Critical density in Early Times

[BillSaltLake]

What was the total mass energy density at the time of recombination? Has it has it been measured by WMAP? (Edit: I think it's 4.28x10^-18kg/m³ from the # of photons/volume in a 2979K blackbody, x baryon:photon ratio x proton mass x (1+ dark:baryon ratio) divided by the 0.755 matter fraction at the time.)

Assuming a flat Universe, in times before dark energy was significant, I think the critical density as a function of time was 3H²/(8 pi G), where
1) H = 1/(2t) during energy-domination (very early times), and
2) H = 2/(3t) during matter-domination.
Thus during matter-domination, the critical density was 1/(6 pi Gt²), and it was 9/16 of that during energy -domination.

I might expect the total energy density (matter + energy) at the time of recombination to be between the matter-dominated and energy-dominated numbers. (Edit: the WMAP appears to be between those numbers. Matter-dominated would be 5.43x10^-18kg/m³) However, when I use a differential eq to solve it directly, I'm only getting 0.291 of the matter-dominated value of 1/(6 pi Gt²) where t is the recombination time. (Edit: I don't do math good. Sorry.)

 

[cepheid]

I didn't look too closely at your work, but here's how I would approach this:

WMAP has measured Ω_m. This matter density parameter is defined as follows:

Ω_m ≡ ρ_m(t₀) / ρ_crit​

where t₀ means now and

ρ_crit ≡ 3H₀² / 8πG​

We also know that

ρ_m(t) = ρ_m(t₀)a^-3.​

So now we just need to know the scale factor of the universe at the time of recombination.

 

[Chalnoth]

so now we just need to know the scale factor of the universe at the time of recombination.

1/1090