- #1

Tomishiyo

- 18

- 1

## Homework Statement

Rewrite Friedmann equation using conformal time and density parameters [itex]\Omega_m[/itex] and [itex]\Omega_r[/itex]. Is there a relation between the two? How many parameters define the problem?

## Homework Equations

Friedmann equation

[tex]\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3c^2}\left(\frac{\epsilon_{0m}}{a^3}+\frac{ε_{0r}}{a^4}\right)[/tex]

Conformal time definition

[tex]dt=a(\eta) d\eta[/tex]

Density parameter:

[tex]\Omega\equiv \frac{ε(t_0)}{ε_c(t_0)}=\frac{3c^2}{8\pi G}H_0^2[/tex]

## The Attempt at a Solution

First part is rather simple: just a matter of changing the variable in Friedmann Equation, noting that:

[tex]\frac{d}{dt}=\frac{d\eta}{dt}\frac{d}{d\eta}=\frac{1}{a}\frac{d}{d\eta}[/tex]

so Friedmann Equations turn out to be:

[tex]\left(\frac{da}{d\eta}\right)^2=\frac{8\pi G}{3c^2}(aε_{m0}+ε_{r0})[/tex]

or in terms of the density parameter:

[tex]\left(\frac{da}{d\eta}\right)^2=H_0^2(a\Omega_m+Ω_r).[/tex]

My trouble starts now. So, normally the densities parameters are constrained due to scale factor normalization, that is to say, they must obey the constrain equation [itex]1=\Omega_m+\Omega_r[/itex]. That relation should hold regardless of the coordinate system I choose to write Friedmann equation, but I cannot see the connection, unless I postulate that there must exist a [itex]\eta_0[/itex] such that [itex]a(\eta_0)=1[/itex], and later find its relation to physical time (and I can only find that once I know [itex]a(\eta)[/itex], i.e, when I solve Friedmann equation). But that does not seem to me as a correct assumption, for the next exercise on my list ask me to make precisely this assumption, indicating that there must exist another way of constraining the parameters. I think, thus, that there must exist a constrain between the density parameters that does not involve normalization of scale factor, but I can't think about anything to solve that. Can anyone help me out?

Thank you!