Interacting neutrinos in cosmology: exact description and constraints

TL;DR

The study develops an exact Boltzmann description of neutrino self-interactions via a four-fermion coupling in the massive-scalar limit and implements it in CLASS to compute CMB observables. It demonstrates that the separable ansatz/RTA reproduces the exact results well, while the $ (c_{ ext{eff}}^2,c_{ ext{vis}}^2)$ parameterization fails to capture the correct scale dependence of the CMB temperature spectrum. Through Planck 2015 and complementary cosmological data with MCMC, the authors find a bimodal posterior for the effective coupling $G_{ ext{eff}}$: a ΛCDM-like mode with $G_{ ext{eff}} o 0$ and an interacting mode with $G_{ ext{eff}} o 0.03 m\,MeV^{-2}$ (approximately $3 imes 10^{9} G_{ m F}$). The interacting mode associates with a shifted scalar spectral index $n_s o 0.935$–$0.94$ and neutrino decoupling near a few eV, suggesting notable implications for inflationary scenarios and neutrino cosmology, while remaining a subdominant improvement over the non-interacting case in overall fit quality.

Abstract

We consider the impact of neutrino self-interactions described by an effective four-fermion coupling on cosmological observations. Implementing the exact Boltzmann hierarchy for interacting neutrinos first derived in [arxiv:1409.1577] into the Boltzmann solver CLASS, we perform a detailed numerical analysis of the effects of the interaction on the cosmic microwave background (CMB) anisotropies, and compare our results with known approximations in the literature. While we find good agreement between our exact approach and the relaxation time approximation used in some recent studies, the popular $\left( c_{\text{eff}}^2,c_{\text{vis}}^2 \right)$-parameterisation fails to reproduce the correct scale dependence of the CMB temperature power spectrum. We then proceed to derive constraints on the effective coupling constant $G_{\text{eff}}$ using currently available cosmological data via an MCMC analysis. Interestingly, our results reveal a bimodal posterior distribution, where one mode represents the standard $Λ$CDM limit with $G_{\rm eff} \lesssim 10^8 \, G_{\rm F}$, and the other a scenario in which neutrinos self-interact with an effective coupling constant $G_{\rm eff} \simeq 3 \times 10^9 \, G_{\rm F}$.

Figures (11)

• Figure 1: Interaction rates per particle (coloured), in comparison with the Hubble expansion rate (black) and the standard weak interaction rate (dashed), assuming, in the left panel, a massive scalar particle and, in the right, a massless scalar particle.
• Figure 2: Left: Magnitudes of the two largest eigenvalues (red and green) at $\ell=0$ as functions of the number of momentum bins $N_{q}$ for $q_{\text{max}}=15$ (dotted), $q_{\text{max}}=21$ (dashed), and $q_{\text{max}}=24$ (solid). Right: Same as the left panel, but at $\ell=1$, and only the largest eigenvalue (blue).
• Figure 3: Regions of validity of the tightly-coupled approximation (TCA) in the $(k,z)$-plane for various values of the effective coupling constant $G_{\text{eff}}$. The black line represents $k = {\cal H}$, i.e., horizon crossing of a wavenumber $k$, while the coloured lines correspond to either $|\dot{\tau}_\nu| = {\cal H}$ if a wavenumber is super-horizon $k < {\cal H}$, or $|\dot{\tau}_\nu| = k$ if a wavenumber is sub-horizon $k > {\cal H}$. For a given $G_{\rm eff}$, the TCA is valid in the region above the corresponding coloured line. In compiling this plot we have used $T_\nu = (4/11)^{1/3} T_\gamma$ for the neutrino temperature, $H \simeq T_{\gamma}^2/m_{\text{Pl}}$ for the Hubble expansion rate at $z \gtrsim 3000$, and $H \simeq (\omega_{\rm cdm}+\omega_b)^{1/2} a^{-3/2}$, assuming the Planck $\Lambda$CDM best-fit values, at $z \lesssim 3000$.
• Figure 4: Neutrino decoupling temperature, as defined in equation (\ref{['eq:Tdec']}), as function of the effective coupling constant $G_{\text{eff}}$. See figure \ref{['TCAplot']} caption for the modelling of the Hubble expansion rate.
• Figure 5: Neutrino energy density contrast $\delta$ (left), velocity divergence $\theta$ (middle), and shear stress $\sigma$ (right) at $k=0.01$ Mpc$^{-1}$ (top), $0.1$ Mpc$^{-1}$ (middle), and $1.0$ Mpc$^{-1}$ (bottom) for different values of the effective coupling constant $G_{\text{eff}}$ (in units of $\mathrm{MeV}^{-2}$), computed from both the exact Boltzmann hierarchy (solid) and the separable ansatz/RTA (short dashes). The free-streaming and fluid limits are also shown for comparison.