Heavy Thermal Relics from Zombie Collisions

Eric David Kramer∗ and Eric Kuflik†

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

Noam Levi‡ and Nadav Joseph Outmezguine§

Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel

Joshua T. Ruderman¶

Center for Cosmology and Particle Physics, Department of Physics,New York University, New York, NY 10003, USA.

We propose a new thermal freezeout mechanism which results in dark matter masses exceeding theunitarity bound by many orders of magnitude, without violating perturbative unitarity or modifyingthe standard cosmology. The process determining the relic abundance is χζ† → ζζ, where χ is thedark matter candidate. For mζ < mχ < 3mζ , χ is cosmologically long-lived and scatters againstthe exponentially more abundant ζ. Therefore, such a process allows for exponentially heavier darkmatter for the same interaction strength as a particle undergoing ordinary 2 → 2 freezeout; orequivalently, exponentially weaker interactions for the same mass. We demonstrate this mechanismin a leptophilic dark matter model, which allows for dark matter masses up to 109 GeV.

Introduction: The observational evidence for the ex-istence and ubiquity of Dark Matter (DM) is well es-tablished, yet its origin and particle nature remains un-known. This puzzle has driven a several decades long ex-ploration into the landscape of potential DM models andcosmological mechanisms for producing the relic abun-dance of DM. Among these, the prospect of DM parti-cles thermally coupled to the Standard Model (SM) inthe early Universe has been especially prominent.

During the cosmological evolution of the Universe, atearly times, the interaction rate between the SM and thedark sector is fast, keeping the two baths in chemicaland thermal equilibrium. When the interaction rate fallsbelow the Hubble expansion rate, chemical equilibriumbetween the SM and the dark sector ceases. Soon af-terwards, the interactions completely stop and the abun-dance of DM is set by a process known as thermal freeze-out. The most widely explored paradigm representingthis concept is the Weakly Interacting Massive Particle(WIMP), where the relic abundance is set by the freeze-out of DM-DM annihilations to the SM. The WIMPis a particularly promising candidate, as it predicts DMmasses at the weak scale with DM annihilation rates onthe order of the weak interactions, thus relating DM pro-duction in the early Universe to theories of the weak scaleand to new physics at the experimental frontier.

Within the WIMP scenario, there is an upper bound onthe mass of DM set by perturbative unitarity of roughlymWIMP ∼ 100 TeV [1]. Are there minimal extensionsto the WIMP paradigm that predict heavier DM massesthan this unitarity bound, leading to qualitatively dif-ferent experimental signatures? Attempts at answeringthis question have focused on out-of-equilibrium dynam-ics with the SM and/or non-standard cosmological histo-ries [2–21]. If DM is a composite object, such as a hadron,the unitarity bound applies to the size of the object and

not the masses of its constituents [1, 22–26].

Thermal freezeout mechanisms considering topologiesbeyond the WIMP have been considered [27–32], but un-til recently none have evaded the unitarity bound. Athermal mechanism that exceeds the unitarity bound,without modifying the standard cosmology, was proposedfor the first time in Ref. [33], requiring a chain of inter-actions but allowing DM masses as high as 1014 GeV.In this Letter we present a new thermal 2-to-2 freeze-out mechanism, that requires just two interactions, andallows for DM masses as heavy as 1010 GeV, without vi-olating unitarity or modifying the standard cosmology.Conversely, this mechanism can achieve thermal weakscale DM, but with much smaller interaction rates thanthe WIMP.

Our setup consists of a DM candidate (χ), a SMportal, and at least one extra interacting degree offreedom in the dark sector (ζ). The dark sector is inequilibrium with the SM at early times, maintained byζζ† annihilations to the SM, and internally in equilib-rium via the process χζ† → ζζ between the two darkparticles. When the masses of the dark sector particlesobey the hierarchy mζ < mχ, heavy DM naturallyarises in our setup, as we detail below. The reason isthat the mass splitting causes ζ to be exponentiallymore abundant than χ in chemical equilibrium, allowingfor χ to be removed by scattering efficiently against aparticle that is exponentially more abundant than itself.Note that a similar process was considered in Ref. [15]for heavy DM within a non-thermal and non-standardcosmological history; in contrast, here we show howto obtain heavy DM thermally and within a standardcosmology.

General Idea: Consider a DM particle, χ, whose num-ber density changes at early times via an interaction of

arX

iv:2

003.

0490

0v1

[he

p-ph

] 1

0 M

ar 2

020

2

χ

ψ† ψ

ψ ψ

ψ†

sm

sm

χ

ζ† ζ

ζ ζ

ζ†

sm

sm

FIG. 1. Schematic representation of the processes setting therelic abundance. On the left, χ is our DM candidate and ζ isa hidden sector zombie particle that turns a χ into a ζ. Theprocess on the right maintains chemical equilibrium betweenthe dark sector and the SM.

the form

L ⊃ χ†ζζζ, (1)

with some field ζ. The DM number density can depletevia the process

χζ† → ζζ . (2)

This process behaves similar to a zombie infection [34],where a ζ particle (the zombie), infects the DM χ (a sur-vivor) and turns it into a zombie. The final DM abun-dance consists of the survivors, χ, that persist after theprocess of Eq. (2) decouples (the outbreak ends; fortu-nately). We refer to this process as a zombie collision.

We restrict ourselves to mζ < mχ < 3mζ , where suchinteractions do not induce (on-shell) χ → ζζζ decays,and the process in Eq. (2) does not enter a forbiddenregime [35, 36]. Whenever mχ > mζ , the processes inEq. (2) can maintain chemical equilibrium longer than ifthe χ particle was annihilating with another χ, becausethe interaction rate is proportional to nζ , which is expo-nentially larger than nχ when χ becomes non-relativistic.Thus, if this process is responsible for χ freezeout, thecorrect relic abundance is obtained for smaller interac-tions than WIMP-like DM, for the same DM mass. Sim-ilarly, since zombie collisions can be very efficient, thisscenario also allows for heavier DM than the WIMP uni-tarity bound, without violating unitarity.

To realize this mechanism we consider the two pro-cesses shown in Fig. 1,

χζ† → ζζ, ζζ† → sm sm , (3)

where ‘sm’ is a light particle which is either part of theSM bath or thermalized with it. The possibility of otherinteractions, such as χχ† annihilations, will be discussedbelow in the context of a specific UV model. Coscat-tering [31, 33, 37, 38] is a different example where theDM abundance can be set by the decoupling of 2-to-2scattering of DM against a lighter state.

The Boltzmann equations governing the evolution ofthe χ and ζ number densities are given by

nχ + 3Hnχ = −〈σχζ→ζζv〉

(nχnζ − n2

ζ

neqχ

neqζ

), (4)

nζ+nχ+3H(nζ+nχ) = −〈σζζ→sm smv〉(n2ζ − n

eqζ

2), (5)

where the superscript ‘eq’ denotes equilibrium abun-dances at zero chemical potential. The density of χ willdepart from its equilibrium distribution (and freeze outsoon after) when the rate of χ number-changing processdrops below the Hubble expansion rate. This happensapproximately when (see e.g. Eq. (5.40) in [39])

nζ(xχ)〈σχζ→ζζv〉 = xχH(xχ) , (6)

where we have defined x ≡ mχ/T and xχ is the temper-ature when χ departs equilibrium. Eq. (6) determinesthis temperature, which will be used to estimate theχ relic abundance. Unlike the WIMP, the freezeoutdynamics depends on whether the zombies, ζ, followan equilibrium distribution or instead have alreadyfrozen out from the thermal bath by the time χ departsequilibrium with the ζ. In what follows we describethese possibilities for the different phases of dark sectorfreezeout.

Equilibrium Phase – Zombies in equilibrium throughoutfreezeout: When the ζζ† ↔ sm sm interactions are effi-cient in maintaining equilibrium of ζ with the SM bath,χ evolves according to the Boltzmann equation

nχ + 3Hnχ = −neqζ 〈σχζ→ζζv〉

(nχ − neq

χ

). (7)

The relic abundance can be estimated using the instan-taneous freezeout approximation, utilizing the fact thatat freezeout the χ-ζ system is still in equilibrium:

nχ(x) ' r3/2 exp

[−r − 1

rx

]nζ(x), (8)

where we defined r ≡ mχ/mζ > 1. The rest of the relicabundance calculation proceeds via standard techniques.Parameterizing the cross-section as 〈σχζ→ζζv〉 ≡ α2

χ/m2χ,

the DM mass required to match the observed abundanceis

mχ '((α2χmpl)

rTeq

) 11+r , (9)

where Teq ' 0.8 eV is the temperature at matter radi-ation equality and mpl is the Planck mass. The sameexpression with r = 1 is the known relationship from thestandard WIMP calculation. The gain in DM mass overthe WIMP is evident from the equation above—the reliccalculation puts an exponentially larger weight on mpl

vs. Teq, leading to exponentially heavier DM for the sameinteraction strength (or conversely, exponentially smallerinteraction strength for the same size DM mass). Thethermal evolution of this phase, for the specific model wedescribe later, is shown in the left panel of Fig. 2.

In most models, one would expect χχ to also annihilateto the SM or zombies ζ. If these annihilations are faster

3

101 10210-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

101 10210-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

FIG. 2. Thermal evolution of the number density of χ (blue), ζ (green), and the zero chemical potential equilibriumdistribution (dashed lines). Left - Equilibrium Phase: ζ is in chemical equilibrium with the SM bath when χ freezes out.Right - Chemical Phase: ζ has departed chemical equilibrium with the SM right before χ freezes out. Both panels show theevolution of yield for dark sector particles, with mχ = 1.9mζ = 106 GeV for interaction rates producing the observed DM relicabundance. The right panel also shows (dotted line) the χ equilibrium distribution after it develops a chemical potential, asdescribed above Eq. (12).

at freezeout than χζ† → ζζ, then the DM will behave asa standard WIMP. Parameterizing the annihilation crosssection as 〈σχχ→sm smv〉 = α2

wimp/m2χ, the condition for

zombie collisions to control the abundance is easily deter-mined by comparing Eq. (9) to the analogous equationfor the WIMP:(

(α2χmpl)

rTeq

) 11+r & αwimp (mplTeq)

12 . (10)

This equation gives the approximate phase boundarybetween the WIMP and the Equilibrium Phase, whichis shown in Fig. 3 for the model realization we discussbelow.

Chemical Phase – Zombies develop chemical potential: Itis possible that ζ freezes out of equilibrium with the SMbath, before χ decouples from ζ. In this case, ζ has a con-stant comoving abundance when χ departs from equilib-rium with it. (A non-zero chemical potential of anotherstate also impacts DM freezeout in Refs. [15, 30, 40–42].)This phase requires a more subtle treatment, since thesudden freezeout approximation will not give a good es-timate of the relic abundance. We show here a qualita-tive analysis of this phase; a more detailed quantitativeanalysis appears in the Appendix.

The ζ distribution departs from equilibrium with theSM bath at a temperature xζ , defined by

neqζ (xζ)〈σζζ→sm smv〉 = xζH(xζ) . (11)

At x > xζ the comoving number density of ζ begins tofreeze out and soon approaches a constant value. Theevolution of the χ density is then calculated by pluggingthe frozen-out density of ζ into Eq. (4). After ζ freezeout,

the χ distribution begins to trace a new equilibrium dis-tribution with a chemical potential dictated by the frozenout abundance of ζ,

neqχ → nζ

neqχ

neqζ

. (12)

Here nζ is the frozen out abundance of ζ that evolvesapproximately as nζ ∼ nζ(xζ)(xζ/x)3. Following sim-ilar steps to those leading to Eq. (9), we show in theAppendix that

mχ ∼[(α2ζmpl)

r+∆Teq

] 11+r+∆ , (13)

where we have defined 〈σζζ→sm smv〉 ≡ α2ζ/m

2χ and

∆ = rxχxζ

+ (r − 1)

(xχxζ− 1

)> 0 , (14)

with xχ and xζ as defined in Eq. (6) and Eq. (11), re-spectively. We see that the Chemical Phase leads to evenhigher DM masses than the Equilibrium Phase, as can beseen by comparing Eq. (13) to Eq. (9) . The thermal evo-lution for this phase is plotted in the right panel of Fig. 2,for the model realization we present below.

The crossover between the Equilibrium Phase and theChemical Phase occurs when the rate for χ to undergoχζ† → ζζ is approximately the same as the rate for ζ toannihilate via ζζ† → sm sm at freezeout. This is simplythe condition:

αζ ' αχ . (15)

Additional phases: Since the zombies can freeze outwith a large abundance, they can come to dominate the

4

energy of the Universe, leading to an early period ofmatter domination. If this happens, the ζ must decayand reheat the radiation bath. The large entropy dumpfrom the decay will change the relic density calculation.We discuss this more in the following section. Finally,if ζ freezes out when relativistic, the model enters anew phase where the relic abundance of χ no longerdepends on when ζ freezes out. Additionally, the χtemperature may not match the SM bath temperature,but the temperatures can be comparable. These twopossibilities—non-thermal DM and large dilution viaentropy dump—were studied in a similar framework inRef. [15]. A main goal of this Letter is to demonstratethe possibility of very heavy thermal DM within per-turbative unitarity, without modifying early cosmology.For this reason, we leave detailed discussion of theseadditional phases for future work.

Unitarity and DM decays: The zombies, ζ, are eitherstable, and themselves constitute a component of darkmatter in addition to χ, or are unstable. If the zombiesare stable, they must satisfy the unitarity bound, appliedto ζζ → sm sm, and therefore χ is at most O(1) heavierby the assumption that mχ < 3mζ . If the zombies areunstable, then both ζ and χ can have masses that farexceed the unitarity bound. In this case the abundanceof ζ will exceed the abundance of χ at freezeout, butthis energy density is removed by decays of ζ. For theremainder of this letter we focus on unstable zombies.

There are two important phenomenological conse-quences of the fact that ζ freezes-out with an abundancelarger than χ and subsequently decays. First, if ζ decays,then χ is unstable via the process χ→ ζζζ, where someor all of the ζ’s are produced off-shell and decay. Thisleads to potentially strong indirect detection signaturesand constraints. Indirect detection constraints on theDM decay lifetime τχ, e.g. from the diffuse gamma-rayspectrum, can be as strong as τχ & 1027 sec [43–45].

Second, if ζ freezes out with a large abundance,it can come to dominate the energy of the Universebefore it decays away. The large entropy dump thataccompanies the decay effectively dilutes χ (see Eq.(5.73) of Ref. [39]), allowing for DM masses beyondthe WIMP unitarity bound. Additionally, a sufficientlylong-lived and abundant ζ can imply that the Universewas matter dominated during BBN, spoiling the success-ful predictions of the standard cosmological scenario.

Example Model: Consider a model, where the SM isextended with a gauged U(1)e−µ lepton number and adark sector containing fermions, χ and ζ, and a scalarfield S, with U(1)e−µ charges qζ = 1, qχ = 3, andqS = −2. The most general renormalizable (and parityinvariant) Yukawa interactions are

Lint = yζSζcζ+yχSζχ+yeHζLe+yµHζ

cLµ+h.c. (16)

10-3 10-2 10-1 110-4

10-3

10-2

10-1

1

101

FIG. 3. α′ vs yχyζm2χ/m

2S , with contours of constant mass

required to match the observed relic abundance. We fix theratios mχ = 1.9mζ and mZ′ = 0.1mχ. We take the lifetimeof χ to be τχ = 1027 sec. Differently colored regions corre-spond to different phases of the model. See the main text foran explanation of each phase. The shaded gray region isexcluded by LEP Z′ searches, while dashed gray indicatesprojected ILC sensitivity for Z′ searches.

where H is the Higgs doublet and L is the lepton dou-blet. We take tree-level kinetic mixing between the newU(1) and hypercharge to be absent, but kinetic mixingis radiatively generated by e and µ loops. The Yukawacouplings generate zombie collisions, χζ† → ζζ, whilethe gauge interactions generate ζζ† → sm sm, where ‘sm’here can be a SM lepton or the e− µ gauge boson. TheYukawa and gauge interactions also generate χχ annihi-lations, which if responsible for χ freezeout, will lead toa WIMP-like scenario. For other models of leptophilicDM, see for example Refs. [46–58].

In Fig. 3 we show curves of constant DM relic abun-dance Ωχ ' 0.27 [59] and the different phases of freeze-out. Here we take mχ = 1.9mζ and mZ′ = 0.1mχ, whereZ ′ is the massive U(1)e−µ gauge boson. Note that forsmaller values of the vector mass mZ′ , the various crosssections will be Sommerfeld enhanced due to the U(1)e−µforce, leading to even heavier DM for the same couplingstrength [60]. We also take mS & O(few)(mχ + mζ) sothat χζ → ζζ is not on the S-resonance and can be ap-proximated by a contact interaction. We fix the lifetimeof χ→ ζHHLL to be τχ = 1027 sec, which is calculatedusing FeynRules [61] and Madgraph [62].

The results of the thermal evolution of our example

5

model contains different phases, depending on therelative size of the interactions. For large U(1)e−µgauge coupling, α′, the annihilations of χχ pairs is veryefficient, and the DM behavior is WIMP-like. This isseen in the top left region of Fig. 3, labeled ‘WIMP-like’,where constant mass lines are horizontal since the relicabundance is insensitive to the Yukawa interactions. Asα′ drops, the χζ† → ζζ process becomes more efficientat late times. If the gauge coupling is still large enoughto maintain chemical equilibrium between the dark andvisible sectors until χ freezeout, then DM enters theEquilibrium Phase, corresponding to the vertical linesin the region labeled ‘Equilibrium Phase’ in Fig. 3. Ifζ freezes out before χ freezes out, the DM will be inthe Chemical Phase. This phase is further separatedinto 3 regions. The first, labeled ‘Chemical Phase’,signals when ζ freezes out when non-relativistic, butnever dominates the energy density of the Universe. Thesecond region, labeled ‘Entropy Dump’, shows where ζfreezes out non-relativistically, but dominates the energydensity before it decays, leading to a non-negligibledilution of the DM density. The final region, labeled‘ζ Relativistic,’ shows where ζ freezes out while stillrelativistic.

Signatures and Constraints: We now discuss thegeneric phenomenological signals of the mechanism andsignatures and constraints of the leptophilic model pre-sented above. Since the focus of this paper is on heavyDM, below we only discuss parameter regions that cor-respond to mχ & 100 GeV. We leave lighter DM phe-nomenology within this framework for a future study.

The mechanism generically predicts an indirect detec-tion signal from dark sector decays. The ζ particles mustdecay before Big Bang Nucleosynthesis (BBN) so as notto obstruct light element formation [63, 64]. On the otherhand, a decay of ζ induces a decay of χ through the in-teraction Eq. (1). Late time χ decays may produce ultrahigh energy cosmic rays (UHECR), detectable by diffusegamma ray satellites [65], high energy neutrino experi-ments [66], and in dedicated UHECR observatories [67].Combined with the BBN bound, the decay rate of ζ mustreside in the window

HBBN . Γζ . Γmax , (17)

where Γmax is the value of Γζ such that the lifetime ofχ is within indirect detection bounds. In Fig. 3 we fixΓ−1χ = (1027 sec)−1. For the plotted parameter ranges,

Γmax > HBBN everywhere, showing the possibility of alarge indirect detection signal for all masses.

The portal between the sectors leads to collider sig-natures. In Fig. 3, we show the constraints from themeasurement of the differential cross-section of leptonpairs at LEP [68], which bounds mZ′/

√α′ > 25 TeV,

and as well as future projections for the ILC with reachmZ′/

√α′ > 200 TeV [53]. Dark production at the LHC

for our model is suppressed because there is no tree-levelinteraction with quarks. However, due to the long-livednature of ζ, the dark sector may be discoverable in ex-periments designed to look for long-lived particles at theLHC, such as AL3X [69], CODEX-b [70], FASER [71],and MATHUSLA [72]. Finally, if a gauged U(1)B−L isconsidered instead of the leptophilic model, the freezeoutwould remain unchanged, but there would be strongercollider signatures. We leave the study of a such a modelto future work.

Thus far, we have set the tree-level gauge kinetic mix-ing between Z ′ and hypercharge to zero. DM-protonscattering is generated at one-loop by the electron andmuon with cross section given by [57, 73–75]

σp =64µ2

χp

πm4Z′α2

emα′2 log2

(me

mµ

), (18)

where µχp is the reduced DM-proton mass. Althoughthis strength of interaction might seem relevant for directdetection experiments, we find that current nucleonrecoil direct detection constraints from XENON1T [76]do not outperform LEP Z ′ searches. In the lower χmass end of our model (mχ . GeV), electron recoilexperiments might be relevant and one would haveto take into account the effect of shielding by theEarth [77], which in our case is dominated by a loopinduced interaction with protons.

Conclusions: In this Letter, we presented a new2-to-2 freezeout mechanism for thermally producedDM, allowing for very heavy DM without altering thestandard cosmology. The DM relic abundance is setvia zombie collisions in the dark sector: χζ† → ζζ.We have shown that the exponential depletion of DMinduced by these interactions can naturally lead to DMmasses as high as mχ ∼ 1010 GeV, without violatingperturbative unitary. This is achieved with only twointeractions, but a chain of zombie collisions, similarto Ref. [33], can potentially allow for even heavier DMmasses. Additionally, higher DM masses may be realizedwithin this framework when considering the dark sectorto be asymmetric, relating leptogenesis to DM freezeout.We leave exploration of these possibilities for future work.

Acknowledgements.— We would like to thank AsherBerlin, Timothy Cohen, Timon Emken, Rouven Essig,Michael Geller, Roni Harnik, Yonit Hochberg, HyungjinKim, Gordan Krnjaic, Sam McDermott, Mukul Shola-purkar and Tomer Volansky for useful discussions. Thework of EDK is supported in part by the Zucker-man STEM Leadership Program. The work of EKis supported by the Israel Science Foundation (grantNo.1111/17), by the Binational Science Foundation(grant No. 2016153) and by the I-CORE Program ofthe Planning Budgeting Committee (grant No. 1937/12).

6

NJO is grateful to the Azrieli Foundation for the award ofan Azrieli Fellowship. NL would like to thank the MilnerFoundation for the award of a Milner Fellowship. JTRis supported by NSF CAREER grant PHY-1554858 andNSF grant PHY-1915409. EDK, EK and NJO wouldlike to acknowledge the GGI Institute for TheoreticalPhysics for enabling us to complete a significant portionof this work. JTR acknowledges hospitality from the As-pen Center for Physics, which is supported by the NSFgrant PHY-1607611.

Appendix

In this appendix we derive an analytic understandingof the Decoupled Phase, when ζ departs chemical equi-librium before χ freezes out. The analysis here is mademuch simpler by defining the yield Y = n/s , the num-ber density normalized by the entropy density. We usex = mχ/T as a clock. In this language the Boltzmannequations for χ and ζ take the form [39]

dYχdx

= −λχx2Yζ

(Yχ − Y eq

χ

YζY eqζ

), (19)

dYζdx

+dYχdx

= −λζx2

(Y 2ζ −

(Y eqζ

)2), (20)

with

λχ =〈σχζ→ζζv〉s(mχ)

H(mχ), λζ =

〈σζζ→sm smv〉s(mχ)

H(mχ)(21)

and the thermally averaged cross-sections are defined as〈σχζ→ζζv〉 ≡ α2

χ/m2χ and 〈σζζ→sm smv〉 ≡ α2

ζ/m2χ. Above

we assumed that the number of relativistic degrees offreedom is constant and defined s(mχ) and H(mχ) as theentropy density and the Hubble parameter, respectively,evaluated at T = mχ. We are interested in cases whereχ freezes out after ζ departs from chemical equilibriumwith the SM bath. This, by virtue of the Boltzmannequations above, happens if αχ & αζ . In that case thefreeze out of ζ is well described by a sudden freeze out,which happens at xζ defined through

λζYeqζ (xζ)

x2ζ

' 1 . (22)

For x > xζ , the inverse process sm sm → ζζ stops, andYζ evolves according to

dYζdx' −λζ

x2Y 2ζ . (23)

Integrating this equation one finds

Yζ(x > xζ) 'x

−xζ + (xζ + 1)xYζ(xζ) . (24)

After ζ departs form its equilibrium distribution, χ fol-lows a new equilibrium distribution given by (similarlyto Eq. (12))

Yχ(x) = Yζ(x > xζ)Y eqχ (x)

Y eqζ (x)

. (25)

This can be understood as a thermal distribution, butwith a chemical potential for χ. The relic abundanceof χ can then be found using standard techniques. Theχ distribution departs from its new equilibrium at xχ,defined through

λχYζ(xχ > xζ)

x2χ

' 1 . (26)

Substituting Yζ in the equation above with its definition(Eq. (24)), one can solve the above equation for xχ:

xχ = xζ

xζ +

√x2ζ + 4

(〈σχζ→ζζv〉〈σζζ→sm smv〉

)2

(1 + xζ)

2 (1 + xζ). (27)

One can easily verify that indeed xχ > xζ for〈σχζ→ζζv〉 > 〈σζζ→sm smv〉. After decoupling χ the in-verse process χζ → ζζ stops, and density of χ evolvesas

dYχdx' −λχ

x2Yζ(x > xζ)Yχ(x) . (28)

For simplicity we assume now xχ xζ , in that case theequation above reduces to

d log Yχdx

' −x2χ

x2. (29)

Integrating this equation from xχ to ∞ one finds

Y∞χ ' Yχ(xχ)e−xχ ' Y∞ζ exp[−xχ

(2− r−1

)]. (30)

Since Y∞ζ ∼ e−x/r but also Y∞ζ ∼ λ−1ζ , by requiring that

χ-radiation equality happens at Teq ∼ 0.8 eV we find

mχ ∼[(α2ζmpl

)r+∆Teq

] 11+r+∆

, (31)

with

∆ = rxχxζ

+ (r − 1)

(xχxζ− 1

)> 0 , (32)

which are Eqs. (13) and (14) in the main text.

∗ ericdavidkramer@gmail.com† eric.kuflik@mail.huji.ac.il‡ noam@mail.tau.ac.il§ Nadav.Out@gmail.com

7

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