Calculating average density of the Universe

[bobo1455]

Q: What was the average density of the universe at the time the light was emitted by the z = 6.56 galaxy?

For the question we know the current average density of the universe and red-shift wavelength z = 6.56. It says to calculate the average density of the universe at the moment when light was emitted by the galaxy at red-shifted wavelength 6.56. I have the linear scale factor but I'm not really sure what to do next. I've read equations about density on the internet but haven't found one that uses a wavelength or linear scale factor. Any help is appreciated.

 

[QuantumQuest]

In order to calculate the densities we use a cubic model of expansion for both.

Density of matter (expanding universe): $\rho(t) = \frac{M}{L(t)^3} = \frac{M}{L_0^3a(t)^3} = \frac{\rho_0}{a(t)^3}$.

where $M$ is mass inside a "cube"
$L(t)$ is expanding side of "cube"
$\rho_0$ is current density
$a(t)$ is scale factor of universe
$\rho(t)$ is density at time t

Energy density of photons (expanding universe): $\rho(t)c^2 = \frac{N E(t)}{L(t)^3} = \frac{\frac{N E_0}{a(t)}}{L_0^3a(t)^3} = \frac{\rho_0c^2}{a(t)^4}$.

where $N$ is number of photons in "cube"
$E(t)$ is photon energy at time t
$\rho_0c^2$ is current density

 

[bobo1455]

Alright I calculate the average density and it is correct because I checked against the book's answer. If I were asked to express the density as the number of Hydrogen atoms per cubic meter, how would I go about it?

What I've done is calculated the number of H atoms that would fit in one cubic meter to be 1.06 x 10^30 and just multiplied it by the density I calculated previously. Am I on the right track or way off?