Universe with matter and radiation | Cosmology

[JD_PM]

Homework Statement:

Given the Friedmann–Lemaitre equation for a universe containing only matter and radiation)

\begin{equation*}
a \dot a^2 = C_r - ka^2 \tag{1}
\end{equation*}


Where $a$ is the dimensionless cosmic scale factor, $C_r$ is simplify a constant and the initial condition (I.C.) is $a(0) = 0$.



a) Solve $(1)$ for a flat universe in terms of the conformal time $\eta$. Then show that the conformal time at matter–radiation equality is given by:


\begin{equation*}
\eta_{eq} = 2\left(\sqrt{2}-1 \right)H_0^{-1}(\Omega_M)_0^{-1/2} a_{eq}^{1/2} \tag{*}
\end{equation*}


b) Use this result to show that the horizon at matter–radiation equality corresponds today
to a scale of $16 ((\Omega_M)_0 h^2)^{-1}$ Mpc, where $h := H_0/(100 \text{km s}^{−1} Mpc^{−1})$.

Relevant Equations:

N/A

a) For a flat universe $(k=0)$, so $(1)$ simplifies to $\dot a^2 = \frac{C_r}{a^2}$. The solution to this first order, separable ODE (given the I.C. $a(0) = 0$) is

\begin{equation*}
a(t) = \left( 4 C_r\right)^{1/4} t^{1/2} \tag{2}
\end{equation*}

We switch to conformal time by means of the change of variables

\begin{equation*}
d\eta = \frac{dt}{a}
\end{equation*}

So $(2)$ takes the form

\begin{equation*}
a(\eta) = \left( C_r\right)^{1/2} \eta \tag{3}
\end{equation*}

But how to go from $(3)$ to $(*)$?

Before discussing b) I need to understand $(*)$

Main sources:

- Chapter 38 in Matthias Blau's GR notes (attached).
- Chapter 1 in Daniel Baumann's notes.

Any help is appreciated.

Thank you