I have a trouble with question that states:

Suppose that the Universe is spatially flat and, at some stage of its evolution, is dominated by a fluid with a so-called 'stiff' equation of state: $p=\rho c^2$.

a) Solve the continuity equation for $\rho$ and obtain the scaling of $\rho$ with respect to the scale factor $a.$

b) Use the above result and solve the Friedmann equation to get the temporal dependence of the scale factor $a=a(t)$, assuming a spatially flat Universe and taking the initial conditions to be $a(t=0)=0$.

c) Once you know $a(t)$ calculate the Hubble parameter $H(t)$ and the density $\rho (t)$ as functions of time.

d) What is the value of the barotropic parameter $w$ for such a fluid?

Relevant equations:

Continuity equation: $\dot{\rho}+3H(\rho+\frac{p}{c^2})=0$

Friedmann equation for flat Universe: $H^2=\frac{8\pi G}{3} \rho$

Hubble parameter: $H=\frac{\dot{a}}{a}$

However, I am stuck at the first task, as I get $\dot{\rho}+6H\rho=0$ and I am not sure what to do next. How should I scale the $\rho$ with respect to $a$? How to get rid of derivative of $\rho$ without solving it for $\rho(t)$ (as it's part of d) point)?


closed as off-topic by Kyle Kanos, Jon Custer, Yashas, John Rennie, Michael Seifert Mar 21 '17 at 16:48

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