Challenging the Stability of Light Millicharged Dark Matter
IIPPP/20/81
Challenging the Stability of Light Millicharged Dark Matter
Joerg Jaeckel and Sebastian Schenk Institut f¨ur theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Institute for Particle Physics Phenomenology, Department of Physics,Durham University, Durham DH1 3LE, United Kingdom (Dated: February 16, 2021)We investigate the cosmological stability of light bosonic dark matter carrying a tiny electriccharge. In the wave-like regime of high occupation numbers, annihilation into gauge bosons can bedrastically enhanced by parametric resonance. The millicharged particle can either be minimallycoupled to photons or its electromagnetic interaction can be mediated via kinetic mixing witha massless hidden photon. In the case of a direct coupling current observational constraints onthe millicharge are stronger than those arising from parametric resonance. For the (theoreticallypreferred) case of kinetic mixing large regions of parameter space are affected by the parametricresonance leading at least to a fragmentation of the dark matter field if not its outright destruction.
I. INTRODUCTION
Very light (sub-eV) dark matter (DM) must consist ofbosonic particles. Common examples are axions, axion-like particles or dark photons [1–5]. By virtue of theirtiny couplings and very low masses, it is often takenfor granted that they make good and cosmologically sta-ble candidates for DM. However, the fact that very lightbosonic DM is long lived is far from trivial. Due to theirlow mass and low velocity, DM made from light bosonsfeatures very high occupation numbers which can dra-matically enhance interaction rates with other particlesand lead to parametric resonance phenomena [6–9]. Forexample, in significant parts of the parameter space thestability of axion-like particles towards their decay intophotons requires a non-trivial interplay of the expansionof the Universe as well as plasma effects [1, 2, 10] (cf. [11–30] for some situations where Bose enhancement fromhigh occupation numbers may lead to interesting signa-tures for axion-like particles). Of course, these particlesdo not carry a conserved charge that would naturally ren-der them stable towards decay. One may therefore won-der what happens if the light DM particles are chargedand only annihilations with suitable antiparticles are pos-sible.To be concrete, in this work, we want to address thequestion of cosmological “stability” for the case that DMcarries a tiny electromagnetic charge [31–35], often called millicharge . Ample motivation for millicharged particlesis provided by Standard Model extensions and, in par-ticular, string theory constructions [36–50]. At the sametime, such scenarios may also offer interesting new op-portunities for direct detection [51] (for an overview ofvarious detection strategies, see [52]).In general, having a conserved electric charge, sta-bility towards particle decay is ensured. However, DMshould not carry any net electric charge . Therefore,it should be composed of an equal number of particles In a scenario where very light bosonic DM does carry an ap- and antiparticles, opening up the possibility of annihila-tions, for example into radiation . Naively, this seems tobe strongly suppressed by the tiny value of the relevantcharge that is required by phenomenology (for a reviewsee, e.g., [55]), independent of the question of cosmolog-ical stability. However, for low masses, enhancementsdue to high occupation numbers may set in. Indeed, inthis work, we argue that the coherent nature of the verylight DM particles can drastically enhance the interac-tion rates with gauge bosons. This, in turn, can cause anannihilation into photons, even for tiny electromagneticcharges.The basic reason for such an annihilation is that theDM coupling acts as an oscillating mass term for the pho-tons in the coherent DM background. This can drive thegauge bosons into a parametric resonance [6–9], such thatcertain momentum modes are excited quite rapidly. Theenhancement of the momentum modes then correspondsto an explosive production of photons. This phenomenoncan lead to an efficient depletion of the DM energy den-sity, that seriously challenges the cosmological stabilityof the DM candidate. This work is structured as follows. In Section II wediscuss the phenomenon of rapid photon production in amillicharged DM background via a parametric resonance.Furthermore, we carefully examine plasma effects in theearly Universe that are able to stop the depletion of the proximately conserved global charge, however, today’s net chargedensity may be non-vanishing [53]. In principle, a spatial separation of positive and negative chargesmay be possible [54]. While we do not have a conclusive argu-ment excluding this possibility, we strongly suspect that such asituation is not viable in the case where the particles in ques-tion carry a gauge charge and are supposed to be the dominantform of DM (e.g. the presence of long range gauge interactionsbetween the regions may modify the equation of state). More-over, we note that this is, by definition, connected to a veryinhomogeneous situation. The amplification of gauge fields in the expanding Universe by aparametric resonance from charged scalars has also been consid-ered as a source of large-scale primordial magnetic fields [56]. a r X i v : . [ h e p - ph ] F e b DM energy density. In Section III we consider the theo-retically preferred situation where the millicharge arisesfrom a hidden (or dark) photon kinetically mixed withits electromagnetic counterpart. A brief summary anddiscussion can be found in Section IV.
II. RESONANT DEPLETION OFMILLICHARGED DARK MATTER
In a scenario where DM carries electromagnetic chargeit is subject to annihilating into visible particles and inparticular photons. In this section, we discuss that thisdepletion of the DM energy density can be drastically en-hanced by parametric resonance phenomena [6–9], evenfor tiny charges. This implies that it is necessary to re-consider the cosmological stability of a millicharged DMcandidate.Let us illustrate the underlying mechanism in a simplesetup where a scalar DM candidate is minimally coupledto the photon, L = − F + ( D µ φ ) † D µ φ − m φ † φ . (1)Here, F µν is the electromagnetic field strength associ-ated to the photon A µ , φ is the DM of mass m andcharge q and D µ = ∂ µ + iqA µ denotes the gauge-covariantderivative . We are interested in very light, possiblysub-eV, DM particles, which, due to their extremelyhigh occupation numbers, can be described by classicalfields. They furthermore require a non-thermal produc-tion which is, for instance, provided by the misalign-ment mechanism [1–5, 57] (but other mechanisms suchas, e.g., [58–77] may also give suitable DM densities).In first approximation the observed DM energy densitycan be understood as coherent oscillations of a spatiallyhomogeneous complex scalar field φ in the expandingUniverse. We can decompose the field, φ = ϕ exp ( iχ ) / √ . (2)In these coordinates, charge neutrality is ensured by triv-ial dynamics for the angular degree of freedom , χ =const, and without loss of generality we can take χ = 0.The radial mode then oscillates with decreasing ampli-tude ϕ ( t ) = ϕ (cid:18) a a ( t ) (cid:19) cos ( m ( t − t ))= Φ( t ) cos ( m ( t − t )) . (3) Here, we absorbed the gauge coupling into the definition of thecharge, as there is only a single field involved. Indeed, in the case of the misalignment mechanism and if thefield already exists during inflation, any fluctuation is stretchedout by inflation, thereby providing for homogeneity. During inflation any non-trivial dynamics of χ is diluted quickly,˙ χ ∝ a − . Here, a ( t ) is the scale factor and t denotes the timetoday with scale factor a . Moreover, ϕ = √ ρ m = 4 . × − eV (cid:18) eV m (cid:19) (cid:18) ρ ρ DM (cid:19) (4)is the (average) oscillation amplitude today where we use ρ DM = 1 . / cm [78] as a reference value. The energydensity associated to the scalar field then dilutes as ρ φ ∼ a − , as appropriate for a cold DM particle. A. Rapid photon production via parametricresonance
Obviously, the requirement that the energy density of φ scales like that of pressureless matter is not entirely suf-ficient for making it the DM. In addition, a viable DMcandidate also has to be cosmologically stable. Crucially,since in our scenario φ carries electromagnetic charge, an-nihilation channels to photons are open, eventually chal-lenging its stability. In the simple theory (1), the mainexample of a depletion mechanism would be the pairwiseannihilation of DM particles, φφ → AA . As we will nowshow, even for tiny electromagnetic charges, the interac-tion rates of this channel can be significantly enhancedby resonance effects that lead to an explosive productionof photons. To see this, let us consider photon modes inthe classical DM background during the evolution of theUniverse. These satisfy¨ A + H ˙ A + (cid:18) k a + q ϕ (cid:19) A = 0 , (5)where A denotes a polarization mode of momentum k and H is the Hubble parameter. A collectively describesthe spatial components of the gauge potential A µ , whilewe fix the temporal components by the Lorentz gaugecondition ∂ µ ( √− gA µ ) = 0. Once the DM field has over-come the Hubble friction, H (cid:46) m , it oscillates accordingto Eq. (3) with amplitude Φ( t ). In this scenario, theequation of motion (5) can be rewritten as a differentialequation of Mathieu type [79],d d x A + ( A k − Q cos (2 x )) A = 0 , (6)with x = mt and we have defined A k = k a m + 34 H m + 2 Q , Q = q Φ m . (7)The interaction with the millicharged DM acts as an os-cillating mass term for the photons. It is therefore pos-sible that some mode functions are enhanced by reso-nance effects, known as parametric resonance [6–9]. Asthe mode functions determine the occupation number ofthe gauge bosons, n k = ω k (cid:32) | ˙ A | ω k + | A | (cid:33) , (8)this process corresponds to a resonant production of pho-tons.Crucially, the solution of (6) contains an exponentialfactor, A ∝ exp ( µ k x ), with Floquet exponent µ k . Thisexponent is in general a complex number which, impor-tantly, can have a positive real part . For the purposeof our work, we will exclusively focus on the case where µ k is purely real and positive. This corresponds to anexponential growth of the respective momentum modes, n k ∝ exp (2 µ k mt ) . (9)That is, the rate of photon production is governed bythe Floquet exponent µ k , which, in general, is a func-tion of A k and Q . Depending on the dynamics of theMathieu equation, a parametric resonance can be con-sidered in two different regimes. In a narrow resonance( Q (cid:28)
1) only very few momentum modes are enhanced,while the opposite is true in a broad resonance (
Q (cid:29)
1. Narrow vs broad resonance
Let us first understand the characteristic behaviour ofthe Mathieu equation, while neglecting the expansion ofthe Universe. This serves as the basis on top of which wecan later include the consequences of expansion.In the narrow resonance regime, where
Q (cid:28)
1, the in-stability bands of the Mathieu equation feature a smallwidth. In our case, this means that only a limited rangeof momentum modes is resonantly enhanced. The firstinstability band is the dominant one, as it contributes tothe exponential growth with the largest Floquet expo-nent. The latter is given by [79] µ k = 12 (cid:115) Q − (cid:18) k m − Q (cid:19) . (10)Furthermore, to good approximation, in momentumspace the resonance bandwidth reads [79]∆ k ∼ m Q . (11)In combination with the requirement Q (cid:28) narrow resonance. Up to corrections of order Q , the Floquet exponent is maximal for momenta of theorder of the DM mass, k ∗ (cid:39) m , such that µ k ∗ = Q / This can, for instance, be read off from the instability chart of A k and Q of the Mathieu equation (see, e.g., [79]). This is essentially the same result that one would ob-tain from a perturbative approach to interaction rates in φφ → AA processes, if Bose enhancement is taken intoaccount (see, e.g., [80]).In the broad resonance regime, where Q (cid:29)
1, the sit-uation is more complicated. Here, exact expressions forthe Floquet exponents are not available. Therefore, wewill use analytic approximations of µ k given in [81, 82].As their precise form is not very enlightening, we do notquote the full expressions here. Instead, we give some ap-proximate numbers and behaviours to facilitate the dis-cussion. That is, typical values of the exponent for a widerange of momenta are µ k ∼ .
15 and it can obtain a max-imum value of µ k ∗ ∼ log (cid:0) √ (cid:1) /π ≈ .
28 [81, 82]. Asa rough approximation one can therefore estimate [81, 82] µ k ∼ . − . , (12)within the resonance bandwidth. Importantly, this doesnot strongly depend on Q . The width of the instabilityband is then typically of the order [8]∆ k ∼ m Q , (13)which can be parametrically large for Q (cid:29) Q between resonance bands of the exponent µ k ( A k ( Q ) , Q ),for momenta of the order k (cid:46) ∆ k , is∆ Q ∼ √Q . (14)The narrow and broad resonance regime can behavevery differently with respect to the depletion of the DMenergy density. Generally speaking, a broad resonance istypically more efficient, as more momentum modes arewithin the resonance band at the same time and theexponent is typically larger. That said, for our discus-sion the dependence of the rate of exponential growth,given by µ k , as well as the width of the resonance ∆ k onthe parameter Q is important. In the narrow resonanceregime, the former is parametrically given by µ k ∼ Q while for a broad resonance it is typically of the order µ k ∼ .
15, largely independent of Q . The bandwidth ofboth regimes is also considerably modified, i.e. ∆ k ∼ m Q for a narrow and ∆ k ∼ m Q / for a broad resonance, re-spectively. As we will see in the following section, bothaspects are crucial for an explosive production of pho-tons.
2. Including the expansion of the Universe
The previous discussion of the dynamics of the Math-ieu equation only applies to a static situation. However,in the early Universe the expansion cannot be neglected.In this case, the parameters A k and Q explicitly dependon the scale factor. Strictly speaking, the concept of(static) resonance bands then ceases to be meaningful.Nevertheless, if the changes are sufficiently slow (com-pared to the time-scale of the DM oscillations) we canstill get a reasonable picture by imagining the move-ment of a given momentum mode k along the trajectory( A k ( t ) , Q ( t )) through the instability chart. For instance,the system might start to evolve within a broad reso-nance, but as Q decreases with time, it eventually endsup in a narrow resonance regime before it terminates.In a situation where Q may change significantly be-tween consecutive oscillations of the driving DM field φ ,one would instead have to move from a parametric reso-nance to a so called stochastic resonance [8]. Here, witha single oscillation of φ the phase of each photon mode isdrastically altered, such that they are practically uncor-related at any stage of photon production. Therefore, dueto interferences, the number of photons produced typi-cally increases but can also decrease with progressing DMoscillations, thereby slightly reducing the efficiency of theresonance. For a detailed discussion of the phenomenonof a stochastic resonance in an expanding Universe, werefer the reader to [8].Such a thorough treatment of stochastic resonance isbeyond the scope of this work. We therefore followthe more intuitive approximate approach outlined above,that has already been pursued in [10]. For our purpose,the most important difference between the static andthe dynamical situation is that the photons experiencea redshift as the Universe expands. Mathematically, theMathieu parameter A k directly depends on the physi-cal momentum of the mode, ∝ k/a , which changes withtime. Each mode therefore only spends a finite amountof time in the resonant region. This can prevent the DMfrom efficiently annihilating into photons, if the latter areshifted out of a resonance quickly enough.After a short amount of time δt , the momentum of aphoton mode is shifted by [10] δkk (cid:39) Hδt , (15)where δt is thought to be differential on cosmologicalscales, δt (cid:28) H − . That is, the momentum modes of thephotons can only grow exponentially for a short amountof time, δt exp ∼ / (2 µ k m ), before they get shifted out ofthe resonance band, δt exp (cid:46) δt . While this is a universalfeature that eventually terminates the resonant produc-tion of gauge bosons, its physical manifestation withinthe Mathieu dynamics has to be established carefully. Inparticular, there can be differences between a narrow anda broad resonance due to the significant modifications ofthe instability chart in these regions. Hence, we will dis-cuss both scenarios separately. a. Narrow resonance In a narrow resonance the Flo-quet instabilities occur at integer values of k/m (see also Fig. 3). In this regime, the lowest instability band, cor-responding to the momentum k ∗ (cid:39) m , is the most effec-tive and we focus on this in our analysis. Hence, effec-tively, there is only a single resonance band which caninduce an exponential growth of the photon modes. Atthe same time the expansion of the Universe can redshiftthe photons out of this instability, thereby preventingtheir resonant enhancement. That is, naively, there isa competition between the characteristic time of expo-nential growth determined by the Floquet exponent andthe time the photon modes spend inside the resonanceband. Reversing this argument, it can be written as anaive condition to avoid the rapid production of photonsin the DM background,12 µ k ∗ m (cid:38) δkk ∗ H . (16)Obviously, this requirement is time dependent. Looselyspeaking, the condition has to be satisfied at all times inthe narrow resonance regime in order to avoid the com-plete fragmentation of the DM field and thereby to guar-antee the cosmological stability of the DM candidate.The stability condition (16) so far only takes into ac-count the growing exponential factor of the photon modefunctions. However, as there can still be a small prefactorin front, a single short burst of rapid photon productionmay not be sufficient to trigger a complete annihilationof the DM field. Instead, the latter is only effective, if asignificant amount of energy is transferred from the DMto the photons, i.e. if ρ A /ρ φ grows sufficiently that itbecomes of order unity . This, in turn, can be used toobtain a more precise condition for its cosmological sta-bility. While the DM energy density schematically reads ρ φ (cid:39) m Φ /
2, the energy density of the photons can beobtained by summing over all modes ρ A = 1(2 πa ) (cid:90) d k ω k n k , (17)where ω k denotes the energy of each momentum mode.Indeed, if the sub-exponential prefactor of n k in (9) issmall, the resonance needs to be active for a considerableamount of time to transfer a significant amount of energyfrom the DM field into photons, δT = ζ µ k m . (18) When ρ A /ρ φ ∼
1, we expect our description of the coherent DMfield to break down and backreaction effects to become impor-tant. We will comment more on this later. For now, we notethat this is the reason we often use the word “fragmentation” in-dicating the loss of the coherent condensate instead of speakingof “annihilation”. Here, we consider one polarization mode. In principle, the twopolarizations of the photons grow equally fast leading to a factorof two in the energy density, which however has only a negligibleeffect on the limits we will derive.
In practice, the factor ζ depends on the initial condi-tions associated to the photon mode functions. Follow-ing [10], the sub-exponential correction can be estimatedvia a saddle-point approximation of (17). It schemati-cally reads ζ ∼ log (cid:18) n (cid:114) mH Φ qm (cid:19) , (19)where n is the initial occupation number of the photonmodes, which can be determined by vacuum fluctuationsor CMB photons, n = 1 / n (cid:39) T CMB /m , respec-tively. The prefactor ζ can then be chosen conservatively,i.e. corresponding to the larger of both options.Finally, requiring that the photons are redshifted outof the resonance quickly enough to avoid a complete frag-mentation of the DM field yields a condition for the sta-bility of the DM candidate, ζ (cid:38) kk ∗ mH µ k ∗ . (20)Using ∆ k/k ∗ = Q and µ k ∗ = Q /
2, this condition reads ζ (cid:38) ( m/H ) Q (see also [8]). Therefore, in a narrowresonance, to avoid a fragmentation of the DM field itselectromagnetic charge q has to satisfylog (cid:18) n (cid:114) mH Φ qm (cid:19) (cid:38) mH (cid:18) q Φ2 m (cid:19) . (21)In principle, to guarantee stability this inequality musthold at all times of the cosmic evolution. However, toensure consistency, the system has to be in the narrowresonance regime, Q (cid:28)
1. In general, this is not neces-sarily the case. For example, as typically ζ ∼ − m/H ∼ O (1) [8]. It is therefore worthwhile toalso consider the broad resonance regime. b. Broad resonance In a broad resonance the insta-bility chart of the Mathieu equation is more complicated.In contrast to the case of a narrow resonance, the instabil-ity bands are not sharply localized around integer valuesof k/m . Instead, for a fixed Q , they can extend from atypical scale of k ∗ ∼ m Q / down to possibly even van-ishing momentum (see also Fig. 3). This means that, asthe Universe expands, a given momentum within a cer-tain instability band is not redshifted out of a single res-onance, but, because Q decreases similarly as Q ∼ a − ,it can cross multiple instability bands before it entersthe regime of a narrow resonance. Therefore, the photonmode can experience multiple resonant enhancements onits trajectory through the instability chart. As an ap-proximation we can model this by summing up all reso-nances that a given k -mode crosses, n k ∝ exp (cid:18) m (cid:90) d t µ k ( t ) (cid:19) . (22) Since the Mathieu parameter Q is now a function of time,in a radiation-dominated Universe the exponent can bewritten as m (cid:90) d t µ k ( t ) = 13 mH (cid:18) q Φ2 m (cid:19) (cid:90) d Q µ k ( Q ) Q . (23)As pointed out above, within the instability bands forthe range of momenta k ≤ ∆ k ∼ m Q / , the Floquet ex-ponent takes typical values of µ k ∼ . − .
28, which aremostly independent of Q to good approximation. In thiscase, the bands can be assumed to be of width and dis-tance of order √Q in Q . Hence, the integration of µ k isdominated by small values of Q . As a rough approxima-tion we can therefore choose the integration boundariesto be Q − = 1 (beginning of the broad resonance region)and Q + = ∞ . Evaluating this numerically we find κ ( k ) = (cid:90) ∞ d Q µ k ( Q ) Q ≈ (cid:40) . , k = 00 . , k = m Q / , (24)with monotonically decreasing values within those limit-ing cases.Similar to the case of a narrow resonance, the fragmen-tation of the DM field is efficient (ultimately leading toa breakdown of our description of the DM oscillations),if a significant fraction of energy is transferred from theDM to the photons, ρ A /ρ Φ ∼
1. Again, this requirementcan be translated into a stability condition for the DMcandidate,log (cid:18) π / √ n (cid:114) mH κq (cid:15) (cid:19) (cid:38) κ mH (cid:18) q Φ2 m (cid:19) . (25)This is the broad resonance equivalent to the stabilityrequirement (21), which is applicable in the narrow res-onance regime.Note that in determining this expression we have madeseveral rough approximations. As already stated abovewe have neglected the dependence on the upper integra-tion boundary in (24). This ignores contributions sup-pressed by an inverse power of Q . We have also usedthat, due to the exponential growth in the modes, thebiggest drain in energy occurs at late times and thereforeevaluated the energy drain only at the end of the broadresonance regime. Moreover, using a fixed κ we haveneglected that during the evolution the physical momen-tum of each mode decreases as k ∼ a − and thereforethe exponent changes. Finally, in line with all these ap-proximations we have simply dropped terms logarithmicin Q . c. Discussion The stability conditions (21) and (25)put strong constraints on the value of the electromagneticcharge of the DM. Before explicitly evaluating them, letus first get some analytical understanding. In principle,in order to avoid the resonant depletion of the DM, bothconditions have to complement each other such that ei-ther one of them is satisfied at all times of the cosmic − − − − m [eV] − − − − − − − q SN1987ASC p u l s a r s C M B m a g n e t i c fi e l d s FIG. 1. Allowed electromagnetic charge q of the scalar DMcandidate as a function of its mass m . The solid blue linecorresponds to the stability condition due to parametric res-onance, evaluated at the time where plasma effects terminatethe latter, i.e. where m A (cid:39) m . The dashed and dash-dottedblue lines illustrate an estimate of where plasma photons with m A > m may be produced due to a broad resonance, Eq. (28),evaluated when m A /m = 1 and m A /m = 100, respectively.For comparison, the dotted blue line shows the narrow reso-nance condition evaluated close to the earliest possible time,1000 t ∗ , while plasma effects are neglected, m A = 0. The ob-servational constraints are given by CMB observations [83],SN1987A [84] and stellar cooling (SC) [85], pulsar timing ar-rays [86] or by interactions with magnetic fields in galaxiesand clusters [87, 88]. In these limits, the dashing indicatesregions where we have used very naive extrapolations fromthe high-mass regime. evolution. As we have pointed out before, we expect thefragmentation of the DM field to first be governed bya broad resonance. Then, as Q decreases with time inan expanding Universe, Q ∼ a − , the system will en-ter the narrow resonance regime before the fragmenta-tion eventually terminates (given that the coherent fieldis not completely destroyed at that point). Therefore,in practice, one has to carefully establish which stabilitycondition gives the correct, i.e. self-consistent, constrainton the millicharge at each time. This depends on thecharge as well as the time when the stability condition isevaluated.In general, it is a priori not at all obvious, what timegives the strongest possible constraint on q . In fact, bothregimes (21) and (25) behave very differently with thescale factor. To see this, we insert the evolution of theDM amplitude, Φ ∼ a − / , and the behaviour of theHubble constant during radiation domination, H ∼ a − ,into the exponential factor (i.e. the right hand side) ofboth stability conditions. For a narrow resonance weobtain mH (cid:18) q Φ2 m (cid:19) ∼ a − , (26) while, in contrast, the broad resonance regime behavesas 2 κ mH (cid:18) q Φ2 m (cid:19) ∼ const . (27)This suggests that in the narrow resonance regime thestrongest constraint arises when evaluating at the earliestpossible time, whereas the broad resonance case appearsto be independent of the scale factor.Let us consider both scenarios. In case of a narrowresonance, Eq. (26) suggests that we should evaluate thestability condition when φ just starts to oscillate, i.e. at t ∗ when H (cid:39) m . However, as noted before, fulfilling (21)with ζ ∼ −
100 is inconsistent with the narrow resonaneregime at t ∗ . Hence, we either have to evaluate at a some-what later time when H (cid:28) m or go into the regime ofa broad resonance. For the strongest self-consistent con-straint on the millicharge in the narrow resonance regimewe can evaluate at t ≈ t ∗ .In contrast, in the broad resonance regime, Eq. (27)suggests that the constraint on the millicharge is inde-pendent of the time when the stability condition is eval-uated. This suggests that the limit does not strengthenmuch when approaching the broad resonance regime .Indeed, we have checked that (25) roughly provides forthe same limit as (21) evaluated at t ≈ t ∗ .A numerical evaluation of the stability condition (21)at t ≡ t ∗ is shown as the dotted blue line in Fig. 1.However, this estimate is probably too optimistic as westill need to include plasma effects, which we will do next. B. Photons inside the early Universe plasma
So far, we have assumed that the photons are mov-ing freely through the Universe. However, during thecosmological evolution, the early Universe is filled witha hot plasma that modifies their propagation. Indeed,for example in the case of axion-like particles, this is thedominant effect ensuring their stability [1, 2, 10]. There-fore, it is sensible to also consider this effect for the case ofmillicharged DM (a discussion of parametric resonance incharged cosmological scalars can also be found in [56] butthey were focused on primordial magnetic fields ratherthan DM).Naively, the photons interact with the charged par-ticles of the medium such that they acquire a modi-fied dispersion relation (and wavefunction renormaliza-tion), see e.g. [89]. Effectively they acquire a mass, m A . In Eqs. (6) and (7) this leads to the replacement k /a → k /a + m A . Therefore, A k is larger and, if theplasma mass is too high, the resonance becomes ineffi-cient. In particular, in a narrow resonance regime the We note that the complete independence of the evaluation timeis, of course, due to the simplistic approximations we employ. instabilities become ineffective if the plasma mass ex-ceeds the mass of the DM candidate, m A (cid:38) m . Fornearly all k the rate of exponential growth µ k becomesimaginary, corresponding to an oscillating rather thana growing mode function. In contrast, in a broad reso-nance the production of photons with masses m A (cid:29) m is in principle possible . However, this process requirescomparatively large couplings in general. In particular,photon production in this regime can only be efficient forcharges satisfying [8] q (cid:38) m A m Φ . (28)Overall, we therefore expect that this possibility will re-sult in a weaker constraint on the electromagnetic mil-licharge (see also the example below).The plasma mass of the photon depends on the temper-ature of the medium. That is, in an expanding Universe,it is time dependent . As noted above, in practice, thecondition m A (cid:46) m sets the earliest time at which the(narrow resonance) stability condition (21) can be evalu-ated and turns out to stabilize the scalar DM candidatein large parts of parameter space of the vanilla theoryof millicharged DM. We show this as a solid blue line inFig. 1.In the broad resonance regime the production of pho-tons with masses above the DM mass is possible. But forthis larger charges are needed. In particular, as a minimalrequirement, the broad resonance must be strong enoughto overcome the mass threshold. This requires fulfill-ing the condition (28). This is usually already a weakerrequirement than evaluating the narrow resonance sta-bility condition at the point at which the photon massis small enough for the narrow resonance to be active.As an example, this is demonstrated by a dashed anda dashdotted blue line in Fig. 1, where we choose timeswhen the plasma mass is of the order of m A /m = 1 and m A /m = 100, respectively.Looking at Fig. 1 we can see the drastic impact of theplasma effects. Comparing the naive estimate that com-pletely neglects plasma effects (dotted blue line) with theconstraint taking into account the plasma effects (solidblue line), the former turns out to be many orders of mag-nitude stronger. Indeed, we observe that in the regime ofvery low masses, the stability condition on millichargedDM is a weaker requirement than current observationalconstraints [83–88] . While this is desirable from a phys-ical point of view, the simple model we have considered We thank Paola Arias, Ariel Arza and Diego Vargas for very use-ful discussions (triggered by the helpful comments of an anony-mous referee for [90]) on this issue in a similar system. We use the cosmological evolution of m A as given in [10] whichis based on [89, 91–94]. As indicated also in the figure, for some constraints we have ex-tremely naively extrapolated to very small masses. Moreover, wenote, that most of the DM constraints have been derived havingat least implicitly particles in mind. It may therefore be worth- in this section is disfavoured from a theoretical point ofview, as quantization of electromagnetic charge would behard to justify. Let us therefore turn to a more realisticand appealing theory involving a hidden photon that iskinetically mixed with the visible sector.
III. MILLICHARGED PARTICLES ARISINGFROM A KINETIC MIXING OF A HIDDENPHOTON WITH THE VISIBLE SECTOR
From a theoretical point of view, a fundamental mil-licharge is unappealing with respect to charge quantiza-tion. A well motivated alternative is provided by kineticmixing [38]. A simple example of this scenario is an ad-ditional massless hidden photon X µ that is kineticallymixed with the electromagnetic photon A µ [38], L (cid:15) = − (cid:15) F µν X µν . (29)For a hidden sector matter particle φ (in our case theDM candidate) carrying a (quantized) charge q under X µ , a small effective electromagnetic charge appears afterdiagonalizing the kinetic term. To see this, one can rotatethe gauge fields by A µ → A µ and X µ → X µ + (cid:15)A µ . Whilethis redefinition leads to canonically normalized kineticterms of the gauge fields, it also appears in the gauge-covariant derivatives of the hidden sector matter field, D µ φ = ( ∂ µ + qgX µ ) φ → ( ∂ µ + qgX µ + (cid:15)qgA µ ) φ , (30)where g is the hidden sector gauge coupling. As a con-sequence, the DM carries an effective electromagneticcharge [38] q eff = (cid:15)qg , (31)where we have again absorbed the factor of the electro-magnetic coupling e into the charge. In this way, a small (cid:15) (and possibly also g ) can lead to a tiny electromagneticcharge, even if q is integer .In general, the amount of kinetic mixing is a free pa-rameter and (cid:15) may even be of order one. However, if weconsider the hidden photon to be part of a hidden sec-tor we usually expect that the mixing is small. For in-stance, the hidden gauge group may be understood as alow-energy remnant of a UV theory with a unified gaugesymmetry broken at some high scale [48]. After sym-metry breaking, some degrees of freedom, for instance a while to rethink and check their validity in the fully wave-likeregime. In this sense the stability constraints may even find somenon-trivial application in this model. In this case the caveats oncoherence and backreaction discussed in the next section should,however, also be taken into account. Such a situation has, for instance, been explored in order tomediate long-range forces between hidden sector particles (see,e.g., [95, 96]). heavy fermion, usually carry a charge both under elec-tromagnetism as well as the hidden gauge group. Quan-tum mechanically, a kinetic mixing between both gaugefields is then induced by a fermion-loop of the UV theory.At low energies, the kinetic mixing parameter is deter-mined by the corresponding one-loop Feynman diagramand parametrically reads (see, e.g., [38, 48]) (cid:15) ∼ eg π log (cid:18) m ψ µ (cid:19) , (32)where m ψ is the mass of the heavy fermion and µ is theregularization scale of the loop integral. This typicallygives a small kinetic mixing, that is particularly tiny ifalso the hidden sector gauge coupling is small, g (cid:28) (cid:15) = 0. Then we argue that the main con-clusions also hold in the phenomenologically more inter-esting case with a small but non-vanishing kinetic mixingparameter. A. Secluded hidden sector
In the case (cid:15) = 0, our discussion in Section II com-pletely carries over. In particular, the DM stability con-ditions (21) and (25) in the narrow and the broad res-onance regime can be applied at all times of the cos-mic evolution. Most importantly, as there is no effectivemass of the hidden photon that could block the resonantenhancements, they can in principle be satisfied at theearliest possible time. As discussed in the previous Sec-tion II, a reasonable estimate is obtained by evaluatingthe stability condition from the narrow resonance regimeat t ≈ t ∗ . This is shown as the solid blue linein Fig. 2, where we display the allowed value of g as afunction of the DM mass m . (Note that here, we havenormalized the field to unit charge, q = 1.) The require-ment of avoiding a resonant depletion of the DM energydensity into hidden photons puts severe constraints onthe hidden gauge coupling for small masses.
1. Backreaction effects
The above evaluation might be an overestimation ofthe constraint posed by the DM stability requirement.This is because, so far, we have neglected the backreac-tion of the parametric resonance on the DM field. Obvi-ously, a first effect is the depletion of the DM field. This − − − − q e ff − − − − m [eV] − − − − g LVSLVS (TeV) c o h e r e n c e w e a k g r a v i t y c o n j e c t u r e FIG. 2. Allowed hidden gauge coupling g to the scalar DMcandidate as a function of its mass m . The blue lines cor-respond to the stability condition due to the parametric res-onance, evaluated close to the time where the field starts tooscillate, t ≈ t ∗ , (solid) and at matter-radiation equality(dashed). The red line is given by a coherence condition, dis-cussed in the main text. For comparison, the right axis showsthe typical corresponding effective millicharge induced by afermion-loop of a UV theory, q eff ∼ (cid:15)g ∼ eg / (6 π ). Alongthe same lines, the light-shaded grey area in the upper regioncorresponds to observational constraints provided by interac-tions with a magnetized intergalactic stellar medium [87, 88],see Fig. 1. is what we have implicitly used to set our constraint,i.e. using energy conservation to determine the depletionfrom the produced gauge bosons. However, if the energydensity in the hidden photons is comparable to that inthe DM field, ρ A ∼ ρ φ , it is conceivable that energy startsto be transferred back to the DM field, slowing down thedepletion. Although this is non-trivial due to the factthat most of the produced hidden photons have momenta k (cid:39) m , processes involving multiple hidden photons maybe possible due to the high occupation numbers and theresonantly enhanced interaction rates. With the presentanalysis we cannot exclude this possibility. A thoroughanalysis of this effect would need to involve some care-ful numerical simulations, which is beyond the scope ofthis work. The overall allowed value of the hidden gaugecoupling might therefore be higher.That said, let us obtain a very conservative estimate ofthe point where the resonance should shut off. Allowingfor the backreaction effect, we nevertheless expect that insuch a situation a significant fraction of the total energyin the DM-hidden photon system, possibly ∼ /
2, willbe in hidden photons and therefore in the form of darkradiation. At matter-radiation equality such a large frac-tion of dark radiation is certainly excluded (cf., e.g. [97]).Therefore, we can evaluate the DM stability condition atmatter-radiation equality. At this stage, at the latest, φ is required to behave as standard cold DM. At the sametime, we expect the system to be in a narrow resonanceregime, such that the stability condition (21) is valid.The resulting constraint on the hidden gauge coupling isshown as a dashed blue line in Fig. 2. It is considerablyweaker than the original estimate, but still affects an ap-preciable region of parameter space, bearing in mind thatthis estimate is probably overly conservative.
2. Non-trivial initial momentum distribution
In addition to the fragmentation of the light scalar DMfield via parametric resonance, there is another physicaleffect that may modify the interaction rate between theDM and the hidden photons. Crucially, in our analysis wetreat φ as a spatially homogeneous classical field. Whilesuch a situation arises naturally in the misalignment [1–5, 57] effect when the field is extremely homogenized byinflation, other production mechanisms (see, e.g., [58–77]) typically feature a non-trivial momentum distribu-tion for the millicharged particles. Hence, in such ascenario, the field exhibits spatial variations. This canbe approximately taken into account by ensuring thatthere is a sufficient amount of coherence.This has been discussed in detail in [90] (see also [17,22, 24, 29, 30] for discussions in the context of DM struc-tures) from which we summarize the main implications.In order to preserve coherence of the hidden photons pro-duced by the resonance, the width of the resonance inmomentum space has to be larger than the momentumspread of the DM, ∆ k (cid:38) ∆ k φ . The latter can be esti-mated to be ∆ k φ ∼ mv mr ( a mr /a ), where we require that φ should be non-relativistic at matter-radiation equality, v mr ∼ − (cf., e.g. [98–101]). Therefore, the conditionfor preserving coherence can be written as∆ k (cid:38) mv mr (cid:16) a mr a (cid:17) . (33)As pointed out in Section II, the width of the resonancebands in momentum space, i.e. the left hand side of thisinequality, depends on the value of the Mathieu param-eter Q . Evaluating (33) at matter-radiation equality wesee that the required width is much smaller than the mass m . Therefore, we can use the narrow resonance regimewhere ∆ k ∼ m Q . This can be immediately translatedinto a constraint on the hidden gauge coupling, which wesimilarly evaluate at matter-radiation equality. This isshown in red in Fig. 2.
3. Discussion
In general, our results, shown in Fig. 2, demonstratethat the stability requirement for a very light DM candi-date in a secluded hidden sector affects sizeable regions Alternatively, this could also be due to the backreaction effect,which likely produces DM particles of non-vanishing momentum. of parameter space. The strongest constraint is posed bythe parametric resonance stability condition evaluatedclose to the time, when the DM field starts to oscillate(solid blue). However, this neglects backreaction effectsand therefore needs to be taken with caution. A moreconservative estimate is given by evaluating the stabilitycondition at matter-radiation equality (dashed blue). Weexpect that a more careful numerical analysis would mostlikely reveal a stability condition that lies in betweenthose possibilities. Aside from backreaction effects alsoa non-trivial initial velocity distribution of the DM par-ticles, possible in some models for their production, mayallow to weaken the constraint as the resonance requiresa sufficient amount of coherence. Taking into accountthat the DM velocity must be small enough to allow forsuccessful structure formation, we find the most conser-vative constraint shown as the red line. Intriguingly, weare still able to probe large regions of parameter space.In fact, it seems challenging to motivate or constructmodels featuring gauge groups with such tiny gauge cou-plings. A famous example allowing for small gauge cou-plings is provided by the large volume scenario (LVS) oftype IIB string compactifications [102, 103]. Here, thegauge theory is supported on D-branes wrapping cyclesof the internal Calabi-Yau manifold of ten-dimensionalspacetime. The volume of these internal cycles, in turn,determines the gauge coupling, g ∼ V − / [104]. There-fore, hyper-weakly coupled gauge theories can be engi-neered by choosing an appropriate Calabi-Yau geometrythat supports large D-brane worldvolumes. In Fig. 2,we show typical values of the gauge coupling achievedin a generic (orange) and low string-scale, M S ∼ B. Non-vanishing kinetic mixing
Along the lines discussed at the beginning of the sec-tion we focus on a situation with small kinetic mixing.In a homogeneous background ϕ , the equations of mo-tion are linear in the photon A and hidden photon X and therefore couple distinct momentum modes of bothfields. In fact, after having redefined the gauge fields by A µ → A µ and X µ → X µ + (cid:15)A µ , their mode functions0 k/m − − − g FIG. 3. Floquet instabilities of the Mathieu equation for X with m = 10 − eV evaluated at t ∗ (blue). The orange dotsillustrate an approximation of the same instabilities for thekinetic mixing case with (cid:15) = 0 . m A /m = 100. Insidethe blue bands, and in between the orange points, the modefunctions can grow exponentially. satisfy¨ A + H ˙ A + (cid:18) k a + m A + (cid:15) g ϕ (cid:19) A = (cid:15)g ϕ X , ¨ X + H ˙ X + (cid:18) k a + g ϕ (cid:19) X = (cid:15)g ϕ A . (34)Here, we have already included an effective mass for thephoton, m A . Naively, the equations of motion imply thatboth the photon as well as the hidden photon modescan be enhanced by a parametric resonance induced bythe oscillating DM background. However, a resonant en-hancement of A is now parametrically weaker as com-pared to X , because its coupling contains an additionalfactor of the mixing parameter (cid:15) . This means that, forinstance, modes of the hidden photons might be growingrapidly due to a broad resonance, while the photon modesalready are in a very narrow resonance regime and notamplified efficiently. At the same time, this amplificationmight also act as an oscillating driving force on the righthand side of (34). Eventually, the growing modes of bothfields will converge to the same resonance frequency aftera certain period of time. Therefore, in general, the DMmay largely annihilate into hidden photons and also, to asmaller fraction, into visible photons which follow shortlyafter.The above observations suggest that an efficient de-pletion of the DM energy density is possible in a theoryfeaturing kinetic mixing. In practice, this is important,as the visible photon can obtain a non-negligible plasmamass, m A (cid:54) = 0, while the hidden photon is still mass-less. However, as the DM mainly annihilates into hiddenphotons, we are able to avoid plasma effects of the vis-ible photons in the early Universe almost entirely. This is illustrated in Fig. 3 where we compare the instabil-ity chart of a completely secluded hidden photon (blue)with that for non-vanishing kinetic mixing, (cid:15) = 0 . m A /m = 100 (orange points denoting the boundary).We obtain these by numerically solving the coupled equa-tions of motion for X and A for different momenta. As anexample, we choose a DM mass of m = 10 − eV and bothinstabilities are evaluated when the DM field starts tooscillate, t ∗ . Inside the instability bands, an exponentialgrowth of the momentum modes of X , i.e. rapid produc-tion of hidden photons, is possible. We can see that theunstable regions are almost identical. The plasma massin the visible sector does not prevent the resonant annihi-lation of DM into hidden photons. We expect this to betrue everywhere in parameter space for kinetic mixing pa-rameters smaller than (cid:15) (cid:46) .
1. Therefore, the allowedvalues of the hidden gauge coupling are almost identicalto what is shown in Fig. 2. To give an impression ofthe constraints of the effective millicharge, we show onthe right hand axis of Fig. 2 indicative values of the mil-licharge obtained by combining Eqs. (31) and (32). Thelight grey region indicates the experimental and observa-tional constraints on the effective charge as also shown inFig. 1. There are large regions where even our most con-servative estimate of the unstable region poses a strongerconstraint on the effective millicharge than current obser-vational bounds.
IV. CONCLUSIONS
The microscopic nature of DM that comprises largeparts of the cosmic fabric remains elusive. As suggestedby its name, so far, there is no experimental evidenceof dark matter (DM) interacting with electromagnetism.While this naively rules out any sizable electric chargeassigned to DM particles, it is still possible that theircharge is tiny, thereby strongly suppressing interactionswith photons. In this work, we have investigated thecosmological longevity of such DM particles in the sub-eV mass regime. In this mass range the DM particlesmust be bosonic and for concreteness we have chosenthem to be scalar.The millicharged particles are either minimally cou-pled to photons or their electromagnetic interaction ismediated via kinetic mixing with a massless hidden pho-ton. In both cases, due to the large occupation numbersof the light DM field, even for tiny charges the DM mayefficiently annihilate into gauge bosons via a parametricresonance [6–9].We find that, in the case of a direct coupling to photonscurrent observational constraints on the millicharge arestronger than those arising from parametric resonance, as We have also checked examples with large kinetic mixing, (cid:15) ∼ ACKNOWLEDGMENTS
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