2
$\begingroup$

My question is: Is it possible to (numerically) solve the Friedmann Equations, starting from some epoch if we do not explicitly know the equation of state $w$?

Let us suppose we have a component, which at an initial time $t_i$ (when the scale factor is $a_i$) has a density $\rho(a_i)$ and pressure $P(a_i)$. I am interested in self-consistently studying the evolution of $\rho(a)$ and $P(a)$ (and hence $w(a)$) as the univese expands.

The way I was proceeding was as follows. First, we take the equations,

$$H^2(t)=\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho,\quad\quad\quad\quad \dot{H}(t)+H^2(t)=\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\big(\rho+3P\big)$$

from which we can get the continuity equation,

$$\dot{\rho}+\frac{3\dot{a}}{a}\big(\rho+P\big)=0$$

Also, the second Friedmann equation can be modified with the help of the first to get,

$$\dot{H}(t)+4\pi G\big(\rho+P\big)=0$$

In the next step, I can convert all $t$ dependencies to $a$ dependencies (primes denote derivatives wrt $a$, $\frac{d}{dt}=aH\frac{d}{da}$),

$$\rho'+\frac{3}{a}\big(\rho+P\big)=0, \quad\quad\quad\quad aHH'+4\pi G\big(\rho+P\big)=0$$

The question is:

Is it possible to solve these equations in a self-consistent manner, or is there too little information? Am I missing something simple, due to which I do not get an expression for $P'$?

$\endgroup$
3
  • $\begingroup$ You have more than two unknowns in those two equations. You need more information. $\endgroup$ Jun 14, 2018 at 4:06
  • $\begingroup$ Thanks a lot. That was what I was thinking, but I was still confused. $\endgroup$ Jun 14, 2018 at 4:11
  • 1
    $\begingroup$ The extra equation is a equation of sate. $p=\omega \rho$ $\endgroup$
    – Nothing
    Apr 9, 2020 at 22:07

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.