Evidence for extra radiation? Profile likelihood versus Bayesian
  posterior
  A number of recent analyses of cosmological data have reported hints for the
presence of extra radiation beyond the standard model expectation. In order to
test the robustness of these claims under different methods of constructing
parameter constraints, we perform a Bayesian posterior-based and a likelihood
profile-based analysis of current data. We confirm the presence of a slight
discrepancy between posterior- and profile-based constraints, with the
marginalised posterior preferring higher values of the effective number of
neutrino species N_eff. This can be traced back to a volume effect occurring
during the marginalisation process, and we demonstrate that the effect is
related to the fact that cosmic microwave background (CMB) data constrain N_eff
only indirectly via the redshift of matter-radiation equality. Once present CMB
data are combined with external information about, e.g., the Hubble parameter,
the difference between the methods becomes small compared to the uncertainty of
N_eff. We conclude that the preference of precision cosmological data for
excess radiation is "real" and not an artifact of a specific choice of
credible/confidence interval construction.
