The 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 measurements is the accurate determination of the effective number of neutrinos $N_{\rm eff}$. Reaching an experimental sensitivity of $\Delta N_{\rm eff} = 0.013$ could indeed falsify the presence of any non-standard 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 $n_{\rm run}$ and a precise determination of the neutron lifetime $\tau_n$. We first show that the constraints on $N_{\rm eff}$ could be significantly biased by the unaccounted presence of a running of the spectral index. Considering the Stage-IV experiment, a negative running of ${\rm d}n/{\rm d}\ln k= - 0.002$ could mimic a positive variation of $\Delta N_{\rm eff} = 0.03$. Moreover, given the current discrepancies between experimental measurements of the neutron lifetime $\tau_n$, we show that the assumption of a conservative error of $\Delta\tau_n \sim 10$s could bring to a systematic error of $\Delta N_{\rm eff} = 0.02$. 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 $N_{\rm eff}$ at percent level.