A composite solution to the EDGES anomaly
AA composite solution to the EDGES anomaly
Anubhav Mathur, Surjeet Rajendran, and Harikrishnan Ramani Department of Physics and Astronomy, Johns Hopkins University,3400 N. Charles St., Baltimore, MD 21218, USA Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA
Subcomponent millicharged dark matter that cools baryons via Coulomb interactions has been in-voked to explain the EDGES anomaly. However, this model is in severe tension with constraints fromcosmology and stellar emissions. In this work, we consider the consequences of these millichargedparticles existing in composite blobs. The relevant degrees of freedom at high temperature areminuscule elementary charges, which fuse at low temperatures to make up blobs of larger charge.These blobs serve as the degrees of freedom relevant in cooling the baryons sufficiently to accountfor the EDGES anomaly. In such a model, cosmology and stellar constraints (which involve high-temperature processes) apply only to the feebly-interacting elementary charges and not to the blobs.This salvages a large range of parameter space for millicharged blobs that can explain the EDGESanomaly. It also opens up new parameter space for direct detection, albeit at low momentumtransfers.
I. INTRODUCTION
The Experiment to Detect the Global Epoch of Re-ionization Signature (EDGES) experiment has reporteda dip in the 21 cm spectrum corresponding to strongabsorption around z = 17 [1]. This can be interpretedas a 3.8 σ deviation from the ΛCDM prediction for thebaryon temperature [2–4]. Dark matter (DM) coolingof the baryonic fluid has been invoked as an explana-tion for this excess [2–4]. A DM model that maximizesthe cross-section around cosmic dawn is sub-componentmillicharged dark matter (mCDM), which has a largercross-section with Standard Model (SM) charges at thelowest relative velocities. However, the millicharge pa-rameter space is extremely constrained due to limits fromCMB and BBN, cooling of SN1987A and stars, and ter-restrial experiments [4]. It has subsequently been shownthat even this limited parameter space results in over-production of mCDM through freeze-in [5, 6].These difficulties have led to two other ways to solvethe EDGES anomaly. The first involves heating theCMB relative to baryons [7–10], while the second in-volves mCDM which is tightly coupled to an additionalcold component that forms the dominant DM which doesthe bulk of the cooling [11]. In this paper, we pointto a third possibility. The mCDM explanations for theEDGES anomaly to date have treated the millichargedparticle (mCP) as elementary without internal structure.As a result, the same mCP is the physical particle at allenergies. In this work, we explore the consequence ofthis mCP being a composite state of elementary mCPswith much smaller mass and charge, glued together by aforce that confines at low temperatures. The elementarycharges are the relevant degrees of freedom at tempera-tures and energies much higher than cosmic dawn. As aresult, in our model, constraints from CMB, BBN, over-closure, stellar and SN cooling as well as colliders allapply only to the elementary charges. We demonstratehere that there is a drastic increase in new parameterspace for mCDM as long as it is in a composite state (blob). Furthermore, we explore the quirky thermal his-tory for the dark sector that involves confinement when T D (the dark temperature) falls below Λ D (the dark con-fining scale) and deconfinement if the dark temperatureincreases subsequently. We point out a novel dark phasewhere thermal contact with the SM results in a thermo-static dark bath, i.e. the dark bath staying at the sametemperature with the heat dump from baryons exactlycancelled by Hubble cooling. II. MODEL
We consider elementary mCP fermions that carry elec-tric charge (cid:15) f and mass m f with a confining force thatconfines at Λ D (cid:46) m f . We also assume that the mCPs arecharged under a dark U (1) with a dark charge g D . Thisdark U (1) accomplishes two goals: first, it allows theelectric charge of the mCP to be generated via kineticmixing and second, it provides Coulomb repulsion thatprevents blobs from getting too big. The mass of thedark photon is unimportant as long as it is sufficientlylong ranged to allow the blobs to interact with baryonsvia Coulomb scattering. We can take this mass to bezero, or low enough to evade direct stellar constraints onthe dark photon.For simplicity we will assume that these elementarymCPs that confine all have the same charge sign. Fornet neutrality, we envision an asymmetric dark “leptonic”component with the opposite charge that is not chargedunder this confining gauge group, just like the leptonsin the SM which are singlets under strong interactions.We further assume that the elementary charges confineto form composite blobs with “atomic number” A suchthat the composites χ carry charge (cid:15) χ = A(cid:15) f . The massof these blobs are roughly m χ = A ( m f − Λ D ) i.e. thedifference between the total mass of the constituents mi-nus the binding energy. Since m f (cid:38) Λ D , we approximatethis to m χ = Am f . These mCP blobs also have non-zero a r X i v : . [ h e p - ph ] F e b size [12] R blob = A Λ D (1)We describe the highlights of the phenomenology of thissetup in sub-section II A. These aspects of the phe-nomenology are quantitatively shown in sections II B andII C. A. Synopsis of Blob Evolution
We start by assuming that the temperature T D of theblobs is lower than the confining scale Λ D . Assumingwe are in the early universe where the baryon tempera-ture T b (cid:29) Λ D . We take the elementary mCPs to have alow enough charge that they are not in thermal equilib-rium in the early universe. The cold elementary mCPswill begin to fuse and form larger blobs. As the blobsbecome bigger, they will interact with the baryons andstart extracting energy from the standard model leadingto heating of the blob sector. But, the temperature ofthis sector cannot get larger than Λ D since the blobs arenot confined above this temperature. This implies thatonly a small fraction of the mCPs are actually able tofuse and become blobs in the early universe. Further,the maximum size of the blobs during this phase is setby Coulomb repulsion—as the blob becomes larger, therepulsion from the dark U (1) grows and it inhibits theability of elementary mCPs to fuse with the blob.This scenario continues until T b drops sufficiently sothat it is unable to transfer enough heat to inhibit blobgrowth. At this stage, there is rapid fusion of the mCPscreating blobs, resulting in most of the mCPs ending upin blobs. The maximum size of the blob in this case isalso set by Coulomb repulsion.Parameters are chosen so that this phase of blob forma-tion occurs around the redshifts of interest to the EDGESexperiment. At this stage, the blobs scatter with thebaryons, cooling the baryons and explaining the EDGESobservations. B. The Size of the Blob
It is necessary to limit the size of the blobs so thatthey can coherently scatter with the baryons, maximiz-ing the heat transfer between the two fluids. The dark U (1) provides the Coulomb repulsion necessary to enforcethis limit. Since g D (cid:29) (cid:15) f , we will ignore the Coulombrepulsion from electromagnetism in this section.How do the blobs form? We follow the prescription de-veloped in [13], but modify it to account for the Coulombrepulsion due to the dark U (1) [14]. Initially, individualpartons merge to form states with atomic number 2. Wecall this the { , } stage. This process can be inefficientsince it requires the emission of some other degree of freedom to make up for the reduced phase space for the2 → Unlike the case considered in [13], inour case as the blobs grow bigger, there is an increasedCoulomb barrier to fusion as treated in [14]. Considertwo cases: the fusion of two large blobs, which we refer toas { A, A } , or a large blob and a small blob which we referto as { A, } . The cross-section for fusion is proportionalto the geometric surface area of the blob. But, as theblobs become larger, this cross-section is suppressed bythe Coulomb barrier. Parametrically, this cross-sectioncan be expressed as: σ A,A (cid:48) = σ (Max( A, A (cid:48) )) P G ( T ) (2)where σ is the base fusion cross-section ∼ Λ − D and P G ( T ) is the Gamow factor [15] which is the temper-ature dependent factor that captures the effects of theCoulomb barrier. This factor is: P G ( T ) = e − G E = e − (cid:114) µQ Q TD (3)Here Q and Q are the charges A , × g D of the twoblobs respectively and µ is their reduced mass. Thus fu-sion freeze-out depends critically on the Gamow factor.From the Gamow factor, it is clear that { A, } fusion willdominate over { A, A } type fusion due to weaker Coulombrepulsion. Moreover, as the blobs grow in size, the num-ber density of larger blobs is lower than that of smallerblobs. Further the cross-section for a smaller blob tomerge with a larger blob is set by the geometric size ofthe larger blob. All of these factors imply that the growthof the blobs in our case is dominated by the mergers ofsmall blobs with larger blobs i.e. processes of type { A, } .Let us now see how big these blobs can get i.e. esti-mate the freeze out of the fusion process. It can easily beverified that for the parameters of interest to this paper,in the absence of the exponentially suppressed Gamowfactor, the rate of the fusion process is very rapid com-pared to Hubble. The size of the blob is then restrictedpurely by the exponential suppression from the Gamowfactor which forces the process to freeze out.Taking T D ≈ µ ≈ Λ D , for { A, } fusion, the Gamowexponent is G E ≈ Ag D (4)This places a bound on fusion growth, A limGamow ≈ g − D (5) The ratio of fusion rate to Hubble rate for { A, A } fusion is nσvH = 10 f D z A (cid:18)
10 KΛ D (cid:19) (cid:115) T D
10 K (cid:18)
10 K m f (cid:19) (cid:29) This limit on the blob size arising from the inhibitionof their growth is stronger than the stability limit [12] A limstab = 1 /g D that can be placed on their size due toCoulomb repulsion. Blob freeze out occurs only due tothe exponential dependence on blob size in the Gamowfactor. Thus, blobs whose sizes are close to, but smallerthan the Gamow limit are rapidly formed. This impliesthat as the universe expands and the temperatures drop,blobs will continue to grow until the Gamow limit isreached.It is also important to consider the heat that is releasedby the fusion process as the blobs grow. Each fusionprocess that occurs in { A, } fusion releases energy ∼ Λ D .Thus in the roughly ∼ A fusion processes that occur toform a blob of size A , approximately A Λ D energy into ∼ A particles is released. Thus the heat released in thefusion cannot change the temperature by more than Λ D and thus does not hinder fusion.While we expect this mechanism to produce a range ofblob masses, the charge to mass ratio of all these blobs isthe same. As seen later in Sec. IV, for most of the rele-vant parameter space, results depend only on the chargeto mass ratio, so it is justified to make a simplifying as-sumption that all blobs are of the same mass m χ . Notethat this analysis of the blob size is independent of thebaryon temperature T b . As we show in the following sec-tion, T b is an important parameter in determining thenumber of mCPs that are fused into blobs but it doesnot determine the maximum size of a blob. C. Heat Transfer
In order to understand heat transfer with the SM bath,we start by deriving the transfer cross-section for blobsand elementary charges to scatter with baryons.The differential cross-section for a mCP with charge (cid:15) to scatter with protons/electrons is [16], dσd cos θ = 2 π(cid:15) α µ v (1 − cos θ ) . (6)with µ the reduced mass and v rel the relative velocity.The forward divergence is cut off by the Debye massof the mediator. For the SM photon, the Debye masssquared is given by Π A = e (cid:18) x e n b T b (cid:19) (7)where x e ≡ n e /n H is the free-electron fraction, deter-mined using [17, 18]. The Debye mass is approximately10 − eV at z = 1000 and 3 × − eV at z = 10. TheDebye mass square of the dark photon isΠ A (cid:48) = g D (cid:18) n D T D (cid:19) (8)For the parameter space we are interested in, g D n D (cid:28) e x E n b and T D ≤ T b , such that Π A (cid:48) (cid:28) Π A . Hence we take only the SM photon Debye mass to regulate thedivergence.Finally, for elementary charges, q max = 2 µv rel , suchthat the θ integral is taken between the limits θ = {− , (cid:15)α √ Π A T b } . For blobs, q max ∼ Min (cid:0) R − , µv rel (cid:1) ,such that θ min = 1 − q µ v . The thermal-averaged trans-fer cross-section in the q (cid:29) Π A limit is given by inte-grating Eqn. 6 over θ , giving, σ T = 2 π(cid:15) α ξµ v (9)with ξ = ln (cid:16) T b π(cid:15) α x e n b q µ v (cid:17) . In the region of interest,it is safe to ignore the factor q µ v since it is inside thelog.Next, we compare the rate of charges scattering offbaryons to the Hubble rate, n b σ T v rel H ≈ − (cid:16) (cid:15) f − (cid:17) z (cid:18)
10 K T D (cid:19) (10)As a result, the elementary charges (cid:15) f which we take toobey stellar-cooling constraints discussed next in Eqn. 15,are never in thermal contact with the SM. We also seethat blobs with charge (cid:15) χ (cid:38) − can interact with theSM bath.At temperatures around T D ≈ Λ D , both elementarycharges and blobs can co-exist. Defining F blob ( z ) as thefraction of millicharges that are in the blob phase, we get˙ T ref D ( F blob ) = − HT D + 23 m χ x e ρ b ( m χ + m b ) F blob σ u χ,b × (cid:40)(cid:114) π ( T b − T D ) (cid:41) (11)Here u χ,b = (cid:113) T D m χ + T b m b is the average relative velocitydue to thermal motion and σ = σ T v . We have verifiedthat the bulk relative velocity between the χ bath and SMfluids does not contribute substantially to the thermalevolution of either fluid.When T D (cid:38) Λ D , the relevant degrees of freedom arethe elementary charges, which have no thermal contactwith the SM such that the dark fluid cools due to Hubbleexpansion. When T D drops below Λ D there is rapid blobformation. These blobs can now interact with the SM andheat up, but the temperature cannot exceed T D ; after all,thermal contact with the SM would immediately be lost.Consequently if the second term in Eqn. 11 dominatesfor F blob →
1, then F blob adjusts to smaller values so asto keep ˙ T D = 0. Thus, we set˙ T D = (cid:40) Min (cid:16) , ˙ T ref D ( F blob = 1) (cid:17) T D ≥ Λ D ˙ T ref D ( F blob = 1) T D < Λ D In the regime where ˙ T D = 0, we can solve for the z dependent fraction in blobs F blob by setting Eqn. 11 to0. We find for T D ≥ Λ D , F blob =Min (cid:32) , HT D × (cid:34) m χ x e ρ b ( m χ + m b ) σ u χ,b × (cid:40)(cid:114) π ( T b − T D ) (cid:41)(cid:35) − (12)We can see that in the limit where T b (cid:29) T D , and wheninteractions are strong enough, the quantity in squarebrackets is much larger than Hubble cooling and hence F blob →
0. This happens because in this limit, blobs thatform immediately break up into elementary charges. Asthe disparity between T b and T D shrinks, F blob → T b = − HT b + 23 m b x e ρ D ( m χ + m b ) F blob f D f He + x e σ u χ,b × (cid:40)(cid:114) π ( T D − T b ) (cid:41) + Γ C ( T CMB − T b ) (13)where f He ≡ n He /n H is the helium fraction and Γ C is theCompton scattering rate.The initial conditions we use are T b ( z = 1000) = T CMB ( z = 1000) ≈ T × T = 2 .
725 K T D ( z = 1000) = 0 K (14)Setting the initial dark temperature to 0 K is not phys-ical, but is accurate because the time evolution rapidlyadjusts the temperature to its correct value just below z = 1000. III. EXISTING LIMITS
As alluded to in the introduction, the constraints oncomposite mCPs can be quite different from elementarymCPs of the same charge. We elucidate further below.
Stellar bounds:
For Λ D (cid:28) (cid:15) f < − for small enough m f . The blobs arenever produced in stellar environments. However thelimit on the elementary charges translates to a limit onblob charge: (cid:15) χ < − m χ Λ D . (15) BBN and CMB N eff : As we have seen in theprevious section, when there is significant thermal contact with baryons and T b (cid:29) Λ D , F blob → (cid:15) f (cid:38) − (cid:0) m χ
10 K (cid:1) .This is more restrictive than stellar constraints onlywhen m f ≈ Λ D ≤ µ eV. Dark photons arising frombremsstrahlung and mesons from dark fusion are pro-duced at the temperature of the dark bath and hence donot contribute appreciably to N eff either. CMB power spectrum:
The effect of mCP scatter-ing on protons was investigated in [4], and constraintsfrom Planck 2015 data effectively ruled out mCPs as asolution to EDGES for f D > . (cid:15) χ m χ , they apply equally to blobs as well aselementary charges. However, it was found in [4] thatno limits exist for f D ≤ . IV. RESULTS
We now display results obtained by numerically solv-ing the coupled differential equations for time evolution.We consider a benchmark blob mass m χ = 1 MeV, andcharge (cid:15) χ = 4 × − and f D = 0 . F blob ( z ) for different Λ D in Fig. 1. For large z , heatfrom the baryonic bath disintegrates the blobs rendering F blob (cid:28)
1. For lower z , cooling due to Hubble expansionbegins to dominate, resulting in larger values of F blob .For the same z , we see that as Λ D is reduced, a smallerfraction of the dark bath exists in blobs as it is easier tobreak them apart.
10 50 100 500 100010 - - - Λ � = � � � � Λ � = � � Λ � = �� � Λ � = ��� � Λ � = ���� � � χ = � ���� ϵ χ = � × �� - � � � � = ��� % FIG. 1. The evolution of the fraction of the dark millichargedbath in blobs as a function of redshift is shown for differentΛ D , the dark confining scale. Smaller Λ D leads to smallerblob fractions. In Fig. 2, the time evolution of the baryonic temper-ature T b and the dark temperature T D are shown fordifferent choices of Λ D , the dark confining scale. TheCMB temperature T CMB and the baryon temperature T b in the absence of interacting DM are shown in black forreference. The colored solid lines correspond to differentchoices of Λ D . We see that models with smaller Λ D stayat the same temperature T D = Λ D for longer. If thesecharges were elementary, the dark temperature wouldbe higher than Λ D in this regime. Instead, for blobs,this is prevented by the rapid break up of blobs withthe resulting elementary charges losing thermal contactwith the SM, cooling rapidly, and forming blobs again.The dashed solid lines track the baryonic temperaturefor different Λ D . The baryonic temperature at z = 17is roughly constant for different Λ D . However, there isa small enough Λ D ≈ . F blob is toosmall even at z = 17, as seen in Fig. 1 and results inlower cooling of T b in Fig. 2.
10 50 100 500 10000.010.101101001000 ����� � � � � � � ( Λ � � � ) � � ( Λ � = � � � � � ) � � ( Λ � = � � � � ) � � ( Λ � = �� � ) � � ( Λ � = � � ) � � ( Λ � = ��� � ) � χ = � ���� ϵ χ = � × �� - � � � � = ��� % FIG. 2. Temperature evolution of the baryonic and DM bathare plotted as a function of redshift z . The CMB temperatureand the baryon temperature without DM are plotted in black.The solid lines track the evolution of the dark temperature T D for different Λ D , the dark confining temperature. Thedashed lines track the baryon temperature T b for differentΛ D with the same color code as T D . The error bar marks thebaryonic temperature at z = 17 as measured by the EDGEScollaboration. We next discuss the contours that explain EDGES inthe (cid:15) χ vs m χ plane and compare it to the parameterspace derived for elementary charges in [19]. Given aDM fraction, elementary charges that explain EDGESobey (cid:15) elem ∝ m elem as seen with the black curve. Thishappens due to the following reason. For a fixed DMfraction, a drop in T b , ∆ T b is associated with an increasein dark temperature ∆ T D ∝ m elem × ∆ T b , i.e. larger el-ementary masses m elem undergo larger temperature gainbecause of equipartition. Another way to see this is thatthe total energy gained is equal to n elem × ∆ T D and thenumber density is inversely proportional to m elem , andhence T D is directly proportional to m elem . Starting withan initially-cold dark bath T D (cid:28) T b , the proportionalityfactor ensures that T D ∝ m elem throughout. This in turnimplies that the elementary charges’ thermal velocity isindependent of the mCP mass. Finally, the heat trans-fer is proportional to the transfer cross-section given inEqn. 9, which is dependent only on the charge to mass ratio since the velocity is mass-independent. Thus, thisbehavior applies to very small masses. It was also pointedout in [19] that for a choice of DM fraction, there is also amaximum mass due to the same equipartition arguments, m elem ≤ µ b f elem Ω c / Ω b . The elementary charge requiredto explain EDGES obeys [19], (cid:15) elem ≈ × − m elem MeV (cid:18) − f elem (cid:19) . (16)It is important to note that the entirety of the elementarycharge solution is ruled out [6].Next we discuss the contours for blobs for different Λ D .In each case, we mark out the unphysical region wherethe elementary charges required to create blobs are ruledout by stellar constraints from Eqn. 15. For the samereason as explained for the elementary charge solution,we observe a linear relationship (cid:15) χ ∝ m χ for the blobs aswell. This linear regime obeys an approximate empiricalrelation (cid:15) χ ≈ − m χ MeV (cid:18) − f χ (cid:19) . . (17)Once again, there exists a cut-off mass, that is now Λ D dependent. For smaller Λ D , the mCP bath stays elemen-tary for longer, i.e. F blob (cid:28) m χ is required to increase heat capac-ity, so as to reach temperatures below Λ D sufficientlysoon. As a corollary, larger Λ D results in an enhancedrange in mass where the EDGES solution is viable. How-ever, larger Λ D translates to stricter stellar constraintsand for large enough Λ D , the charge required to explainEDGES, Eqn. 17 is ruled out by Eqn. 15.The blobs in most of the parameter space shown inFig. 3 do not survive galaxy formation. The parameterspace for which blobs do not break up in the galaxy isgiven in Eqn. 19 and can be recast as, m gal χ (cid:46)
86 eV Λ D D ≈
10 K, the blobs resize themselves tomasses below 1 keV, making prospects for direct detec-tion tricky.
V. CONCLUSION
Making mCDM inherently composite is a simple nu-ance with parallels in SM baryons. In this work, we haveconsidered this possibility and explored its myriad conse-quences with specific emphasis on explaining the EDGESanomaly.The DM degrees of freedom are blobs at temperaturesbelow the confining scale and elementary charges at tem-peratures above it. For an appropriately chosen confin-ing scale Λ D , the elementary charges are the degrees of - - - - - - - - - - ↑ ↑ � � � �� � � � � � �� � � � ↑ ↑↑ ↑ � � � �� � � � � � �� � � � ↑ ↑ Λ � = � � � � Λ � = � � � � � � � � � � � � � � � � � � � � � � � � � � [ � � � � � � � � � � ] � � = ���� % - - - - - - - - - - ↑ ↑ � � � �� � � � � � �� � � � ↑ ↑↑ ↑ � � � �� � � � � � �� � � � ↑ ↑↑ ↑ � � � �� � � � � � �� � � � ↑ ↑ Λ � = � � � � Λ � = � � Λ � = � � � � � � � � � � � � � � � � � � � � � � � � � � � [ � � � � � � � � � � ] � � = ��� % FIG. 3. The contours that explain the EDGES anomaly in the blob charge (cid:15) χ vs blob mass m χ plane are shown for differentchoices of the confining scale Λ D for mCP bath fractions of f D = 0 .
04% (left) and f D = 0 .
4% (right). Also shown are stellarcooling constraints from Eqn. 15. The elementary charge solution from [19] is shown in black. freedom during BBN, CMB and in the interior of stars.The elementary charges are chosen to be feeble enoughto evade all these constraints. However, at temperaturesbelow the confining scale, these rapidly fuse into blobsincreasing in size till they reach a size determined bystability considerations due to repulsion. These blobsnow have large enough charges that coherently scatterwith baryons at temperatures around z = 17, relevantfor physics during the dark ages, without suffering fromthe strict stellar and cosmology constraints that apply toelementary mCPs. Thus, we find a large unconstrainedparameter space for mCP blobs for f D ≤ . v vir ≈ − . Self- interactions are large enough to break up the blobs oncemore if the kinetic energy exceeds the confining scale.Thus the blobs stay intact till today only if m χ v (cid:46) Λ D . (19)Hence, for large enough blob masses m χ , there is sig-nificant fission in galaxies, the blobs are resized intosmaller ones that obey Eqn. (19) which are present inthe galaxy today. These smaller blobs should neverthe-less be present in the galaxy today since the dark pho-ton sets the range for self-interactions [25] and cuts offlong-range galactic processes such as evacuation from thegalactic disk [26, 27] and retention in galactic magneticfields [28], and prevents the mCP from being blown awayby the solar wind [27, 29].This parameter space increases the scope of directdetection experiments sensitive to masses lower than1 MeV, albeit at momentum transfers smaller than R − to retain coherence. Experiments such as SENSEI [30],DAMIC [31], super-CDMS [32], and even future pro-posals [33–36] are not sensitive to momentum transfers q ≤ Λ D ≈ meV. Instead, manipulation with electric andmagnetic fields [37] is a promising detection strategy. Forlarge enough blob charge, terrestrial accumulation andsubsequent detection [38] might be a viable avenue. ACKNOWLEDGMENTS
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