Sterile Neutrinos as a Dynamical Cosmological Fluid: Implications for the Expansion History and Matter-Radiation Equality

Abstract

Sterile neutrinos arise naturally in extensions of the Standard Model and can affect cosmological evolution even with subdominant abundance. Their impact is often described by a constant shift in the effective number of relativistic species, Delta Neff, assuming a radiation-like equation of state. However, for finite mass sterile neutrinos with incomplete thermalization, the equation of state evolves with time. In this work, we develop an analytic framework treating sterile neutrinos as a dynamical cosmological fluid with a time-dependent equation of state. Starting from the Boltzmann equation in an expanding Friedmann-Lemaitre-Robertson-Walker background, we show that suppressed active-sterile oscillations lead to a reduced Fermi-Dirac distribution characterized by a thermalization parameter less than unity. We compute the resulting energy density and pressure and incorporate them into the Friedmann equations. We identify distinct regimes, including a relativistic phase, a transition phase, and a matter-like behavior. For GeV scale sterile neutrinos, their contribution at matter-radiation equality is effectively matter-like, shifting the equality epoch in proportion to their energy fraction. Observational constraints indicate that this fraction remains small. This framework connects microscopic production physics to cosmological expansion and goes beyond the standard Delta Neff description.

Figures (4)

• Figure 1: Parameter space for sterile neutrinos in the $(m_s,\sin^2\theta)$ plane (schematic). The solid line denotes the full thermalization boundary defined by $\Gamma_s/H \sim 1$, with $\sin^2\theta \propto m_s^{-1}$ scaling consistent with the Dodelson--Widrow production rate Dodelson1994. The dashed line corresponds to the requirement of decay before BBN, with $\sin^2\theta \propto m_s^{-5}$ from the hadronic decay width Dolgov2002. The green shaded region represents the regime of partial thermalization ($\alpha < 1$), where sterile neutrinos behave as a dynamical cosmological fluid with a time-dependent equation of state. The red shaded region denotes the overproduced/excluded parameter space above the full thermalization boundary.
• Figure 2: Fractional modification to the Hubble expansion rate, $\Delta H/H$, induced by a partially thermalized sterile neutrino component ($\alpha = 0.1$). Three distinct regimes are visible: (I) a relativistic plateau at early times where $\Delta H/H \approx \mathrm{const}$, (II) a transition regime near $a \sim a_\mathrm{nr}$ where the equation of state evolves rapidly, and (III) a matter-like phase where the sterile contribution grows relative to radiation. The vertical dashed line marks $a_\mathrm{nr}$ and the dotted vertical line marks $a_\mathrm{eq}$. This behavior cannot be captured by a constant $\Delta N_\mathrm{eff}$ parameterization.
• Figure 3: Fractional shift in the matter--radiation equality scale factor, $\Delta a_\mathrm{eq}/a_\mathrm{eq}$, as a function of the sterile energy fraction $f_s$ at equality. For GeV-scale sterile neutrinos the population is deeply non-relativistic at equality, implying $\eta \ll 1$ and yielding $\Delta a_\mathrm{eq}/a_\mathrm{eq} \simeq -f_s$. The shaded region denotes parameter values disfavoured by Planck constraints on the equality scale. Consistency with observations requires $f_s(a_\mathrm{eq}) \lesssim \mathcal{O}(10^{-2})$. The vertical dashed line marks $f_s = 0.023$, corresponding to $\Delta N_\mathrm{eff} = 0.3$ from Planck 2018 Planck2018.
• Figure 4: Evolution of the sterile neutrino equation-of-state parameter $w_s(a)$. At early times ($a \ll a_\mathrm{nr}$), the sterile population is relativistic and behaves as radiation with $w_s \approx 1/3$. As the Universe expands, the particles transition to non-relativistic behavior near $a \sim a_\mathrm{nr}$, leading to $w_s \to 0$. The vertical dashed line marks $a_\mathrm{nr} = 10^{-3}$. This continuous evolution underlies the dynamical fluid description and cannot be captured by a fixed $\Delta N_\mathrm{eff}$ offset.