Fluid Equation (Cosmology)

[cosmoshadow]

1. The problem statement, all variables and given/known data

The fluid equation in cosmology is given as:

\dot{\epsilon} + 3*(\dot{a}/a)*(\epsilon+P) = 0

Where \epsilon is the energy density and a(t) is a scale factor.

Using the equation of state, P = w*\epsilon, show how \epsilon change with a(t).


2. Relevant equations

\dot{\epsilon} + 3*(\dot{a}/a)*(\epsilon+P) = 0
P = w*\epsilon


3. The attempt at a solution

I can solve for the equation to the point where I re-arrange it to look like this:

\dot{\epsilon}/\epsilon = -3*(1+w)*(\dot{a}/a)

I do not know how to proceed from here. I know that this equation is supposed to end up like this,

\epsilonw(a) = \epsilonw,0*a-3*(1+w)

but I do not know how to get to this point. Can someone assist me please?

[cosmoshadow]

can someone take a look at this? i'm pretty sure its a simple operation that i'm failing to realize.

[cosmoshadow]

bump?

[scottie_000]

You have your equation
\frac{\dot\epsilon}{\epsilon} = -3(w+1)\frac{\dot a}{a}
From here you can eliminate the time-dependence
\frac{d\epsilon}{\epsilon} = -3(w+1)\frac{da}{a}
and this is a differential equation involving just \epsilon and a you can solve by integrating both sides