Density of radiation at CMB I was trying to find the radiation density at the CMB time.
For two different methods, the first one is using the Planck Law and the data for the actual measured temperature $T_{0}=2.7255° K$.
$$ \rho_{r}=\rho_{\gamma}+\rho_{\nu}$$
$$\rho_{\gamma}=\frac{a_{b}}{c^2}T_{0}^{4}$$
$$\rho_{\nu}=3.046\frac{7}{8}(\frac{4}{11})^{4/3}\rho_{\gamma}$$
$$\frac{a_{b}}{c^2}=8.41805*10^{-33}$$
The second way to find it is using the  $\Lambda CDM$ model, we know that the equation for density of radiation in this model is given by
$$\rho_{r}=c(1+z)^4$$
The constant C is determined to be the actual value for radiation density i.e $c=7.85846*10^{-31} kg\, m^{−3}$
Know with this two formulas we can calculate the radiation density at CMB in two different ways:- 
1. Using the value of the themperature at CMB $T_{CMB}=3000°K$. 
2. Using the redshift at CMB $z_{CMB}\approx 1100$
For the first method I obtain $$\rho_{r}^{CMB}=1.15355*10^{-18}kg\, m^{−3}$$
For the second method
$$\rho_{r}^{CMB}=8.51682*10^{-26}kg\, m^{−3}$$
Clearly I am doing something wrong, maybe Planck's Law is not a good aproximation at CMB time?
 A: If you look the wiki link on neutrino decoupling,

In Big Bang cosmology, neutrino decoupling refers to the epoch at which neutrinos ceased interacting with other types of matter , and thereby ceased influencing the dynamics of the universe at early times. Prior to decoupling, neutrinos were in thermal equilibrium with protons, neutrons and electrons, which was maintained through the weak interaction. Decoupling occurred approximately at the time when the rate of those weak interactions was slower than the rate of expansion of the universe. Alternatively, it was the time when the time scale for weak interactions became greater than the age of the universe at that time. Neutrino decoupling took place approximately one second after the Big Bang, when the temperature of the universe was approximately 10 billion kelvins, or 1 MeV.

In your first method you assume the same temperature at CMB time for the neutrino flux too, and this must be wrong because decoupling means no thermal equilibrium with other matter.
If you want to include the neutrino flux at the the time of the CMB  I think you  should start from the 1 second when it decouples, going forwards.
