Density of a univrse filled with x-matter

[hadoque]

Homework Statement

Consider a flat (k=0) universe filled up with X-matter, with an equation of state: $$p_X/c^2=w\rho_X$$, where -1<w<1. Find the expressions for the evolution of density $$\rho_X(a),a(t) and \rho(t)$$, where a is the evolving scale factor

Homework Equations

$$\left ( \frac{da}{dt} \frac{1}{a} \right ) ^2 = \frac{8 G \pi \rho}{3}-\frac{k}{a^2} \\ \dot{\rho}}=-3 \frac{\dot{a}}{a}(\rho + p)$$

The Attempt at a Solution

The big question is, am I on the right track to solve this? I started by solving $$\left ( \frac{da}{dt} \frac{1}{a} \right ) ^2 = \frac{8 G \pi \rho}{3}-\frac{k}{a^2}$$, which gives me: $$a=Ce^{\pm \sqrt{\frac{8 G \pi \rho}{3}}t}$$
Now I have a(t), and should be able to solve $$\rho(t)$$ by solving the next equation. But this one has the two a's in it, which are time dependant. This gives me a pretty tricky equation to solve. So, is this the right approach, or is there a better way? I only want hints, not a solution.

Thanks/ Johan


I hope the latex-script comes out right...

 

[anathema]

Hello Johan,

Your approach is definitely on the right track. To find the evolution of density, you can use the second equation given: \dot{\rho}}=-3 \frac{\dot{a}}{a}(\rho + p). You can substitute in the equation of state for p to get:

\dot{\rho}}=-3 \frac{\dot{a}}{a}(\rho + w\rho)

From here, you can rearrange and integrate to get an expression for \rho(a). To find \rho_X(a), you can use the fact that X-matter is the only type of matter present in the universe, so \rho_X(a) = \rho(a). Then, you can use the expression for a(t) that you found to get an expression for \rho_X(t).

I hope this helps! Let me know if you have any more questions.

 

[m2003]

Your approach is on the right track. The first equation you solved gives the evolution of the scale factor a(t), which is a crucial component in solving for the density \rho(t). The second equation relates the time derivative of the density to the scale factor and the pressure. Since we are dealing with X-matter, the pressure is related to the density through the equation of state given in the homework statement. So you can substitute that in and use the expression for a(t) that you have already solved for to get an equation for \dot{\rho}. From there, you can integrate and solve for \rho(t). Just make sure to keep track of the constants and signs in your integration.