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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.

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

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