# Calculating average density of the Universe

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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
Gold Member
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

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?