Thermal Dark Matter Below an MeV

TL;DR

This paper explores models where thermal dark matter below 1 MeV can thermalize with the Standard Model after neutrino-photon decoupling, affecting cosmological constraints and opening new experimental avenues.

Contribution

It introduces a model with fermionic dark matter and a light scalar mediator that allows sub-MeV thermal dark matter consistent with observations.

Findings

Dark matter can be as light as a few keV.

Thermalization below neutrino decoupling alters cosmological constraints.

Supports new experiments for sub-MeV dark matter detection.

Abstract

We consider a class of models in which thermal dark matter is lighter than an MeV. If dark matter thermalizes with the Standard Model below the temperature of neutrino-photon decoupling, equilibration and freeze-out cools and heats the Standard Model bath comparably, alleviating constraints from measurements of the effective number of neutrino species. We demonstrate this mechanism in a model consisting of fermionic dark matter coupled to a light scalar mediator. Thermal dark matter can be as light as a few keV, while remaining compatible with existing cosmological and astrophysical observations. This framework motivates new experiments in the direct search for sub-MeV thermal dark matter and light force carriers.

Equations

$$m_{\chi }\sim {\left({\alpha }_{\chi }/{\alpha }_{\chi }^{\prime }\right)}^{1/3}{\left(T_{\text{eq}}/m_{\mathrm{Pl}}\right)}^{1/6}{T}^{\chi \text{eq}},$$

$$(g_{\ast }^{\nu }{\xi }_{\nu }^4+g_{\ast }^{\chi }{\xi }_{\chi }^4)/(g_{\ast }^{\gamma }{)}^{4/3}=\text{constant}$$

$$(g_{\ast }^{\nu }{\xi }_{\nu }^3+g_{\ast }^{\chi }{\xi }_{\chi }^3)/g_{\ast }^{\gamma }=\text{constant}$$

$$N_{\text{eff}}\simeq 3\,\bigg{[}\bigg{(}\frac{\xi_{\nu}}{\xi_{\nu}^{\mathrm{SM}}}\bigg{)}^{4}+\Theta(T_{\chi}-m_{\chi})~{}\frac{g_{*}^{\chi}}{g_{*}^{\nu}}~{}\bigg{(}\frac{\xi_{\chi}}{\xi_{\nu}^{\mathrm{SM}}}\bigg{)}^{4}\,\bigg{]}~{},$$

$$1.T^{\text{BBN}}<T^{\nu \text{dec}}<T^{\chi \text{eq}}$$

$$2.T^{\text{BBN}}<T^{\chi \text{dec}}<T^{\chi \text{eq}}<T^{\nu \text{dec}}$$

$$3.T^{\chi \text{dec}}<T^{\text{BBN}}<T^{\chi \text{eq}}<T^{\nu \text{dec}}$$

$$4.T^{\chi \text{dec}}<T^{\chi \text{eq}}<T^{\text{BBN}}<T^{\nu \text{dec}}.$$

$$N_{\text{eff}}\simeq\begin{cases}(a)~{}3\,\big{(}1+\frac{g_{*}^{\chi}}{g_{*}^{\nu}}~{}\xi_{\chi}^{0\,4}\big{)}\\ (b)~{}3\left(1+\frac{g_{*}^{\chi}}{g_{*}^{\nu}}\right)^{1/3}\left(1+\frac{g_{*}^{\chi}}{g_{*}^{\nu}}\,\xi_{\chi}^{0\,4}\right)~{},\\ \end{cases}$$

$$\mathcal{L}\supset \phi \left({\lambda }_{\chi }{\chi }^2+{\lambda }_{\nu }{\nu }^2\right)+\text{h.c.},$$

Full text

The expectation of new physics at the electroweak scale has motivated the search for dark matter (DM) in the form of weakly interacting massive particles (WIMPs) with mass $10\text{ GeV}\lesssim m_{{}_{\text{DM}}}\lesssim 1\text{ TeV}$. However, no direct signs of WIMPs have been observed to date Aad:2015zva ; Khachatryan:2014rra ; Tan:2016zwf ; Akerib:2015rjg ; Akerib:2016vxi ; Agnese:2015nto . From a bottom-up perspective, astrophysical observations constrain the mass of DM to lie anywhere between $10^{-31}\text{ GeV}$ and $10^{58}\text{ GeV}$. The lack of evidence for the WIMP paradigm thus leaves an overwhelmingly vast range of viable DM masses. However, the WIMP is only a specific example of DM that is thermally generated from the Standard Model (SM) bath, and in such models the acceptable range of masses is significantly reduced.

If DM acquires its abundance through thermal contact with the SM, perturbative unitarity of the theory dictates that $m_{{}_{\text{DM}}}\lesssim\mathcal{O}{(100)}\text{ TeV}$ Griest:1989wd . Although no similar theoretical inconsistencies arise for small masses, $m_{{}_{\text{DM}}}\gtrsim\mathcal{O}{(1)}\text{ MeV}$ is often quoted as a robust lower bound on the mass of thermal DM. It has been argued that any sub-MeV relic that is in thermal contact with the SM below the temperature of neutrino decoupling necessarily leads to measurable deviations in the expansion rate of the universe Ho:2012ug ; Steigman:2013yua ; Boehm:2013jpa ; Nollett:2013pwa ; Nollett:2014lwa ; Steigman:2014uqa ; Green:2017ybv ; Serpico:2004nm . During radiation domination, the expansion rate is determined by the number of relativistic degrees of freedom, or, equivalently, by the effective number of neutrino species, $N_{\text{eff}}$, at low temperatures. The value of $N_{\text{eff}}$ is constrained by the successful predictions of Big Bang nucleosynthesis (BBN) and observations of the cosmic microwave background (CMB). Thermal DM lighter than an MeV can substantially alter the observed value, $N_{\text{eff}}\simeq 3$, by heating or cooling neutrinos relative to photons or by contributing directly to the energy density as dark radiation.

In this Letter, we propose a novel mechanism to generate sub-MeV thermal relics that can be applied to a large class of models. Our key observation is that if a light state enters thermal equilibrium with the SM after neutrino-photon decoupling, then the constraints from measurements of $N_{\text{eff}}$ during BBN and recombination are significantly relaxed.111Delayed thermalization has been used to alleviate cosmological constraints on light hidden sectors Bartlett:1990qq ; Chacko:2003dt ; Chacko:2004cz and sterile neutrinos, e.g., in regards to the LSND anomaly Dasgupta:2013zpn ; Mirizzi:2014ama ; Cherry:2014xra ; Chu:2015ipa ; Cherry:2016jol .

If this light state couples only to neutrinos, equilibration draws heat from the SM bath without changing $N_{\text{eff}}$. Later, when the species decouples, the neutrino bath is heated. Alternatively, if the light state couples only to photons, equilibration and decoupling increases and lowers $N_{\text{eff}}$, respectively. In either case, modifications to $N_{\text{eff}}$ are reduced at late times. Although this idea can be applied to any light thermal relic, we will focus on the production of DM. In this case, thermal DM as light as the warm DM limit, $\sim\text{keV}$, is possible. Our study strengthens the case for new technologies in the search for sub-MeV thermal DM, such as superconducting detectors Hochberg:2015pha ; Hochberg:2015fth ; Schutz:2016tid ; Knapen:2016cue .

In the standard WIMP freeze-out paradigm, DM, denoted by $\chi$, enters equilibrium with the SM at early times and acquires a large thermal abundance. As the universe cools below the DM mass, its number density decreases as long as it remains in chemical equilibrium with the SM. If annihilations into SM particles are responsible for maintaining chemical equilibrium, the rate is proportional to the thermally-averaged cross-section $\langle\sigma v\rangle\sim\alpha_{\chi}^{2}/m_{\chi}^{2}$. Thermal DM lighter than a GeV necessarily requires the presence of a new light mediator Hut:1977zn ; Lee:1977ua . The DM and SM sectors can be equilibrated through mediator decays and inverse decays. The corresponding rate is given by $\Gamma_{\chi\,\text{eq}}\sim\alpha_{\chi}^{\prime}\,m_{\chi}^{2}/T$, where $m_{\chi}/T$ is a relativistic time-dilation factor and we have assumed similar masses for the DM and the mediator. Equilibrium is achieved when $\Gamma_{\chi\,\text{eq}}$ overcomes the Hubble expansion rate, $H\sim T^{2}/m_{\mathrm{Pl}}$, at a photon temperature of $T=T^{\chi\,\text{eq}}$, where $m_{\mathrm{Pl}}$ is the Planck mass. A rephrasing of the standard WIMP-miracle estimate implies that $\chi$ saturates the observed relic abundance for

[TABLE]

where $T_{\text{eq}}\simeq 0.8\text{ eV}$ is the temperature at matter-radiation equality. Neutrinos decouple from the photon plasma at a temperature of $T^{\nu\,\text{dec}}\sim 2\text{ MeV}$ Enqvist:1991gx . Equation (1) implies that DM enters thermal equilibrium after neutrino decoupling ($T^{\chi\,\text{eq}}\lesssim T^{\nu\,\text{dec}}$) for $m_{\chi}\lesssim\text{ keV}\times(\alpha_{\chi}/10^{5}\,\alpha_{\chi}^{\prime})^{1/3}$. In this case, $\alpha_{\chi}\gg\alpha_{\chi}^{\prime}$ is required to evade warm DM limits, which exclude $m_{\chi}$ below a few keV. This hierarchy of couplings is possible if DM annihilations are enhanced relative to the decays of the light mediator into SM particles, e.g., through resonant processes or if the mediator interacts feebly with the SM, as in models of secluded DM Pospelov:2007mp .

Most investigations into the effects of additional light degrees of freedom assume that the new species is already in thermal equilibrium with the SM bath before neutrino-photon decoupling at $T^{\nu\,\text{dec}}\sim 2\text{ MeV}$. We consider the complementary case in which a sub-MeV DM species, $\chi$, enters thermal equilibrium with the SM at temperatures below $T^{\nu\,\text{dec}}$. New light particles that couple to electromagnetism are severely constrained by the anisotropies of the CMB and the observed cooling rates of stars and supernovae Raffelt:1996wa ; Ade:2015xua . For simplicity, we will therefore focus on the case in which $\chi$ couples exclusively to SM neutrinos. Even then, this new species can change the expansion history of the universe and thus may disrupt nucleosynthesis and recombination.

The formation of light nuclei occurs when the temperature of the photon bath falls in the range $50\text{ keV}\lesssim T\lesssim 1\text{ MeV}$. The presence of additional relativistic degrees of freedom during nucleosynthesis alters the predictions for the primordial abundances of light nuclei, such as Helium-4 and deuterium. For example, if $N_{\text{eff}}>3$, neutrons freeze out earlier due to the increased expansion rate of the universe, resulting in a larger neutron to proton ratio at the onset of BBN and hence an enhanced Helium-4 yield. If the baryon density is fixed by the observed abundances of Helium-4 and deuterium, detailed considerations of BBN alone bound $N_{\text{eff}}\simeq 2.85\pm 0.28$ during nucleosynthesis Cyburt:2015mya . Observations of the CMB angular power spectrum by the Planck satellite are also sensitive to the total radiation energy density at the time of recombination. Current measurements restrict the effective number of neutrino species at the time of photon decoupling with unprecedented precision, $N_{\text{eff}}\simeq 3.15\pm 0.23$, in impressive agreement with the standard $\Lambda\text{CDM}$ prediction of $N_{\text{eff}}\simeq 3.046$ Ade:2015xua . Future CMB-S4 observations will be sensitive to deviations at the level of $\Delta N_{\text{eff}}\simeq\pm 0.027$ Abazajian:2016yjj . The effects of a light dark sector on the expansion rate during these two well-studied epochs can therefore be encapsulated in the time evolution of $N_{\text{eff}}$.222Neutrino heating or cooling also directly affects the rate of the lingering neutron-to-proton conversion during BBN. We ignore this effect since we are most interested in times well after the freeze-out of weak interactions Nollett:2013pwa ; Nollett:2014lwa .

The temperature evolution of the neutrino and $\chi$ populations can be derived from the conservation of the sum of their comoving energy or entropy densities. The effective number of relativistic degrees of freedom, $g_{*}^{i}$, in each population, $i=\nu,\chi,\gamma$, determines the entropy density, $s_{i}\equiv(2\pi^{2}/45)\,g_{*}^{i}\,T_{i}^{3}$, and energy density, $\rho_{i}\equiv(\pi^{2}/30)\,g_{*}^{i}\,T_{i}^{4}$, where $T_{\gamma}\equiv T$. Note that $g_{*}^{\chi}$ includes contributions from additional particles in the DM sector. For three generations of SM neutrinos, we have $g_{*}^{\nu}\equiv(7/8)\times 3\times 2=21/4$.

If $\chi$ equilibrates with the SM neutrinos below $T^{\nu\,\text{dec}}$ while relativistic, the energy density of the $\chi$-neutrino bath, $\rho_{\nu+\chi}\equiv\rho_{\nu}+\rho_{\chi}$, scales as $a^{-4}$, where $a$ is the scale factor Green:2017ybv . This observation, together with entropy conservation of the photon bath, implies

[TABLE]

before and immediately after equilibration, where $\xi_{i}\equiv T_{i}/T$ is the temperature of species $i$ normalized to the photon temperature Feng:2008mu . We also define $\xi_{\nu}^{\mathrm{SM}}$ as the value of $\xi_{\nu}$ in the standard cosmology, such that $\xi_{\nu}^{\mathrm{SM}}(T\gtrsim m_{e})\equiv 1$, $\xi_{\nu}^{\mathrm{SM}}(T\lesssim m_{e})\equiv\left(4/11\right)^{1/3}$, in the instantaneous decoupling approximation. The value of $\xi_{\chi}$ before electron decoupling and DM-neutrino equilibration is denoted as $\xi_{\chi}^{0}$ and encodes the UV-sensitivity of our model. Below $T^{\chi\,\text{eq}}$, the entropy densities of the $\chi$-neutrino bath, $s_{\nu+\chi}\equiv s_{\nu}+s_{\chi}$, and of the photons, $s_{\gamma}$, separately scale as $a^{-3}$. This implies that

[TABLE]

before and after $\chi$ becomes non-relativistic. Equations (2) and (3) can be used to determine the temperature evolution of the neutrinos and the dark sector. The effective number of neutrino species is then given by

[TABLE]

where the step function encodes the instantaneous decoupling of $\chi$ when its temperature drops below its mass, $T_{\chi}\lesssim m_{\chi}<\text{MeV}$ Feng:2008mu ; Berlin:2016gtr . Note that Eq. (4) reduces to $N_{\text{eff}}\simeq 3$ in the standard cosmology with $g_{*}^{\chi}=0$ and $\xi_{\nu}=\xi_{\nu}^{\mathrm{SM}}$. We have approximated $3.046\simeq 3$, neglecting the partial heating of neutrinos from $e^{\pm}$ annihilations Mangano:2001iu ; Mangano:2005cc .

The contribution of the DM sector to the observed value of $N_{\text{eff}}$ depends on the ordering of the temperatures at which $\chi$ enters and exits equilibrium with the SM ($T^{\chi\,\text{eq}}\gg m_{\chi}$ and $T^{\chi\,\text{dec}}\sim m_{\chi}$, respectively) relative to $T^{\nu\,\text{dec}}\sim 2\text{ MeV}$ and the temperature at which nucleosynthesis has effectively concluded, $T^{\text{BBN}}\sim 50\text{ keV}$. The possible orderings are

[TABLE]

The first scenario above corresponds to standard weak-scale WIMPs, where $\chi$ enters equilibrium with the SM before neutrino-photon decoupling. In the following three cases, $\chi$ thermalizes with the SM neutrinos after they have decoupled from photons. The values of $N_{\text{eff}}$ in the latter scenarios are readily estimated using Eqs. (2)-(4). The effective number of neutrino species is given by

[TABLE]

before/during and after $\chi$ is in kinetic equilibrium with the neutrino bath, respectively. In scenario 2 of Eq. (Thermal Dark Matter Below an MeV), $\chi$ enters and exits kinetic equilibrium with the neutrino bath during nucleosynthesis and $N_{\text{eff}}$ evolves through (a) and (b) in Eq. (6) during BBN. In scenarios 3 and 4 of Eq. (Thermal Dark Matter Below an MeV), $\chi$ decouples after nucleosynthesis has concluded. In this case, $N_{\text{eff}}$ only takes the form of (a) in Eq. (6) during BBN. For every temperature ordering of Eq. (Thermal Dark Matter Below an MeV), $N_{\text{eff}}$ is given by (b) in Eq. (6) at the time of recombination, relevant for measurements of the CMB. In Figs. 1 and 2, we show the temperature evolution of $\xi_{\nu,\chi}$ and $N_{\text{eff}}$, respectively, for scenario 3 of Eq. (Thermal Dark Matter Below an MeV) in the DM model described below. These results were obtained by solving Boltzmann equations for the neutrino and DM energy and entropy densities without relying on the instantaneous decoupling approximation. This numerical treatment agrees well with the approximate results of Eq. (6) and justifies their use in the remainder of this work.

The analytic results of Eq. (6) show that for $\xi_{\chi}^{0}\lesssim 1$, constraints on $N_{\text{eff}}$ are relaxed. If the DM sector is colder than the SM before equilibration ($\xi_{\chi}^{0}\ll 1$), then $N_{\text{eff}}\simeq 3\left(1+g_{*}^{\chi}/g_{*}^{\nu}\right)^{1/3}$ at late times. This is significantly reduced compared to the standard result, $N_{\text{eff}}\simeq 3\left(1+g_{*}^{\chi}/g_{*}^{\nu}\right)^{4/3}$, when $T^{\chi\,\text{eq}}\gtrsim T^{\nu\,\text{dec}}$, as shown in Fig. 2. Also in this limit, $N_{\text{eff}}\simeq 3$ while $\chi$ is relativistic and in equilibrium with the neutrino bath, as required by conservation of comoving energy. Our scenario is different from the usual “loophole” where a cold dark sector does not equilibrate with the SM during DM freeze-out. If the initial $\chi$ population is cooler than the SM plasma, then equilibration with the neutrino population drains neutrinos of thermal energy, lowering $\xi_{\nu}$ below the standard prediction. This effect also increases $\xi_{\chi}$, which counteracts large changes to $N_{\text{eff}}$ (see Eq. (4)). Once $\chi$ decouples, conservation of comoving entropy demands that it heats the neutrino population close to its original temperature (relative to photons) for sufficiently small $\xi_{\chi}^{0}$. This can be seen explicitly in Figs. 1 and 2. Note that this cyclic behavior of $\xi_{\nu}$ would not be possible if $\chi$ equilibrated with the SM before neutrino-photon decoupling, as equilibration and decoupling would effectively borrow heat from the photon bath before pumping it into the neutrino sector.

In Fig. 3, we show the viable parameter space as a function of $\xi_{\chi}^{0}$ and $g_{*}^{\chi}$ for the different temperature orderings of Eq. (Thermal Dark Matter Below an MeV). While a detailed simulation of nucleosynthesis is beyond the scope of this work, we conservatively demand that the maximum value of $|\Delta N_{\text{eff}}|$ obtained during BBN does not exceed the observed value by more than $2\sigma$. The shaded regions denote parameters consistent with observations, assuming that $\chi$ equilibrates with the SM neutrinos after neutrino-photon decoupling. Also shown are regions consistent with $N_{\text{eff}}\simeq 3.15\pm 0.23$ and $N_{\text{eff}}\simeq 3.3\pm 0.3$, as derived from Planck data and its combination with local measurements of $H_{0}$ Ade:2015xua ; Riess:2016jrr ; Brust:2017nmv . CMB-S4 observations will be sensitive to the entire parameter space shown.

In Fig. 3, we also highlight representative model points, which correspond to a dark sector consisting of a DM scalar or Majorana fermion, $\chi$, coupled to a real scalar mediator, $\varphi$, such that $g_{*}^{\chi}=2$ or $g_{*}^{\chi}=(7/8)\times 2+1=2.75$, respectively. For concreteness, we will take our DM candidate, $\chi$, to be a Majorana fermion. Note that if $\chi$ was initially in thermal contact with the SM bath through the exchange of a heavy mediator, but decoupled above the electroweak scale, then conservation of comoving entropy dictates that $\xi_{\chi}^{0}\simeq(10.75/106.75)^{1/3}\simeq 0.5$. This is a viable model of thermal DM if the mediator also couples to the SM neutrinos, $\nu$, allowing the dark sector to re-enter equilibrium with the SM below $T^{\nu\,\text{dec}}$.

The low-energy effective Lagrangian is given by

[TABLE]

where $\chi$ and $\nu$ are two-component Weyl fermions and a sum over neutrino flavors is implied. We take $m_{\chi}>m_{\varphi}$, such that $\chi$ freezes out through annihilations into pairs of $\varphi$ particles, $\chi\chi\to\varphi\varphi$, after which $\varphi$ promptly decays into pairs of SM neutrinos.333Direct annihilations into neutrinos ($\chi\chi\to\varphi\to\nu\nu$) are also viable, but require tuning $m_{\varphi}\sim 2m_{\chi}$. The rate for the annihilation process is $p$-wave suppressed by the relative $\chi$ velocity in the non-relativistic limit. Since this deposits a negligible amount of energy into the photon plasma at the time of recombination, there are no relevant limits derived from observations of CMB anisotropies Ade:2015xua . The neutrino-$\varphi$ coupling, $\lambda_{\nu}$, can arise from higher-scale interactions with right-handed neutrinos, $N$. For instance, in the standard seesaw mechanism, an interaction of the form $\varphi\,N^{2}$ naturally generates a SM neutrino coupling, $\lambda_{\nu}\sim m_{\nu}/m_{N}$, after electroweak symmetry breaking Minkowski:1977sc ; Yanagida:1979as ; Mohapatra:1979ia ; GellMann:1980vs ; Schechter:1980gr . Such interactions also arise in Majoron models of neutrino masses Chikashige:1980ui ; Gelmini:1980re ; Georgi:1981pg ; Aulakh:1982yn ; Bertolini:1987kz ; Babu:1991we . Direct couplings to electrons are generated at one loop, but they are strongly suppressed by $\sim(g_{2}^{2}m_{e}m_{\nu}^{2})/(16\pi^{2}m_{W}^{2}m_{N})$.

In this model, various processes can establish kinetic equilibrium between the DM and SM sectors. Examples include elastic scattering between $\chi$ and $\nu$ through $\varphi$ exchange, and decays and inverse-decays of $\varphi$ into pairs of SM neutrinos Feng:2010zp . We find that as long as $\chi$ and $\varphi$ are chemically coupled, the latter process, $\varphi\leftrightarrow\nu\nu$, dominates kinetic equilibration between the DM and the neutrino bath since it is suppressed by fewer powers of small couplings.

In Fig. 4, we illustrate the viable parameter space of the representative toy model as a function of the DM mass, $m_{\chi}$, and mass ratio, $m_{\chi}/m_{\varphi}$, for $\xi_{\chi}^{0}\lesssim 0.5$ (corresponding to the Planck bound in Fig. 3). For each value of $m_{\chi}$ and $m_{\varphi}$, the coupling $\lambda_{\chi}$ is fixed to generate an abundance of $\chi$ that is in agreement with the observed DM energy density. In the limit that $m_{\chi}/m_{\varphi}\gg 1$, this requires $\lambda_{\chi}\sim 10^{-5}\times(m_{\chi}/\text{keV})^{1/2}$. We also demand that the DM sector equilibrates with the SM neutrinos while relativistic. This is accomplished by fixing $\lambda_{\nu}$ such that the rate for $\varphi\leftrightarrow\nu\nu$ overcomes the Hubble parameter at $T^{\chi\,\text{eq}}\simeq 3m_{\chi}/\xi_{\chi}^{\text{eq}}$, corresponding to $\lambda_{\nu}\sim 10^{-11}\times(m_{\chi}/m_{\varphi})(m_{\chi}/\text{keV})^{1/2}$. General constraints on $N_{\text{eff}}$ during nucleosynthesis and recombination are only relevant for $T^{\chi\,\text{eq}}\gtrsim T^{\nu\,\text{dec}}$ and hence exclude sub-MeV DM masses greater than $\sim(2\xi_{\chi}^{\text{eq}}/3)\text{ MeV}\simeq 400\text{ keV}$. As shown in previous studies, thermal DM heavier than several MeV is also viable as it is sufficiently non-relativistic at the time of neutrino decoupling Ho:2012ug ; Steigman:2013yua ; Boehm:2013jpa ; Nollett:2013pwa ; Nollett:2014lwa ; Steigman:2014uqa ; Green:2017ybv ; Serpico:2004nm . For $m_{\chi}\gtrsim 100\text{ keV}$ and $m_{\chi}/m_{\varphi}\lesssim 10$, $\chi$ and $\varphi$ become non-relativistic at the tail-end of nucleosynthesis. Pending a detailed simulation of the light nuclei abundances, Fig. 3 suggests that these masses are possibly in slight disagreement with the predictions of BBN. If $\chi$ is instead a real scalar boson, any possible tension is completely alleviated, while the general phenomenological picture in Fig. 4 remains qualitatively unchanged. We also note that while the parameter space shown is consistent with existing cosmological observations for $\xi_{\chi}^{0}\lesssim 0.5$, it will be decisively tested by CMB-S4 experiments Abazajian:2016yjj .

Also shown in Fig. 4 are constraints from considerations of warm DM. Once $\chi$ decouples from $\varphi$, it freely diffuses and suppresses matter perturbations below the free-streaming length, $\lambda_{\text{fs}}$. Limits derived from observations of the Lyman-$\alpha$ forest imply that $\lambda_{\text{fs}}\lesssim\mathcal{O}{(1)}\text{ Mpc}$ Viel:2013apy ; Baur:2015jsy . Such studies generally assume that DM has decoupled from the SM bath at early times, in which case $\lambda_{\text{fs}}\simeq 1.2\text{ Mpc}\times(m_{{}_{\text{DM}}}/\text{keV})^{-1}$ Abazajian:2017tcc . We translate the $2\sigma$ lower bound on the warm DM mass from Ref. Viel:2013apy , $m_{{}_{\text{DM}}}\gtrsim 3.3\text{ keV}$, into the upper bound $\lambda_{\text{fs}}\lesssim 0.36\text{ Mpc}$. We then compare this to the calculated value of the free-streaming length in our model, assuming that $\chi$ begins diffusing once the rate for $\chi\varphi\leftrightarrow\chi\varphi$ falls below the Hubble expansion rate. Observations of the 21-cm hydrogen line during the cosmic dark ages could potentially improve the limits on $\lambda_{\text{fs}}$ and $m_{\chi}$ by an order of magnitude Sitwell:2013fpa ; Sekiguchi:2014wfa ; Shimabukuro:2014ava .

DM self-scattering, $\chi\chi\to\chi\chi$, via $\varphi$ exchange, is constrained by studies of merging galaxy clusters. Following the discussion in Refs. Tulin:2013teo ; Tulin:2012wi , we calculate the transfer scattering cross-section per DM mass, $\sigma_{T}/m_{\chi}$, at the relevant virial velocities for galaxy clusters and demand that this does not exceed $1\text{ cm}^{2}/\text{g}$. As seen in Fig. 4, this excludes values of $m_{\chi}/m_{\varphi}\gtrsim\mathcal{O}{(10)}-\mathcal{O}{(100)}$ for $m_{\chi}\sim\mathcal{O}{(10)}\text{ keV}-\mathcal{O}{(100)}\text{ keV}$, respectively. Additional cosmological and astrophysical constraints include modifications of the CMB from neutrino self-interactions Cyr-Racine:2013jua ; Lancaster:2017ksf , diffusion-damped oscillations in the matter power spectrum from DM-neutrino scattering Mangano:2006mp ; Serra:2009uu ; Wilkinson:2014ksa ; Boehm:2000gq ; Boehm:2004th ; Boehm:2014vja ; Schewtschenko:2015rno ; Boehm:2003xr ; Boehm:2001hm , and supernovae cooling Choi:1989hi ; Kachelriess:2000qc ; Farzan:2002wx ; Heurtier:2016otg . We find that these limits are negligible compared to the DM free-streaming and self-scattering constraints presented above.

If the mediator-neutrino coupling is generated from interactions with right-handed sterile neutrinos, $N$, then at low-energies $\lambda_{\nu}\sim m_{\nu}/m_{N}$. Therefore, for SM neutrino masses at the level of $m_{\nu}\sim 0.1\text{ eV}$, the values of $\lambda_{\nu}$ favored in Fig. 4 motivate sterile neutrinos near the GeV scale. It is intriguing to note that future direct searches for sterile neutrinos, such as the proposed SHiP experiment, could have sensitivity to couplings as small as $\lambda_{\nu}\sim 10^{-10}$ Alekhin:2015byh .

The minimal model presented above opens the cosmological window for sub-MeV thermal relics, but it does not give rise to detectable signals at proposed low-threshold direct detection experiments Hochberg:2015pha ; Hochberg:2015fth ; Schutz:2016tid ; Knapen:2016cue . However, simple scenarios that build upon our mechanism are within the projected reach of these technologies. For instance, interactions between DM and hadrons are present at low-energies if $\varphi$ also couples to a heavy generation of vector-like quarks, as in Ref. Knapen:2017xzo . As a result, $\varphi$ can be produced in supernovae, but leaves their cooling rates unchanged if it becomes trapped, implying a lower bound on the $\varphi$-nucleon coupling, $\lambda_{n}\gtrsim 10^{-7}$. In the early universe, DM can equilibrate with the neutrino or photon bath as described in our minimal model. The nucleon coupling does not equilibrate the DM and SM sectors before neutrino-photon decoupling provided that $\lambda_{n}\lesssim 10^{-5}$ and the reheat temperature of the universe is $T_{\text{RH}}\sim 10\text{ MeV}$. Furthermore, for nucleon couplings of this size and $m_{\chi}\sim 100\text{ keV}$, the DM-nucleon elastic scattering cross-section is $\sim 10^{-40}\text{ cm}^{2}$, which is well within the projected reach of proposed experiments Battaglieri:2017aum .

In this Letter, we have considered a class of models in which sub-MeV thermal DM equilibrates with the SM neutrino bath after neutrino-photon decoupling. Dark sector equilibration and decoupling cools and reheats the neutrino sector close to its original temperature. Constraints derived from measurements of the effective number of neutrino species at the time of nucleosynthesis and recombination are significantly alleviated. While we have explicitly focused on DM, this mechanism could be more generally applied to any light thermal relics Bartlett:1990qq ; Chacko:2003dt ; Chacko:2004cz , such as those in Ref. Beacom:2004yd . Contrary to the standard lore, DM that acquires its abundance through thermal contact with the SM is viable for masses ranging from a keV to an MeV.

Acknowledgments

We would like to thank Kevork Abazajian, Jonathan Feng, Alex Friedland, Anson Hook, Manoj Kaplinghat, Simon Knapen, Gordan Krnjaic, Tongyan Lin, Andrew Long, Gustavo Marques-Tavares, Brian Shuve, and Kathryn Zurek for valuable conversations. AB and NB are supported by the U.S. Department of Energy under Contract No. DE-AC02-76SF00515.