The Cosmology of Sub-MeV Dark Matter (cid:7) (cid:7) Daniel Green and Surjeet Rajendran (cid:7) Department of Physics, University of California, Berkeley, CA 94720, USA 7 1 0 2 r Abstract a M Light dark matter is a compelling experimental target in light of stringent constraints on heavier 0 WIMPs. However, for a sub-MeV WIMP, the universe is sufficiently well understood at tempera- 2 tures below 10 MeV that there is no room for it to be a thermal relic. Avoiding thermalization is ] itself a strong constraint with significant implications for direct detection. In this paper, we ex- h p plore the space of models of sub-MeV dark matter with viable cosmologies. The parameter space - of these models that is also consistent with astrophysical and lab-based limits is highly restricted p e for couplings to electrons but somewhat less constrained for nuclei. We find that achieving nu- h [ clear cross-sections well-above the neutrino floor necessarily predicts a new contribution to the 2 effective number of neutrino species, ∆Neff = 0.09 that will be tested by the next generation of v CMB observations. On the other hand, models with absorption signatures of dark matter are 0 5 less restricted by cosmology even with future observations. 7 8 0 . 1 0 7 1 : v i X r a Contents 1 Introduction 1 2 Requirements for a Thermal Abundance 2 3 Direct Detection through Scattering 5 3.1 Simplified Model 5 3.2 Electron Interactions 6 3.3 Nucleon Couplings and Dark Radiation 9 4 Dark Matter Absorption 13 4.1 Stability through Symmetry 14 4.2 Cosmological Constraints 15 4.3 Absorption Signatures 16 5 Conclusions 17 A Dark Radiation from Mediator Decay 19 B Dark Matter Abundance 20 C Dark Matter Sub-Component 21 References 23 1 Introduction The identity of the dark matter is one of the great mysteries in physics. Direct detection ex- periments to date have been particularly effective in the search for the classic models of WIMP dark matter with masses in the 1 GeV to 10 TeV range. Null results from a number of experi- ments [1–5] rule out large regions of parameter space that were previously compatible with this simple picture. The lack of evidence for dark matter in this mass range has motivated, in part, the search for dark matter at lower masses and/or with different detection signatures [6]. Decreasing the mass of the dark matter can evade many conventional direct searches by low- ering the recoil energy. One might therefore imagine sub-MeV dark matter is weakly constrained down to the warm dark matter limit of (10) keV [7] and requires dedicated searches or new O experiments. However, the universe becomes transparent to neutrinos at temperatures below a few MeV and undergoes big bang nucleosynthesis (BBN) at temperatures of hundreds of keV. Measurements of the neutrino energy density in the cosmic microwave background (CMB), via N , and primordial abundances strongly constrain physics at times when a thermal sub-MeV eff mass particle is relativistic. In fact, if the dark matter is a relic of thermal equilibrium with the Standard model, this low mass regime is already excluded by observations [8–13]. Cosmological bounds are, of course, sensitive to assumptions about the physics of the early universe. Our inference of physics at MeV temperatures is somewhat indirect and one might imagine that the limits on light dark matter are highly model-dependent. However, dark matter is necessarily of cosmological origin and simply neglecting the cosmological constraints is not a viable alternative to the model-dependence. The abundance of dark matter may arise from a non-thermal mechanism or even as an initial condition set by (or before) reheating. However, for scenariostobeviable,wemustensurethatitissufficientlyweaklycoupledtotheStandardModel to have never been in equilibrium. Otherwise, the thermal abundance of dark matter would over- closetheuniverse(iffreeze-outoccurswhenthedarkmatterisrelativistic)orcauseunacceptably large changes to the neutrino and/or photon energy densities (non-relativistic freeze-out). Given thatcurrentmeasurementsdetectathermalcosmicneutrinobackgroundathighsignificance[14– 16], observations require that a sub-MeV dark matter particle should be more weakly coupled to the Standard Model than neutrinos (at MeV energies). In this paper, we will explore the space of viable models for sub-MeV dark matter and the implications for direct detection experiments. We will consider two classes of direct detection signatures: scatteringandabsorption. Thelargestpossiblescatteringcross-sectionsforlightdark matter require the presence of a light mediator. Introducing a new light particle coupled to the StandardModelishighlyconstrainedbylab-based,astrophysicalandcosmologicalmeasurements. Astrophysical constraints are particularly stringent for couplings to electrons and photons. The couplings to nuclei are less severely constrained by astrophysics and can give rise to larger elastic cross-sections than with the electrons. When mediator couplings to nuclei are large enough to bring the dark matter cross-section well-above the neutrino floor, the mediator was necessarily in thermal equilibrium prior to the QCD phase transition. In this region, the mediator must decay to dark radiation to avoid over-closing the universe, thereby producing an increase in N eff of at least ∆N = 0.09. This contribution will be tested with both Stage III and IV CMB eff experiments [17] and excluding this abundance of dark radiation would push the cross-section 1 close to the neutrino floor, despite the weakness of other constraints in this parameter range. Dark matter absorption is a more accessible experimental signature for light dark matter. Furthermore, for sub-MeV masses, kinematics forbids dark matter decay to the charged fermions of the Standard Model. While kinematics allows decays to photons or neutrinos, these decays can be highly suppressed if that dark matter is protected by a non-linearly realized non-Abelian symmetry. It is natural to embed this symmetry in the flavor symmetry of the Standard Model to allow for preferred coupling to specific Standard Model fermions. These couplings allow for dark matter absorption by either electrons or nuclei. However, the dark matter is necessarily of non-thermal origin and the couplings to the Standard Model must be sufficiently small to avoid thermalization. While these cosmological constraints are significant, they do not push the absorption cross-sections below the region that can be experimentally accessible. This paper is organized as follows: In Section 2, we review the observation that thermal sub-MeV dark matter is excluded by current observations. In Section 3, we discuss the class of models that would be observable through scattering with electrons and nuclei. We show that these cross-sections are already highly constrained by a combination of lab, astrophysical and cosmological bounds. Upcoming cosmological observations will probe the regions of currently allowed parameter space with the most experimentally accessible cross-sections with nuclei. In Section 4, we discuss models with dark matter absorption by electrons and nuclei and explain the strong limits from cosmology. The paper is supplemented by three appendices: Appendix A computes the change to N for eff a thermalized mediator under various circumstances. Appendix B discusses the origin of dark matter and the implications for the elastic cross-section. Appendix C discusses the bounds on the cross-sections for sub-components of the dark matter that avoid the bullet cluster constraints on dark matter self-interactions. 2 Requirements for a Thermal Abundance The most basic requirement for a model of dark matter is that it reproduces the observed abun- dance Ω h2 = 0.12. For a dark matter particle of mass m , the energy density is given by c dm ρ m n , where n is the number density. To be compatible with observations, therefore dm dm dm dm ≈ (cid:18) (cid:19) 1 1MeV n = Ω ρ 1.2 10−3cm−3 (2.1) dm c cr m ≈ × × m dm dm For masses m > 10 eV, the number density of dark matter particles is much smaller than dm the number of photons, n 4 102cm−3. Regardless of how the dark matter was produced, γ ≈ × the final population was not simply determined by number density in thermal equilibrium at T m , when n n . dm dm γ (cid:29) (cid:39) For m > 10 MeV, the necessary suppression is easily achieved through freeze-out at a tem- dm perature T < m . When the temperature falls below the mass of the particle, the Boltzmann F dm suppression of the number density in equilibrium can naturally explain the observed dark matter abundance. However, as wewill nowreview, whenm 10MeV, thermalequilibriumwiththe dm (cid:28) StandardModel(atT m )isexcludedbycurrentobservations[8–13]. Inessence,throughthe dm ∼ freeze-out of neutrinos and the generation of the primordial abundances of nuclei during BBN, 2 cosmological evolution for T < 10 MeV does not leave room for even one additional degree of freedom. 0.34 0.32 0.30 Y p 0.28 0.26 NBBN=2.85 0.28 0.24 eff ± 0.22 2.0 2.5 3.0 3.5 4.0 N eff Figure 1. Predictions for N -Y for thermal equilibrium with photons (black) and neutrinos (blue) eff p assuming g =1. The length of the bands along the N -axis allows for additional dark radiation to be DM eff added, which for photons can compensate the dilution of neutrinos from dark matter annihilation. The width along the Y -axis allows for the dark matter to be relativistic or non-relativistic at BBN, with the p larger Y corresponding to smaller m . Current Planck 1 and 2σ contours are shown in indigo [14]. p dm The red region shows the range of Y that is predicted by BBN using NBBN in the range consistent with p eff primordial abundance measurements NBBN = 2.85 0.28 [18]. The direct measurement of Y yields a eff ± p weaker constraint of Y = 0.2465 0.0097. We see that none of the hatched regions are consistent with p ± both CMB and abundance measurements. The white contours show forecasts for CMB Stage IV [16] (see also [17, 19]). We see that future CMB data could completely rule out the entire space without the need for abundance measurements. Figure adapted from [16]. Thefundamentaldifficultywithm < 1MeVisthatneutrinosbegintodecoupleatT = 1 10 dm − MeV. The dark matter must annihilate at temperatures below its mass and will either dilute or enhance the neutrino abundance which is out of equilibrium. At these energies, the dark matter may couple to (1) photons1, (2) neutrinos or (3) both. It is easy to see that all of these scenarios are in tension with current observations: 1. Darkmatterannihilatingtophotonsdilutesthenumber/energydensityinneutrinosrelative to the photons. From the conservation of comoving entropy: (cid:18) (cid:19)3 T 2 ν = . (2.2) Tγ 2+ 72 +gDM 1Couplings to electrons may or may not be a viable option depending on the mass of the dark matter. If m >m thentheannihilationtoelectronsiseffectivelythesameaannihilatingtophotonsbecausetheelectrons dm e and photons are in equilibrium. If m (cid:28)m , then the dark matter cannot annihilate efficiently when T ∼m dm e dm and will over-close the universe. 3 The resulting change to energy density in neutrinos is expressed as (cid:18) (cid:19)4/3 (cid:32) (cid:33)4/3 8 11 ρ 1 ν N = = 3 2.4 . (2.3) eff 7 4 ργ × 1+ 121gDM ≤ where we used g 1 for the final inequality. More details of the calculation of N in DM eff ≥ this and other scenarios can be found in Appendix A. 2. Thermal equilibrium with neutrinos would heat the neutrinos after being diluted by elec- trons such that (cid:18)Tν(cid:19)3 = 4 241 +gDM (2.4) T 11 21 γ 4 which means that (cid:18) (cid:19)4/3 4 N = 3 1+ g 3.78 , (2.5) eff DM × 21 ≥ where again we used g 1. DM ≥ 3. Finally, if everything is in equilibrium with the dark matter until after the electrons and positrons annihilate and then the dark matter annihilates in equilibrium, then T = T and ν γ (cid:18) (cid:19)4/3 11 N = 3 11.5 . (2.6) eff × 4 ≈ These numbers are all excluded at more than 3σ for the current limit N = 3.04 0.18 [14]. eff ± Therefore, the only loopholes either require further modifications of the model to cancel these effects or to dial the couplings such that non-equilibrium production of neutrinos and/or photons is just right (notice that there is no way to make it work in equilibrium). Fortheparticularcaseofcouplingtophotons, onemightconcludethatwecaneasilyevadethe boundsbyaddingsomeadditionalformofradiationtocompensateforthedilutedcontributionof the neutrinos. However, one cannot add this radiation without introducing significant changes to BBN. Because m < 1 MeV, the dark matter will contribute significantly to the expansion rate dm when T > m and therefore alters primordial abundances. Furthermore, when the dark matter dm doesannihilateitwilldilutenotonlytheneutrinosbutalsoanyadditionaldarkradiationaswell. As a result, in order to add ∆NCMB to N at recombination to avoid CMB constraints, we must eff eff introduce ∆NBBN (1+ 2 g )4/3∆NCMB to N during BBN. Adding dark radiation during eff ≈ 11 DM eff eff BBN alters the Helium fraction, Y because it changes the expansion after neutron decoupling as p well as other primordial abundances such as Deuterium. The predictions for such a scenario are shown in Figure 1 and are excluded by at least 2σ over the entire parameter space. Summary: Dark Matter with a mass m < 1 MeV cannot get its abundance from thermal dm equilibrium with the Standard Model at temperatures T m . Under minimal assumptions, dm ∼ it is excluded by CMB measurements of N for equilibrium with photons, neutrinos or both eff photons and neutrinos. Simple attempts to evade these constraints by adding extra sources of radiation are excluded by measurements of primordial abundances. While this description is based on the qualitative features, precise limits on the mass have been derived for each of these scenarios in [8–13] . 4 3 Direct Detection through Scattering In a conventional picture of thermal freeze-out, it is necessary that the dark matter is coupled to theStandardModel. Toachievetheobservedabundance, thesecouplingsarerequiredtobelarge enough that they make an enticing target for direct detection experiments. On the other hand, for sub-MeV dark matter, one must exclude couplings large enough to bring the dark matter into thermal equilibrium. These upper-limits on the couplings of dark matter to the Standard Model leave uncertain the experimental prospects for viable models of sub-MeV dark matter. In this section, we will explore these constraints and how they impact the experimental scattering cross-sections of dark matter with electron or nuclear targets. WewilldeferdiscussionoftheoriginofdarkmattertoAppendixBandevaluateconstraintson theelasticcross-sectionbetweendarkmatterχandtheStandardModelassumingthatthecorrect abundance is generated but not necessarily by a thermal mechanism. Elastic scattering of light dark matter will deposit tiny amounts of energy in a detector and thus inelastic processes might be preferable for direct detection experiments. But, the inelastic scattering cross-section can be expressedasaproductoftheelasticscatteringcross-sectionandaninelasticformfactor,typically suppressing the inelastic cross-section. Thus, our limits on the elastic scattering cross-section should be viewed as an upper bound, satisfying all cosmological, astrophysical and laboratory constraints. 3.1 Simplified Model Large experimental cross-sections require mediators that are light enough to be produced at relatively low temperatures in the early universe and/or in stars (even if they can be integrated- out for the purpose of direct detection). To discuss cosmological/astrophysical constraints, our model must include both the dark matter (χ) and mediator (φ) particles. Our results can be summarized in terms of the model 1 1 1 m λφχ2+g φNN¯ +g φEE¯ m2 χ2 m2φ2 , (3.1) L ⊃ 2 dm N e − 2 dm − 2 φ which describes the (real-scalar) dark matter χ scattering off either a nucleon (N, coupling g ) N or electron (E, coupling g ). E The couplings g and g are constrained by a variety of bounds. This includes short distance e N forceexperiments,colliderboundsandstellar/cosmologicalconstraints. Theboundsong andg N e are strong functions of the mass m of the mediator. Short distance force experiments dominate φ form < 100 eV,stellarboundsarestrongform < 10keV,cosmologicalconstraintsandcollider φ φ bounds are important for heavier masses. The interaction cross-section between dark matter and a Standard Model fermion can be written as λ2g2 m2 σ = f dm (3.2) χf→χf 4π (cid:16)(m v)2+m2(cid:17)2 dm φ where g is either g or g depending upon the fermion (electron, nucleon) that scatters with f e N the dark matter. This cross-section is valid in the limit m m ,m , which is the case for dm e N (cid:28) sub-MeV dark matter. Naively, it would seem that the largest direct detection cross-section 5 would be obtained by taking m 0. But since the bounds on g are strongly dependent φ f → on m , this is not the case. For elastic/inelastic scattering, in light of proposed experimental φ technologies2, our primary interest is for dark matter with mass keV. To obtain the maximum ≥ allowed cross-sections, we will thus concentrate on mediators with mass 1 eV, below which the ≥ direct detection scattering cross-section does not change, while bounds on the long range force mediated by φ between Standard Model particles gets considerably stronger. Bounds on dark matter self-interaction constrains λ independently of g . The self-interaction f scattering cross-section is constrained to be (cid:47) 10−25 cm2/GeV from observations of the bullet cluster [20]. This bound is stringent and further suppresses the direct detection cross-section. However, if χ is less than 10 percent of the dark matter abundance, this bound does not apply. ∼ The experimental and astrophysical limits on g are of course independent of λ and constrain f the experimental cross-sections for even such a sub-component. In Appendix C, we will show the limits on the cross-section when the self-interaction bound is not applied, with the caveat that the larger cross-section in this case is only possible for a sub-component of dark matter. 3.2 Electron Interactions Cosmology and astrophysics are the dominant bounds on g for light mediators with m 10 e φ (cid:28) MeV. In the range MeV – GeV, bounds are placed by beam dump experiments while colliders are relevant for masses above GeV. For mediator masses in the range MeV - GeV, sub-MeV dark matter is also constrained by SN1987A. If we focus on the region m < 10 MeV, the dominant bounds on g arise from cosmology and φ e astrophysics. Specifically, this region is dominated by the two constraints: A thermal abundance of φ from thermal equilibrium with electrons and positrons would • contribute ∆N = 2.2 and is easily excluded with current data. Avoiding thermal equilib- eff rium with electrons sets a limit g < 2 10−10 [21]. For m > 10 MeV, the mediator could e φ × decay before neutrino decoupling but cannot bring the dark matter into equilibrium (see Appendix B). White dwarf cooling constrains g < 8.4 10−14 [22] when the mediator is light enough to e • × be produced from e+e e+e+φ using k 400 keV. f → ≈ For m > 10 MeV, the thermalization of φ does not disrupt neutrino freeze-out or BBN, but we φ can impose the collider constraint B-factory searches for direct axion production limit [23] g < 10−3 for m < 10 GeV . e φ • Finally, we have the constraint on λ from the bullet cluster [20] 1 m2 σ = λ2 dm < 1cm2/g m . (3.3) χ 8π (m2 +v2m2 )2 × dm φ dm From these four constraints, we find the largest cross-section possible as a function of m and φ m as shown in Figure 2. dm 2Note that the cosmological bound of m > 10 keV [7] does not apply given that the dark matter is not dm assumed to be of thermal origin. 6 The qualitative feature of the limits on the electron-dark matter cross-section is that stellar constraintsdominateforlowermasses,m < 400keV.Avarietyofconstraintsfromstellarcooling φ exist at this level but the strongest limits on the electron coupling arise in white dwarfs. This is also important that due to the large density of electrons, the Fermi-momentum of 400 keV is much higher than the thermal momentum of 1 keV. As a result, the electrons are effectively relativistic. This is not true of nucleons which is largely responsible for the qualitative differences between bounds on the electron and nucleon couplings. Including the constraint from the bullet cluster, when vm < m < 400 keV, the cross section limit scales as σ m−2. dm φ ∝ φ As the mediator mass approaches the Fermi-momentum, the mediator-induced cooling be- comes exponentially suppressed and quickly weakens the bounds until we reach the limit set by cosmological constraints. When m < m < 10 MeV, forbidding equilibrium at T = m (rather e φ φ than T = m ) requires that g < 2 10−10(m /m )1/2. Naively the cosmological limits do not e e φ e × apply above 10 MeV since the mediator would annihilate before neutrino decoupling. However, as described in Appendix B, if λ is sufficiently large to bring χ into equilibrium then the entropy carriedbythedarkmatterwillproduceachangetoN thatisalreadyexcludedbyobservations. eff As a result, Figure 2 shows the cross-section above 10 MeV is still strongly contained because of this additional bound. For mediators heavier than GeV, one could try to avoid these cosmological bounds by ∼ reheating the universe below 100 MeV. In this case, for sub-MeV dark matter, there are two important bounds. First, the dark matter would be emitted in SN1987A. For mediators of mass (cid:38) GeV, at SN1987A temperatures 20 MeV, the interaction between electrons and the dark ∼ matter is the same higher-dimension operator as observed in direct detection experiments. If the produced dark matter escapes the supernova, it can lead to enhanced cooling. The elastic cross-sectionnecessarytoavoidthisboundwouldbe(cid:47) 10−50 cm2,wellbeyondthescopeofdirect detection. However, if the dark matter is more strongly coupled, it can thermalize within the supernova. In this case, there might be an additional bound from requiring that the produced dark matter does not cause too many events in Super Kamiokande or contribute to a diffuse background of hot dark matter particles (much like the diffuse supernova neutrino background) that may have lead to events in experiments such as XENON. Moreover, this region is also constrained by LEP bounds on light dark matter [24]. These bounds jointly squeeze the available parameter space in this window for sub-MeV dark matter. A more detailed analysis3 is necessary to figure out if there is a sliver of parameter space that is still allowed by these bounds. We note that these constraints also rule out exotic electron - neutrino interactions that have been proposed to explain DAMA [25]. These models typically require the coupling of a massive vector boson (e.g. B-L) with gauge couplings g (cid:39) 10−8 and mass less than MeV. These vector bosons will be in thermal equilibrium with the Standard Model at temperatures MeV, after ∼ neutrino decoupling. As we have seen, the entropy in these bosons cannot be easily removed. Moreover, for B-L bosons with mass less than MeV and gauge couplings (cid:39) 10−6, electrons ∼ and neutrinos would be coupled more strongly than the weak interactions at MeV temperatures, changing the neutrino temperature at decoupling. This additional entropy would appear as dark radiation and is similarly constrained. 3P.W. Graham, S. Rajendran and G. M. Tavares, in progress 7 10 43 − 10 44 − 10 45 − 10 46 − ) 10 47 2 − m (c 10−48 stic 10−49 a σel 10−50 10−51 103 eV 10 52 104 eV − 105 eV 10 53 − 106 eV 10 54 −10 1 100 101 102 103 104 105 106 107 108 109 1010 1011 − m (eV) φ Figure 2. Limits on the dark matter-electron elastic cross-section as a function of mediator mass for variousdarkmattermassesintherangekeV–MeV,asindicatedinthelegend. Thesharpfeatureatm φ ∼ 105−6 eV arises when m approaches the white dark fermi-momentum, k 400 keV and the additional φ f ≈ cooling rate falls exponentially. A second feature at m 10 MeV indicates where thermalization of φ ≈ the mediator becomes possible because it can decay to electrons or neutrinos before neutrino decoupling. Neutrinodecouplingisagradualprocessthatbeginsaround10MeVandthereforethepreciselocalization of this feature requires more care. The most stringent bounds in these scenarios apply directly to the mediator φ and not on the light dark matter particle χ. Once φ satisfies cosmological and stellar constraints, the production of χ would be sufficiently suppressed to avoid these bounds. While this is typically true of any model of light mediators, the argument fails for one specific case - when the mediator is a hidden photon that is kinetically mixed with electromagnetism. In this case, the Lagrangian (in the mass basis) can be expressed as F2 +G2 +m2B2 +g JµB +eJµ(A +(cid:15)B ) (3.4) µν µν µ χ χ µ e µ µ L ⊃ where F is the gauge field strength of electromagnetism (A ), G is the gauge field strength µν µ µν of the hidden photon B . In this case, as is well known, the effects of the mediator decouple with µ its mass m and thus bounds on the mediator are weakened in the limit m 0. However, the → dark matter χ does not decouple when m 0 - it is in fact charged under electromagnetism with → a charge (cid:15)g . In processes where the momentum transfer q m, χ behaves as a milli-charged χ ∼ (cid:29) particle and would thus be constrained by cosmology and stellar bounds. These bounds imply that the effective charge g (cid:15) (cid:47) 10−13 for m (cid:47) 10 keV (stellar bounds) and g (cid:15) (cid:47) 10−11 for χ χ χ m (cid:39) 10 keV (cosmology), leading to cross-sections that are at least as small as 10−42 cm2 dm (m (cid:39) 10 keV) and 10−45 cm2 (m (cid:47) 10 keV) (the long range coulombic enhancement has been χ dm cut-off at the angstrom scale, beyond which electric fields are screened in high density matter), without including additional inelastic or phase-space suppressions. 8