Cosmological constraints on neutrino self-interactions with a light
  mediator
  If active neutrinos undergo non-standard (`secret') interactions (NS$\nu$I)
the cosmological evolution of the neutrino fluid might be altered, leaving an
imprint in cosmological observables. We use the latest publicly available CMB
data from Planck to constrain NS$\nu$I inducing $\nu-\nu$ scattering, under the
assumption that the mediator $\phi$ of the secret interaction is very light. We
find that the effective coupling constant of the interaction, $g_\mathrm{eff}^4
\equiv \langle \sigma v\rangle T_\nu^2$, is constrained at $<
2.35\times10^{-27}$ (95\% credible interval), which stregthens to
$g_\mathrm{eff}^4 < 1.64\times10^{-27}$ when Planck non-baseline small-scale
polarization is considered. Our findings imply that after decoupling at
$T\simeq 1$ MeV, cosmic neutrinos are free streaming at redshifts $z>3800$, or
$z>2300$ if small-scale polarization is included. These bounds are only
marginally improved when data from geometrical expansion probes are included in
the analysis to complement Planck. We also find that the tensions between CMB
and low-redshift measurements of the expansion rate $H_0$ and the amplitude of
matter fluctuations $\sigma_8$ are not significantly reduced. Our results are
independent on the underlying particle physics model as long as $\phi$ is very
light. Considering a model with Majorana neutrinos and a pseudoscalar mediator
we find that the coupling constant $g$ of the secret interaction is constrained
at $\lesssim 7\times 10^{-7}$. By further assuming that the pseudoscalar
interaction comes from a dynamical realization of the see-saw mechanism, as in
Majoron models, we can bound the scale of lepton number breaking $v_\sigma$ as
$\gtrsim (1.4\times 10^{6})m_\nu$.
