Cosmological Expansion: Estimating Present Horizon Length

[Kyrios]

Homework Statement

If light traveled a distance L = $H_{eq}^{-1}$ at M-R equality, how large does this distance expand to at present? (in Mpc)

Homework Equations

$z_{eq} = 3500$
$\Omega_m = 0.32$ at present
$\rho_c = 3.64 \times 10^{-47} GeV^4$ present critical density

The Attempt at a Solution

Not entirely certain where to begin for this one. I think it's asking for the horizon length at present, so perhaps need to use the equation
$$L =a(t) \int \frac{da}{a^2 H}$$

 

[Orodruin]

Since the problem quotes $L = H_{\rm eq}^{-1}$, I suspect that what they want you to do is to compute (roughly) the present size of a region that was in causal contact at the time of matter-radiation equilibrium.

 

[Kyrios]

So would this be done by calculating $H_{eq}$ at equality, and then expanding with scale factor, $L(z=0) = L_{eq} (1 + z_{eq})$ ?
If I do that, it gives a value a little under 150 Mpc.

 

[Orodruin]

This is the approach I would take - assuming that my interpretation of the problem is correct.

 

[paranormal]

where a(t) is the scale factor and H is the Hubble parameter. We can use the given values for z_eq and \Omega_m to calculate the scale factor at M-R equality. From there, we can use the current critical density to calculate the Hubble parameter at present. Plugging these values into the equation for L should give us the present horizon length in Mpc. However, without more information or context, it's difficult to provide a more specific response. It would also be helpful to know what the M-R equality refers to and how it relates to the cosmological expansion.