Universe with matter and radiation | Cosmology

In summary, to go from equation (3) to equation (*), we use the relation between conformal time and cosmic time and perform a change of variables to simplify the integral.
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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 :biggrin:
 
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  • #2


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.
 

1. What is the Universe made of?

The Universe is made up of matter and radiation. Matter is anything that has mass and takes up space, while radiation is energy that travels through space in the form of waves or particles.

2. How did matter and radiation come into existence?

The current theory is that the Universe began with the Big Bang, a massive explosion that occurred approximately 13.8 billion years ago. This explosion created the fundamental particles of matter and energy, which eventually formed into atoms and eventually stars, galaxies, and all the structures we see in the Universe today.

3. What is dark matter and how does it affect the Universe?

Dark matter is a type of matter that does not interact with light or other forms of electromagnetic radiation, making it invisible to telescopes. Scientists believe that dark matter makes up about 85% of the total matter in the Universe and plays a crucial role in the formation and evolution of galaxies.

4. How does radiation shape the Universe?

Radiation plays a significant role in shaping the Universe. In the early stages of the Universe, radiation was the dominant form of energy, and it helped to smooth out the irregularities caused by the Big Bang. As the Universe expanded, radiation cooled and formed into the cosmic microwave background, which can still be observed today.

5. What is the future of the Universe?

The future of the Universe is still uncertain and is a topic of ongoing research and debate among scientists. Some theories suggest that the Universe will continue to expand indefinitely, while others propose that it will eventually collapse in on itself in a "Big Crunch." It is also possible that the expansion of the Universe will accelerate, leading to a "Big Freeze" where all matter and energy are spread out and the Universe becomes a cold, dark place.

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