Radius by t *)
   rh[a_] := c/H[a];                                  (* Proper Hubble Radius by a *)
   RH[t_] := c/H[a[t]]/a[t];                        (* Comoving Hubble Radius by t *)
   Rh[a_] := c/H[a]/a;                              (* Comoving Hubble Radius by a *)
   
   t0 = Re[t/.FindRoot[a[t]-1, {t, 10 Gyr}]]; ti = t Gyr; τi = τ Gyr;
   "t0"->t0/Gyr "Gyr"                                              (* Current Time *)
   
   ã[η_] := Quiet[FindRoot[Rp[Ã]/Glyr-η,     (* Scale Factor a by Conformal Time η *)
   {Ã, 0.00001}, WorkingPrecision->wp, MaxIterations->1000][[1, 2]]];
   ā = Quiet[Interpolation[Join[{{0, 0}},
   ParallelTable[{((Sin[η π/ηmax-π/2]+1) ηmax/2),
   ã[((Sin[η π/ηmax-π/2]+1) ηmax/2)]}, {η, ηmax/im, ηmax, ηmax/im}]]]];
   
   Ť[η_] := Quiet[FindRoot[RP[τ Gyr]/Glyr-η,                          (* t by η *)
   {τ, 1}, WorkingPrecision->wp, MaxIterations->1000][[1, 2]]]
   (* ţ = Quiet[Interpolation[Join[{{0, 0}},
   ParallelTable[{((Sin[η π/ηmax-π/2]+1) ηmax/2),
   Ť[((Sin[η π/ηmax-π/2]+1) ηmax/2)]}, {η, ηmax/im, ηmax, ηmax/im}]]]]; *)
   
   rpN = rp[1]/Glyr;
   
   "PROPER DISTANCES, f(t)"
   
   pt = Quiet[Plot[Evaluate[
   {rH[τi]/Glyr, rP[τi]/Glyr, rE[τi]/Glyr}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot1[t_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {rL[ti, τi]/Glyr, -rL[ti, τi]/Glyr}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], pt]], 90 Degree]}}]];
   
   Do[Print[plot1[t]], {t, {t0/Gyr}}]
   
   plot2 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[a[τ Gyr] n^(7/2)/250, {n, 1, 55, 1}]]],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]
   
   "COMOVING DISTANCES, f(t)"
   
   ct = Quiet[Plot[Evaluate[
   {rH[τi]/(a[τi]Glyr), rP[τi]/(a[τi]Glyr), rE[τi]/(a[τi]Glyr)}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot3[t_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {rL[ti, τi]/(a[τi]Glyr), -rL[ti, τi]/(a[τi]Glyr)}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], ct]], 90 Degree]}}]];
   
   Do[Print[plot3[t]], {t, {t0/Gyr}}]
   
   plot4 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[n, {n, 10, 100, 10}]]],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]
   
   "PROPER DISTANCES, f(a)"
   
   pa = Quiet[Plot[Evaluate[
   {rh[α]/Glyr, rp[α]/Glyr, re[α]/Glyr}],
   {α, 0, prmax Gyr/t0}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, prmax Gyr/t0}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot5[å_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {rl[å, α]/Glyr, -rl[å, α]/Glyr}],
   {α, 0, prmax Gyr/t0}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, prmax Gyr/t0}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], pa]], 90 Degree]}}]];
   
   Do[Print[plot5[å]], {å, {1}}]
   
   plot6 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[α n^(7/2)/250, {n, 1, 55, 1}]]],
   {α, 0, prmax Gyr/t0}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, prmax Gyr/t0}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]
   
   "COMOVING DISTANCES, f(a)"
   
   ca = Quiet[Plot[Evaluate[
   {rh[α]/Glyr/α, rp[α]/Glyr/α, re[α]/Glyr/α}],
   {α, 0, prmax Gyr/t0}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, prmax Gyr/t0}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot7[å_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {rl[å, α]/Glyr/α, -rl[å, α]/Glyr/α}],
   {α, 0, prmax Gyr/t0}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, prmax Gyr/t0}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], ca]], 90 Degree]}}]];
   
   Do[Print[plot7[å]], {å, {1}}]
   
   plot8 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[n, {n, 10, 100, 10}]]],
   {α, 0, prmax Gyr/t0}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, prmax Gyr/t0}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]
   
   "COMOVING DISTANCES, f(η)"
   
   cη = Quiet[Plot[Evaluate[
   {Rh[ā[Ct]]/Glyr, Ct, Rε[ā[Ct]]/Glyr}],
   {Ct, 0, ηmax}, Frame->True, AspectRatio->prmax/ηmax,
   FrameTicks->None, PlotRange->{{0, ηmax}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot9[η_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {η-Ct, Ct-η}], {Ct, 0, ηmax},
   Frame->True, AspectRatio->prmax/ηmax,
   FrameTicks->None, PlotRange->{{0, ηmax}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], cη]], 90 Degree]}}]];
   
   Do[Print[plot9[η]], {η, {rpN}}]
   
   plot10 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[n, {n, 10, 100, 10}]]],
   {Ct, 0, ηmax}, Frame->True, AspectRatio->prmax/ηmax,
   FrameTicks->None, PlotRange->{{0, ηmax}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]

Big Rip & Big Crunch

Verfasst: Mi 4. Dez 2024, 11:28
von Yukterez
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Big Crunch

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Evolution of a big crunch universums with Ωr=2, Ωk=-1 and H=H0·√[Ωr/a⁴+Ωk/a²] (Ωm and ΩΛ were set to 0). Initial conditions at a=1: H0=67.15 → the big crunch happens at a=√2 and t=41.1856 billion years after the big bang (the Hubbleparameter changes its sign at t=20.5928). Spacetime diagrams in proper, comoving & conformal coordinates, the light cone origin is at a=√2:

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↑ proper, r(t) ◉ ↓ comoving, R(t)

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↑ comoving, R(t) ◉ ↓ conformal, R(η)

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Code: Alles auswählen

   (* | Evolution of a Big Crunch Universe | Simon Tyran | http://yukterez.net/f | *)
   
   set = {"GlobalAdaptive", "MaxErrorIncreases"->100};         (* Integration Rule *)
   if[F_] := If[ΩΛ<=0, Nothing, If[ΩK>0, Nothing, F]]       (* Event Horizon Check *)
   n = 100;                                                     (* Recursion Depth *)
   int[f_, {x_, xmin_, xmax_}] :=                                      (* Integral *)
   Quiet[NIntegrate[f, {x, xmin, xmax},
   Method->set, MaxRecursion->n, WorkingPrecision->wp]];
   wp = MachinePrecision;                                     (* Working Precision *)
   im = 180;                                                         (* Image Size *)
   prmax = 70; ptmax = tmax/Gyr;                             (* Regular Plot Range *)
   
   c = 299792458 m/sek;                                              (* Lightspeed *)
   G = 667384*^-16 m^3 kg^-1 sek^-2;                            (* Newton Constant *)
   Gyr = 10^7*36525*24*3600 sek;                                  (* Billion Years *)
   Glyr = Gyr*c;                                             (* Billion Lightyears *)
   Mpc = 30856775777948584200000 m;                                  (* Megaparsec *)
   kB = 13806488*^-30 kg m^2/sek^2/K;                        (* Boltzmann Constant *)
   h = 662606957*^-42 kg m^2/sek;                               (* Planck Constant *)
   ρc[H_] := 3H^2/8/π/G;                                       (* Critical Density *)
   ρR = 8π^5 kB^4 T^4/15/c^5/h^3;                             (* Radiation Density *)
   ρΛ = ρc[H0] ΩΛ;                                          (* Dark Energy Density *)
   T = 2725/1000 K;                                             (* CMB Temperature *)
   kg = m = sek = 1;                                                   (* SI Units *)
   
   ΩR = 2;                             (* Radiation Proportion including Neutrinos *)
   ΩM = 0;                              (* Matter Proportion including Dark Matter *)
   ΩΛ = 0;                                               (* Dark Energy Proportion *)
   ΩT = ΩR+ΩM+ΩΛ;                           (* Total Density over Critical Density *)
   ΩK = 1-ΩT;                                                 (* Curvature Density *)
   
   H0 = 67150 m/Mpc/sek;                                        (* Hubble Constant *)
   H[a_] := H0 Sqrt[ΩR/a^4+ΩM/a^3+ΩK/a^2+ΩΛ]                   (* Hubble Parameter *)
   Ж[t_] := (Sqrt[ΩR]+H0(t-t ΩR))/(t(2Sqrt[ΩR]+H0 (t-t ΩR)));
   
   a[t_]  := Re[Sqrt[H0 t (2Sqrt[ΩR]+H0 (t-t ΩR))]];   (* Scale Factor a by Time t *)
   т[a_]  := Re[(2H0 Sqrt[ΩR]-Sqrt[-4a^2 H0^2 (-1+ΩR)+4H0^2ΩR])/(2H0^2(ΩR-1))];
   rP[t_] := Re[a[t] int[c/a[т], {т, 0, t}]];      (* Proper Particle Horizon by t *)
   rp[a_] := Re[a int[c/A^2/H[A], {A, 0, a}]];     (* Proper Particle Horizon by a *)
   RP[t_] := Re[int[c/a[т], {т, 0, t}]];         (* Comoving Particle Horizon by t *)
   Rp[a_] := Re[int[c/A^2/H[A], {A, 0, a}]];     (* Comoving Particle Horizon by a *)
   rE[t_] := Nothing;                                 (* Proper Event Horizon by t *)
   re[a_] := Nothing;                                 (* Proper Event Horizon by a *)
   RE[t_] := Nothing;                               (* Comoving Event Horizon by t *)
   Rε[a_] := Nothing;                               (* Comoving Event Horizon by a *)
   rL[t0_, t_] := Re[a[t] int[c/a[т], {т, t, t0}]];      (* Proper Light Cone by t *)
   rl[a0_, a_] := Re[a int[c/A^2/H[A], {A, a, a0}]];     (* Proper Light Cone by a *)
   RL[t0_, t_] := Re[int[c/a[т], {т, t, t0}]];         (* Comoving Light Cone by t *)
   Rl[a0_, a_] := Re[int[c/A^2/H[A], {A, a, a0}]];     (* Comoving Light Cone by a *)
   rH[t_] := c/Abs[Ж[t]];                             (* Proper Hubble Radius by t *)
   rh[a_] := c/Abs[H[a]];                             (* Proper Hubble Radius by a *)
   RH[t_] := c/Abs[Ж[t]*a[t]];                      (* Comoving Hubble Radius by t *)
   Rh[a_] := c/Abs[H[a]*a];                         (* Comoving Hubble Radius by a *)
   
   t0 = Quiet[Re[t/.FindRoot[a[t]-Sqrt[2], {t, 10 Gyr}]]]; ti = t Gyr; τi = τ Gyr;
   "t0"->t0/Gyr "Gyr"                                              (* Current Time *)
   
   tmax = 2*t0;
   rpN = Rp[tmax/2]/Glyr;
   ηmax = RP[0.9999999 tmax]/Glyr;

   Ť[η_] := Quiet[FindRoot[RP[τ Gyr]/Glyr-η,                             (* t by η *)
   {τ, ((Sin[η π/ηmax-π/2]+1)/2) ptmax}, WorkingPrecision->wp,
   MaxIterations->1000][[1, 2]]]
   
   "PROPER DISTANCES, f(t)"
   
   pt = Quiet[Plot[Evaluate[
   {rH[τi]/Glyr, rP[τi]/Glyr, Nothing}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot1[t_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {rL[ti, τi]/Glyr, -rL[ti, τi]/Glyr}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], pt]], 90 Degree]}}]];
   
   Do[Print[plot1[t]], {t, {ptmax/2}}]
   
   plot2 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[a[τ Gyr] n^(7/2)/250, {n, 1, 55, 1}]]],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]
   
   "COMOVING DISTANCES, f(t)"
   
   ct = Quiet[Plot[Evaluate[
   {rH[τi]/(a[τi]Glyr), rP[τi]/(a[τi]Glyr), Nothing}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]},
   {Darker[Green], Thickness[0.005]}, {Purple, Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot3[t_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {rL[ti, τi]/(a[τi]Glyr), -rL[ti, τi]/(a[τi]Glyr)}],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], ct]], 90 Degree]}}]];
   
   Do[Print[plot3[t]], {t, {ptmax/2}}]
   
   plot4 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[n, {n, 10, 100, 10}]]],
   {τ, 0, ptmax}, Frame->True, AspectRatio->prmax/ptmax,
   FrameTicks->None, PlotRange->{{0, ptmax}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]
   
   "COMOVING DISTANCES, f(η)"
   
   cη = Quiet[Plot[Evaluate[{RH[Ť[Ct] Gyr]/Glyr, Ct}],
   {Ct, 0, ηmax}, Frame->True, AspectRatio->prmax/ηmax,
   FrameTicks->None, PlotRange->{{0, ηmax}, {0, prmax}},
   PlotStyle->{{Thickness[0.005]}, {Darker[Green], Thickness[0.005]}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}]];
   
   plot9[η_] := Rasterize[Grid[{{Rotate[Quiet[Show[Plot[Evaluate[
   {η-Ct, Ct-η}], {Ct, 0, ηmax},
   Frame->True, AspectRatio->prmax/ηmax,
   FrameTicks->None, PlotRange->{{0, ηmax}, {0, prmax}},
   PlotStyle->{{Orange, Thickness[0.005]}, {{Orange, Thickness[0.005]}, Dashed}},
   ImageSize->im, Filling->Top, FillingStyle->Opacity[0.1], ImagePadding->1,
   GridLines->{{}, {}}], cη]], 90 Degree]}}]];
   
   Do[Print[plot9[η]], {η, {rpN}}]
   
   plot10 = Rasterize[Grid[{{Rotate[Quiet[Plot[Evaluate[
   Join[{0}, Table[n, {n, 10, 100, 10}]]],
   {Ct, 0, ηmax}, Frame->True, AspectRatio->prmax/ηmax,
   FrameTicks->None, PlotRange->{{0, ηmax}, {0, prmax}},
   PlotStyle->Table[{Dashing->Large, Thickness[0.005],
   Gray}, {n, 1, 100}], ImageSize->im, ImagePadding->1]], 90 Degree]}}]]

Metric, tensors and equations of motion: click here, for a static big bang universe see here.
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