I am trying to solve problem 1.20 in the book 'Physical Foundations of Cosmology' from Mukhanov.

In the problem, we should show that the scale factor $a$ for a flat universe filled with cosmological constant and radiation is given by, $$ a(t)=a_0(\sinh2H_\Lambda t)^{1/2},\tag{1} $$ wherein $H_\Lambda=(8\pi G\epsilon_\Lambda/3)^{1/2}$.

As a hint, we are given, $$ a^{\prime\prime}+ka=\frac{4\pi G}{3}\left(\epsilon-3p\right)a^3,\tag{2} $$ where prime denotes the derivative with respect to conformal time $\eta$.

The equation of state for the cosmological constant $\Lambda$ is $p=-\epsilon$ whereas the equation of state for radiation is $p=\frac{1}{3}\epsilon$. Using these equations the right-hand side of the former equation yields, $$ \frac{4\pi G}{3}4\epsilon_\Lambda a^3=2H_\Lambda a^3.\tag{3} $$ As for a flat universe $k=0$ we arrive at, $$ a^{\prime\prime}=2H_\Lambda a^3.\tag{4} $$

If we multiply both sides with $a^\prime$ we can write, $$ \frac{1}{2}\frac{d}{d\eta}(a^\prime)^2=a^{\prime\prime}a^\prime=2H_\Lambda a^3a^\prime=\frac{1}{2}H_\Lambda \frac{d}{d\eta}a^4,\tag{5} $$ and we can integrate both sides with respect to $\eta$.

Dropping the integration constant and taking the square root of both sides, $$ a^\prime=\pm H_\Lambda^{1/2}a^2,\tag{6} $$ we can solve this by separation and of variables and find, $$ \eta=\int d\eta=\pm H^{-1/2}_\Lambda\int \frac{da}{a^2}=\mp H^{-1/2}_\Lambda \frac{1}{a}.\tag{7} $$

We need the scale factor to be positive, therefore, $$ a(\eta)=H^{-1/2}_\Lambda \eta.\tag{8} $$

From the definition of the conformal time, $$ \eta=\int\frac{dt}{a(t)},\tag{9} $$ we find, $$ t=\int dt=\int d\eta a(\eta)=\frac{1}{2}H^{-1/2}_\Lambda \eta^2,\tag{10} $$ which solved for $\eta$ can be used to express the scale factor in proper time $t$, $$ a(t)=\sqrt{2 t}H_\Lambda^{3/4},\tag{11} $$ which obviously is very different from the actual result.

What did I wrong?


I see some mistakes:

  • First of all, according to your definition of $H_{\Lambda}$, in the right-hand side of your equation you should have $H_{\Lambda}^{2}$ and not $H_{\Lambda}$.

  • When you solve by separation of variables and leave $a$ in terms of $\eta$, you obtain $a \propto \eta^{-1}$ and not $a \propto \eta$ as you wrote.

  • At some point in your derivation you should take into account the integration constant, that is how you will make the $a_0$ appear in the final result.
  • I'm not sure that you have switched correctly from conformal to proper time. I think that the best thing to do is to reduce the second-order equation to a first-order equation (just as you did) and then switch to proper time by using the chain rule and $d\eta = dt/a(t)$.

Finally, I would recommend to enumerate your equations, in this way it is much easier to correct your work.

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  • $\begingroup$ @GuillermoFranvoAbelián Thanks, I added references to the equations. Can you elaborate a bit what you mean in your second point? $\endgroup$ – bodokaiser Jul 21 '19 at 11:04
  • $\begingroup$ @bodokaiser By following all the steps correctly, your equation 7 should look like $\eta = H_{\Lambda}^{-1}/a$. Therefore $ a = H_{\Lambda}^{-1} / \eta$, and not $ a= H_{\Lambda}^{-1} \eta$ as you wrote in your equation 8. $\endgroup$ – Guillermo Franco Abellán Jul 21 '19 at 11:35

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