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

 

[mark726]

Hello,

To go from equation (3) to equation (*), we need to use the relation between conformal time and cosmic time. In particular, we have the following relation:

\begin{equation*}
\eta = \int \frac{dt}{a} \tag{1}
\end{equation*}

Substituting equation (3) into equation (1), we get:

\begin{equation*}
\eta = \int \frac{dt}{\left( C_r\right)^{1/2} \eta} = \frac{1}{\left( C_r\right)^{1/2}} \int \frac{dt}{\eta} \tag{2}
\end{equation*}

Now, using the relation between conformal time and cosmic time, we can rewrite equation (2) as:

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

Substituting equation (3) into equation (2), we get:

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

Finally, rearranging equation (4) gives us equation (*).

I hope this helps. Let me know if you have any further questions.