# Scale factor for flat universe filled with radiation and cosmological constant [closed]

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.

• @GuillermoFranvoAbelián Thanks, I added references to the equations. Can you elaborate a bit what you mean in your second point? – bodokaiser Jul 21 '19 at 11:04
• @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. – Guillermo Franco Abellán Jul 21 '19 at 11:35