Is the ratio of matter to radiation density constant in the universe? Or does it vary over time, as matter annihilates to radiation and vice-versa?
As the universe expands, both radiation and matter "thin out", but at different rates. Expansion of the universe decreases the density of radiation more rapidly than it decreases the density of matter. If ##a \left(t\right)## is the linear scale of the universe (so ##a \left(t\right)## increases as time ##t## increases), then the volume that a given amount of matter occupies is proportional to ##a \left(t\right)^3##, and thus matter densities scales as ##1/a \left(t\right)^3##. Radiation is made of photons, and photon density also scales as ##1/a \left(t\right)^3##, but the expansion of the universe also scales the wavelength by another factor of ##a \left(t\right)##, so radiation energy density scales as ##1/a \left(t\right)^4##.
Thank you for your reply. While I am aware of how matter and radiation vary with the scale factor, this is what I am trying to ascertain: is the total amount of matter and total amount of radiation in the universe constant, or does it vary because of annihilation of matter to radiation and vice-versa?
Let ##\rho_r \left(t\right)## be the density of radiation and ##\rho_m \left(t\right)## be the density of matter. I think that you are asking "Is $$\frac{\rho_m \left(t\right) a\left(t\right)^3}{\rho_r \left(t\right) a \left(t\right)^4}$$ constant?" I think that this ratio is now fairly constant.
Yes, that is what I am trying to ascertain. Could you please clarify what do you mean by "fairly constant"? While I should not plug my own work, recently I published a paper online on dark energy where I argue for a correlation between inertial mass density and the cosmological constant. Thus, if total matter density were not to be constant, that would affect the rate of acceleration of the universe, and thereby provide a way to test the correlation. You may like to check out the paper at http://article.sapub.org/10.5923.j.astronomy.20140301.02.html