Impact of theoretical assumptions in the determination of the neutrino effective number from future CMB measurements

One of the major goals of future cosmic microwave background (CMB) measurements is the accurate determination of the effective number of neutrinos N-eff. Reaching an experimental sensitivity of Delta N-eff = 0.013 could indeed falsify the presence of any nonstandard relativistic particles at 95% C.L. In this paper, we test how this future constraint can be affected by the removal of two common assumptions: a negligible running of the inflationary spectral index n(run) and a precise determination of the neutron lifetime tau(n). We first show that the constraints on N-eff could be significantly biased by the unaccounted presence of a running of the spectral index. Considering the Stage-IV experiment, a negative running of dn/d ln k = -0.002 could mimic a positive variation of Delta N-eff = 0.03. Moreover, given the current discrepancies between experimental measurements of the neutron lifetime tau(n), we show that the assumption of a conservative error of Delta tau(n) similar to 10 s could cause a systematic error of Delta N-eff = 0.02. Complementary cosmological constraints on the running of the spectral index and a solution to the neutron lifetime discrepancy are therefore needed for an accurate and reliable future CMB bound of N-eff at the percent level.