A natural expectation would be that the mass arises from some dynamics, and there would be other particles and interactions, such as self-interactions, connected to it. The consequences of such an interaction can be significant. Light fermions with large mixings can have their oscillations suppressed in the early universe [3][4][5][6][7], changing the cosmological constraints significantly. In the presence of a self-interaction, regions of parameter space arise where a ∼keV fermion with small mixings can be dark matter [8][9][10][11]. In contrast, absent self-interactions, direct production of such dark matter through weak interactions, is excluded by a combination of x-ray data and the presence of small scale structure [12] (a famous loophole exists when SM neutrinos have chemical potentials and a lepton asymmetry [13]). Thus it is clear that a dark fermion with interactions is qualitatively different from the "unnaturally minimal" scenario of an inert dark state. Upcoming CMB and large scale structure (LSS) observations will probe the ΛCDM desert, motivating a broader exploration of such models.
In this Letter, we study the equilibration of dark sectors with the SM neutrinos after BBN and neutrino decoupling. Equilibration relies on the dark sector containing at least one neutral fermion which can mix with SM neutrinos and has interactions in the dark sector. For concreteness, we consider a single dark fermion ν d which mixes with a SM neutrino by an amount sin θ 0 in vacuum. We assume that ν d has a self-interaction mediated by a force carrier ϕ with m ϕ ≪ m νd and coupling strength α d . We find two important results: (i) The dark sector comes into equilibrium with the neutrinos over a very large parameter space roughly bounded only by θ 2 0 α 2 d M Pl > m νd , allowing mixing angles ranging from 1 to 10 -13 . (ii) Over most of the parameter space the temperature at which ν d equilibrates is α d independent and given by
Thus even though the range of allowed values of θ 0 and α d is huge, ν d naturally equilibrates at temperatures near m νd , and at most a few orders of magnitude higher, because of the 1=5 power. Consequently, dark sectors with light (< MeV) fermions often equilibrate after BBN and are therefore unconstrained by primordial light element abundances.
The simplest thermal history is sketched in Fig. 1. After neutrino decoupling and electron self-annihilation at T ∼ MeV, the dark sector ϕ and ν d come into equilibrium with the SM neutrinos. At the lower temperature T ∼ m νd , the dark fermions ν d annihilate away. This causes the SM neutrinos to decouple and become free-streaming again, and the entropy of ν d is shared between ϕ and the SM neutrinos.
Importantly, dark sector equilibration with SM neutrinos after neutrino decoupling does not change the relativistic energy density because the total energy in neutrinos þ dark sector is conserved in the equilibration process. Thus N eff is unchanged during equilibration, and constraints on N eff from the CMB and LSS do not a priori constrain it.
However, if equilibration occurs prior to 100 keV, BBN can be modified. If ν e (rather than ν μ or ν τ ) equilibrates with ν d , then ν e is cooled, suppressing n → p conversion. When T ∼ m νd there is a "step" [14][15][16] in the total relativistic energy density (i.e., N eff increases) as ν d annihilates away. This can affect BBN as well [17] if it occurs before 100 keV. We leave a detailed study of this for future work.
For later equilibration, BBN is unaffected. However, prior to T ∼ m νd , the νν dϕ fluid is tightly coupled. This, combined with the step in N eff , leaves an inevitable imprint on the density perturbations of the universe.
Should other particles have couplings to ϕ and ν d , they, too, will come into equilibrium with the SM neutrinos below T ∼ MeV. As a result, there is a possibility for other interesting dynamics within a dark sector to affect cosmology, such as the thermalization and freeze-out of dark matter, the presence of a second "step" [14][15][16] in the energy density of the dark sector due to the annihilation of additional massive particles into lighter ones. Alternatively, in a minimal scenario with m νd ≲ eV, self-interactions in (a portion of) the relativistic energy density may arise only at late times, near recombination. Neutrino-dark sector equilibration after BBN thus has a very interesting and modeldependent impact on the CMB and structure formation with possible implications for H 0 and S 8 , all of which will be probed by a wide range of upcoming experiments.
Interactions and dark sector equilibration.-A generic dark sector which contains a fermion ν d that mixes with the SM neutrinos can equilibrate with the SM neutrinos very efficiently by the combined effect of νν d oscillations and scattering. The relevant formalism is well developed; see Refs. [18][19][20]. For simplicity we consider the case of one dark fermion oscillating with one SM neutrino. The rate of conversion of a SM neutrino into a dark fermion can be written as
where we assume averaging over many oscillations, Γ int is the rate of scattering, θ m is the in-medium mixing angle between the SM neutrino and the dark fermion, and both depend on the incoming neutrino energy E. The process of dark sector equilibration is the usual competition between the production rate in Eq. (2) and Hubble. The mixing angle is generally suppressed by the presence of large diagonal effective thermal masses and thus the overall conversion rate grows rapidly as T declines.
The in-medium mixing angle is given by
where θ 0 is the in-vacuum angle that parametrizes the mixing between the SM neutrino and the dark fermion, Δm 2 ≃ m 2 νd is the mass-squared difference between the two mass eigenstates and is dominated by the dark fermion mass, and
The effective potential of ν from the SM weak interactions is well known [1] and given by V SM eff ≃ -c V G 2 F T 4 ν E where c V ≃ 22 (for mixing with ν μ or ν τ ), and we assume vanishing lepton asymmetry [13]. The dark sector effective potential arises due to scattering with light particles and a light mediator in the dark thermal bath and can be parametrized as 2EV DS eff ≡ α d T 2 d [5]. In
FIG. 1. Thermal history of a universe with dark sector thermalization from neutrino mixing after BBN. The dark sector initially has negligible energy density (dashed line). After neutrino decoupling and electron annihilation it equilibrates with the SM neutrinos at T equil . After ν d annihilation at T ∼ m νd the SM neutrinos redecouple and free-stream. what follows we take this as the definition of α d . The expression for the effective potential (and dark interaction rate) assumes that the dark sector is self-equilibrated with temperature T d and vanishing chemical potentials (see discussion below). The exact expression can vary with Dirac/Majorana, internal symmetries, and other model dependencies which amount to an overall OðfewÞ rescaling of α d . The precise mapping onto a specific model Lagrangian is straightforward and not important for our discussion. We ignore a possible shift of the scalar expectation value in the thermal background which would change the mass of ν d . The scattering rate is the sum of the SM weak interaction
, and the scattering rate of the dark fermions which we parametrize as
This assumes that the cross section scales as
Here κ is a number greater than one, which allows for the presence of additional dark states which scatter via ϕ exchange. For simplicity, we set κ ¼ 3, and in general it would shift the precise region of parameter space but not make it much larger or smaller.
Finally, averaging the conversion rate Γ over the thermal distribution of the SM neutrinos approximately replaces
We can now determine if and when the dark sector equilibrates with the neutrinos by comparing Γ with the expansion rate, H ≃ T 2 ν =M Pl . There are two important limits to consider. First, in the Dodelson-Widrow (DW) [1] limit of vanishing dark sector interactions, α d ¼ 0, the maximum conversion rate occurs when G F T 3 ν =m νd ∼ 0.1. This peak temperature is above an MeV so that full equilibration from DW would yield a thermalized dark sector before BBN which is excluded. The dark sector equilibrates if Γ ¼ H at the peak; therefore, we obtain the constraint (in the DW limit) that θ 2 0 m νd M Pl G F ≲ 100. A qualitatively different solution is obtained when the dark sector interactions dominate over the weak interactions. Then hΓi=H grows monotonically with decreasing temperature, and we can solve for the equilibration temperature (when
It is remarkable both that this is independent of α d and the dependence on θ 2 0 M pl is mild because of the 1=5 power. Thus for a very broad range in parameter space the dark sector equilibrates with the neutrinos, and it does so at a temperature which is at most a few orders of magnitude above the dark fermion mass. This yields the important qualitative result that in the presence of a light (≪ MeV) fermion, the natural equilibration scale is below the BBN scale, but also above recombination (a similar phenomenology can be achieved in models of neutrinos which couple to a Majoron, and resonantly produce dark matter at late times [21]).
This intuition is borne out by a numerical calculation. Integrating the Boltzmann equations for the phase space distribution functions of dark sector particles against energy and summing over dark sector species, we obtain an evolution equation for the total energy density in the dark sector
where ρ DS is the total energy density in the DS, which we solve numerically. The evolution of the dark sector temperature is shown in Fig. 2. Details on the calculation of the dark sector temperature evolution are found in the Supplemental Material [22]. Our primary result is contained in Fig. 3 which shows the large regions of parameter space where the dark sector comes into equilibrium with the SM neutrinos at some point before T ν ¼ m νd and where equilibration is reached below T ν ¼ MeV, i.e. after neutrino decoupling and BBN. Note that the small "fin" regions on the right of Fig. 3 correspond to space in which α d T 2 equil =m 2 νd < 1. For the purposes of this figure we define the equilibration temperature T equil as the temperature at which ρ DS crosses ρ ν g DS Ã =g ν Ã with ρ DS obtained from solving Eq. ( 6) with the backreaction term omitted.
It is worth noting that because of mixing of the SM neutrinos, for most of parameter space all three SM neutrinos equilibrate with the DS in rapid succession. That only a single SM neutrino equilibrates with the DS can occur for special regions in parameter space. Either the couplings of ν d are tuned such that it only couples to a single SM neutrino mass eigenstate, or the dark parameters are such that equilibration with the first of the SM neutrinos occurs at a temperature just above m νd so that νν d conversion shuts off because m νd is reached before another SM neutrino can equilibrate.
Discussion.-One of the simplest extensions of the standard model is to include a massive neutral fermion that mixes with the SM neutrino. It is natural-perhaps expected-that it should come with its own interaction, as well. In the presence of such an interaction, we find that even for very small couplings and mixings, a new eV-MeV mass fermion is equilibrated with the neutrino bath at a temperature within a few orders of magnitude of its mass, and often much less. Consequently, it typically equilibrates after BBN, leaving no imprint on light element abundances. Its implications for the CMB and LSS, however, can be significant. Once the dark fermion equilibrates at T equil , a whole series of additional particles can come into equilibrium as well, including dark matter, which can have mass above T equil , including above an MeV.
Although the equilibration of the dark sector does not immediately increase the energy density in radiation, it can transform some or all of the radiation into an interacting fluid. The associated mass threshold can change the relative amount of relativistic radiation, turn on or off interactions in a dark sector, and provide a basis for equilibrating a broader dark sector which may contain part or all of the dark matter.
At high values of 100 eV ≲ m νd ≲ MeV, the dark sector equilibrates with neutrinos and then goes through the mass threshold of the dark fermion before the CMB is directly sensitive to the transition. One consequence is the increase in N eff by
, where N eq is the number of neutrinos that come into equilibrium with the dark sector, and g UV Ã ðg IR Ã Þ is the total number of effective degrees of freedom above (below) the mass threshold, including the thermalizing neutrinos. The relativistic energy below this threshold could be interacting, noninteracting, or a combination.
At intermediate values of Oð1Þ eV ≲ m νd ≲ 100 eV, equilibration typically happens before 100 eV, but the mass threshold occurs in a period which is directly probed by the CMB and LSS. This can have important implications for many observables, including H 0 [14,15] and S 8 [16].
At very low values of m νd , the equilibration can happen below 100 eV, and the signal could appear as a transition of the relativistic energy from free-streaming to strongly interacting. This transition would occur sequentially for the three SM neutrino mass eigenstates and would lead to observable signals in the CMB if it occurred at times near recombination. These implications for the CMB are beyond our scope and warrant their own study.
It is interesting to consider what might be a minimal setup, where a single dark Majorana fermion comes into equilibrium with all three SM neutrinos after BBN, but then annihilates away into a real scalar ϕ before the CMB or LSS are directly sensitive. The late universe would have N eff ≃ 3.30 with ð1 -fÞN eff ¼ 2.78 free-streaming neutrinos and fN eff ¼ 0.53 interacting particles (arising from ϕ). Even in this minimal model, the resulting radiation (ΔN eff ≃ 0.26) is within the bounds from Planck [23] but is well above the sensitivity of Simons Observatory [24] and CMB-S4 [25]; and the fraction f ¼ 1=ð1 þ 3 • 7=4Þ of the "neutrinos" that is interacting can be measured from phase shifts of the CMB peaks [26][27][28][29][30][31][32][33]. If additional particles couple to ν d or ϕ, they, too, will equilibrate at after T equil and the thermal history can be yet richer. If additional light particles are present, then the requirement that m ϕ ≪ m νd is no longer necessary for a viable cosmology. Instead only m ϕ ≪ T equil is needed for our calculations to hold, and in this case the neutrinos would become free-streaming again at m ϕ rather than m νd . With additional stable particles, dark matter could be produced through thermal processes. For freeze-out, in particular, the dark matter can have masses which are above T equil , and dark matter would have naturally strong couplings to a radiation bath, at least for some period. In all of these cases, ΔN eff can be found simply by an appropriate counting of degrees of freedom in the UV and IR (and intermediate steps, if needed).
In summary, we have considered the thermal history of dark fermions which mix with the SM neutrinos and have self-interactions through a light (m ϕ ≪ m νd ) mediator. We find that such particles equilibrate at temperatures near their mass, and thus typically at late times. This implies that later universe observables, such as LSS and the CMB, are independent probes when compared to BBN for such models. This can have important implications for models attempting to address cosmological tensions. As we look forward to upcoming results from CMB telescopes such as SPT, ACT, Simons Observatory, CMB-S4 as well as studies from LSS measurements KiDS, DES, HSC, and future galaxy surveys with Rubin, Roman, and UNIONS, such models provide an example of natural late-universe phenomena which may have significant impact. Should such particles populate the ΛCDM desert, these upcoming studies may show striking deviations from ΛCDM expectations.