Reviving Millicharged Dark Matter for 21-cm Cosmology

The existence of millicharged dark matter (mDM) can leave a measurable imprint on 21-cm cosmology through mDM-baryon scattering. However, the minimal scenario is severely constrained by existing cosmological bounds on both the fraction of dark matter that can be millicharged and the mass of mDM particles. We point out that introducing a long-range force between a millicharged subcomponent of dark matter and the dominant cold dark matter (CDM) component leads to efficient cooling of baryons in the early universe, while also significantly extending the range of viable mDM masses. Such a scenario can explain the anomalous absorption signal in the sky-averaged 21-cm spectrum observed by EDGES, and leads to a number of testable predictions for the properties of the dark sector. The mDM mass can then lie between 10 MeV and a few hundreds of GeVs, and its scattering cross section with baryons lies within an unconstrained window of parameter space above direct detection limits and below current bounds from colliders. In this allowed region, mDM can make up as little as $10^{-8}$ of the total dark matter energy density. The CDM mass ranges from 10 MeV to a few GeVs, and has an interaction cross section with the Standard Model that is induced by a loop of mDM particles. This cross section is generically within reach of near-future low-threshold direct detection experiments.


I. INTRODUCTION
Dark Matter (DM)-baryon interactions can affect the thermal history after recombination [1,2], possibly leading to observable deviations in the 21-cm global signal compared to the ΛCDM prediction [3].Moreover, if the scattering cross section is enhanced at small relative velocities, its impact on the cosmological history is most prominent between recombination and reionization, when the baryons are at their coldest.The most optimistic scenario for identifying new physics in 21-cm observables is therefore realized if the DM-baryon interactions are Rutherford-like, i.e. with a cross section scaling as v −4  rel , where v rel is the relative velocity between particles.This may result from an exchange of a light mediator with negligible mass compared to the typical exchange momentum in the scattering process.
In most scenarios, however, the presence of a new light mediator is strongly constrained by a combination of fifthforce and stellar cooling bounds [4][5][6].Accounting for these constraints leaves little room for observable effects from DMbaryon scattering in 21-cm cosmology [7], with the only exception being a DM that carries an effective electric charge Q, inducing DM interactions with the SM 1.Even in this so-called millicharged dark matter (mDM) scenario, which has been studied extensively in the literature [1,2,[7][8][9][10][11][12][13], several stringent experimental constraints [7,8,10] apply.Most notably, measurements of the CMB anisotropy power spectrum [14][15][16][17][18] and N eff constraints [10,19] limit the mDM to a small region of parameter space, with mass 10 MeV m m 40 MeV, 1 In this paper, Q is the charge measured in units of the electron charge so that the coupling of the DM to the SM photon reads eQAµJ µ DM .In the nearly massless dark photon case, one obtains Q ∼ α D 2 /α EM , with being the parameter controlling the mixing between the visible and dark sectors, while α D = g 2 D /4π is defined by the minimal coupling to the DM current, g D A µ J D .Alternatively, the DM electromagnetic charge can be assigned by coupling it directly to the SM photon.charge 10 −5 Q 10 −4 , and energy density between 0.01% and 0.4% of the total dark matter energy density.These severe restrictions make simple models of mDM unattractive as a significant source of cooling for baryons in the early universe.
In this paper, we revisit the possibility of baryon cooling by a millicharged component of DM.We show that if mDM also has an additional long-range interaction with the rest of the dark matter (which forms a cold dark matter (CDM) bath), it may remain cold for longer, thereby greatly increasing the effective heat capacity of the hidden sector.This substantially improves the efficiency of baryon cooling, extending the region of parameter space in which we expect large effects on 21-cm cosmology.In particular, the setup, which we depict in Fig. 1, can explain the recent sky-averaged 21-cm spectrum measurement presented by the EDGES collaboration [20], while opening up the higher mass and smaller density regions of the mDM parameter space.This can be accomplished while ensuring that the momentum transfer between mDM and CDM remains small enough to circumvent the stringent CMB constraints.
The allowed parameter space in this scenario leads to precise predictions that will be tested in upcoming direct detection and collider experiments.The large heat capacity needed to cool the gas implies an upper bound of several GeVs on the CDM mass (see Fig. 4), while BBN constraints require it to be larger than roughly 10 MeV [21].Interactions between CDM and SM particles are radiatively generated by loops of mDM particles, and the corresponding CDM-baryon cross section lies below the current direct detection limit from Xenon10 and SENSEI [22][23][24], but well within the reach of future runs of low threshold direct detection experiments like SENSEI and DAMIC [25].Interesting constraints on the nature of CDM self-interactions can be inferred from colliding (sub-)clusters [26,27] and from cluster ellipticity [28,29], even though the reliability of the latter has been questioned [30][31][32].The allowed parameter space for the mDM component in our new framework is greatly enlarged compared to the standard scenario, leading to a rich phenomenology that can motivate many upcoming beam dump and direct detection G n u L u 3 f 3 B Y O j p u 6 j h V D B s s F r F q B 1 S j 4 B I b h h u B 7 U Q h j Q K B r W B 0 P / V b T 6 g 0 j + W j G S f o R 3 Q g e Z 8 z a q x U x 1 6 p 7 F b c G c g q 8 X J S h h y 1 X u m n G 8 Y s j V A a J q j W H c 9 N j J 9 R Z T g T O C l 2 U 4 0 J Z S M 6 w I 6 l k k a o / W x 2 6 I S c W y U k / V j Z k o b M 1 L 8 T G Y 2 0 H k e B 7 Y y o G e p l b y r + 5 3 V S 0 7 / 1 M y 6 T 1 K B k 8 0 X 9 V B A T k + n X J O Q K m R F j S y h T 3 N 5 K 2 J A q y o z N Z m F L E E S X 2 j 4 2 x H B S t O l 4 y 1 m s k u Z V x b O 8 f l 2 u 3 u U 5 F e A U z u A C P L i B K j x A D R r A A O E F X u H N e X b e n Q / n c 9 6 6 5 u Q z J 7 A A 5 + s X Y j + X N w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S 4 + a + H f a o u A l t o B A F v H W c u P n a 7 4 = " > A A A C A X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 S N g V Q Y 9 B L x 4 T M A 9 I l j A 7 2 0 m G z M 4 u M 7 N C W H L y 6 F U / w p t 4 9 U v 8 B n / C S b I H k 1 j Q U F R 1 0 9 0 V J I J r 4 7 r f z t r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 j h V D B s s F r F q B 1 S j 4 B I b h h u B 7 U Q h j Q K B r W B 0 P / V b T 6 g 0 j + W j G S f o R 3 Q g e Z 8 z a q x U x 1 6 p 7 F b c G c g q 8 X J S h h y 1 X u m n G 8 Y s j V A a J q j W H c 9 N j J 9 R Z T g T O C l 2 U 4 0 J Z S M 6 w I 6 l k k a o / W x 2 6 I S c W y U k / V j Z k o b M 1 L 8 T G Y 2 0 H k e B 7 Y y o G e p l b y r + 5 3 V S 0 7 / 1 M y 6 T 1 K B k 8 0 X 9 V B A T k + n X J O Q K m R F j S y h T 3 N 5 K 2 J A q y o z N Z m F L E E S X 2 j 4 2 x H B S t O l 4 y 1 m s k u Z V x b O 8 f l 2 u 3 u U 5 F e A U z u A C P L i B K j x A D R r A A O E F X u H N e X b e n Q / n c 9 6 6 5 u Q z J 7 A A 5 + s X Y j + X N w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S 4 F F X s g r e X O e n X f n w / m c t u a c 2 c w h m Y P z 9 Q s X 3 p t + < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k S K z M u q U U S I Z u F F X s g r e X O e n X f n w / m c t u a c 2 c w h m Y P z 9 Q s X 3 p t + < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k S K z M u q U U S I Z u F F X s g r e X O e n X f n w / m c t u a c 2 c w h m Y P z 9 Q s X 3 p t + < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k S K z M u q U U S I Z u x c M V J t 3 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 L m U L p j W T q L x c M V J t 3 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 L m U L p j W T q L x c M V J t 3 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 L m U L p j W T q L O e n X f n w / m c t q 4 4 s 5 k j M g f n 6 x f T 4 5 t U < / l a t e x i t > (1 f m ) < l a t e x i t s h a 1 _ b a s e 6 4 = " p x 0 K r p U 3 x p w 8 S e 7 x 7 D 4 0 l + r v a c g = " l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p x 0 K r p U 3 x p w 8 S e 7 x 7 D 4 0 l + r v a c g = " l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p x 0 K r p U 3 x p w 8 S e 7 x 7 D 4 0 l + r v a c g = " • atoms: • ions: x H,He  experiments.The mDM fraction, f m , should be within 10 −8 f m 0.004, where the upper bound comes from present CMB constraints [14][15][16][17] and the lower bound is the minimal fraction required to explain EDGES while still being consistent with constraints from collider and beam dump experiments.The mDM electric charge lies in the range 10 −5 Q 1, such that the interactions between the mDM component and the baryons are strong enough to lead to sufficient cooling, while still being consistent with collider searches.The mDM mass, m m , is found to be in the range 10 MeV m m 200 GeV, where the lower bound again stems from BBN constraints2, while the upper bound is derived by requiring that cooling is efficient enough to explain EDGES while not violating CMB constraints.
A large array of future experiments can potentially discover the mDM component in our scenario.The large-Q region is constrained by present collider searches [35][36][37][38] and will be further probed by a combination of future collider experiments such as milliQan [38,39], MicroBooNE [40], LDMX [41] and dedicated detector proposals in the NuMI or DUNE beamline [42,43].Ground-based direct detection of mDM is challenging, due to mDM energy losses in the atmosphere, in the Earth's crust and in the shielding material of a given experiment [44,45].For this reason, the best sensitivity to mDM in this coupling regime may be achieved by surface runs of low threshold detectors (e.g.Ref. [46]), which can detect the soft tail of the DM velocity distribution after atmospheric scatterings (see Ref. [47] for an updated reach of these experiments for mDM).Pushing the reach to higher Q can be achieved by either lowering the threshold of the detectors [48][49][50][51][52][53][54][55][56][57][58] or reducing the atmospheric shielding using balloon-and satelliteborne detectors [47].
The paper is organized as follows.In Sec.II, we summarize the relevant 21-cm cosmology formalism and discuss the 2 For the cross sections we consider here, both CDM and mDM are in thermal equilibrium with the SM in the early universe.In the simplest setup, the freeze out of CDM and mDM into light dark radiation can be in tension with N eff measurements at CMB and BBN posing a more stringent lower bound on both the mDM and CDM masses (see for example [33,34]).This constraint can be circumvented easily for the mDM coupling to the SM by directly assigning a charge Q under the SM electromagnetism.For the CDM-mDM mediator, the N eff bounds can be easily circumvented in non-minimal models and are ignored in the following.
EDGES measurement.In Sec.III, we explain why it is difficult for an mDM component with no interaction with the dominant CDM component to cool baryons efficiently.Along the way, we develop an analytic understanding of the parametric behaviour of DM cooling after recombination, which qualitatively explains most of the features of the full numerical treatment, based on Ref. [59].In Sec.IV, we discuss quantitatively the advantages of the setup summarized in Fig. 1.The allowed parameter space is derived by fitting the EDGES signal and accounting for other cosmological constraints.Sec.V is dedicated to the phenomenological predictions of our setup and the future prospects for excluding or discovering both the mDM and the CDM components.Sec.VI contains our conclusions.This paper is supplemented by four appendices in which we summarize many useful results that are at the core of both our analytic and numerical treatment.App.A summarizes a derivation of the thermodynamic equations for a system of two baths, first derived in Refs.[1,2].The relevant rates are formulated in a model independent fashion in terms of the DM-baryon interaction form factors as usually done in the direct detection literature [60].In App.B, we compute the relevant form factors in the Born approximation.We discuss in particular the ones necessary to include for DM scattering with hydrogen and helium atoms in 21-cm cosmology.We also assess the regions of parameter space in which the Born approximation breaks down, leaving a full quantum mechanical computation of the rates for future work.In App.C, we discuss more quantitative details about the implementation of the framework described in Fig. 1.We discuss the thermodynamic equations for a system of three baths and show how the standard mDM scenario previously discussed in Refs.[7,8,10] is parametrically extended as a function of the coupling between mDM and CDM.We then discuss in more detail the cosmological constraints from CMB on our scenario, relying on Refs.[14][15][16][17][18]. Lastly, App.D discusses concrete models realizing the scenario of Fig. 1 which are used to derive the CDM component direct detection prospects.

II. THE EDGES OBSERVATION
The brightness temperature of the 21-cm hydrogen line, T 21 , measures the contrast between redshifted 21-cm radiation and the CMB blackbody spectrum at the same frequency.As a function of redshift, this can be written as [61] where T γ is the CMB temperature3, T s is the spin temperature of the 21-cm line and τ (T s ) its optical depth.The latter two quantities are defined as where H is the Hubble parameter at a given redshift, λ 21 = 1/E 21 21 cm and A 10 2.9 × 10 −15 s −1 is the Einstein spontaneous emission coefficient for the 21-cm transition.The spin temperature T s characterizes the ratio between the number densities of the triplet and singlet hyperfine sub-levels of the hydrogen 1s state.When T s is larger (smaller) than T γ , we expect to see an emission (absorption) feature in T 21 .
The EDGES experiment claimed the first measurement of the brightness temperature at the cosmic dawn (z 17) [20].During this epoch, the first stars begin to emit Ly-α radiation, affecting the spin temperature as Ly-α photons pass through hydrogen clouds by driving spin-flip transitions between the hyperfine sub-levels via the Wouthuysen-Field effect [65,66].The rate of these transitions quickly become much faster than downward transitions through stimulated emission by CMB photons, driving the temperature of the baryons T b , the spin temperature T s and the temperature of the Ly-α spectrum to a common value.
One can thus quite generally expect T b T s T γ , with the ratio T s /T b depending on the strength of the Wouthuysen-Field effect4.The strongest absorption peak in the brightness temperature defined in Eq. ( 1) is obtained by saturating the inequality on the left (T s = T b ), giving The ΛCDM prediction for T b and T γ at z = 17 gives T ΛCDM 21 | z=17 = −0.22K.This lower bound is in 3.8σ tension with the central value of the EDGES measurement [20], 3 Throughout this paper we assume that Tγ is given by the CMB temperature.
See Refs.[62][63][64] for examples in which this assumption is relaxed, giving rise to important alterations to the 21-cm global signal.4 Our computation of Ts (and the subsequent determination of T 21 in Eq. ( 4)) neglects astrophysical effects that could lead to extra heating of the gas, diminishing the absorption feature at cosmic dawn.Even under the assumption that X-ray heating is turned off at z 17 [20] (see [61,67,68] for a quantitative discussion of X-ray heating), a full self consistent computation should include the hydrogen heating from its scattering with the Ly-α photons [67] and the CMB heating first computed in Ref. [69].The latter has been shown to be the dominant effect leading to up to 50% deviations in the 21-cm brightness temperature.Since the magnitude of these effects is not enough to invalidate our picture, we defer a careful study of these, as well as of the shape of the absorption peak, for future studies.
whose central value and 99% confidence intervals are This observed tension, if it persists, would point to a deviation from standard cosmology, and has drawn significant attention.Previous work in the literature has focused on two possible solutions: (i) either 21-cm photons at z = 17 were hotter than the CMB due to astrophysical sources [62,63,70] or new physics [64], or (ii) the gas temperature at z 17 was colder than the one predicted by ΛCDM due to cooling induced by the scattering of baryons with some fraction of the cold DM bath [1,2,[7][8][9][10][11][12].In this paper we continue to investigate the latter option.
Using Eqs. ( 4) and ( 5) and assuming that T γ is simply the CMB temperature, the gas temperature inferred from the EDGES measurement is with the deviation from the ΛCDM prediction being given by The DM bath must therefore cool the baryons by at least 1.7 K to a temperature of 5.1 K in order to accommodate the EDGES result within the 99% CL.This is the amount of cooling that will be conservatively required throughout this paper.
Before leaving this section it is worth pointing out that the 21-cm global signal measurement are experimentally intricate and face significant background subtraction challenges.For this reason, it has been suggested that the significance of the EDGES result could be appreciably reduced depending on the background subtraction procedure [71][72][73][74].Future and ongoing experiments are likely to provide more insights and possibly independent tests on the EDGES result [75][76][77][78][79][80].Nevertheless, we believe it is an important theoretical question to classify the classes of dark sectors that could give signals in 21-cm cosmology without being already excluded by other complementary probes.In the remainder of the paper, we will explore this question, focusing on DM cooling the baryons through elastic collisions.

III. STANDARD MILLICHARGED DARK MATTER
Having presented the EDGES result, let us now discuss the presence of an mDM component that interacts with the baryonic gas.We shall argue that such an addition to ΛCDM is insufficient to explain the EDGES result consistently with the existing cosmological bounds.
We begin by estimating the mDM-baryon interaction rate required for the mDM to account for the necessary gas cooling presented above in Eq. (7).To do so we study the evolution of the gas temperature T b , given by5 (see App.A for a detailed derivation and the full expression) The first term on the right hand side accounts for adiabatic cooling due to the expanding universe while the second term describes the cooling of the gas due to the mDM-baryon interactions.T m is the mDM temperature, x b is the fraction of baryons interacting with mDM, µ bm is the reduced mass of the interacting baryon and the mDM, and is the mDM-baryon interaction rate, with σ bm T the transfer cross section (see Eqs. ( 10) and (A5)) and v rel the relative velocity between the colliding mDM and baryon particles.We define f m ≡ ρ mDM /ρ DM to be the energy density fraction of the mDM.The symbol . . .indicates thermal averaging over the velocity distributions of both fluids.
We calculate the mDM-baryon transfer cross section assuming that the Coulomb potential is regulated by the Debye thermal mass of the photon, m 2 γ α EM n e /T b .Since (i) m γ is much smaller than the typical momentum transfer q µ bm v rel , and (ii) for the relevant charges Q considered here, m γ µ bm v 2 rel /Qα EM , σ bm T should be evaluated in the classical regime described in [81][82][83][84], that is, We stress that this expression has a different log factor compared to the one commonly used in the literature [8,85,86].The cross section in Eq. ( 10) is evidently enhanced as v −4 rel at low relative velocities.Substituting this expression into Eq.( 8), we see that the effect of mDM-baryon scattering on the temperature evolution grows rapidly as redshift decreases.
In the absence of the second term on the RHS of Eq. ( 8), baryons cool only adiabatically.This is insufficient (by an order one amount) for the temperature evolution to be consistent with the EDGES measurements.Consequently, in order for significant cooling to occur, the rate of energy transfer must be at least as large as the energy loss from adiabatic expansion, but not much larger.We therefore require the cooling rate to be comparable to Hubble at around z = 17 in order to explain the EDGES discrepancy in Eq. ( 7).This gives where a much smaller rate will not be able to cool the baryons, and a much larger one would cause an exponential decrease in the baryon temperature.
The mDM temperature at z = 17 must lie between where Ω b /Ω DM 0.18.The upper bound on T m ensures that the mDM bath can cool the gas, while the approximate lower bound can be derived by assuming that the additional heat per unit volume removed from the baryons is transferred to a zero temperature mDM fluid, so that n b |∆T EDGES b | ∼ n m T m .Using the lower bound for T m in Eq. ( 11), we get an estimate of the smallest rate required to explain the EDGES measurement.In the limiting case where T m is exactly equal to T b in Eq. ( 12), we find The upper bound on m m occurs at the point where the number density of mDM is too small to absorb the required amount of heat from the baryons.CMB constraints for a typical mDM charge required for sufficient cooling exclude f m 0.4% [16,18,86].Fig. 2 shows in dark green the value of Q as a function of m m required to fit the EDGES signal with f m = 0.4%, computed using a full numerical treatment.The turnaround is qualitatively explained by Eq. ( 13): at higher mDM masses the heat capacity of the mDM bath becomes insufficient to cool the baryonic gas.The full numerical treatment includes the heating of the baryons from Compton scattering with CMB photons, explaining why the turnover in Fig. 2 happens at a slightly smaller mDM mass than predicted by Eq. ( 13).
Given the upper bound in Eq. ( 13) and the required couplings to baryons in Eq. ( 11), the range of possible m m and f m that can explain the EDGES result is severely restricted by constraints on the effective number of relativistic degrees of freedom from BBN and CMB.These are notoriously difficult to accommodate given a particle freezing out at temperatures below the QCD phase transition.Even though some caveats to this argument can be constructed (see Ref. [10] for a discussion), these difficulties substantially reduce the appeal of this minimal mDM scenario as an explanation of the EDGES result.

IV. MILLICHARGED DARK MATTER INTERACTING WITH A COLD DARK MATTER BATH
In the remainder of this paper, we demonstrate how a new long-range interaction between a small fraction of mDM and the rest of the DM bath significantly improves upon the standard mDM scenario in explaining the EDGES observation.We first begin with an overview of our framework in Sec.IV A. We then explain how the mDM parameter space is extended in Sec.IV B, discuss the cosmological constraints on the dark sector in Sec.IV C, and finally describe the importance of mDM-neutral atom scattering in our setup in Sec.IV D. A more detailed discussion of each of these aspects of our model is included in the appendices.

A. The Framework
Before diving into a detailed analysis, we wish to introduce the main features of our new framework, illustrated in Fig. 1 Parameter space of the scenario described in Fig. 1 in the plane (mm, Q) where we fix mC = 10 MeV to maximize the heat capacity of the CDM bath and the maximal αmαC allowed by CMB bounds [14][15][16][17][18].The dark blue contours give the mDM fraction fm required for a given (mm, Q) point to fit the upper value of the 99% CL interval of the EDGES measurement in the setup of Fig. 1.For a fixed fm the entire region above the dark blue contour can be probed by reducing αmαC (see text for details).For comparison, the dark green contour shows the standard mDM case where 0.4% of mDM alone provides the baryonic cooling.The light blue region for mm < 10 MeV is robustly excluded by BBN contraints on Neff [7,8,10,19], the two dotted lines distinguish between the case in which mDM is a scalar or a Dirac fermion.The gray shaded area is a collection of different constraints taken from Refs.[37,87], plus limits on millicharge particles from milliQ at SLAC [35], searches for low ionizing particles in CMS at the LHC [88] and the new constraints from LSND and MiniBooNE derived in Ref. [40].The region on the left of the blue line is excluded by CMB constraints on Neff only when mDM couples to a dark photon with coupling gD = 0.1.
The green region is excluded by present direct detection experiments as shown in Ref. [47].The green dashed line indicates our extrapolation of the results in Ref. [47] to higher masses (see discussion in the text).The red/black/magenta lines indicate the Fermilab/SLAC/CERN effort to probe mDM.Solid/dashed/dotted lines give a rough sense of the short/medium/long time scale of the proposal.Solid red is the ArgoNeut sensitivity derived in Ref. [43], dashed red is the sensitivity of the Fermini proposal at NuMI [42] (see Ref. [43] for a more conservative reach based on ArgoNeut at NuMI), dotted red is the DUNE reach [40] while dotted black is the LDMX reach [41].Dashed magenta is the milliQan reach as [38] while dotted magenta is the SHiP sensitivity [40].The dash-dotted/dotted green lines indicate the reach of a SENSEI-like dark matter detector on a balloon/satellite with 0.1 gram-month exposure [47].
constitutes a fraction f m of the total DM energy density.The mDM-baryon long-range interaction is controlled by the mDM charge Q, which may or may not stem from the presence of a new light mediator.The novelty in our setup is that the mDM fraction also interacts with the remaining CDM component, of mass m C , through a distinct long-range hidden interaction controlled by the coupling g m g C .The same interaction also induces a CDM self-interaction proportional to g 2 C .The two long-range interactions of our setup imply the existence of one or two new light mediators with masses below a keV, which is the typical size of the exchange momentum in scattering collisions during the cosmic dawn.
The long-range force between mDM and CDM opens up the mDM parameter space at higher masses (up to m m 200 GeV) and smaller dark matter fraction (down to f m 10 −8 ).This is because the cooling is now driven by the CDM bath, with the mDM acting as a mediator between CDM and the baryons.As we show in Sec.IV B, the CDM mass m C must lie below a few GeV in order to have a large enough heat capacity to cool the gas sufficiently.
The allowed parameter space of our framework is mostly determined by ensuring that the mDM-baryon and mDM-CDM couplings are consistent with CMB constraints, as discussed in Sec.IV C. In Fig. 2, we show three contours on the m m -Q plane where sufficient cooling of the baryonic bath is achieved in our framework in order to explain the EDGES result for f m = 10 −4 , 10 −6 and 10 −8 .We have fixed m C = 10 MeV and g m g C ; these two parameters can vary over a broad range of values without affecting our qualitative results, as we will explain in greater detail in Sec.IV C below.The white region of Fig. 2 represents the available mDM parameter region, bounded between present collider bounds [35,37,40,87,88] and direct detection bounds [47].We also show the prospects of exploring this region of parameter space with upcoming experiments.This, together with the direct detection prospects for the CDM component, which has a loop-induced interaction with the SM, will be discussed in Sec.V.

B. Cooling Parametrics
The long-range interaction between CDM and mDM induces a Rutherford-like transfer cross section between the two baths given by where we defined α m = g 2 m /4π, α C = g 2 C /4π and again we assume the mediator mass m φ satisfies m φ µ mC v 2 rel / √ α m α C as in Eq. ( 10).The exact nature of the mediator (scalar or vector) will not modify the discussion here.The explicit example of a vector mediator will be discussed in both Sec.V and Appendix D.
As in the standard mDM case, due to the strong velocity dependence, the cooling of baryons is largest at low redshifts, where the velocity of the baryonic bath is the smallest.The crucial difference is that in our framework, the cooling is achieved dominantly by an mDM fluid that is coupled to the CDM bath.In a manner similar to Eq. ( 11), sufficient cooling of baryons to explain the EDGES measurement is approximately achieved when with the important difference from Eq. (11) being that the rate now depends on T C /T b , where T C is the CDM temperature, instead of T m /T b .
Following the derivation of Eq. ( 12) closely, we find that the CDM temperature at z = 17 is now bounded by where the lower bound is set by the minimal amount of heat that CDM absorbs from cooling the baryons through the mDM fraction.Saturating the inequalities above results in an upper bound on m C , where the gain compared to the standard mDM bound in Eq. ( 13) is given by the fact that the heat capacity of the dark sector is now set by the number density of CDM, which is much larger than that of the mDM component.In deriving the approximate upper bound in Eq. ( 17), we have neglected Compton heating from the CMB photons.For m C values well below the bound in Eq. ( 17), the heat capacity of the CDM is always large enough to absorb heat from the mDM fluid efficiently, and as a result the contours shown in Fig. 2 are insensitive to the exact value of m C .In Fig. 3, we show a typical temperature history in our framework where baryons are cooled sufficiently to give the EDGES result.The dark sector coupling g m g C has been chosen so that the mDM bath is coupled to baryons at recombination; this is important for CMB constraints on our setup, which we will discuss below.For simplicity, in Fig. 3 we also neglect the drag force between the mDM and CDM fluids, which induces an extra heating source for the baryons at high redshift.These effects are included in our full numerical solution, and have a qualitative impact on some of our parametric estimates.A complete discussion of our full numerical treatment can be found in Appendix C 1.

C. Cosmological Bounds
Like the pure mDM model discussed in Sec.III, this setup faces important cosmological constraints from the CMB power spectrum measurement (which constrains both the momentum transfer between baryons and CDM as well as N eff ), and from the primordial abundance of light nuclei produced during Big Bang nucleosynthesis (BBN).
As we have mentioned above, for f m 0.4% the CMB power spectrum bounds on mDM vanishes, since it can be considered to be a component of the baryonic fluid whose contribution to the total baryon energy density Ω b lies within the uncertainty of its observed value.The addition of a new CDM-mDM interaction leads to a momentum drag between the baryonic and the CDM fluid, which is constrained by measurements of the CMB anisotropy power spectrum [14][15][16][17].While a dedicated analysis of our framework is beyond the scope of this paper, a simple recasting of existing constraints from Refs.[14][15][16][17] can be performed as long as the mDM is tightly coupled to the baryons before recombination; this can only be achieved if the mDM coupling to the CDM bath is sufficiently weak (see App. C 2 for derivation): Under the above assumption, the standard constraint for mDM interacting with the baryons can be rescaled to obtain an upper limit on the CDM-mDM interaction.One finds (see App. C 2 for details) where V mC is the bulk relative velocity between the CDM and both the mDM bath and the baryons, which we take to be 29 km s −1 before kinetic decoupling at z = 1010; this is roughly the rms value expected in ΛCDM [89,90].Note that by virtue of Eq. ( 14), the LHS depends only logarithmically on V mC .A more careful analysis of the CMB constraints 3000, the mDM temperature is close to the baryon temperature while the CDM bath is significantly colder.As the free electron fraction xe drops, the mDM decouples from the baryons, thermalizing with the CDM while cooling the baryonic gas.These parameters are consistent with known experimental constraints on Q, as well as CMB constraints on gmgC.They also produce sufficient cooling to be consistent with the upper limit of the EDGES 99% CL (arrow, black).For the range of mDM masses considered here, the cooling of the baryons from both scattering with the ionized fraction (solid, orange) and with hydrogen and helium atoms (solid, green) are important.At low redshifts (i.e.low relative velocities), the Born approximation for scattering between mDM and atoms breaks down (gray, dotted).In particular, the formation of mDM-baryon bound states could increase the cooling rate from mDM scattering with hydrogen and helium at lower redshift.Since this effect would only enlarge the parameter space presented here, we leave an improved calculation of this effect for future work.We also neglect the effect of the drag forces between the baths; this will modify the temperature evolution, introducing an O(1) change in the required mDM charge for sufficient cooling.This effect does not modify the qualitative features of our solution, but it is accounted for in Fig. 2. A quantitative assessment of this effect can be found in Appendix C 1.
on our scenario is expected to lead to weaker bounds on the mDM-CDM interaction strength.Unlike the standard mDM scenario presented in Sec.III, the CMB constraints can be relaxed by reducing f m and/or choosing m m + m C m p : this is one of the key reasons why the range of viable mDM masses can be so large.In our numerical scans over m m , m C and f m we choose the largest possible coupling which controls the CDM-mDM cross section allowed by the CMB bound in Eq. ( 19).This choice of coupling also allows the mDM component to thermally couple to the CDM bath before z 17, which is important for efficient cooling of the baryons in most of the parameter space.Given the upper bound on α m α C from the CMB constraint and the minimal required value to recouple the mDM bath back to the CDM bath at low redshift, we can explore the allowed range of α m α C in the m m -Q plane, for a given CDM mass and mDM fraction.This is shown in Fig. 5 for f m = 10 −4 and m C = 100 MeV.The wide range of couplings over which we can explain the EDGES measurement shows that our scenario does not rely on any special fine-tuning between Q and α m α C in most of the parameter space.
BBN and CMB constraints on N eff set a lower bound on the masses of both mDM and CDM.For the couplings under consideration, both mDM and CDM are in thermal equilibrium with the SM at high redshift.Moreover, mDM and CDM annihilations into the light states mediating the long-range interactions between mDM and the baryons in Eq. (10) and between CDM and mDM in Eq. ( 14) would inject new rela-tivistic degrees of freedom, which are highly constrained by N eff limits set by the CMB and the primordial abundance of nuclei.
To avoid any modifications to BBN, we take both masses to be above 10 MeV.For mDM, the value of N eff resulting from annihilation into dark photons has been studied in detail in Ref. [34], leading to a stronger lower bound on the mDM mass (m m 400 MeV) shown in Fig. 2 as a light blue shaded region.This bound assumes g D = 0.1 where the mDM charge is given by Q = g D /g EM and is the mixing between the dark photon and the SM photon (see also footnote 1).This bound can be circumvented in the case of a pure millicharged DM without a dark photon, where the mDM annihilates only into SM photons and no extra degrees of freedom is introduced.For CDM, both BBN and N eff limits are model dependent; in the most conservative scenario, a mass of m C 400 MeV is sufficiently large for CDM to freeze out early enough to avoid any significant contribution to N eff .To access lower masses, a less minimal model of CDM freeze-out should be explored.
Throughout this paper we have neglected heating of any of the three fluids from the annihilation of dark sector particles.Although previous work has shown that the annihilation of millicharged DM can have a non-negligible heating effect on the baryon temperature [11][12][13], the annihilation of mDM is suppressed by f 2 m , and the value of f m examined here is small enough that no appreciable heating of the baryons is anticipated.Dark sector annihilations into other dark sector particles are also unlikely to have any effect on the temperature of the dark sector fluids.We refer the reader to Appendix C 2 for more details on this topic.

D. Neutral Hydrogen/Helium Scattering
With the range of viable mDM masses extended to 200 GeV, interactions between mDM and neutral hydrogen and helium become important to consider.Having an mDM mass heavier than the proton mass leads to a higher momentum transfer in the mDM-gas collision compared to the standard mDM setup where the bound in Eq. ( 13) applies.This enhances the mDM scattering with hydrogen and helium when the momentum transfer is larger than the inverse Bohr radius, resulting in an unscreened interaction with the nuclei of these atoms at energies much smaller than their ionization energies.In fact, the momentum transfer to helium atoms is comparable to that of hydrogen atoms despite the smaller abundance of helium, due to the larger nuclear charge and more diffuse electron wavefunction.The mDM scattering rates with atoms have a non-trivial velocity dependence, exhibiting a different behavior from the interactions of mDM with the ionized fraction.Including these interactions in our numerical calculation leads to changes of order O(5) in the value of Q required to fit the EDGES result for mDM masses above 100 MeV.The relative importance of scattering with neutral (solid green line) vs. charged particles (solid orange line) for one particular choice of parameters is shown in the right panel of Fig. 3.
For the mDM charges considered here, the Born approximation breaks down at small enough velocities, and we expect resonance effects to enhance the low-redshift scattering of mDM with the atoms.The parametrics of when we expect this to happen is discussed in Appendix B 2, and corresponds to the dotted-gray line in Fig. 3.At correspondingly small redshifts, the naive estimation of the atomic-mDM interaction rate should not be trusted, and large corrections may exist.Since this effect is unlikely to change our results qualitatively, and will only enhance the mDM-baryon scattering rate, thereby enlarging the viable parameter space further, we defer a more complete computation of this effect to future work.For a detailed discussion of how mDM-neutral atom scattering is included in our calculation, see Appendix B.

V. PHENOMENOLOGY
The phenomenological consequences of our setup are very distinctive, and a large portion of the available parameter region will be probed in the near future.Beam dump and collider experiments can probe the region of parameter space with larger values of Q by directly producing mDM, while direct detection experiments are sensitive to lower values of Q for both mDM and CDM.The latter acquires a non-zero scattering cross section with the SM via a loop of mDM particles.

A. Millicharged Dark Matter Component
The constraints on the mDM component are summarized in Fig. 2, with the viable parameter space shown in white.Within that region the mDM charge, Q, is too small to be already excluded by present collider searches [35,37,87,88] and too large for direct detection experiments to be sensitive.The green region in Fig. 2 shows the most recent determination of direct detection experimental sensitivity to particles with large Q, based on Ref. [47].The sensitivity of direct detection constraints in the large Q region for mDM masses above 10 MeV goes down to fraction as small as f m = 10 −9 and therefore applies to the full parameter space considered here.Due to energy losses of mDM particles in the Earth overburden, the maximal reach for high-Q is achieved by surface runs of the various experiments.For charges above that, the energy of most mDM particles reaching the Earth surface is below the energy threshold of single-electron scattering experiments due to atmospheric attenuation.At masses above 1 GeV, we extrapolate the determination of Ref. [47] using their approximate analytic treatment of the energy losses6, which happens to reproduce the high-mass Monte Carlo results accurately.
Present collider and beam dump experiments can probe the existence of millicharged particles regardless of their cosmological abundance.CMS can set constraints on particles with charges above Q = 1/3 by looking for soft tracks created by their interactions with the tracking material [88].However, at lower Q, mDM pairs will depart the detector without interacting.The only way to constrain them would be to look at the missing energy given by their recoil against a hard object (either a photon or a jet) from initial state radiation.At the LHC, the reach of these search strategies is limited because of large backgrounds (see for instance the discussion in Ref. [38]) and improvements are only possible at future lepton colliders.
More interestingly, a robust search program for millicharged particles has been developed at beam dump experiments, following up on the first dedicated experiment on millicharged particles performed at SLAC [35] .The potential reach of the various proposed searches are summarized in Fig. 2.These can be broadly divided into three classes: ionization-based experiments [35,39,42,43], electron scattering experiments [40] and missing momentum experiments [41].The idea of most of these setups is to produce millicharged particles in SM particle collisions that are then detected downstream in scintillators [35,39], liquid argon detectors [43], or by exploiting their scattering with electrons [40].Alternatively, missing momentum experiments such as LDMX [41] can be sensitive to the emission of mDM pairs by measuring the time evolution of the momenta of electrons and muons as they pass through the detector.On short timescales, the ArgoNeut collaboration [91] can almost completely exclude the EDGES explanation for an mDM fraction of f m 10 −8 , assuming that the result in Ref. [43] is confirmed.In the near future, the installation of milliQan at CERN [92] can cover a large portion of the EDGES explanation up to a fraction f m 10 −6 .Larger mDM fractions (and therefore smaller electric charges) can also be probed in the future with the installation of the Fermini detector on the NuMI beamline [42], or in more ambitious future experimental facilities like DUNE and SHiP [40,43].
As seen in Fig. 2, the beam dump program has a limited reach for mDM charges smaller than ∼ 10 −3 and for mDM masses above 10 GeV.The direct detection program can in principle complement the collider effort in probing the mDM explanation of the EDGES anomaly completely.The main hurdle to overcome in this case is the mDM energy losses in the atmosphere.The atmosphere shielding can in principle be reduced by installing single-electron-threshold detectors on a balloon or on a satellite.In Fig. 2, we show the projected sensitivity for a SENSEI-like dark matter detector installed on a balloon/satellite, with a 0.1 gram-month exposure taken from Ref. [47].Constraints from previous rocket experiments like XQC [93] are not sensitive to the small mDM fractions required to fit EDGES and hence do not appear in the plot.We also do not include the exclusion from RRS [94] which has not been properly recast for mDM.Alternatively, the experimental threshold of direct detection experiments can be reduced such that they become sensitive to the very slow mDM particles reaching the surface of the Earth.Given the existing proposals for low-threshold experiments below single-electron recoil energy [48][49][50][51][52][53][54][55][56][57][58] and the flourishing ongoing activity in this direction, it would be interesting to study the ability of these experiments to probe the strongly interacting window for mDM.
Further constraints on the mDM parameter space can come from astrophysical probes.Two examples in this direction are the constraints on mDM-baryon scattering which can be deduced from the heating of very cold gas clouds in the interstellar medium [95,96] or in dwarf galaxies [97].These types of probes are affected by large systematic uncertainties, but have in principle the sensitivity to probe a large portion of the mDM window displayed in Fig. 2.Moreover, it is worth mentioning that the mDM component will also have non-trivial interactions with the magnetic fields in the interstellar medium and could also be accelerated in supernova shocks like regular cosmic rays [98].A quantitative assessment of these effects, their astrophysical systematics and their dependence on the details of the dark sector is then necessary to fully understand the direct detection prospects in the parameter space shown in Fig. 2 [99].
Finally, another interesting problem would be to determine further consequences of our setup at lower redshifts.An example would be to understand to what extent the CDM distribution in the Milky Way could be modified by the presence of a small fraction of mDM strongly interacting with the baryons.Indeed, in the absence of mDM-CDM interactions, the mDM would thermalize with the baryons and be dragged into the Milky Way as first noticed in Ref. [100].Adding mDM-CDM interactions can substantially alter the standard mDM picture, allowing mDM to be coupled back to the CDM halo.A quantitative assessment of the consequences of our framework for structure formation goes beyond the scope of this work and is left for future studies.

B. Cold Dark Matter Bath
In our setup, the CDM bath interacts with mDM via a longrange interaction.This implies the presence of long-range CDM self-interactions mediated by the same force carrier.Accounting for the most conservative constraints on CDM self-interaction from Refs.[32,102], we require We emphasize, however, that the robustness of such constraints is still under debate (see e.g.Ref. [30]).CDM-SM interactions are radiatively generated through a loop of mDM pairs.For a given choice of mDM mass, mDM fraction, and mDM-SM and mDM-CDM couplings (possibly to explain the EDGES result) one expects a minimal CDM-SM cross section.To illustrate this point, we adopt a simple benchmark model where a light vector boson mediates the mDM-CDM interactions.The direct detection cross section of CDM is induced by the mixing between the new vector boson and the SM photon generated by loops of mDM particles, and is proportional to Q 2 α C α m .For a choice of mDM mass, m m , and fraction f m , and fixing the value of g m g C to saturate the CMB constraint, the EDGES signal fixes Q, as shown in Fig. 2. Using this value, we get a prediction for the direct detection cross section of CDM.For an mDM sector containing two Dirac fermions with masses ∼ m ± , the direct detection cross section defined in Eq. (B3) reads where we neglect the log-enhancement of the Rutherford scattering.When m + m − we recover, the UV dependence of the cross section in the case of a single Dirac pair (see Appendix D).If instead we define ∆m m = (m + − m − )/2 and m m = (m + + m − ), we see that log(m + /m − ) ∼ ∆m m /m m in the limit ∆m m m m and the direct detection cross section is then suppressed as (∆m m /m m ) 2 for small splittings between the mDM Dirac pairs.The detailed derivation of the CDM cross section and its dependence on the structure of the mDM sector as well as on g m g C is discussed in Appendix D.
In Fig. 4 we show the CDM direct detection cross section for f m = 10 −4 with the maximal g m g C allowed by CMB and different choices of the mDM mass, with Q chosen as a function of m m and m C to account for the EDGES signal.On the left panel, we consider the simplest model, where the UV cut-off is fixed at m + = 10 TeV and m − ≡ m m .In this scenario, for m m masses larger than 100 MeV, the direct detection cross section of CDM is already excluded by Xenon-10 data [22].For mDM masses below 100 MeV, the direct detection of CDM cross section is within the reach of the future SENSEI with10 gram of detector material and 1 month of exposure.On the right panel, we show a scenario where ∆m m /m m 10 −2 , which is of  4. Prospects for the direct detection of the dominant CDM component that couples to mDM via a light vector and acquires a coupling to the baryons through a loop of mDM.Gray shaded regions show the current bounds from Xenon-10 recast for electron scattering [22] and the nuclear recoil bounds taken from Ref. [101].The vertical dotted lines at low masses indicate the ∆Neff constraints from BBN for a Dirac fermion and a complex scalar extracted from [101].All the sensitivities in the plot correspond to three signal events under the assumption of zero background.The dashed (dashed-dotted) purple line is the reach of SENSEI 10 g with 1 month exposure (100 g with 1 year exposure).The red dashed-dotted line gives the reach of DAMIC with 1 kg and 1 year of exposure.The thick blue lines are predictions for the direct detection cross section of the CDM component fitting EDGES for different mDM masses at fixed mDM fraction fm = 10 −4 and fixed gmgC to be the maximal coupling satisfying the CMB constraint in Eq. ( 19) (see App. C 2 for further details).On the left figure we assume a minimal model with a single light millicharged particle, where in Eq. ( 21) we fix m+ = 10 TeV and m− = mm (see Appendix D for details).On the right we show the case of a Dirac pair of millicharged particles with a mass splitting ∆mm/mm = 10 −2 .Correspondingly, σe is suppressed by (∆m/m) 2 = 10 −4 with respect to the naive log-enhanced expectation (see Appendix D for details).
the order of the splitting between charged and neutral pions.In this case, the direct detection cross section is suppressed by four orders of magnitude; all of the mDM masses below 10 GeV give a currently viable cross section that is within the reach of future low-threshold DM experiments.

VI. CONCLUSION
We present a new setup where millicharged DM-baryon scattering produces large effects that can be observed in 21-cm cosmology, without being constrained by any current cosmological, terrestrial or astrophysical probe.This work has been motivated by the recent EDGES measurement [20], which exhibits an anomalously large absorption peak in the 21-cm global spectrum at the cosmic dawn.In our model, the cooling capability of the millicharged DM (mDM) is enhanced by introducing a long-range interaction between the cold dark matter (CDM) component and the mDM fraction, thereby allowing the CDM component to act as a heat sink while only interacting weakly with the SM.As a consequence, and in contrast to previous attempts at explaining the EDGES result with mDM, the mDM parameter space extends to mDM masses as high as 200 GeV, with a dark matter energy density fraction as small as 10 −8 , thereby relaxing the existing bounds considerably.
Interestingly, fitting EDGES in our setup provides a surprising connection between a low-redshift cosmological signal and the direct detection of the CDM bath.We have shown how forthcoming direct detection experiments sensitive to electron recoils, such as SENSEI and DAMIC [25], will be able to probe a large portion of the CDM bath parameter space which explains the EDGES result.In particular, the viable EDGES parameter space for the simplest realization of our setup should be probed in the near future by SENSEI-10g and SENSEI-100g.
Beyond providing a potential explanation for the EDGES result, we have shown that the inclusion of the scattering of mDM with both hydrogen and helium in the Born approximation dominates the cooling rate for sufficiently heavy mDM, and should be included to correctly compute the 21-cm signal [2].Extending our treatment beyond the Born approximation and accounting for the effects of mDM-SM resonances that are expected to appear goes beyond the scope of this paper and is postponed to future work.Including this effect together with a more complete modelling of astrophysical effects at low redshifts [61,[67][68][69] would be necessary to better assess the predictions of our setup for the absorption peak shape at cosmic dawn.We hope to come back to this issue in the near future.
To present this in a more model-independent fashion, we introduce the cross section parametrization adopted by direct detection experiments [60]: where σ is a constant and F (q 2 ) is the interaction form factor, further decomposed into DM and SM contributions, written as f DM (q 2 ) and f SM (q 2 ) respectively.Using this notation together with the definition of σ T in Eq. (A6), and changing the order of integration so that we integrate first over v rel and then over q 2 , we arrive at the simple 1D integrals below: This is as far as we develop the formalism on model independent grounds; to proceed further, one should specify a model and calculate the form factor F , as we will do in the next section.For completeness, we give the expressions for the integrals above in the limit where V rel → 0 and V rel → ∞ in which only thermal motion and bulk motion matters respectively:

Appendix B: Millicharged Dark Matter Form Factors
We now provide the complete form factors needed to correctly compute the temperature evolution of both the baryons and the mDM after recombination.The baryonic bath after recombination is composed of 93% hydrogen and 7% helium by number density, with roughly 10 −4 of hydrogen being ionized in the form of H + and e − .The interactions of mDM with the SM bath will then depend on the behavior of the form factors of the different components.We begin by presenting the form factors derived using the Born approximation and then discuss the range of validity for this approximation.

Born Approximation a. Ionized Fraction
The differential mDM − I low energy scattering cross section with I = e, p is given by where µ I is the I-mDM reduced mass and m φ is the mediator mass, which we always take to be much smaller than the typical momentum exchange q.Using the direct detection experiments notation introduced in Ref. [60], we have where µ e is the e-mDM reduced mass.This allows us to write We can thus identify the DM form factor defined in Eq. (A17) as while the SM form factor is equal to one since we are dealing with point particles.

b. Atoms
To calculate the mDM-atom interaction, we will treat each atom as a positively charged point particle surrounded by a negative charge density, where ψ A,Z is the electron wave function which we assume to be hydrogen-like, and Z, A are the atomic number and mass respectively of the atom under consideration.Note that the total charge when integrated over all space vanishes, but the negative charge is distributed over the support of the wave function.Within the Born approximation, this charge density leads to a differential cross section which can be factorized into an interaction with a point particle of mass Am p and charge Z times a form factor.The latter is the Fourier transform of the squared density above, Here a eff is the effective Bohr radius, i.e. a 0 ≡ (α EM m e ) −1 for hydrogen and roughly a 0 /1.69 for helium [104].Since the mediators we consider are very light and will always obey a 0 m φ 1, we can safely set m φ = 0 for all mDM-atom interactions.

Beyond the Born Approximation a. Ionized Fraction
For scattering in a Yukawa potential, at low momentum transfer compared to the Yukawa potential energy, the Born approximation no longer faithfully describes the scattering process [84,102].As was further shown in Ref. [84], when the momentum transfer is also much larger than the Yukawa potential length-scale, the full non-perturbative quantum mechanical calculation reduces to the classical scattering calculation of Refs.[81,82].The authors of Refs.[81,82] calculated the center-of-mass scattering process for an attractive Yukawa potential and found that for Qα EM m φ µv 2 , the transfer cross section defined in Eq. (A5) is very precisely approximated by We have reproduced this calculation also for a repulsive potential and obtained the same result in the same limit.Comparing this with the Born approximation transfer cross section we see that the classical differential cross section can be obtained from the Born approximation through where e 2.71828 is Euler's number.From this result we can extract the classical form factor in Eqs.(A18) -(A19): The scattering of mDM with hydrogen-like atoms requires extra care.The classical central potential between an mDM particle and the Hydrogen-like charge density of Eq. (B5) is given by where the ± sign corresponds to a repulsive and an attractive interaction respectively, and a eff is the effective radius defined below Eq. (B7).When a particle is close enough to feel the potential, the above potential is of order Qα EM /a eff .If this potential energy is larger than the kinetic energy in the scattering process we expect resonances to appear; this happens whenever where µ m is the reduced mDM-atom mass.While the above inequality identifies when resonances are expected in the spectrum, by computing when the second term in the Born series becomes as important as the first, we get a self-consistency condition for the Born approximation to hold [105] This can be calculated analytically for the potential in Eq. (B13) and in the limit µ m v m a eff 1 reduces to During the cosmic dawn, the relative velocity between the mDM and the baryonic bath can be as low as v rel ∼ 10 −6 .As a consequence the Born approximation breaks down for a large portion of the allowed parameter space in Fig. 2. A typical example of when the Born approximation breaks down is indicated in Fig. 3. Including a full quantum computation of the scattering mDM-atoms would enhance the cooling rate at low redshift, further enlarging the parameter space presented in our study.We defer a detailed study of this effect for a future publication.

Appendix C: Details of the Framework
We now present the precise equations at the heart of this paper governing the evolution of all the relevant physical quantities in our model.We start by generalizing the thermal evolution equations presented in App.A to include three different fluids and study the different regimes.We discuss in particular how our new solution is parametrically connected to the standard mDM scenario.Lastly, we discuss several important constraints on these parameters, including a simple way of recasting CMB constraints for our model.

Three Fluid System and Parameter Space
We are interested in the thermodynamic evolution equation of three coupled fluids: CDM, mDM and baryons.For simplicity we will assume throughout that the different constituents of the baryonic fluid always maintain thermal equilibrium.The thermodynamic variables in that case are the temperature of each fluid T C , T m and T b , and two relative bulk velocities V bm and V mC .We assume for simplicity that the two relative velocities lie along the same direction, which is justified by our boundary conditions, which assume mDM to be tightly coupled to the baryonic fluid at high redshift.The thermodynamic evolution is then given by where Γ Comp is the Compton rate of interaction between photons and baryons, which is dominated by electrons.We have neglected the Compton heating term from T m : this is justified if the thermal decoupling from the CMB occurs before recombination at z ∼ 1100, so that after this redshift, the Compton heating rate is negligible compared to the adiabatic cooling rate.The redshift of thermal decoupling for mDM occurs at [12] ( where T γ,0 is the CMB temperature today.We therefore see that for thermal decoupling to occur at z 1100, we require which is always satisfied when the contours shown in Fig. 2 lie within the current experimentally allowed region.
Using the formalism already introduced in App.A we can write the above equations more explicitly in terms of a few simple 1D integrals: ) The index j runs over all the components of the baryonic bath that interact with the mDM, with m j and µ jm being the mass of the jth particle and the j-m reduced mass respectively.The quantities I T /D XX are the thermal averaged integrals defined in Eqs.(A18) and (A19), using the appropriate form factors from App. B; these quantities may also carry an index j for their dependence on u 2 j = T b /m j + T m /m m .The sixth equation accounts for the dynamics of the ionized fraction x e that is coupled to the other variables we solve for.The temperature of the CMB photons is assumed to be decoupled from matter for all practical purposes and redshifts as a relativistic fluid throughout.
It is instructive to study some limiting cases to get some intuition for the set of equations above.The mDM-CDM interaction is controlled by α C α m while the mDM-baryon interaction is controlled by Q 2 α 2 EM .In the limit where α C α m → 0, the equations above reduce to the two-fluid system presented in Appendix A and the analysis is the usual one performed in recent literature [7,8,10,16].Conversely, in the limit α C α m → ∞, mDM and CDM are tightly coupled and the equations reduce Contours of Q mm that produce sufficient cooling of baryons to satisfy the EDGES 99% CL limit for different values of RCm, defined in Eq. (C14), while also remaining consistent with the CMB bounds in Eq. ( 19).Here, fm = 10 −4 , mC = 100 MeV.We have also set Vmb = 0 initially, and show contours for both VmC = 29 km s −1 at z = 1010 (solid) and VmC = 0 (dotted).For all values of RCm and initial conditions, the contour shown in the region where mm < 1 MeV (blue) is allowed, corresponding to the case where the mDM fluid alone can cool the baryons sufficiently (at such small mm, VmC does not significantly affect the thermal history [2]).In the region 1 MeV mm 10 MeV, the mDM fluid does not have sufficient heat capacity to cool baryons alone, but also does not recouple to the CDM fluid sufficiently early.As a consequence, the CDM fluid is unable to assist in cooling.Our setup opens up the parameter space for mm 10 MeV, when the mDM fluid cools below the baryon temperature and equilibrates with the CDM fluid after recombination, allowing for an enhanced cooling of baryons.Contours representing parameter values for RCm = 10 −8 (green), 10 −4 (orange) and 1 (red) are shown, showing that a large range of αCαm is possible in our model.again to a two-fluid system.If we neglect the mass difference between m C and m m , the resulting equations are exactly those of 100% millicharged DM with an effective DM electric charge equal to √ f m Q.The first limit correspond to the standard mDM explanation of the EDGES measurement which has been shown to be challenged by a number of cosmological, astrophysical and ground based constraints [7,8,10,16,19].The second one is ruled out by the CMB constraints alone [16,18,86].
The intermediate scenario is the one of interest for this paper: • α C α m must be small enough to not be excluded by the CMB constraints on the CDM momentum drag on the baryonic fluid.
In App.C 2 we will show how the existing CMB bounds on mDM [16,18,86] can be extended with a simple rescaling to our scenario under the assumptions that mDM is thermally coupled to the baryons before recombination, while CDM is decoupled.We will work under this assumption to simplify the CMB analysis of our paper, deferring a thorough study of the exact CMB limits for future studies.In our treatment, the maximal value of the couplings, α C α m | max , must also ensure that mDM remains tightly coupled to baryons, as discussed later in App.C 2.
• α C α m must be large enough such that CDM drives the cooling of the baryonic bath necessary to explain the EDGES observation.α C α m is then fixed for a given value of Q, m m and m C .For the same choice of couplings, in many regions of the parameter space, mDM will couple back to the CDM bath after recombination (when x e drops rapidly).
These two conditions identify an interval for α C α m whose boundaries can be parametrized by the ratio R Cm = 0 corresponds to α C α m → 0 which brings us back to the standard mDM scenario, while R Cm = 1 corresponds to the boundary of the parameter space, where α C α m required to cool the baryons is equal to the maximum allowed value by CMB constraints in the tight coupling approximation between mDM and baryons.Fig. 5 shows contours of R Cm on the m m -Q plane with f m = 10 −4 and m C = 100 MeV.Notice that R Cm can also be used as a measure of the allowed model space of our construction, as we can see for f m = 10 −4 most of the parameter space allows for R Cm 1, which corresponds to a wide range of possible choices for α m α C to fit the EDGES signal.
• mDM tightly coupled to the baryons.When the mDM fraction is tightly coupled to the baryonic fluid before recombination it can effectively be considered as a component of the baryonic fluid at CMB.The CDM interaction with the baryons is then dominated by the CDM interactions with the mDM component.Since the CMB constrains mainly the drag exerted on the baryonic fluid, we want to compute the drag of CDM on the mDM component.For m m m p the relative fluid velocity is dominated by the relative bulk motion so that following the derivation of Eq. (A9) and (A20) one finds where σ T,mC is the transfer cross section for the CDM-mDM interaction.This result allows us to recast the constraint on the cross section from Ref. [18] as Notice that forecast constraints from CMB stage-4, according to Ref. [18], would improve this constraint only by a factor of 2. In terms of coupling constants, this reads In order for mDM to be tightly coupled to baryons before recombination as assumed in deriving Eq. (C17), we need the energy transfer rate between baryons and mDM to be significantly larger than the energy transfer rate between CDM and mDM.This gives an upper bound on α C α m for a given value of Q.In order to get an analytical approximation for this, we can observe from Eq. (C10) that T m ∼ T b if the heat transfer terms from the CDM fluid is much smaller than the heat transfer terms from the baryons with T m → 0; explicitly, this means we require where we have assumed that the energy transfer from CDM to mDM is dominated by the bulk relative velocity for simplicity.
Using the limiting expressions found in Eqs.(A20)-(A21) and neglecting log factors, we find For concreteness, we take the maximum value of α C α m for tight coupling between mDM and baryons to be This has been found numerically to be sufficient to ensure that T m ∼ T b at recombination.In reality, this is simply a heuristic bound to guarantee the accuracy of recasting of the CMB limits; a full analysis would likely relax our assumptions significantly.

b. Dark Matter Annihilation
Annihilation of either mDM or CDM particles can lead to energy injection into any of the three fluids we consider here.In what follows we argue that the effects of the various annihilation processes are always small in the parameter space of interest.
• mDM/CDM to SM Annihilations.These processes produce high-energy particles that ultimately heat up the baryons, possibly counteracting any cooling processes [12].To determine if this has a significant effect on the baryon temperature evolution, we can simply compare the energy injection rate with the cooling due to adiabatic expansion.
For mDM annihilations into SM particles, we can take the annihilation cross section to be σv ∼ Q 2 α 2 EM /m 2 m .The energy injected per baryon per unit time is given by fields to be fermions for concreteness, but we do not expect our arguments to be modified in the scalar case.In the simplest model, we can write Notice that in this simple case there is a residual symmetry acting on the fermions and the gauge fields under which the full Lagrangian is invariant: Computing a loop of χ's we get the natural size of the mixing parameter κ κ(q 2 ) = Qq m eg V 4π 2 1 0 dx 3x(x − 1) log Λ 2 m 2 m + q 2 (x − 1)x where we expanded at low momenta with respect to the mDM masses and the mixing is logarithmically sensitive to the cut-off Λ.In this theory, the direct detection cross section of CDM with electrons reads (up to O(1) logarithmic terms) where we defined For the region of couplings required to fit the EDGES result, this cross section is within the reach of present/near-future direct detection experiments as we show in the left panel Fig. 4 in Sec.V.
Considering the next-to-minimal model we can see that the mixing between CDM and the SM can be arbitrarily suppressed via an accidental symmetry acting on the fermions and the gauge fields.The model introduces two Dirac-like pairs χ ± -χ± in the mDM sector which have opposite charges ±q m under the dark gauge field V .The vector currents and the gauge invariant masses are in this case: In this theory there is now an accidental symmetry under which only the mixing operator F F V is odd: This symmetry is only softly broken by the mass difference between the two mDM Dirac-like pairs ∆m m ≡ m − − m + . (D13) For ∆m m = 0, this symmetry is exact so we should find κ = 0.By explicit computation we find indeed κ(q 2 ) = Qq m eg V 4π 2 We can therefore arbitrarily suppress the direct detection cross section of CDM by requiring a small splitting in between the dirac-like pairs of mDM.In the case of m + m − we recover the result in Eq. (D7) if we identify the cut-off Λ with m + .In the left/right panel of Fig. 4 in Sec.V, we show the prospects for discovering the CDM bath that explains EDGES for m + = 10 TeV/(∆m m /m) = 10 −2 .The direct detection cross section is fixed there by choosing α m α C equal to its maximal value allowed by CMB.In Fig. 7 we show for a fixed choice of m m = 100 MeV how much the direct detection cross section changes by lowering the value of α m α C .Since a smaller α m α C should always be compensated by a higher Q, the direct detection cross section does not change by more than two orders of magnitude as shown in Fig. 7.
arXiv:1908.06986v1 [hep-ph] 19 Aug 2019 BARYONS Millicharged-DM Qe < l a t e x i t s h a 1 _ b a s e 6 4 = " i h 3 V 9 S B k C x K x T K k l R s X a Z M a A M N k = " > A A A C A n i c b V D L S g N B E J y N r x h f U Y 9 e B o P g Q c K u C H o M e v G Y i H l A s o T Z 2 d 5 k y M z s M j M r h C U 3 j 1 7 1 I 7 y J V 3 / E b / A n n C R 7M I k F D U V V N 9 1 d Q c K Z N q 7 7 7 R T W 1 j c 2 t 4 r b p Z 3 d v f 2 D 8 u F R S 8 e p o t C k M Y 9 V J y A a O J P Q N M x w 6 C Q K i A g 4 t I P R 3 d R v P 4 H S L J a P Z p y A L 8 h A s o h R Y q z 0 0 I B + u e J W 3 R n w K v F y U k E 5 6 v 3 y T y + M a S p A G s q J 1 l 3 P T Y y f E W U Y 5 T A p 9 V I N C a E j M o C u p Z I I 0 H 4 2 u 3 S C z 6 w S 4 i h W t q T B M / X v R E a E 1 m M R 2 E 5 B z F A v e 1 P x P 6 + b m u j G z 5 h M U g O S z h d F K c c m x t O 3 c c g U U M P H l h C q m L 0 V 0 y F R h B o b z s K W I B A X 2 j 4 2 h H B S s u l 4 y 1 m s k t Z l 1 b O 8 c V W p 3 e Y 5 F d E J O k X n y E P X q I b u U R 0 1 E U U R e k G v6 M 1 5 d t 6 d D + d z 3 l p w 8 p l j t A D n 6 x c K S Z e S < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "i h 3 V 9 S B k C x K x T K k l R s X a Z M a A M N k = " > A A A C A n i c b V D L S g N B E J y N r x h f U Y 9 e B o P g Q c K u C H o M e v G Y i H l A s o T Z 2 d 5 k y M z s M j M r h C U 3 j 1 7 1 I 7 y J V 3 / E b / A n n C R 7 M I k F D U V V N 9 1 d Q c K Z N q 7 7 7 R T W 1 j c 2 t 4 r b p Z 3 d v f 2 D 8 u F R S 8 e p o t C k M Y 9 V J y A a O J P Q N M x w 6 C Q K i A g 4 t I P R 3 d R v P 4 H S L J a P Z p y A L 8 h A s o h R Y q z 0 0 I B + u e J W 3 R n w K v F y U k E 5 6 v 3 y T y + M a S p A G s q J 1 l 3 P T Y y f E W U Y 5 T A p 9 V I N C a E j M o C u p Z I I 0 H 4 2 u 3 S C z 6 w S 4 i h W t q T B M / X v R E a E 1 m M R 2 E 5 B z F A v e 1 P x P 6 + b m u j G z 5 h M U g O S z h d F K c c m x t O 3 c c g U U M P H l h C q m L 0 V 0 y F R h B o b z s K W I B A X 2 j 4 2 h H B S s u l 4 y 1 m s k t Z l 1 b O 8 c V W p 3 e Y 5 F d E J O k X n y E P X q I b u U R 0 1 E U U R e k G v6 M 1 5 d t 6 d D + d z 3 l p w 8 p l j t A D n 6 x c K S Z e S < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "i h 3 V 9 S B k C x K x T K k l R s X a Z M a A M N k = " > A A A C A n i c b V D L S g N B E J y N r x h f U Y 9 e B o P g Q c K u C H o M e v G Y i H l A s o T Z 2 d 5 k y M z s M j M r h C U 3 j 1 7 1 I 7 y J V 3 / E b / A n n C R 7 M I k F D U V V N 9 1 d Q c K Z N q 7 7 7 R T W 1 j c 2 t 4 r b p Z 3 d v f 2 D 8 u F R S 8 e p o t C k M Y 9 V J y A a O J P Q N M x w 6 C Q K i A g 4 t I P R 3 d R v P 4 H S L J a P Z p y A L 8 h A s o h R Y q z 0 0 I B + u e J W 3 R n w K v F y U k E 5 6 v 3 y T y + M a S p A G s q J 1 l 3 P T Y y f E W U Y 5 T A p 9 V I N C a E j M o C u p Z I I 0 H 4 2 u 3 S C z 6 w S 4 i h W t q T B M / X v R E a E 1 m M R 2 E 5 B z F A v e 1 P x P 6 + b m u j G z 5 h M U g O S z h d F K c c m x t O 3 c c g U U M P H l h C q m L 0 V 0 y F R h B o b z s K W I B A X 2 j 4 2 h H B S s u l 4 y 1 m s k t Z l 1 b O 8 c V W p 3 e Y 5 F d E J O k X n y E P X q I b u U R 0 1 E U U R e k G v6 M 1 5 d t 6 d D + d z 3 l p w 8 p l j t A D n 6 x c K S Z e S < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " i h 3V 9 S B k C x K x T K k l R s X a Z M a A M N k = " > A A A C A n i c b V D L S g N B E J y N r x h f U Y 9 e B o P g Q c K u C H o M e v G Y i H l A s o T Z 2 d 5 k y M z s M j M r h C U 3 j 1 7 1 I 7 y J V 3 / E b / A n n C R 7 M I k F D U V V N 9 1 d Q c K Z N q 7 7 7 R T W 1 j c 2 t 4 r b p Z 3 d v f 2 D 8 u F R S 8 e p o t C k M Y 9 V J y A a O J P Q N M x w 6 C Q K i A g 4 t I P R 3 d R v P 4 H S L J a P Z p y A L 8 h A s o h R Y q z 0 0 I B + u e J W 3 R n w K v F y U k E 5 6 v 3 y T y + M a S p A G s q J 1 l 3 P T Y y f E W U Y 5 T A p 9 V I N C a E j M o C u p Z I I 0 H 4 2 u 3 S C z 6 w S 4 i h W t q T B M / X v R E a E 1 m M R 2 E 5 B z F A v e 1 P x P 6 + b m u j G z 5 h M U g O S z h d F K c c m x t O 3 c c g U U M P H l h C q m L 0 V 0 y F R h B o b z s K W I B A X 2 j 4 2 h H B S s u l 4 y 1 m s k t Z l 1 b O 8 c V W p 3 e Y 5 F d E J O k X n y E P X q I b u U R 0 1 E U U R e k G v6 M 1 5 d t 6 d D + d z 3 l p w 8 p l j t A D n 6 x c K S Z e S < / l a t e x i t > e < l a t e x i t s h a 1 _ b a s e 6 4 = " S 4 + a + H f a o u A l t o B A F v H W c u P n a 7 4 = " > A A A C A X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 S N g V Q Y 9 B L x 4 T M A 9 I l j A 7 2 0 m G z M 4 u M 7 N C W H L y 6 F U / w p t 4 9 U v 8 B n / C S b I H k 1 j Q U F R 1 0 9 0 V J I J r 4 7 r f z t r 6 x u b W d m y 2 w 6 d T M L M j b S E / I F L t / o R 7 s S t P + E 3 + B N O 2 y x s 6 4 E L h 3 P u 5 d 5 7 / F h w j Y 7 z b a 2 s r q 1 v b B a 2 i t s 7 u 3 v 7 p Y P D h o 4 S x a D O I h G p l k 8 1 C C 6 h j h w F t G I F N P Q F N P 3 h 7 c R v P o H S P J I P O I 7 B C 2 l f 8 h 5 n F I 3 0 O

FIG. 1 .
FIG.1.The structure of the dark sector studied in this paper.A fraction of millicharged DM (mDM), fm, which carries electromagnetic charge Q, interacts with the baryonic bath which is composed of hydrogen (H), helium (He) with fraction xH and xHe respectively and an ionized fraction xe composed of e − , H + .The latter goes from being order one at CMB to 10 −4 at z 17.The mDM also interacts with the rest of the cold DM (CDM) through a new dark long-range force, which couples with strength gm to mDM and gC to CDM.The same long-range force induces CDM self-interactions.
FIG.2.Parameter space of the scenario described in Fig.1in the plane (mm, Q) where we fix mC = 10 MeV to maximize the heat capacity of the CDM bath and the maximal αmαC allowed by CMB bounds[14][15][16][17][18].The dark blue contours give the mDM fraction fm required for a given (mm, Q) point to fit the upper value of the 99% CL interval of the EDGES measurement in the setup of Fig.1.For a fixed fm the entire region above the dark blue contour can be probed by reducing αmαC (see text for details).For comparison, the dark green contour shows the standard mDM case where 0.4% of mDM alone provides the baryonic cooling.The light blue region for mm < 10 MeV is robustly excluded by BBN contraints on Neff[7,8,10,19], the two dotted lines distinguish between the case in which mDM is a scalar or a Dirac fermion.The gray shaded area is a collection of different constraints taken from Refs.[37,87], plus limits on millicharge particles from milliQ at SLAC[35], searches for low ionizing particles in CMS at the LHC[88] and the new constraints from LSND and MiniBooNE derived in Ref.[40].The region on the left of the blue line is excluded by CMB constraints on Neff only when mDM couples to a dark photon with coupling gD = 0.1.The green region is excluded by present direct detection experiments as shown in Ref.[47].The green dashed line indicates our extrapolation of the results in Ref.[47] to higher masses (see discussion in the text).The red/black/magenta lines indicate the Fermilab/SLAC/CERN effort to probe mDM.Solid/dashed/dotted lines give a rough sense of the short/medium/long time scale of the proposal.Solid red is the ArgoNeut sensitivity derived in Ref.[43], dashed red is the sensitivity of the Fermini proposal at NuMI[42] (see Ref.[43] for a more conservative reach based on ArgoNeut at NuMI), dotted red is the DUNE reach[40] while dotted black is the LDMX reach[41].Dashed magenta is the milliQan reach as[38] while dotted magenta is the SHiP sensitivity[40].The dash-dotted/dotted green lines indicate the reach of a SENSEI-like dark matter detector on a balloon/satellite with 0.1 gram-month exposure[47].

FIG. 3 .
FIG.3.Temperature evolution (left) together with ionization evolution and heating/cooling rates Ṫb/HTb of relevant processes (right) for benchmark parameters given by Q = 6 × 10 −4 , mm = 2 GeV and fm = 10 −4 , with mC = 100 MeV and αmαC = 4 × 10 −16 setting the interactions between mDM and CDM.At z 3000, the mDM temperature is close to the baryon temperature while the CDM bath is significantly colder.As the free electron fraction xe drops, the mDM decouples from the baryons, thermalizing with the CDM while cooling the baryonic gas.These parameters are consistent with known experimental constraints on Q, as well as CMB constraints on gmgC.They also produce sufficient cooling to be consistent with the upper limit of the EDGES 99% CL (arrow, black).For the range of mDM masses considered here, the cooling of the baryons from both scattering with the ionized fraction (solid, orange) and with hydrogen and helium atoms (solid, green) are important.At low redshifts (i.e.low relative velocities), the Born approximation for scattering between mDM and atoms breaks down (gray, dotted).In particular, the formation of mDM-baryon bound states could increase the cooling rate from mDM scattering with hydrogen and helium at lower redshift.Since this effect would only enlarge the parameter space presented here, we leave an improved calculation of this effect for future work.We also neglect the effect of the drag forces between the baths; this will modify the temperature evolution, introducing an O(1) change in the required mDM charge for sufficient cooling.This effect does not modify the qualitative features of our solution, but it is accounted for in Fig.2.A quantitative assessment of this effect can be found in Appendix C 1.
FIG.4.Prospects for the direct detection of the dominant CDM component that couples to mDM via a light vector and acquires a coupling to the baryons through a loop of mDM.Gray shaded regions show the current bounds from Xenon-10 recast for electron scattering[22] and the nuclear recoil bounds taken from Ref.[101].The vertical dotted lines at low masses indicate the ∆Neff constraints from BBN for a Dirac fermion and a complex scalar extracted from[101].All the sensitivities in the plot correspond to three signal events under the assumption of zero background.The dashed (dashed-dotted) purple line is the reach of SENSEI 10 g with 1 month exposure (100 g with 1 year exposure).The red dashed-dotted line gives the reach of DAMIC with 1 kg and 1 year of exposure.The thick blue lines are predictions for the direct detection cross section of the CDM component fitting EDGES for different mDM masses at fixed mDM fraction fm = 10 −4 and fixed gmgC to be the maximal coupling satisfying the CMB constraint in Eq. (19) (see App. C 2 for further details).On the left figure we assume a minimal model with a single light millicharged particle, where in Eq. (21) we fix m+ = 10 TeV and m− = mm (see Appendix D for details).On the right we show the case of a Dirac pair of millicharged particles with a mass splitting ∆mm/mm = 10 −2 .Correspondingly, σe is suppressed by (∆m/m) 2 = 10 −4 with respect to the naive log-enhanced expectation (see Appendix D for details).

m m 3 ( 1 +
FIG.7.The thick blue band encloses the predictions for the direct detection cross section of the CDM component fitting EDGES for fixed mDM mass mm = 100 MeV and mDM fraction fm = 10 −4 lowering αmαC from its maximal value satisfying the CMB constraint down to the entire interval allowed by fitting EDGES (see Fig.5).The cross section does not change by more than 2 orders of magnitude by lowering the coupling up to eight orders of magnitude, showing the robustness of our direct detection prediction.Details about the present constraints and prospects can be found in the caption of Fig.4.