  We investigate how inflation model selection is affected by the presence of
additional free-streaming relativistic degrees of freedom, i.e. dark radiation.
We perform a full Bayesian analysis of both inflation parameters and
cosmological parameters taking reheating into account self-consistently. We
compute the Bayesian evidence for a few representative inflation scenarios in
both the standard $\Lambda$CDM model and an extension including dark radiation
parametrised by its effective number of relativistic species $N_\mathrm{eff}$.
Using a minimal dataset (Planck low-$\ell$ polarisation, temperature power
spectrum and lensing reconstruction), we find that the observational status of
most inflationary models is unchanged. The exceptions are potentials such as
power-law inflation that predict large values for the scalar spectral index
that can only be realised when $N_\mathrm{eff}$ is allowed to vary. Adding
baryon acoustic oscillations data and the B-mode data from BICEP2/Keck makes
power-law inflation disfavoured, while adding local measurements of the Hubble
constant $H_0$ makes power-law inflation slightly favoured compared to the best
single-field plateau potentials. This illustrates how the dark radiation
solution to the $H_0$ tension would have deep consequences for inflation model
selection.
