Ast402/830: Cosmology

Age of the Universe
28 January 1998

Friedmann's equation \(TraditionalForm
      \`\((da\/\(\(H\_0\) dt\))\)\^2 - 
          \(Ω\_0\) \(\(a\^2\)(ϱ\/ϱ\_0)\) = 
        1 - Ω\_0\) can be integrated to find the expansion parameter as a function of time. With a(t), we can find the age of the universe when various events occurred, for example when the light from a distant galaxy was emitted.
Let us compute a(t) specifically for pressureless matter for which ϱ/(Subscript[ϱ, 0]) = (a^(-3)).  Then  \(TraditionalForm
      \`\((da\/\(\(H\_0\) dt\))\)\^2 - 
          \(Ω\_0\) a\^\(-1\) = 1 - Ω\_0\).
In class we interpreted this equation as KE+PE=E.  For negative total energy or (Subscript[Ω, 0]) > 1,  the expansion proceeds until the kinetic term reaches zero at  \(TraditionalForm
      \`a = Ω\_0\/\(Ω\_0 - 1\)\), and then the universe collapses.
Now let's integrate Friedmann's equation.  Let us change variables to \(TraditionalForm
      \`\(u = a\ \(Ω\_0 - 1\)\/Ω\_0, \)\) so that u runs from 0 to 1 for (Subscript[Ω, 0]) > 1.

integrand = 
      dd/Sqrt[1 - o + o/a] /. dd -> o/(o - 1) /. 
            a -> u o/(o - 1) /. o - 1 -> od /.  1 - o -> -od

Integrate[
      ((o (od^(-3/2)))/(Sqrt[-1 + (1/( up))])), {up, 0, u}]

Let us examine the conditions in the answer.  I will not bother with the boundary  a = 0.  The second condition is that the universe expands to u = 1and then contracts if (Subscript[Ω, 0]) > 1. Thus the integral is found for attainable values of  a.

ageSupercritical = %[[2]]

For (Subscript[Ω, 0]) < 1, let \(TraditionalForm
      \`u = a \(\ 1 - Ω\_0\)\/Ω\_0\).

integrand = 
      dd/Sqrt[1 - o + o/a] /. dd -> o/(1 - o) /. 
            a -> u o/(1 - o) /. o - 1 -> -od /.  1 - o -> od

Integrate[((o (od^(-3/2)))/(Sqrt[1 + (1/( up))])), {up, 0, u}]

tmpSubcritical = %[[2]]

Let's see if we get back the limit for expansion at constant speed:

ageEmpty = 
      Limit[tmpSubcritical /. u -> a (1 - o)/o /. od -> 1 - o, 
        o -> 0]

For a universe with critical density,

For a universe with twice the critical density,

ageTwiceCritical = 
      ageSupercritical /. u -> a (o - 1)/o /. od -> o - 1 /. 
        o -> 2

This expression for the age has singularities at (Subscript[Ω, 0]) = 0 and 1 and is complex for  (Subscript[Ω, 0]) < 1. Let us define a function age without such defects for use in computations.

Remove[age];

\(age[a_, o_ /; o == 0] := a; 

    age[a_, o_ /; o == 1] := 2  a^\((3/2)\)/3; 

    age[a_, o_ /; o > 1] := 
      Module[{u}, u = a \((o - 1)\)/o; 
        o \((ArcSin@Sqrt[u] - Sqrt[u - u^2])\)/\((o - 1)\)^\((3/2)\)]; 

    age[a_, o_ /; o < 1] := 
      Module[{u}, u = a \((1 - o)\)/o; 
        \(-o\) \((\ ArcSinh@Sqrt[u] - Sqrt[u + u^2])\)/
            \((1 - o)\)^\((3/2)\)]; \)

\($TextStyle = "Helvetica"; 

    ParametricPlot[{{age[a, 0], a}, {age[a, 1], a}, {age[a, 2], a}}, {a, 0, 
        1}, AxesLabel -> {t\ H\_0, a}, PlotLabel -> "From the Big Bang", 
      GridLines -> Automatic]; \)

ParametricPlot[{{age[a, 0] - age[1, 0], a}, {age[a, 1] - age[1, 1], 
          a}, {age[a, 2] - age[1, 2], a}}, {a, 0, 1}, 
      AxesLabel -> {t (Subscript[H, 0]), a}, PlotLabel -> "From the Present", 
      GridLines -> Automatic];

Plot[age[1, om], {om, 0, 2}, 
      AxesLabel -> {(Subscript[Ω, 0]), t (Subscript[H, 0])}, 
      PlotLabel -> "Age of the Universe", GridLines -> Automatic];

Comoving coordinate vs expansion parameter
30 January 1998

We want to know the radial coordinate of a source that emitted the light that we now see. The source is at expansion parameter a and comoving coordinate r. We are at a = 1 and r = 0. For a light ray, \(TraditionalForm
      \`d\ t = \(\(a(t)\)\ d\ r
            \)\/\((1 - \((Ω\_0 - 1)\) \(H\_0\^2\) r\^2)\)\^\(1/2
              \)\), or \(TraditionalForm
      \`\(\(H\_0\) d\ t\)\/\(a(t)\) = 
        \(d\ u\)\/\((1 - \((Ω\_0 - 1)\) u\^2)\)\^\(1/2\)\), where u = (Subscript[H, 0]) r.  We use Friedmann's equation to convert the left-hand side to  \(TraditionalForm
      \`\(d\ a\)\/\(a(1 - Ω\_0 + Ω\_0/a)\)\^\(1/2
            \)\).

rhs = Integrate[1/Sqrt[1 - (o - 1) up^2], {up, 0, u}]lhs = Integrate[1/(ap Sqrt[1 - o + o/ap]), {ap, a, 1}]

rhs = rhs Sqrt[1 - o]lhs = lhs Sqrt[1 - o] // Simplify

rhs = Sinh[rhs]/Sqrt[1 - o]lhs = FullSimplify@TrigExpand@Sinh[lhs]/Sqrt[1 - o]

We arrive at \(TraditionalForm
      \`r = \(H\_0\^\(-1\)\) 2\ 
          \(\(Ω\_0\^\(-2\)\)(
            \(-2\)\ a\ \((\(-1\) + Ω\_0)\) + 
              Ω\_0 + 
              a\ \((\(-2\) + Ω\_0)\)\ 
                \@\(1 + \((\(-1\) + 1\/a)\) Ω\_0\))\)\). This is called Mattig's formula.

For nearby galaxies, where a -> 1, this should be equivalent to Hubble's Law.

Simplify@Series[lhs, {a, 1, 2}]

Therefore for nearby galaxies, r = ((Subscript[H, 0])^(-1))(1 - a). Because a = (Subscript[λ, e])/(Subscript[λ, r]),  \(TraditionalForm
      \`r = \(\(\(H\_0\^\(-1\)\)(λ\_r - λ\_e)\)/
            λ\_r = \(H\_0\^\(-1\)\) \(v.\)\)\)

Improved behavior at a->1

The behavior at a -> 1is not evident in Mattig's formula, but we can make it so. The problem with Mattig's formula is that many terms cancel in this limit. We can write it so that the cancellations are explicit. Let

substitution = sfactor -> Sqrt[a^2 + a (-a + 1) o]substitution2 = sfactor2 ->  a^2 + a (-a + 1) o

lhs1 = 
      ((2 (-a (-2 + o) - a o + o +  
              (-2 + o) sfactor))/(o^2));

Are the two expressions the same?

(lhs1 /. substitution) - lhs // Simplify

lhs1 = 
      (1/(o^2)) 
            (2 (
                  o (1 - a) + 
                    (-2 + o) 
                      (-a^2 +  sfactor2)/(a + sfactor))) /. 
          substitution2 // Simplify

lhs = 
      -((2 (-1 + a) 
                  (a o + (-a^2 + sfactor2)/(a + sfactor))
                    )/(o (a + sfactor))) /. substitution2 // 
        Simplify

lhs = 
      -((2 (-1 + a) a (1 + sfactor))/(a^2 + 
                  2  a sfactor + sfactor2)) /. substitution2 // Simplify

lhs = 
      (((-1 + a) 
          )/(Factor@(a (-2 + o) - o + 2)/2/
              (1 + sfactor) - 1)) /. substitution

This is a cleaner version of Mattig's formula:  \(TraditionalForm
      \`r = \(\(H\_0\^\(-1\)\)(1 - a)\)/
          \((1 + \((1 - a)\) \((1 - Ω\_0/2)\) 
                \((1 + \@\(a\^2 + \((1 - a)\)\ a\ Ω\_0\))
                    \)\^\(-1\))\)\)

We can write it in terms of redshift z = (((1 + a))^(-1)) also.

lhs1 = 
      (z/(1/a + ((z (-1 + o/2))/((1 + a (Sqrt[ a + z o])))))) /. 
        a -> 1/(1 + z)

The better form of Mattig's formula in terms of the redshift is 						\(TraditionalForm
      \`r = \(H\_0\^\(-1\)\) 
          z/\((1 + z + 
                \(z(1 - Ω\_0/2)\) 
                  \((1 + \(\((1 + z)\)\^\(-1\)\) 
                          \@\(\(Ω\_0\) z + 
                              \((1 + z)\)\^\(-1\)\))\)\^\(-1\))\)\)

Check the limit:

Series[lhs1 /. z -> 1/a - 1, {a, 1, 2}]Series[lhs1, {z, 0, 2}]

Radial distance

The dimensionless radial distance is the result of the first integral,

rd = 
      Log[2 + ((2 - o)/(Sqrt[1 - o]))]/(Sqrt[1 - o]) - 
        Log[((-2 a (-1 + o) + o)/(Sqrt[1 - o])) + 
              2 a (Sqrt[1 + (-1 + (1/a)) o])]/(Sqrt[1 - o]);

Three simple cases

There are three simple cases:  (Subscript[Ω, 0]) = 0, 1, and 2. For these the radial coordinate is

r012
       = {lhs /. o -> 0 // PowerExpand // Simplify // Apart, 
        lhs /. o -> 1 // PowerExpand // Simplify, 
        lhs /. o -> 2 // PowerExpand // Simplify}

and the radial distance is (Mathematica needed help with Ω=2)

rD012
       = {Limit[rd, o -> 0], Limit[rd, o -> 1] // PowerExpand, 
        rd /. o -> 2 // Sin // Simplify // ArcSin}

Galaxies on two light rays

Two light rays, separated by 0.5 radian, hit us.  The light rays have traversed a lot of space on their way toward us.  What is the present location of the regions they have visited? 
We shall make a plot with  (Subscript[Ω, 0]) = 0, 1,  and  2.  We will put markers on our plot at expansion parameter a = {.67, .50, .2, .1}.   At a = 0.67,  galaxies can be viewed with CCD cameras in the rest-frame V band.  At a = 0.50, the light from galaxies viewed with CCD cameras was emitted in the ultraviolet. The most distant galaxies discovered are at a = 0.2.
The unit on the plot is the Hubble distance ((Subscript[H, 0])^(-1)).

phi = .25;

\(aa = {.75, .5, .2, .1}; \  (*\ Markers\ on\ our\ plot\ *) 

    addedPoints = 
      Map[Table[{Point[{phi\ r012[\([#]\)], rD012[\([#]\)]}], 
                  Point[{\(-phi\)\ r012[\([#]\)], rD012[\([#]\)]}]} /. 
                a -> aa[\([i]\)], {i, Length[aa]}]&, {1, 2, 3}] // Flatten; 
    \)

$TextStyle = "Helvetica"; 
    ParametricPlot[{{phi r012[[1]], rD012[[1]]}, {
          phi r012[[2]], rD012[[2]]}, {phi r012[[3]], 
          rD012[[3]]}, {-phi r012[[1]], rD012[[1]]}, {
          -phi r012[[2]], rD012[[2]]}, {-phi r012[[3]], 
          rD012[[3]]}}, {a, .1, 1}, AspectRatio -> Automatic, 
      PlotRange -> All, Epilog -> {PointSize[.015], addedPoints}, 
      PlotLabel -> "Galaxies on Two Light Rays", MaxBend -> 5];