temperatures of the neutrino and the photon backgrounds, respectively. The effective number of neutrinos is constrained to be ${N_\mathrm{eff}}=3.04$, if there are no extra relativistic degrees of freedom besides three active neutrinos [@Neff], and has an impact mainly on the primordial nucleosynthesis and the time of matter-radiation equality. Analyses of light element abundances [@NeffBBN] and of the time of equality [@wmap5komatsu] produce results that agree with this value.

When neutrinos become non-relativistic, their velocities begin to decrease from $c$ as $T_\nu(t)$, due to momentum conservation. Assuming that the mass states are degenerate ($m_1=m_2=m_3\equiv m_\nu$), the velocity is given by $v_\mathrm{th}=3k_BT_\nu(t)/m_\nu c$ or, since $T_\nu(z)\simeq 1.9(1+z)$ K, by $$\label{eq:vth}
v_\mathrm{th}= 150\,(1+z)(m_\nu/1\,\mathrm{eV})^{-1}~\mathrm{km/s}.$$ The transition occurs at $$\label{eq:ztrans}
1+z_\mathrm{tr}= 2\times 10^3\left(\frac{m_\nu}{1\,\mathrm{eV}}\right).$$ For masses smaller than $\sim$ 0.6 eV, the transition occurs well into the matter-dominated era and after recombination. Therefore, massive neutrinos affect the anisotropies of the CMB temperature only through the background evolution. However CMB data strongly constrain other cosmological parameters that are degenerate with the neutrino mass, and is thus invaluable to joint analyses. The result would slightly differ for non-degenerate masses, for the normal or inverted hierarchies, but at a negligible level for a total mass above 0.2 eV [@lesgourgueshepph].

