7. Relativity and Cosmology

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Needs["Calculus`VectorAnalysis`"]

Needs["Graphics`PlotField3D`"]

<< "D:\Mma\Inits\ch7init.m"

Clear["Global`*"];

7.2 Problems

7.2.2  General Relativity and Cosmology Problems

Problem 8:  Zero-pressure cosmological models

The equations for the zero-pressure Robertson-Walker models can be written as:
		 R''[t]==(1/3) R[t] λ - 
          (4/3)  G π ((Subscript[R, 0])^3) ρ[t]/R[t]^2
	or
		((R'[t])^2)== -κRW + (1/3) λ R[t]^2 + 
          (8/3) G π ((Subscript[R, 0])^3) ρ[0]/R[t]
Parameters directly related to observation are:
	1.  The Hubble expansion rate (Subscript[H, 0]) = R'[0]/R[0] ;
	2.  The dimensionless deceleration parameter (Subscript[q, 0]) = -R''[0]/ (((Subscript[H, 0])^2) R[0]) ;
	3.  The dimensionless density parameter σ = 4 G π ρ[0] /  (3 ((Subscript[H, 0])^2)) ; and
	4.  The dimensionless cosmological parameter L = λ / (3 ((Subscript[H, 0])^2)) .

Part (a)  Show that (Subscript[q, 0]) and (Subscript[κ, RW]) satisfy the following equations:
		(Subscript[q, 0]) = - L + σ
		(Subscript[κ, RW]) =  -(Subscript[H, 0]) R[0] (1 - L - 2σ)

eq1 = R''[t] == (λ R[t])/3 - (4 G π R[0]^3 ρ[0])/(3 R[t]^2)

eq2 = R'[t]^2 == -κRW + (λ R[t]^2)/3 + (8 G π R[0]^3 ρ[0])/(3 R[t])

parameterRule = {R'[0] -> Ho R[0]
				,R''[0]-> - Ho^2 qo R[0]
				,ρ[0] -> 3 Ho^2 σ/(4 G π)
				,λ -> 3 L Ho^2 }

qRule = Solve[{eq1,eq2}/.t->0/.parameterRule,{qo,κRW}]//Flatten//Simplify

Part (b)  Derive an analytical expression for the cosmological age of the universe as a function of (Subscript[q, 0]) when the cosmological parameter L is zero.  Plot the results.

change = {R'[t]->X'[t] R[0], R[t]->X[t] R[0]}

eq4 = (eq2/.parameterRule/.qRule/.change//Solve[#,X'[t]]&)[[2]]

integrand = (1/X'[t]/.eq4/.{L->0,Ho->1,σ->qo}//Together)

??Integrate

age0 = Integrate[integrand,{X[t],0,1},GenerateConditions -> False]//PowerExpand//Simplify

Re[Evaluate[age0/. qo->.5001]]

label = Graphics[
	{PointSize[.03],Point[{.5,Re[Evaluate[age0/. qo->.500001]]}]
	,Text[FontForm["k=0 Flat",{"Times-Roman",10}],{.77,.67}]
	,Text[FontForm["k=1 Closed",{"Times-Roman",10}],{1.4,.58}]
	,Text[FontForm["k=-1 Open",{"Times-Roman",10}],{.44,.82}]}];

agePlot[Ho_] := Plot[age0,{qo,.00001,2}
				,Frame->True
				,PlotLabel->"Age for zero cosmological constant"
				,FrameLabel->{"10^-29 gm/cc","Age"
					,"Density","Billion Years"}
				,FrameTicks->
					{ {0,{1/2, .000196 Ho^2}
					    ,{1,2*.000196 Ho^2},{1.5,3*.000196 Ho^2}}
					,{ {.5,.5*9.78*(100/Ho)},{.6,.6*9.78*(100/Ho)}
					  ,{.7,.7*9.78*(100/Ho)},{.8,.8*9.78*(100/Ho)}
					  ,{.9,.9*9.78*(100/Ho)},{1,1*9.78*(100/Ho)}}}
				,GridLines->Automatic
				,Epilog->label[[1]]	]

agePlot[50];

Part (c)  Consider models with a non-zero cosmological parameter and make a table of ages for different values of σ and (Subscript[q, 0]).

ageL[q_,dimmass_] := NIntegrate[(1/X'[t]/.eq4/.
		{Ho->1}/.{L->dimmass-q}/.{qo->q}/.
		{σ->dimmass}/.{X[t]->x}),{x,0,1}]//N[#,4]&

ageTable[Ho_] := Table[N[ 9.78*(100/Ho) ageL[q,σ],3],
							{q,0,2,.5},{σ,0,2.5,.5}]

TableForm[ageTable[50],TableSpacing->{0,3,3}
		,TableHeadings->{{"qo=0.0","qo=0.5","qo=1.0","qo=1.5","qo=2.0"}
				,{"σ=0.0","σ=0.5","σ=1.0","σ=1.5","σ=2.0","σ=2.5"}}]