Cosmic microwave background constraints on secret interactions among sterile neutrinos

Secret contact interactions among eV sterile neutrinos, mediated by a massive gauge boson $X$ (with $M_X \ll M_W$), and characterized by a gauge coupling $g_X$, have been proposed as a mean to reconcile cosmological observations and short-baseline laboratory anomalies. We constrain this scenario using the latest Planck data on Cosmic Microwave Background anisotropies, and measurements of baryon acoustic oscillations (BAO). We consistently include the effect of secret interactions on cosmological perturbations, namely the increased density and pressure fluctuations in the neutrino fluid, and still find a severe tension between the secret interaction framework and cosmology. In fact, taking into account neutrino scattering via secret interactions, we derive our own mass bound on sterile neutrinos and find (at 95% CL) $m_s < 0.82$ eV or $m_s < 0.29$ eV from Planck alone or in combination with BAO, respectively. These limits confirm the discrepancy with the laboratory anomalies. Moreover, we constrain, in the limit of contact interaction, the effective strength $G_X$ to be $ < 2.8 (2.0) \times 10^{10}\,G_F$ from Planck (Planck+BAO). This result, together with the mass bound, strongly disfavours the region with $M_X \sim 0.1$ MeV and relatively large coupling $g_X\sim 10^{-1}$, previously indicated as a possible solution to the small scale dark matter problem.