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.
