Question

Problem 5: Cosmology (15 points) In the 1990s, physicists generally assumed that ?,-0, that ?,-0 at the p possible that the universe contained enough dark matter so that 1. Assume that in th "golden oldie" universe model that the measured value of Ho-13.9 Gy resent. It was also is (a) Is this model's spatial geometry spherical, flat, or saddle-shaped? Explain. 2/3 (b) Assume that a O at t 0, show that a(o) - Hot) (c) Find the age of the universe in this model, and compare with 13.2 Gy, which is the best independent estimate of the age of the oldest known star in our galaxy. Is there a problem here? (d) Show that conformal time t (12H2t)/3 in this model. (e) Consider a galaxy whose redshift we measure to be z 3. Use the results above to determine what the model implies about (1) how long light emitted by this galaxy has been traveling to reach us now, and (2) how far the galaxy is from us at the present time (i.e. what a tape measure stretched between these galaxies would read at the present).

Answer 1

I have followed Cosmology books of Scott Dodelson and Barbara Ryden and to a little extent Loeb's book of "How did the first stars and galaxies form?" You can look them up for details. Always feel free to ask if you re stuck anywhere.

As nothing is mentioned about Dark energy, I have assumed that in this model universe it is absent.

Because  $\Omega_t = \Omega_r+\Omega_\nu+\Omega_m=1$ ,

The universe in this model is flat

$P_w = w \epsilon_w$ is the equation of state of any component in the universe. For non-relativistic matter (as considered in the model here),

$\epsilon_w$ is the energy of the component with equation of state described by

Fluid equation,

$\dot{\epsilon_w}+3(P_w+\epsilon_w)\frac{\dot{a}}{a}=0$

Substituting for $P_w$ and integrating,

$\epsilon_w= \epsilon_{w0}a^{-3(1+w)}$$\epsilon_{w0}$ is the current energy density of the ^th component

For a flat universe, with Friedmann-Robertson-Walker metric gives:

$\left(\frac{\dot{a}}{a}\right)^2= \frac{8\pi G}{3c^2}\sum_{w}\epsilon_w= \frac{8\pi G}{3c^2}\sum_{w}\epsilon_{w0}a^{-3(1+w)}$

Here summation is over all the components of the universe in the model

For a flat single component universe,

$\dot{a}^2= \frac{8\pi G}{3c^2}\epsilon_{0}a^{-(1+3w)}$

Let $a(t) = \left(\frac{t}{t_0} \right )^q$

You can go ahead and substitute this in the Ordinary Differential equation above to get

$q = \frac{2}{3(1+w)}$ and   $t_0 = \frac{1}{1+w}\sqrt{\frac{c^2}{6\pi G \epsilon_0}}$

$H(t) = \frac{\dot{a}}{a}= \frac{2}{3(1+w)t}$

$H_0 = \frac{2}{3(1+w)t_0}$

Conformal time $\bar{t}= \int_0^t\frac{dt'}{a} = \int_0^t{\left(\frac{t'}{t_0} \right )^{-\frac{2}{3(1+w)}}dt'}$

Now coming to matter dominated universe as described in the question,

$a(t) = \left(\frac{t}{t_0} \right )^\frac{2}{3}$ and   $H_0 = \frac{2}{3t_0}$

$\therefore a(t) = \left(\frac{3 }{2}H_0 t \right )^\frac{2}{3}$

Age of the universe is $t_0$

$t_0 = \frac{2}{3H_0}$

Current value of Hubble Constant is

Age comes around billion years

Conformal time $\bar{t} = \int_0^t{\left(\frac{t'}{t_0} \right )^{-\frac{2}{3}}dt'}=3t_0\left(\frac{t}{t_0} \right )^\frac{1}{3}=(12 H_0^{-2}t)^\frac{1}{3}$   

I have used the value of  $t_0$ found previously

$a = \frac{1}{1+z}$

$\therefore z=3 \implies a = \frac{1}{4}$

$t = \left(\frac{1}{4} \right )^\frac{3}{2}t_0=1.2 byr$ byr means billion years

Conformal time is $\bar{t} = \left(\12H_0^{-2} 1.2byr\right )^\frac{1}{3}=14.4 byr$

Proper distance of the galaxy is speed of light times Conformal time

$d_p = \frac{c}{H_0}$  

So the Galaxy is 14.4 billion light years away.

EXTRA:

You can go ahead and try to show that $d_p = \frac{2c}{H_0}\left(1-\frac{1}{\sqrt{1+z}}\right)$ for a matter dominated universe