Radiation and luminous matter

One contribution that can be easily estimated is that of the cosmic background radiation (CBR) discovered in 1964 by Penzias and Wilson which has been extremely well evaluated by the COBE satellite(3) as a black body radiation of $T=2.728\pm 0.002\;K$. From the Stefan-Boltzmann law and using the Appendix

$$\rho _R\sim T^4\sim (2\times 10^{-4}eV)^4\simeq 10^{-33}gr\,cm^3,$$ (9)

which corresponds to 400 photons per $cm^3$ and is much smaller than the critical density. Of course, since the decrease of $\rho_R$ is faster with the universe expansion than that of nonrelativistic matter

$$\rho _R\propto \frac 1{R^4}, \hspace{0.1cm} \rho _M\propto \frac 1{R^3},$$ (10)

radiation must have dominated in the past.

It is usual to refer the different contributions of density to the critical value so that at present from Eqs.(9) and (7)

$$\Omega _R=\frac{\rho _R}{\rho _c}\sim 10^{-4}$$ (11)

The next obvious contribution to density is that of luminous matter. This is evaluated summing over all the masses $M_i$ in a volume $V$ using a typical mass to luminosity ratio $M/L\simeq 5M_\odot /L_\odot$ related to that of sun.

$$\rho _L=\frac ML\sum_i\frac{L_i}V\simeq 5\frac{M_{\odot }}{L_{\odot }}\lambda .$$ (12)

From the measures of fluxes coming from bright regions of galaxies,

$$\lambda =(2\pm 0.6)\times 10^8L_{\odot }h\,Mpc^{-3},$$ (13)

where the $h$ factor comes because the distances are calculated through the Hubble law which gives the redshift

$$z=H_od.$$ (14)

Inserting the mass of the sun $M_\odot$ into Eq.(12) one obtains

$$\rho _L=(0.7\pm 0.2)\times 10^{-31}h\,gr\,cm^{-3},$$ (15)

which again gives a small fraction of the critical density

$$\Omega _L=(0.003\pm 0.001)h^{-1}\sim 0.005.$$ (16)