We use the Wilkinson Microwave Anisotropy Probe ( WMAP ) data on the spectrum of cosmic microwave background anisotropies to put constraints on the present amount of lepton asymmetry L , parameterized by the dimensionless chemical potential ( also called degeneracy parameter ) \xi and on the effective number of relativistic particle species .
We assume a flat cosmological model with three thermally distributed neutrino species having all the same mass and chemical potential , plus an additional amount of effectively massless exotic particle species .
The extra energy density associated to these species is parameterized through an effective number of additional species \Delta N ^ { \mathrm { eff } } _ { \mathrm { others } } .
We find that 0 < | \xi| < 1.1 and correspondingly 0 < |L| < 0.9 at 2 \sigma , so that WMAP data alone can not firmly rule out scenarios with a large lepton number ; moreover , a small preference for this kind of scenarios is actually found .
We also discuss the effect of the asymmetry on the estimation of other parameters and in particular of the neutrino mass .
In the case of perfect lepton symmetry , we obtain the standard results .
When the amount of asymmetry is left free , we find \sum m _ { \nu } < 3.6 eV at 2 \sigma .
Finally we study how the determination of |L| is affected by the assumptions on \Delta N ^ { \mathrm { eff } } _ { \mathrm { others } } .
We find that lower values of the extra energy density allow for larger values of the lepton asymmetry , effectively ruling out , at 2 \sigma level , lepton symmetric models with \Delta N ^ { \mathrm { eff } } _ { \mathrm { others } } \simeq 0 .