Hydrogen Mixing as a Novel Mechanism for Colder Baryons in 21 cm Cosmology
HHydrogen Mixing as a Novel Mechanism for Colder Baryons in 21 cm Cosmology
Lucas Johns
1, 2, ∗ and Seth Koren
3, 4, † Departments of Astronomy and Physics, University of California, Berkeley, CA 94720, U.S.A. Department of Physics, University of California, San Diego, CA 92093, U.S.A. Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, U.S.A. Department of Physics, University of California, Santa Barbara, CA 93106, U.S.A.
The anomalous 21 cm absorption feature reported by EDGES has galvanized the study of scenariosin which dark matter (DM) siphons off thermal energy from the Standard Model (SM) gas. In adeparture from the much-discussed models that achieve cooling by DM scattering directly with SMparticles, we show that the same end can be achieved through neutral atomic hydrogen H mixingwith a degenerate dark sector state H (cid:48) . An analysis of in-medium H – H (cid:48) oscillations reveals viableparameter space for generic types of H (cid:48) –DM interactions to provide the requisite cooling. Thisstrategy stands in stark contrast to other proposals in many respects, including its cosmologicaldynamics, model building implications, and complementary observational signatures. Before the first stars cast their light, the cosmos wasa dim but dynamic place. The long period known as thecosmic dark ages saw the formation of halos and galaxiesas the universe sowed the seeds of its later illumination.But while dark, the era was not pitch black. Photonsfrom the cosmic microwave background (CMB) streamedeverywhere, interacting with the hyperfine transition ofhydrogen and reaching us, today, as messengers.In 2018 the Experiment to Detect the Global Epoch ofReionization Signature (EDGES) announced a detectionof 21 cm hyperfine absorption at redshifts 15 (cid:46) z (cid:46) . σ [9, 10]. If this is correct, theEDGES measurement demands a reassessment of thephysics of the dark ages.An enticing explanation is that the gas lost some of itsthermal energy to dark matter (DM) in the lead-up tocosmic dawn [1, 10]. In this paper we adopt that interpre-tation and develop the hydrogen mixing portal —mixingbetween neutral atomic hydrogen H and a dark sectorstate H (cid:48) —as a new mechanism for transferring heat fromthe Standard Model (SM) sector to the dark one.The mechanism stands in contrast to the well-knownproposal that the cooling was facilitated by scatteringbetween DM and SM particles [9–17]. Although modelsof the latter type are tenable, the most elegant of themare strongly constrained by fifth force experiments, cos-mology, and other observations. In thinking about howelse the gas might be cooled by new particle physics, weare guided by the hint that the EDGES anomaly appearsin connection with neutral hydrogen. One of the specialattributes of a neutral particle is that it can oscillate intoa dark state without violating SM gauge symmetry. Inthis sense, H – H (cid:48) mixing is in kinship with, for example,the neutrino portal (between SM and sterile neutrinos)and the kinetic mixing portal (between SM and dark pho-tons).To get a sense of the relevant energy scale of mixing,we make note of the following. For a hydrogen atom in the late dark ages to undergo ∼ δ mustbe comparable to the hydrogen scattering rate Γ H at therelevant redshifts. For z = 30, this condition translatesto δ ∼ O (10 − ) GeV.It is immediately apparent that H – H (cid:48) mixing isexquisitely sensitive to any physics that distinguishesthe two states. This means that hydrogen oscillationsare not likely to show up in any terrestrial experiment—interactions easily swamp the mixing term—but it alsoindicates that even a tiny mass splitting can throttle thehydrogen portal. We therefore assume that H and H (cid:48) areexactly degenerate. In a companion paper [18] we discuss H (cid:48) as mirror hydrogen , a composite particle related to H by an exact Z symmetry. While we mention that exam-ple a couple times below, we note that all the cosmologywe discuss herein is independent of the identity of H (cid:48) past its mass degeneracy with H .How exactly does mixing bring about cooling of thegas? For a single atom, the sequence can be thought ofas occurring in steps: H ( p ) osc −−→ H (cid:48) ( p ) H (cid:48) – X −−−−→ H (cid:48) ( p (cid:48) ) osc −−→ H ( p (cid:48) ) . (1)A SM hydrogen atom with momentum p oscillates intomirror hydrogen, scatters with DM (which we call X ),and oscillates back into SM hydrogen, now possessing asmaller momentum p (cid:48) .Figure 1 shows a conservative parameter space inwhich this sequence thermally equilibrates H (cid:48) and X and cools the gas to the temperature inferred from theEDGES measurement. The free parameters are δ andthe momentum-transfer cross section ¯ σ H (cid:48) − X . Text la-bels in Fig. 1 summarize the major considerations thatare explicated below. We observe that the viable rangein δ is indeed at roughly the scale estimated a few para-graphs up. While a careful analysis of the mixing dynam-ics is of course more nuanced than the simple estimate,their agreement indicates that the preferred region of δ is picked out fairly straightforwardly by cosmology. Co-incidentally, the region is also at an interesting scale for a r X i v : . [ h e p - ph ] D ec - - - - - - - - - - - -
10 Log δ GeV L og σ - H ' - X ( z = ) σ H - H n = n = - Figure 1. Viable parameter space for cooling through H – H (cid:48) mixing, adopting H (cid:48) – X cross section ¯ σ H (cid:48) − X = σ v nm with n =0 (blue) and n = − σ H − H = 4 πa . Consistency with EDGES( Cooling ) implies lower limits on ¯ σ H (cid:48) − X for both cases shown.Indirect H – X coupling at z (cid:38)
200 (
CMB ) generally bounds δ from above but leaves open a swath for n = −
4. Other labeledregions are excluded for reasons described in the main text.This analysis assumes that H (cid:48) and X thermally equilibrate,from which it follows that m X ∼ high energy physics. In the case where H (cid:48) is identifiedwith mirror hydrogen in a Z -symmetric model, we mayestimate [18] δ ∼ (4 π ) Λ Λ a , (2)where Λ is the ultraviolet scale at which the hydrogen-mixing interactions are generated and the factors of theBohr radius a arise from the wavefunction overlap be-tween the electron and proton. One finds automaticallythat δ ∼ − GeV arises from new physics appearingat a scale Λ ∼
100 GeV − z ∼ H (cid:48) density is vanishingly small. Oscillations begin once neu-tral atoms form and the universe enters the dark ages,but efficient cooling through H (cid:48) – X scattering must waitat least until z ∼ H and H (cid:48) . With the hydro-gen species in chemical ( i.e. , mixing) equilibrium and H (cid:48) and X in thermal equilibrium, the SM hydrogen densitydilutes by half ( n H = n H (cid:48) ) and all three temperaturesreach a common value ( T H = T H (cid:48) = T X ). We restrictour analysis to the regime where this is fully attained bycosmic dawn.The timeline also marks z ∼
17, the redshift at whichthe EDGES signal reaches its maximum amplitude andastrophysical sources begin to significantly heat the gas.At this stage, the SM sector still only contains half thebaryons it had at recombination. It would be flagrantlyunacceptable for this to remain true down to the presentday: the cosmic baryon density η is known to high preci-sion, and more than 50% of it is accounted for by low- z observations [19–23]. A large part of the H (cid:48) populationthat is produced must ultimately return to our own sec-tor if the scenario is not to run afoul of the low- z baryontally. This reconversion occurs naturally if mixing is ef-ficient during the epoch of reionization ( z ∼ n H (cid:48) to track n H (the neutral density) as it declines.Other versions of the timeline are conceivable—DMmight never fully thermalize with H (cid:48) , for example—butwe focus only on the one just described, which is thesimplest of them.The H – H (cid:48) density matrix ρ evolves according to thequantum kinetic equation i dρdt = [ H , ρ ] + i C , (3)where H is the Hamiltonian and C is the collision term.Since H and H (cid:48) are degenerate in vacuum, the Hamilto-nian is H = (cid:18) ∆ V δδ − ∆ V (cid:19) . (4)The potential ∆ V = ( V H − V H (cid:48) ) / k experiences an index of refraction n = 1 + (cid:88) i πn i k f i (0) , (5)where n i and f i (0) are the target number density andelastic forward scattering amplitude associated with pro-cess i .We parametrize the mixing in terms of an in-mediumoscillation frequency ω m = (∆ V + δ ) / and an in-medium mixing angle given by sin θ m = δ /ω m . Equi-libration of the mixing channel then occurs on a timescaleΓ − , with Γ osc ∼ Γ c θ m c / ω m ) , (6) H’ mostlyreconvertedNegligible H’ density SM gasthermallycoupled to CMB Mixingequilibrium,DM cooling Gas heated byastrophysicalsources z ~ 1100 z ~ 200 z ~ 17 z ~ 7Dark Ages Cosmic Dawn ReionizationRecombination Figure 2. Cosmic timeline schematically marking the significant epochs in the proposed scenario. Sometime between 20 (cid:46) z (cid:46) H – H (cid:48) mixing comes into (or continues to be in) equilibrium and both species are cooled by interactions between H (cid:48) anddark matter (DM). Mixing equilibrium is maintained during the epoch of reionization, and the H (cid:48) population depletes alongwith the neutral abundance of SM hydrogen. where Γ c = Γ H + Γ H (cid:48) is the sum of the H and H (cid:48) scat-tering rates. Eq. (6) is often used in the literature onsterile neutrino dark matter, going back to Refs. [24–26]and others. It follows quite immediately from applying arelaxation-time approximation to the evolution of ρ [27].During the dark ages, the most important interactionfor mixing is H – H scattering. It gives a good approxi-mation of the total collisional rate of hydrogen:Γ H ∼ n H (cid:104) σv (cid:105) H − H ∼ (cid:0) × − GeV (cid:1) (1 + z ) . (7)The same is true of the forward-scattering potential: V H ∼ πn H a m H ∼ (cid:0) × − GeV (cid:1) (1 + z ) . (8)In the equations above, (cid:104) σv (cid:105) is the velocity-averaged H – H cross section and a is the Bohr radius. Contribu-tions from other processes are considered in Ref. [18] andshown to be subdominant. For the sake of completeness,we include them in the results presented here.Figure 3 presents the evolution over redshift of Γ osc for three different values of δ . Feedback on the mixingfrom H (cid:48) – H (cid:48) scattering—an unimportant effect for thepurposes of estimating timescales—is not included. Moresignificantly, the figure assumes a scenario of minimalmixing , by which we mean that H (cid:48) – X scattering does notsubstantially change the mixing dynamics. Although thisstipulation is not a constraint per se, increasing ¯ σ H (cid:48) − X above the region of minimal mixing (and into the regionlabeled with that name in Fig. 1) makes oscillations lesseffective.The curves in Fig. 3 peak at ∆ V ( z ) ∼ δ , transitioningfrom a mixing regime in which the medium suppresses H – H (cid:48) conversion (high redshift) to a regime in whichthe conversion rate is independent of δ (low redshift).From Eqs. (6), (7), and (8), we see that Γ osc scales like z − on the high- z side of the peak, being suppressed notonly by sin θ m but by the quantum Zeno effect, i.e. , the
50 100 20010 - - - - - - Γ osc , H [ GeV ] z Figure 3. Hubble rate H (dotted) compared with the mix-ing equilibration rate Γ osc at δ = 5 × − GeV (thin),5 × − GeV (medium), and 5 × − GeV (thick). Γ osc captures the effects of forward scattering and quantum deco-herence on vacuum oscillations. The dominant interaction is H – H scattering. prevention of quantum coherence development by rapidscattering. On the low- z side of the peak, Γ osc drops offlike z .For the hydrogen portal to be consistent with cos-mology, it must also be in equilibrium as the universereionizes. Rather than modeling the radiation field as itevolves over the epochs of cosmic dawn and reionization,we choose z = 7 as a representative redshift and assumethat there are O (10) ionizing photons per baryon at thistime. While the influence of photons on ∆ V continuesto be marginal, photoionization greatly enhances Γ c be-cause of its large cross section σ I ∼ σ T near threshold.( σ T is the Thomson cross section.)Figure 4 displays the ratios Γ osc /H and δ/H at reion-ization ( z = 7) as functions of δ . (In calculating Γ osc , weimplicitly adopt a cosmology in which intergalactic mag- - - - - δ ≫ Γ c , | Δ V | Γ c ≫ δ ≫ | Δ V | δ [ GeV ] Figure 4. Comparison of rates during the epoch of reioniza-tion: Γ osc /H (blue) and δ/H (purple) evaluated at z = 7.Dashed lines show the limiting behaviors. At large values of δ , Γ osc ∼ Γ c /
4. At smaller values, Γ osc ∼ δ / Γ c . Within theparameter space of Fig. 1, the model is in the quantum Zenoregime, but mixing remains efficient on the Hubble timescale. netic fields originate from astrophysical processes and donot strongly affect H – H (cid:48) mixing until after reionization.)Mixing is suppressed by the quantum Zeno effect for δ (cid:46) Γ c , thus implying a minimum δ such that Γ osc (cid:38) H at reionization. That cutoff is shown in Fig. 1 as the bor-der of the region marked Reion.
It happens to be a factorof ∼
25 above the lower limit that comes from requiringthat mixing equilibrate before z ∼ H and H (cid:48) rapidly interconverting, thermal equi-libration of H (cid:48) and X at best brings the gas temperaturedown to T g ∼ T g n H + n H (cid:48) n H + n H (cid:48) + n X ∼ T g
11 + m X , (9)relative to the gas temperature T g in the standard sce-nario of adiabatic cooling [10]. The thermal energy ofDM has been neglected in the rightmost expression inEq. (9). Realistically, T g cannot quite reach this ide-alized limit because the bulk relative velocities of thegas and DM are dissipated into heat by H (cid:48) – X scattering[28, 29]. The approximation is adequate for our purposes,however. Matching to the EDGES anomaly thus implies m X ∼ σ H (cid:48) − X = (cid:90) d cos θ (1 − cos θ ) dσ H (cid:48) − X d cos θ = σ | (cid:126)v m | n (10)in terms of the relative velocity (cid:126)v m , the heating rate ofthe gas becomes [28–30]˙ Q g ∼ − ρ X σ m H ( m H + m X ) (cid:18) T g m H (cid:19) n +12 n Γ (cid:0) n (cid:1) √ π T g . (11) Eq. (9) is valid provided that | ˙ Q g | H − is at least on theorder of the gas thermal energy during a period in which H – H (cid:48) mixing is in equilibrium. This condition is requiredso that thermal equilibrium may be attained.The lower bounds on ¯ σ H (cid:48) − X shown in Fig. 1 derivefrom this condition. Two benchmark cases are shown: n = − n = 0. The first of these is comparable tohydrogen-cooling models where SM–DM scattering is fa-cilitated by a light mediator. In those models, the crosssection needs to exhibit a steep velocity-dependence inorder to avoid frequent interaction during the recombi-nation era [9, 13–17]. The second of the two cases illus-trates how that constraint on scattering can effectivelybe exchanged, in the hydrogen portal scenario, for a con-straint on mixing ( i.e. , not allowing mixing to equilibratebefore z ∼ δ shown in Fig. 1.No such constraint applies to n = − σ H (cid:48) − X isless than ∼
4% of σ H − H . At these relatively weaker crosssections, heat transfer between H (cid:48) and X is inefficientprior to z ∼ CMB . Values of δ far above the range plottedin Fig. 1 may eventually affect recombination enough tobe ruled out, but we do not extend the analysis up tothis regime. Larger δ implies a lower natural scale forUV physics generating the mixing from Eq. (2), but thesensitivity is mild as d log Λ / d log δ = − / T g to the temper-ature inferred from the EDGES absorption trough. It isworth emphasizing that our analysis has included manysimplifying assumptions to focus on the simplest regionof parameter space to study, and it would be interestingto relax these in a variety of directions. In particular, thefocus here has been on thermal DM scenarios in which H (cid:48) – X heat transfer is complete. This restriction impliesDM mass near the GeV scale, but lighter masses are pos-sible as well if these assumptions are relaxed.Hydrogen mixing raises a number of issues that wehave not yet addressed. On the high energy side, it re-mains to establish what underlying physics might giverise to the mixing—or even if doing so is plausible at all.Unlike neutrinos or photons, hydrogen atoms are com-posite particles. Coupling H to a new state H (cid:48) seemscontrived unless the latter knows about the SM struc-ture in some way.Introducing a mirror sector that parallels the SM parti-cle content is a well-studied model building framework inwhich to address this concern. Ref. [18] shows that a mir-ror model with an effective dimension-12 partonic mix-ing operator implements the mechanism discussed hereinwith its features symmetry-protected. Furthermore it es-tablishes leptoquarks as a plausible origin of the mixingoperator in a toy UV completion. As shown in that work,the threat of proton decay can be easily avoided despitethe violation of baryon number.On the cosmological side, having half the baryon den-sity during some stretch of cosmic history is an eyebrow-raising modification. There are bound to be effects onstar formation, likely with other consequences for 21 cmcosmology. Detailed study is warranted here.One reasonable prediction is that some mirror hydro-gen clouds are destined for cooling and collapse, possiblyresulting in black holes and mirror stars [18]. Regardlessof outcome, some fraction of the baryon density almostcertainly finds itself stuck in the mirror sector even afterreionization. In other words, the hydrogen portal pre-dicts missing baryons in the low- z universe, and to ourknowledge is the first model to do so.21 centimeter cosmology has truly opened a new win-dow to the universe, and promises to illuminate for usthe heretofore-dark ages. Not only will we learn aboutthe formation of large scale structure and the timeline ofreionization, but with some luck the 21 cm sky may alsoprovide a surprise glimpse of new fundamental physics.Forthcoming experiments will test the EDGES result andfurnish further insight, and we may look forward to see-ing the fruits of outstanding observational efforts cometo bear in the coming years.We thank Samuel Alipour-fard, Guido D’Amico,Robert McGehee, Paolo Panci, and Yiming Zhong forcomments on a draft of this manuscript. LJ thanksAnna Schauer for insights into first-star formation andfor suggesting a connection to direct-collapse black holes.SK thanks Vera Gluscevic for presenting an enlighteningseminar on 21cm cosmology at the KITP in December2019.The work of LJ was supported by NSF Grant No.PHY-1914242 and by NASA through the NASA Hub-ble Fellowship grant ∗ NASA Einstein Fellow ([email protected]) † EFI Oehme Fellow ([email protected])[1] J. D. Bowman, A. E. E. Rogers, R. A. Monsalve, T. 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