# Friedmann equations and the age of the universe

We have Friedmann equations of the form $$\begin{equation} \left(\frac{a'}{a}\right)^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3}. \end{equation}$$ We also assume that the stress-energy tensor has $$T^{00} = \rho$$ with all other components zero, so that $$\rho \sim 1/a^3$$ by conservation of energy. Assuming that the cosmological constant is small $$|\Lambda| \ll 3H_0^2$$, I've been asked to show that the age of the universe is given by: $$\begin{equation} \tau_0 = \frac{2}{3H_0}\left[ 1+\frac{\Lambda}{9H_0^2} + \mathcal{O}\left( \frac{\Lambda^2}{H_0^4} \right) \right]. \end{equation}$$ So far I haven't had much success. I assume the approach is to rewrite the Friedmann equations as: $$\frac{1}{a'} = \frac{1}{a} \left( \frac{8\pi G}{3}\rho + \frac{\Lambda}{3}\right)^{-1/2 }$$ and then have: $$\tau_0 = \int\frac{1}{a} \left( \frac{8\pi G}{3}\rho + \frac{\Lambda}{3}\right)^{-1/2 }da$$ with which I can do some kind of expansion. I just don't know how to get rid of $$\rho$$ properly or plug in $$H_0$$ in a natural way. Also I'm a bit unclear about which limits I should be using for the integration. Any help would be really appreciated. Thanks in advance.

• I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. – user4552 May 29 '19 at 20:28
• You haven't actually used the fact $\rho \sim a^{-3}$ anywhere. Maybe that will help? – jacob1729 May 29 '19 at 21:15