Equivalent Neutrinos, Light WIMPs, and the Chimera of Dark Radiation

TL;DR

This work analyzes how the cosmological parameter ${\rm N}_{\rm eff}$, used to quantify relativistic energy density, can be affected not only by true dark radiation but also by light WIMPs and the decoupling history of equivalent neutrinos. By deriving analytic relations for ${\rm N}_{\rm eff}$ as functions of the neutrino decoupling temperature $T_{\nu d}$, the equivalent-neutrino decoupling temperature $T_{\xi d}$, and the WIMP mass $m_{\chi}$ (including cases where WIMPs couple to photons, neutrinos, or both), the paper reveals extensive degeneracies among these parameters. The key results show that Neff>3 does not guarantee the presence of Delta${\rm N}_\nu$ and that Neff≈3 remains compatible with additional light species; conversely, Neff<3 can arise even with dark radiation if a sufficiently light WIMP heats photons after neutrino decoupling. Furthermore, the analysis demonstrates how these degeneracies modify neutrino mass constraints and can mimic or obscure sterile-neutrino signatures, including the provocative possibility of “dark radiation without dark radiation” in the truly weak WIMP limit. These insights are crucial for interpreting Planck/ACT/SPT and LSS observations, informing how to disentangle new light degrees of freedom from early-Universe thermal histories.

Abstract

According to conventional wisdom, in the standard model (SM) of particle physics and cosmology the effective number of neutrinos is Neff=3 (more precisely, 3.046). In extensions of the standard model allowing for the presence of DeltaNnu equivalent neutrinos (or dark radiation), Neff is generally >3. The canonical results are reconsidered here, revealing that a measurement of Neff>3 can be consistent with DeltaNnu=0 (dark radiation without dark radiation). Conversely, a measurement consistent with Neff=3 is not inconsistent with the presence of dark radiation (DeltaNnu>0). In particular, if there is a light WIMP that annihilates to photons after the SM neutrinos have decoupled, the photons are heated beyond their usual heating from e+- annihilation, reducing the late time ratio of neutrino and photon temperatures (and number densities), leading to Neff<3. This opens the window for one or more equivalent neutrinos, including sterile neutrinos, to be consistent with Neff=3. By reducing the neutrino number density at present, this allows for more massive neutrinos, relaxing the current constraints on the sum of the neutrino masses. In contrast, if the light WIMP only couples to the SM neutrinos and not to the photons, its late time annihilation heats the neutrinos but not the photons, resulting in Neff>3 even in the absence of equivalent neutrinos or dark radiation. A measurement of Neff>3 is thus no guarantee of the presence of equivalent neutrinos or dark radiation. In the presence of light WIMPs and/or equivalent neutrinos there are degeneracies among the light WIMP mass and its nature (fermion or boson, as well as its couplings to neutrinos or photons), the number and nature (fermion or boson) of the equivalent neutrinos, and their decoupling temperature (the strength of their interactions with the SM particles). There's more to a measurement of Neff than meets the eye.

Figures (13)

• Figure 1: The effective number of relativistic degrees of freedom, N$_{\rm eff}$, as a function of $T_{\nu d}$.
• Figure 2: A zoomed in version of Fig. \ref{['fig:neffvstnud1']} for $1 \leq T_{\nu d} \leq 10\,{\rm MeV}$ (for a linear temperature scale). Notice that N$_{\rm eff} = 3$ (lower horizontal, magenta line) when $T_{\nu d} \approx 8.3\,{\rm MeV}$ (dashed, vertical blue line). For $T_{\nu d} = 2\,{\rm MeV}$, N$_{\rm eff} = 3.018$ (dashed, vertical red line), while for $T_{\nu d} = 3\,{\rm MeV}$, N$_{\rm eff} = 3.012$ (dashed, vertical green line). In the instantaneous decoupling approximation, N$_{\rm eff} = 3.046$ (upper horizontal, magenta line) when $T_{\nu d} \approx 1.3\,{\rm MeV}$.
• Figure 3: The ratio of the sum of the neutrino masses to its canonical value (assuming instantaneous neutrino decoupling and $(T_{\nu}/T_{\gamma})^{3}_{0} = 4/11$), $\Sigma m_{\nu}' /\Sigma m_{\nu}$, as a function of the neutrino decoupling temperature, $T_{\nu d}$. If the upper bound to $\Sigma\,m_{\nu}$ were $1\,{\rm eV}$, the curve would show the upper bound to the sum of the SM neutrino masses ($\Sigma\,m_{\nu}'$), in eV.
• Figure 4: Analogous to Fig. \ref{['fig:neffvstnud1']}, N$_{\rm eff}$ is shown as a function of the equivalent neutrino decoupling temperature, $T_{\xi d}$, for one equivalent neutrino, $\Delta{\rm N}_\nu$ = 1 (black curve). The blue curve is the contribution to N$_{\rm eff}$ from the equivalent neutrino and the red curve is the contribution to N$_{\rm eff}$ from the three SM neutrinos .
• Figure 5: N$_{\rm eff}$ is shown as a function of the equivalent neutrino decoupling temperature, $T_{\xi d}$, for one equivalent neutrino, $\Delta{\rm N}_\nu$ = 1, a Majorana fermion (solid curve; the black curve in Fig. \ref{['fig:neffvstd']}). The long-dashed curve shows N$_{\rm eff}$ for a scalar equivalent neutrino, $\Delta{\rm N}_\nu$ = 4/7. The horizontal band is the $\pm 1\,\sigma$ region allowed by WMAP9 hinshaw.