Dark matter from an even lighter QCD axion: trapped misalignment
DDESY 21-011IFT-UAM/CSIC-20-144FTUAM-20-21
Dark matter from an even lighter QCDaxion: trapped misalignment
Luca Di Luzio a , Belen Gavela b, c , Pablo Quilez a , Andreas Ringwald a a Deutsches Elektronen-Synchrotron DESY,Notkestraße 85, D-22607 Hamburg, Germany b Departamento de Fisica Teorica, Universidad Autonoma de Madrid,Cantoblanco, 28049, Madrid, Spain c Instituto de Fisica Teorica, IFT-UAM/CSIC,Cantoblanco, 28049, Madrid, Spain
Abstract
We show that dark matter can be accounted for by an axion that solves thestrong CP problem, but is much lighter than usual due to a Z N symmetry. Thewhole mass range from the canonical QCD axion down to the ultra-light regimeis allowed, with 3 ≤ N (cid:46)
65. This includes the first proposal of a “fuzzy darkmatter” QCD axion with m a ∼ − eV. A novel misalignment mechanism occurs– trapped misalignment – due to the peculiar temperature dependence of the Z N axion potential. The dark matter relic density is enhanced because the axion fieldundergoes two stages of oscillations: it is first trapped in the wrong minimum,which effectively delays the onset of true oscillations. Trapped misalignment is moregeneral than the setup discussed here, and may hold whenever an extra source ofPeccei-Quinn breaking appears at high temperatures. Furthermore, it will be shownthat trapped misalignment can dynamically source the recently proposed kineticmisalignment mechanism. All the parameter space is within tantalizing reach of theexperimental projects for the next decades. For instance, even Phase I of CASPEr-Electric could discover this axion. E-mail: [email protected] , [email protected] , [email protected] , [email protected] a r X i v : . [ h e p - ph ] F e b ontents Z N axion dark matter 9 Z N axion potential . . . . . . . . . . . . 93.2 Z N axion misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 One stage of oscillations: simple ALP regime . . . . . . . . . . . . 143.2.2 Two stages of oscillation: trapped misalignment . . . . . . . . . . 163.2.2.1 Pure trapped misalignment: ˙ θ tr ∼ θ tr . . . . . . . 203.2.3 Solving both the strong CP problem and the nature of DM . . . . 233.3 Trapped vs. non-trapped mechanisms . . . . . . . . . . . . . . . . . . . . 25 g ∗ and g s
36B Anharmonicity function 38C Maximal relic density in the trapped+kinetic regime 39 Introduction
Were an axion-like particle (ALP) to be ever discovered, it would be compelling to explorewhether it has something to do with the strong CP problem [1–4], and/or whether it canbe a viable dark matter (DM) candidate [5–7]. The canonical QCD axion (aka “invisibleaxion”) satisfies m QCD a f a = m π f π √ z (1 + z ) , where z = m u m d (1.1)and m QCD a , f a , f π , m π , m u and m d denote respectively the axion mass, the axion scale,the pion decay constant, and the pion, up and down quark masses [8–13]. Eq. (1.1) holdswhenever QCD is the only confining group of the theory, irrespectively of the ultraviolet(UV) model details. A departure from this m a - f a relation always requires to extend thegauge confining sector beyond the Standard Model (SM) QCD group.Axions that solve the strong CP problem but are heavier than the canonical QCDaxion have been explored since long and revived in the last years [14–30]. In contrast,solutions to the strong CP problem with lighter axions were uncharted territory untilvery recently.The goal of this work is to determine the implications for DM of a freshly proposeddynamical –and technically natural– scenario [31, 32], which solves the strong CP problemwith an axion much lighter than the canonical QCD one. Its key point was to assume thatNature is endowed with a Z N symmetry realized non-linearly by the axion field [31, 32]. N mirror and degenerate worlds linked by the axion field would coexist with the samecoupling strengths as in the SM, with the exception of the effective θ k -parameters, L = N − (cid:88) k =0 (cid:20) L SM k + α s π (cid:18) θ a + 2 πk N (cid:19) G k (cid:101) G k (cid:21) + . . . . (1.2)Here, L SM k denotes exact copies of the SM total Lagrangian excluding the topological G k ˜ G k term, θ a ∈ [ − π, π ) is defined in terms of the axion field a , θ a ≡ a/f a , and the dotsstand for Z N -symmetric portal couplings among the SM copies.The solution to the strong CP problem of this Z N scenario required N to be odd.Overall, the ∼
10 orders of magnitude tuning required by the SM strong CP problem istraded by a 1 / N adjustment, where N could be as low as N = 3 (viable solutions werefound with 3 ≤ N (cid:46)
47 for f a (cid:46) . × GeV, and any N for larger values of f a [32]).The resulting axion is exponentially lighter than the canonical QCD one, because the non-perturbative contributions to its potential from the N degenerate QCD groups conspireby symmetry to suppress each other. This can be intuitively understood from the large N limit of a non-linearly realized Z N shift symmetry, which is a continuous global U (1)symmetry: the axion acts as the Goldstone boson of the discrete symmetry [33]. Its massthus vanishes asymptotically for large N as befits a U (1) Goldstone boson. Indeed, ithas been shown [32] that in the large N limit the total axion potential is given in allgenerality by V N ( θ a ) (cid:39) − m a f a N cos( N θ a ) , (1.3) The SM is identified from now on with the k = 0 sector and this label on SM quantities will be oftendropped in the following. m a (cid:39) m π f π f a √ π (cid:114) − z z N / z N , (1.4)which is exponentially suppressed ( ∝ z N ) in comparison to ( m QCD a ) . The crucial properties of such a light axion are generic and do not depend on thedetails of the putative UV completion. An important byproduct of this construction is anenhancement of all axion interactions which is universal , that is, model-independent andequal for all axion couplings, at fixed m a . The detailed exploration of the Z N paradigmand of the phenomenological constraints which do not require the axion to account forDM can be found in Ref. [32].We will determine here instead the parameter space that solves both the strong CPproblem and the nature of DM within the Z N framework under discussion. The questionof DM is of strong experimental interest and very timely, in particular given the plethoraof projects targeting light DM candidates, down to the region of “fuzzy” DM [34]. Forinstance, CASPEr-Electric [35–37] probes directly the anomalous gluonic coupling ofthe axion via an oscillating neutron electric dipole moment (nEDM): the strength ofthat coupling cannot be modified for the canonical QCD axion irrespective of the modeldetails, unlike other axion-to-SM couplings that can be selectively enhanced [38–44]. Weshow here that, in contrast, a hypothetical early discovery at CASPEr-Electric could beinterpreted as a solution to the strong CP problem via a Z N reduced-mass axion, becauseof the same-size enhancement of all axion couplings in such a scenario.It will be shown that the cosmology of the Z N axion exhibits several novel aspects.The cosmological impact of hypothetical parallel “mirror” worlds has been studied atlength in the literature (for a review, see e.g. [45]). In particular, the constraints onthe number of effective relativistic species N eff present in the early Universe imply thatthe mirror copies of the SM must be less populated –cooler– than the ordinary SMworld [46]. This requires in turn that the SM has never been in thermal contact withits mirror replica. Fortunately, mechanisms that source this world-asymmetric initialtemperatures while preserving the Z N symmetry may arise naturally in the cosmologicalevolution [47–49]. It has also been suggested that DM could be simply constituted bymirror matter, for which relevant constraints apply given the differences in temperatures.Note, nevertheless, that in most cases only one replica of the SM was considered, whilelarge N could significantly modify those analyses. Furthermore, an axionic nature of DMhas not been previously considered in the mirror world setup with a reduced-mass axion. While gravity and axion-mediated interactions are naturally small enough, the impact ofother possible ( Z N -symmetric) interactions on the thermal communication between theSM and its mirror copies will be discussed.The evolution of the Z N axion field and its contribution to the DM relic abundancewill be shown to depart drastically from both the standard case and the previouslyconsidered mirror world scenarios. Due to the peculiar temperature dependence of the Z N axion potential, the production of DM axions in terms of the misalignment mechanismis modified. The scenario results in a novel type of misalignment, with a large value of Note that the value of m a for N = 1 should be equal to m QCD a given in Eq. (1.1), while Eq. (1.4)only holds in the large N limit. An axion heavier than the QCD one has been contemplated as DM, within a Z setup which realizedlinearly the symmetry [50]. trapped in the wrong minimum (with θ = π ), which effectively delays the onset ofthe true oscillations and thus enhances the DM density. We will call this new productionmechanism trapped misalingment . Note that inflation played a crucial role in previousdynamical setups advocated to drive the initial misalignment to π [53–55], while ourmechanism results directly from temperature effects.Furthermore it will be shown that, in some regions of the parameter space, trappedmisalignment will automatically source the recently proposed kinetic misalignment mech-anism [56]. In the latter, a sizeable initial axion velocity is the source of the axionrelic abundance as opposed to the conventionally assumed initial misalignment angle.The original kinetic misalignment proposal [56] required an ad-hoc Peccei Quinn (PQ)-breaking non-renormalizable effective operator suppressed by powers of the Planck mass, M Pl . In contrast, we will show that the early stage of oscillations in the Z N axion frame-work naturally flows out into kinetic misalignment.The interplay of the different mechanisms is studied in detail. Moreover, althoughgeneral phenomenological consequences of a large misalignment angle were studied inRef. [57], we will identify the main novel consequences that follow from the scenario un-der discussion. On the phenomenological arena, we discuss the implications of the Z N reduced-mass axion for axion DM searches, namely those experiments that rely on thehypothesis that an axion or ALP sizeably contributes to the DM density. More specif-ically, the experimental prospects to probe its coupling to photons, nucleons, electronsand the nEDM operator are considered. The analysis sweeps through the whole massrange down to ultra-light axions (with masses m a (cid:28) − eV), within the Z N axionframework under discussion. The present and projected sensitivity to the number of pos-sible mirror world N will be determined, and the constraints obtained will be comparedand combined with those stemming from experiments which are independent of whetheraxions account or not for DM [32].The structure of the paper can be easily inferred from the Table of Contents. The existence of a hypothetical parallel mirror world, with microphysics identical to thatof the observable world and connected to the latter only gravitationally, has a non-trivialimpact on cosmology (for a review, see e.g. [45]). In particular, extra relativistic speciesin the early Universe (mirror photons, electrons and neutrinos) affect the number ofeffective neutrino species N eff that can be measured through: i) the abundances of lightelements –in particular Helium– at the time of Big Bang nucleosynthesis (BBN), i.e. at T ∼ While this manuscript was being written, Ref. [51] appeared where a trapping mechanism was usedin a different context, i.e. to reduce the axion DM abundance. Conversely, an alternative enhancementmechanism has also been proposed recently [52]. T ∼ .
26 eV. Present data yield [58–60] BBN: N eff = 2 . ± . , CMB: N eff = 2 . +0 . − . . (2.1)In order to satisfy these constraints in the presence of one mirror world, its temperature T (cid:48) needs to be smaller than that of the SM, T . In the Z N scenario under discussion thedifferent k > T k < T , and hencedifferent energy densities ρ k and entropies s k , ρ k = π g ∗ ( T k ) T k , s k = 2 π g s ( T k ) T k , (2.2)where g ∗ and g s denote the effective degrees of freedom related to energy and entropydensity, respectively. In principle, the functional forms g ∗ , g s could have an extra k -dependence through the respective θ k parameters, but the overall impact is expectedto be minor and will be disregarded in the following. Moreover, we assume here thatthermally produced axions give a negligible contribution to N eff . This is typically thecase for f a (cid:38) GeV [61–63].The different sectors evolve in time with separately conserved entropies, therefore theratio of entropy densities γ k ≡ ( s k /s ) is constant, while the ratio of temperatures is givenby T k T = γ k · (cid:20) g s ( T ) g s ( T k ) (cid:21) / ≡ γ k · b k ( T, T k ) , for k ≡ , . . . , N − , (2.3)where from now on T will denote the temperature of the SM copy, T ≡ T . The Hubbleexpansion rate depends on the total energy density of the Universe. In a radiationdominated era with N mirror worlds it reads H ( t ) = 4 π N − (cid:88) k =0 g ∗ ( T k ) T k M ≡ π T M g ∗ ( T ) , (2.4)with the number of effective degrees of freedom g ∗ ( T ) given by g ∗ ( T ) = g ∗ ( T ) (cid:32) N − (cid:88) k =1 c k γ k (cid:33) , (2.5)and c k ( T, T k ) ≡ [ g ∗ ( T k ) /g ∗ ( T )] · [ g s ( T ) /g s ( T k )] / .Let us discuss next the implications for N eff . In the SM, g ∗ ( T = 1 MeV) = 5 . N SMeff = 10 .
83 where N SMeff = 3 .
046 is the effective number of neutrino species. Anal-ogously, at recombination g ∗ ( T = 0 .
26 eV) = 3 .
93. It follows from Eq. (2.5) that thecontribution of the ensemble of N worlds to ∆ N eff = N eff − N SMeff at BBN and CMB isgiven byBBN: ∆ N eff = 47 g ∗ ( T = 1 MeV) N − (cid:88) k =1 c k γ k = ⇒ N − (cid:88) k =1 c k γ k < . , (2.6) For BBN, the constraint corresponds to
BBN + Y p + D in Table V of Ref. [58], translating their68.3% CL result for the number of neutrino species N ν to a 95% CL value N ν = 2 . ± .
28, and N eff = 1 . N ν . For CMB, the combination of TT+TE+EE+lowE+lensing+BAO data by Planck2018 [59, 60] is considered here; note that due to the disagreement between local and CMB measurementsof the Hubble expansion parameter H , the constraint on ∆ N eff would weaken to N eff = 3 . ± .
30 if H measurements were included. N eff = 47 g ∗ ( T = 0 .
26 eV)
N − (cid:88) k =1 c k γ k = ⇒ N − (cid:88) k =1 c k γ k < . , (2.7)where the bounds on the right-hand side stem from the constraints in Eq. (2.1). Theseexpressions illustrate an interesting prediction of the mirror world(s) scenario: the devia-tion in the number of measured effective neutrino species with respect N SMeff is not constantbut varies with temperature.
That is, it may vary with time as certain mirror species willbecome non-relativistic at different times in the evolution, because of the different tem-peratures of the mirror worlds. c k (cid:39) The temperature dependence of the function g s is quite flat in most regions of interest, g s ( T k (cid:54) =0 ) (cid:39) g s ( T ). Within this approximation b k (cid:54) =0 (cid:39) c k (cid:54) =0 (cid:39)
1, and the parameter γ k directly gives the ratios of temperatures γ k (cid:39) T k T , (2.8)which remain constant throughout the Universe evolution, with γ k < k (cid:54) = 0.This approximation is valid as far as the temperatures involved correspond to the sameplateau of the N eff distribution (see e.g. [64]). In other words, as long as the number ofrelativistic degrees of freedom does not change between T k and T and thus c k (cid:39) k because allexotic worlds are cooler than the SM one. For temperatures around the BBN region, wehave checked that the approximation is still good as long as γ k is not too small, e.g. betterthan 25% for γ k (cid:54) =0 (cid:38) .
1, which does not change noticeably the analytical expressions.We will work in this framework all through the rest of the paper, except for the numericalresults. More details can be found in Appendix A.In general, the
N − T max ,BBN: T max T < . , CMB: T max T < . . (2.9)If all mirror images of the SM are instead assumed to have the same temperature, T k ∼ T (cid:48) for k (cid:54) = 0, the most restrictive bounds follow,BBN: T (cid:48) T < . N − / , CMB: T (cid:48) T < . N − / . (2.10)It is interesting that BBN data set the most constraining bound on N eff , in spite of CMBmeasurements being more precise. This is due to the temperature dependence of themirror worlds contribution. Moreover, this non-trivial dependence represents a smokinggun for the existence of the latter, as it generically predicts incompatible measurements of N eff from BBN and CMB. Specifically, the scenario predicts in all generality the followingdiscrepancy: N BBNeff − N CMBeff = 3 . N − (cid:88) k =0 c k γ k (cid:39) . N − (cid:88) k =0 γ k . (2.11)If such a difference were to be experimentally established, it would allow to predict thetemperature of the mirror worlds e.g. in the two limiting cases in Eqs. (2.9)–(2.10).7 onstraints on portal couplings The SM must avoid thermal contact with its mirror copies all through the (post-inflation)history of the Universe so as to fullfil the condition T k (cid:54) =0 (cid:28) T . This implies that the in-teractions between the SM and its copies need to be very suppressed. Non-renormalizableinteractions such as gravity and axion-mediated ones are naturally small enough, whilethe Higgs and hypercharge kinetic portal couplings can potentially spoil the condition T k (cid:54) =0 (cid:28) T . For instance, in the Z mirror case T (cid:48) /T (cid:46) . L ⊃ κ | H | | H (cid:48) | + (cid:15)B µν B (cid:48) µν , where H and H (cid:48) ( B µν and B (cid:48) µν ) denote respectivelythe SM Higgs doublet (hypercharge field strength) and its mirror copy, and κ and (cid:15) aredimensionless couplings, to respect κ , (cid:15) (cid:46) − [65]. Even smaller couplings are neededin the Z N case with N >
2. This can suggest a ‘naturalness’ issue for the Higgs andkinetic portal couplings, as they cannot be forbidden in terms of internal symmetries.Nevertheless, such small couplings may be technically natural because of an enhanced
Poincare symmetry [66, 67]: in the limit where non-renormalizable interactions are ne-glected, the κ, (cid:15) → P N symmetry (namely, an independent space-time Poincar´e transformation P in each sec-tor). This protects those couplings from radiative corrections other than those inducedby the explicit P N breaking due to gravitational and axion-mediated interactions, whichare presumably small. The microphysics responsible for the evolution of the early SM Universe and of its mirrorcopies is almost the same. Which mechanisms can then source different temperatures forthe SM and its replicae?One difference in the microphysics of our setup is the axion coupling to the G µν ˜ G µν pseudo-scalar density in Eq. (1.2): the effective value of the θ parameter differs for eachsector k , θ k = 2 πk/ N (and thus relaxing to zero in the SM with probability 1 / N – seeSect. 3). This implies that nuclear physics would be drastically different for the SM andits mirror copies. Indeed, the one-pion scalar exchange parametrized by the effectivebaryon chiral Lagrangian, L χ PT ⊃ c + N − (cid:88) k =0 m u m d sin θ k [ m u + m d + 2 m u m d cos θ k ] / π ak f π N k τ a N k , (2.12)where π k and N k denote respectively the pion and nucleon fields and c + is an O (1) low-energy constant, induces long-range forces (i.e. not spin-suppressed in the non-relativisticlimit) among nucleons in all worlds but the SM one [68]. This may lead to differentthermalization histories of the mirror copies, modulated by their ‘distance’ k from theSM sector ( k = 0). Qualitatively, nucleons are kept longer in thermal equilibrium in themirror replicas of the SM due to this extra interaction channel, which could play a rolein generating different temperatures. However, a quantitative estimate is complicated bythe fact that the interaction in Eq. (2.12) becomes relevant only around the QCD phasetransition.Previously in the literature, mirror world scenarios leading to sufficiently differenttemperatures relied on specific inflation implementations. Some of them could also holdin our Z N scenario. For instance, in the so-called asymmetric inflation it is argued thatthe initial condition after inflation may be very different for the SM and the various8irror copies, even if they have very similar microphysics dynamics. This mechanismwas proposed in the context of an exactly Z symmetric mirror world [47, 48]. Twoinflaton fields would be present: φ for the SM and φ (cid:48) for its Z mirror copy, respectivelyslow-rolling towards the minimum of their potentials (the generalization to the case of Z N is straightforward). Inflation ends when the potential steepens and the inflaton fieldpicks up kinetic energy so that it starts to oscillate towards the minimum of its potential.In this process, quantum fluctuations can be such that the two inflaton fields do notnecessarily reach their minimum at the same time, even in the same spatial region. Inregions where φ (cid:48) reaches its minimum first, any mirror particles produced due to itsoscillations are diluted by the remaining inflation driven by the second field φ . Hence,by the time when φ reheats the SM sector, the density of mirror matter will be verysmall, and in consequence T (cid:48) (cid:28) T . Quantitatively, whether such asymmetric inflationscenario can be realized or not depends on the shape of the inflaton potential [48]. Analternative idea to achieve different temperatures is that the inflaton itself transformsnon-linearly under the Z N symmetry [49]: this automatically implies different valuesfor the inflaton fields in each mirror world, and thus different thermal histories. In thepresent paper we will not commit ourselves to a specific mechanism realizing differentSM/mirror temperatures, but simply assume as a working hypothesis T k (cid:54) =0 < T .Mirror baryonic matter has been advocated to explain partially or even totally thenature of DM, in the context of the Z -symmetric mirror world (see e.g. [45]). This maybe achieved, in spite of the lower temperature of the mirror world, by means of a properbaryogenesis mechanism in the mirror sector [46]. In the following, we will not dwell withmirror baryonic DM (which is assumed to give a negligible contribution due to the lowertemperature of the SM mirror copies) and focus instead on axionic DM. Z N axion dark matter One of the key ingredients in order to compute the axion relic density is the temperatureat the onset of oscillations around the true minimum, since it determines for how longaxion oscillations are damped (and therefore to what extent the relic density is diluted).In the traditional scenario, that corresponds to the temperature at which the Hubbleparameter is of the order of the axion mass. We analyze here the intrinsic differencesbetween the Z N scenario with a reduced-mass axion and the canonical QCD axion.An important question is whether the Z N axion can account for the observed DMdensity, without requiring any further adjustment other than the inherent 1 / N tuningneeded to solve the strong CP problem, i.e. to choose θ a = 0 as the vacuum for the SMworld among the N possible θ a values, θ a = {± π(cid:96)/ N } with (cid:96) = 0 , , . . . , ( N − /
2. Itwill be demonstrated in this section that this is indeed the case.We will focus here on the study of the axionic zero mode, assuming spatial homogene-ity and the pre-inflationary PQ breaking scenario. Z N axion potential At temperatures well below the critical temperature of the QCD phase transition, T (cid:28) T QCD (cid:39)
150 MeV, the customary QCD axion potential is well approximated by thezero temperature chiral potential. However, the chiral expansion breaks down close to T QCD . Lattice computations of the topological susceptibility χ ( T ) are instead availablefor the cross-over regime, from which the QCD axion mass is estimated. As the exact9emperature dependence is not known, we will parametrize the QCD axion potential atfinite temperature V ( θ a , T ) by weighing the zero temperature expression of the chiralaxion potential V ( θ a ) by a h ( T ) factor V ( θ a , T ) ≡ V ( θ a ) × h ( T ) = − m π f π (cid:115) − m u m d ( m u + m d ) sin (cid:18) θ a (cid:19) × h ( T ) , (3.1)where h ( T ) ≡ (cid:40) T < T
QCD (cid:16) T QCD T (cid:17) α for T > T
QCD . (3.2)Constant terms in the potential have been left implicit in this expression, as irrelevantfor the axion mass and cosmological evolution. The coefficient α parametrizes the power-law behaviour at high temperatures. Unless otherwise stated, α = 8 will be assumed,as suggested by the dilute instanton gas approximation (DIGA) and in agreement withsome lattice computations [69]. Note, however, that the potential in Eq. (3.1) differsfrom the DIGA-motivated cosine potential commonly assumed for the QCD case: ourparametrization ensures that V ( θ a , T ) is a continuous function of the temperature notonly around the minimum but for any field value θ a .The generalization of the ansatz above to the total Z N axion potential is simply givenby the sum over the N different potentials, V N ( θ a , T ) = − N − (cid:88) k =0 m π f π (cid:115) − m u m d ( m u + m d ) sin (cid:18) θ a πk N (cid:19) × h ( T k ) , (3.3)where the power-law temperature suppression is assumed to be k -independent and asgiven in Eq. (3.2), that is, α k = α for all k . Furthermore, h ( T k ) = h ( b k γ k T ) (cid:39) h ( γ k T ),see Eq. (2.3) and Appendix A. Irrespective of the specific γ k (cid:54) =0 values, three temperatureregimes can be distinguished which are qualitatively different. They are determined bythe minimum and maximum value of γ k (cid:54) =0 , γ min ≤ γ k ≤ γ max , (3.4)corresponding respectively to the coolest ( γ min T ) and the hottest ( γ max T ) mirror copiesof the SM, see Fig 1. Low temperatures
T < T
QCD
The potential for all N replicas is well approximated by the –exponentially suppressed–zero-temperature potential in Eqs. (1.3) and (1.4), V LT N ( θ a , T ) = V N ( θ a ) , (3.5)with minima at 2 πk/ N for N odd. This potential depends on all the temperatures of the different mirror worlds, but the simplifiednotation V N ( θ a , T, T . . . T N − ) ≡ V N ( θ a , T ) is used here for simplicity. This notation also reflects thefact that the SM temperature T –being the largest– is the dominant driver of the Universe evolution. Z N axion potential. The acronyms HT, M T and LT stand for high, medium and lowtemperatures, respectively. Medium temperatures T QCD < T < T
QCD /γ max In this regime, only the SM contribution to the total potential is temperature suppressed.It is weighed down by a factor (cid:0) T QCD /T (cid:1) α , while all other contributions are still wellapproximated by their zero-temperature potential, V MT N ( θ a , T ) (cid:39) (cid:18) T QCD T (cid:19) α V ( θ a ) + N − (cid:88) k =1 V ( θ a + 2 πk/ N )= (cid:20)(cid:18) T QCD T (cid:19) α − (cid:21) V ( θ a ) + N − (cid:88) k =0 V ( θ a + 2 πk/ N ) (cid:39) (cid:20)(cid:18) T QCD T (cid:19) α − (cid:21) V ( θ a ) − m π f π √ π (cid:114) − z z N − / ( − N z N cos ( N θ a ) T (cid:29) T QCD −−−−−−→ − V ( θ a ) , (3.6)where in the last step the term ∝ z N has been neglected compared to the size of V ( θ a ).It follows that the total axion potential is well approximated in this regime by minus theQCD zero-temperature axion potential, that is, the single k = 0 world contribution withthe opposite sign. The potential thus depends only on f a and its minimum is located at θ a = ± π . High temperatures
T > T
QCD /γ min The temperature effects are now important for all N replicas. In the simplified case withall N − T k> ≡ T (cid:48) , γ k> ≡ γ (cid:48) , b k> ≡ b (cid:48) ) V HT N ( θ a , T ) (cid:39) (cid:18) T QCD T (cid:19) α V ( θ a ) + (cid:18) T QCD T (cid:48) (cid:19) α N − (cid:88) k =1 V ( θ a + 2 πk/ N )11 (cid:29) −−−−→ − (cid:18) T QCD
T γ (cid:48) b (cid:48) (cid:19) α V ( θ a ) (cid:2) − ( γ (cid:48) b (cid:48) ) α (cid:3) γ (cid:48) (cid:28) −−−−→ − (cid:18) T QCD
T γ (cid:48) b (cid:48) (cid:19) α V ( θ a ) , (3.7)where in the second step we neglected again the exponentially suppressed Z N contributionat T = 0 and the definition T (cid:48) ≡ T γ (cid:48) b (cid:48) was used. As for the medium temperature regime,the total axion potential depends essentially only on f a and its minimum is located at θ a = ± π . In the more general case in which the mirror copies of the SM have differenttemperatures, only the coldest world will give a sizeable contribution to the potential,and the minimum will no longer be at ± π although typically not too far from it, seeFig. 2. - π - π π π θ a = a/f a . . . . . V N = ( θ a , T ) / m π f π T ( G e V ) - π - π π π θ a = a/f a . . . . . V N = ( θ a , T ) / m π f π T ( G e V ) Figure 2: Temperature dependence of the Z axion potential. Left panel: all mirrorworlds have the same temperature, γ k = γ = 0 .
5. Right panel: mirror worlds havedifferent temperatures, γ = 0 . , γ = 0 .
25. The low and medium temperature potentialsdo not depend on γ k , as it can be appreciated by comparing the two plots.In summary, temperature effects lead to regimes in which the minimum of the to-tal axion potential is displaced to θ a = ± π or similar large field values. This high-temperature shift of the minimum parallels that at finite density, previously identified inRefs. [32, 70, 71] and recently employed as well in Ref. [72]. This phenomenon is rich inconsequences. In particular, it follows that the axion mass at finite temperature, m a ( T ) = 1 f a (cid:118)(cid:117)(cid:117)(cid:116) d V N ( θ a , T ) dθ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) min , (3.8)is given –in the three temperature regimes discussed above– by m a ( T ) (cid:39) m π f π f a √ π (cid:114) − z z N / z N / for T < T
QCD LT √ z − z for T QCD < T < T
QCD /γ max MT √ z − z (cid:18) T QCD γ min T b min (cid:19) α for T QCD /γ min < T HT . (3.9)The LT expression for m a ( T ) equals that for the zero-temperature mass m a in Eq. (1.4),as expected. In contrast, for the intermediate temperature case it corresponds to the12achyonic mass of the QCD axion at the top of its potential, m QCD a,π ≡ f a (cid:115) d V ( θ a ) dθ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ a = π = (cid:114) z − z m QCD a , (3.10)which differs from m QCD a by a factor ∼ . m a ( T ) (cid:39) m a for T < T
QCD LT m QCD a,π for T QCD < T < T
QCD /γ max MT m QCD a,π (cid:18) T QCD γ min T b min (cid:19) α for T QCD /γ min < T HT . (3.11)In summary, at low temperatures the Z N axion mass depends both on f a and N and isexponentially suppressed by a factor z N / . In contrast, at medium and high temperaturesit only depends on f a in the large N limit, and is of the order of the canonical QCD axionmass in vacuum and at high temperature respectively. Z N axion misalignment We analyze here the consequences of the temperature-dependent Z N axion potential foraxion DM production. The axion relic density can be estimated from the solution to theequation of motion (EOM) for a classical, non-relativistic and homogeneous scalar field θ a in an expanding Universe: ¨ θ a + 3 H ˙ θ a + 1 f a V (cid:48) ( θ a ) = 0 , (3.12)where V (cid:48) ( θ a ) = dV ( θ a ) /dθ a and the spatial gradients have been neglected. For a fieldoscillating near its minimum, V (cid:48) ( θ a ) (cid:39) m a ( T ) f a θ a is a good approximation and thepotential matches that of a damped harmonic oscillator. The behaviour of the solutiondepends then on the interplay between two variables, the axion mass m a ( T ) and theHubble parameter H ( T ), which both vary with temperature.We discuss next the evolution of the axion mass as a function of T , f a and N . Asthe Universe cools down, the axion mass progressively grows from its initial vanishinglysmall value at high temperatures, until it suddenly drops down around T QCD (feeling the z N exponential suppression of the potential) and reaches then a constant value. This isshown in Fig. 3, where the Z N trajectories are depicted in blue for different values of N .The figure illustrates the N -independence of m a ( T ) at high and medium temperatures (inthe large N limit), while it becomes N -dependent at low-temperature, see also Eq. (3.9)and Fig. 2. At some point in that trajectory, the mass overcomes the Hubble friction–depicted in green – and the axion starts to oscillate.There are two qualitatively distinct regimes, depending on whether this crossing hap-pens after or before the axion mass abruptly drops at T QCD (as illustrated in Fig. 3).13 − − − − T ( GeV) − − − − − − − − − m a ( T ) v s . H ( T ) T QCD T QCD γ m crit m QCD a,π N =1 N =9 N =17 N =25 N =33 (a) f a = 4 × GeV > f crit a : Simple ALP − − − − T ( GeV) − − − − − − − − − m a ( T ) v s . H ( T ) T QCD T QCD γ m crit m QCD a,π N =1 N =9 N =17 N =25 N =33 (b) f a = 1 × GeV < f crit a : Trapped mis. Figure 3: Axion mass vs. Hubble temperature dependence. Left panel: “simple ALP”misalignment regime (i.e. scalar with constant mass) corresponding to one stage of oscil-lations. Right panel: trapped misalignment regime with two stages of oscillations. Notethe degeneracy in N of the trajectories in the medium and high-temperature regimes.The value γ k> ≡ γ (cid:48) = 1 /
10 has been used.Let us denote by f crit a the value of f a for which the canonical QCD axion would start tooscillate precisely at T QCD , f crit a ≡ m π f π m crit √ z z (cid:39) × GeV , where m crit = 1 . H ( T QCD ) (cid:39) . × − eV . (3.13)Whenever m a > m crit ( m a < m crit ) Hubble crossing takes place just (more than) oncein the cosmic trajectory. The factor 1 .
67 for m crit stems from our criteria for the onsetoscillations that better fits the numerical solution to the EOM, as argued further be-low. Strictly speaking, the value of f crit a for the Z N axion is slightly larger than that inEq. (3.13), see Eq. (3.10); this small difference will be taken into account in the numericalresults and figures, but disregarded in the discussion below for the different regimes ofoscillation. In Fig. 3, f a = f crit a corresponds to the case in which the ( green ) Hubbleline would cross the canonical QCD axion trajectory (i.e. N = 1, orange ) exactly at theflattening point of the latter, i.e. ∼ T QCD . For f a > f crit a the T = 0 potential is already developed when the axion starts to oscillate,see left panel of Fig. 3. Oscillations only take place around the true minimum θ a = 0,and the relic density corresponds to that of a “simple ALP” with constant mass. Thissingle regime of oscillations with constant mass applies as well to the standard QCDaxion scenario (i.e. N = 1 in Fig. 3) for large enough f a . The evolution of the axion fieldis straightforward and goes as follows: T > T . At high temperatures the axion field is frozen at a given and arbitrary initialmisalignment angle θ due to Hubble friction, H (cid:29) m a ( T ). As the Universe cools downthe Hubble parameter diminishes, until the friction term intercepts the m a ( T ) trajectory14t a temperature T such that 1 . H ( T ) = m a ( T ) ≡ m . (3.14)Since T < T QCD , the true vacuum potential has already developed when the axion fieldstarts to oscillate. In consequence, its mass at T is already the zero-temperature mass: m = m a , with θ a ( T ) = θ , ˙ θ = 0 . (3.15) T > T . The axion fields keeps oscillating around θ a = 0 until nowadays and its massis almost constant, ˙ m a (cid:39)
0. It is adequate to apply the WKB approximation provided m a (cid:29) (cid:8) ˙ m a /m a , ˙ H/H (cid:9) , i.e. provided the oscillations are much faster than both theexpansion rate of the Universe and the rate at which the mass changes, which is the casehere. The adiabatic approximation predicts the existence of a conserved quantity [5–7], anadiabatic invariant N a that can be interpreted as the comoving number of non-relativisticaxion quanta, defined for a generic axion mass m a as N a ≡ ρ a a m a = const . , with ρ a ≡
12 ˙ θ a f a + 12 m a f a θ a , (3.16)where a denotes the scale factor and ρ a the axion energy density. For the regime underdiscussion, it follows from N a conservation between T and today that the current energydensity ρ a, can be expressed as ρ a, = m a N a a (cid:39) m a ( θ f a ) (cid:18) a a (cid:19) , (3.17)where a and a denote respectively the value of the scale factor today and at the onset ofoscillations, a ≡ a ( T ). The latter can be expressed in turn in terms of the axion massusing Friedmann equations and the conservation of entropy, a ∝ / √ m a M Pl , allowingus to obtain the ratio of the axionic relic density to the current DM abundance ρ DM as afunction of m a , θ and f a , ρ a, ρ DM (cid:39) . (cid:114) m a eV (cid:18) θ f a GeV (cid:19) F ( T ) , (3.18)where F ( T ) ≡ ( g ∗ ( T ) / . ( g s ( T ) / . − is an O (1) factor and ρ DM (cid:39) .
26 keV / cm from Planck 2018 data [60]. For the Z N scenario under discussion, it follows from Eq. (1.4)that this ratio can be rewritten in the large N limit as ρ a, ρ DM (cid:39) . N / z N / θ (cid:18) f a GeV (cid:19) / F ( T ) . (3.19) Whenever the initial misalingment angle is large, the relic density in Eq. (3.17) needs to be correctedby the so-called anharmonicity factor, see Appendix B. The slight discrepancy with respect to the result in Ref. [73] is due to the use of the updated valueof ρ DM (cid:39) .
26 keV / cm from Planck 2018 data [60] and a different choice for the onset of the oscillationsthat better fits the numerical result (1 . H = m a instead of 3 H = m a , see e.g. Refs. [74, 75]). θ can a priori take any value in the [ − π, π ) intervaland the axion field will roll down to the closest minimum among the N possibilities.Thus it is with probability 1 / N that the final θ -parameter will correspond to the CP-conserving point θ = 0. In other words, in order for the simple ALP regime to solve thestrong CP problem and also account for DM it is necessary and sufficient that the initialmisalignment angle lies in the interval θ ∈ [ − π/ N , π/ N ) within the simple ALP regime,for any f a > × GeV.Eq. (3.17) also indicates that the simple ALP solutions which account for the relicDM obey (see Fig. 7) m a f a ∝ const . , (3.20)to be compared with the Z N axion mass relation, that for a given N predicts m a f a ∝ const . This behaviour is depicted in the { m a , /f a } plane in Fig. 6 by continuous su-perimposed lines (for different values of θ ), together with some experimentally excludedregions as a reference (while the complete set of present and projected sensitivity regionsis depicted in Fig. 11). It follows from Fig. 6 that the simple ALP Z N solutions can solveboth the strong CP problem and account for DM mass down to the region of fuzzy DM( m a ∼ − eV) and for any value of N ≥ For f a < f crit a , two different potential minima develop during the Universe evolutionfrom high to low temperatures, see Fig. 3 (right panel). In a nutshell: • A first phase of oscillations takes place around θ a = π at temperatures above T QCD ,because temperature effects are important then and the axion mass is unsuppressed. • At T QCD the true minimum θ a = 0 develops (while π becomes a maximum). Theaxion mass suddenly becomes exponentially suppressed due to the Z N symmetry.Oscillations around θ a = 0 will start whenever the kinetic energy becomes smallerthan the height of the potential barrier.These two stages are thus separated by a drastic (non-adiabatic) modification of thepotential. This novel axion DM production mechanism is named trapped misalignment .Its cosmic evolution is illustrated in Figs. 4 and 5 for toy-model trapped trajectories( blue ), versus that of a QCD-like axion ( orange ) with the same zero temperature mass.Let us expatiate next on the evolution details. T > T . In this period, the behaviour is alike to that for a simple ALP, see Eq. (3.14).Before Hubble crossing at T , the axion field is frozen and remains constant at an arbitraryinitial misalignment angle θ . T > T > T QCD . The finite temperature effective potential is relevant in this range,see Eqs. (3.6) and (3.7), and the initial axion velocity is ˙ θ = 0. Thus, the axion “istrapped” in this first stage of oscillations around θ a ∼ π from T until T QCD . Its mass is Note that the relevant axion field displacement that should enter in Eq. (3.19) corresponds to thefield distance to the closest minimum and it is only when the closest minimum is θ a = 0 that the fielddistance coincides with θ . Z N breaking by the thermal background: m a ( T ) raisesprogressively until it stabilizes at ∼ m QCD a .Trapping causes a delay of the start of oscillations around the true minimum θ a = 0 , as compared to T for the canonical QCD axion and the simple ALP scenarios. Theconsequence is an enhancement of the DM density , as the dilution time is thus shortened.In Figs. 4 and 5 the evolution for a QCD-like axion (i.e. N = 1, orange ) is shownto oscillate around θ a = 0 since well before T QCD . In contrast, for
N ≥ blue ) theoscillations around θ = π take place from T down to T QCD . The crucial impact of thetrapped stage is thus to delay the onset of oscillations around the true minimum, and toset the initial value of θ a and ˙ θ a at T QCD , θ tr ≡ θ a ( T QCD ) ∼ π, ˙ θ tr ≡ ˙ θ a ( T QCD ) . (3.21)It is not possible to predict the exact value of ˙ θ tr . However, an order of magnitudeestimate stems from the energy density when the temperature approaches T QCD , applying N a conservation to the adiabatic interval [ T , T QCD ), ρ a, QCD ≡ ρ a ( T QCD ) = m QCD a,π N a a = 12 m m QCD a,π (cid:18) a a QCD (cid:19) ( θ − π ) f a , (3.22)which shows that the precise value of ρ a, QCD depends on the arbitrary initial misalignmentangle with respect to the high-temperature minimum in π : ( θ − π ). This dependence isinherited by the mean velocity at the end of this period, which follows from the equalityof the mean kinetic and potential energies for a harmonic oscillator, (cid:113) (cid:104) ˙ θ (cid:105) = 1 √ (cid:113) m m QCD a,π (cid:18) a a QCD (cid:19) / | θ − π | . (3.23) T = T QCD . At this point, the low-temperature potential for the Z N axion scenariodevelops (see Eqs. (1.3), (1.4) and (3.5)), and thus θ a = π becomes a maximum. Theabrupt exponential suppression of the axion mass was shown in Fig. 3 for different valuesof N . It illustrates that the two stages of oscillation are separated by a sudden, inherentlynon-adiabatic, modification of the potential as ˙ m a /m a (cid:29) m a . The energy density of theaxion just after the transition at T QCD reads ρ a, tr = 12 ˙ θ f a + m a f a N (cid:2) − cos( N θ tr ) (cid:3) (cid:39)
12 ˙ θ f a + 2 m a f a N , (3.24)that is, the total energy density at this point is larger or equal than the maximum potentialenergy (the height of the barrier 2 m a f a / N ). What happens next depends crucially onthe value of ˙ θ tr , which acts as initial condition for the subsequent period. It determinesat which temperature the second stage of oscillations begins: • For very small axion velocity, ˙ θ tr (cid:28) m a / N , oscillations around one minimum maystart closely after T QCD . This case will be denoted below “pure trapped misalign-ment”. • For large enough ˙ θ tr so that the kinetic energy dominates ρ a, tr , ˙ θ tr (cid:29) m a / N , theaxion field will roll over the top of the barriers for a long time before its oscillationsstart at a temperature sensibly lower than T QCD . This case will be denoted as“trapped+kinetic misalignment”. 17 .2.2.1 Pure trapped misalignment: ˙ θ tr ∼ T QCD > T . Were the axion velocity exactly zero at T QCD , ˙ θ tr = 0, the setup would notbe viable: the axion would simply roll to the closest minimum θ a = π N ( N − θ (cid:46) − .Nevertheless, ˙ θ tr may be very small but not zero. Even a tiny kinetic energy maysuffice to allow the axion to go over some potential barriers, because the misalignmentset by the trapping corresponds to a maximum of the zero temperature potential. InSect. 3.2.3 the minimum kinetic energy necessary to roll over O ( N ) potential barriers will be estimated and shown to be attained in most of the parameter space. The axionfield will then end up in the CP-conserving minimum θ a = 0 with 1 / N probability.An estimation of the time required for the axion to go from one maximum to the nextis δt ∼ π/m a . To overfly only ∼ N barriers a very short time is required because in thisregime m a (cid:28) H , i.e. the difference between T QCD and the temperature at the onset ofoscillations around the true minimum is negligible. In conclusion, for ˙ θ tr (cid:28) m a / N thekinetic energy is not large enough to extend beyond ∼ T QCD the delay time accumulatedduring the trapped period (see Eq. (3.24)). The value of θ a when it starts to oscillatefrom the top of the barrier closest to θ a = 0 is θ ∼ π N . (3.25)The evolution is illustrated in Fig. 4, which depicts the numerical solution for a toyexample exhibiting pure trapped misalignment ( blue ) and compares it with that fora QCD-like axion ( orange ) with the same zero temperature mass. The figure reflectsthe enhancement of the relic DM density in the trapped case. The latter can be easilycomputed analogously to that for the usual misalignment mechanism. After the transitionat T QCD , the adiabatic approximation is valid again since ˙ m a (cid:39) m a (cid:29) H . From the N a conservation law –Eq.(3.16), and the initial conditions at the onset of the second stageof oscillations (cid:8) θ (cid:39) π/ N , ˙ θ (cid:39) (cid:9) , it follows that the current axionic relic density ρ a, is given in the pure trapped case by ρ a, (cid:12)(cid:12) tr = m a N a a = 12 m a (cid:16) π N f a (cid:17) (cid:18) a QCD a (cid:19) , (3.26)where a QCD ≡ a ( T QCD ). For a given final vacuum mass m a , it demonstrates an en-hancement of the relic density in the scenario with a Z N trapped axion vs. that witha simple ALP, for two main reasons: i) the relative scale dependence is larger by afactor ( a QCD /a ) and does not depend on the axion mass; ii) the initial misalignmentangle is fixed to θ tr ∼ π/ N , while for the simple ALP it is arbitrary within the in-terval [ − π/ N , π/ N ). Note that the current relic density in the pure trapped scenariobecomes independent of the initial misalignment θ , unlike the case of the simple ALPand QCD-like pre-inflationary scenarios.For the Z N reduced-mass axion under discussion, the product m a f a –and thus ρ a, in Eq. (3.26)– is a function of just one unknown: N , see m a in Eqs. (1.4) and (3.9).Substituting this dependence in the ratio ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr (cid:39) . (cid:114) m a eV (cid:18) m a m crit (cid:19) / (cid:18) π N f a GeV (cid:19) F ( T QCD ) , (3.27) Strictly speaking, about N / θ a = 0 minimum, but wewill obviate this finesse as it does not change the order of magnitude estimates intended here. π π π θ a θ a = cte st Osc. Trapped in θ a = π nd Osc. WKB, N a = cteθ a = cte Osc. WKB, N a = cte δθ i = 3 δθ i = 3 − ˙ θ a Delay onset osc.Pure trapped Mis.QCD axion-like Mis. − ρ a ∝ a − T T QCD − like1 T QCD
High Temp Low Temp − m a ( T ) v s . H . H ( T ) m a ( T ) for Z N axion m a ( T ) for QCD-like axion Figure 4: Axion evolution for the
Pure Trapped misalingment vs. that for a canonicalQCD axion, illustrating the relic density enhancement for the same final zero-temperaturemass m a .it follows that ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr ∼ . × z N √N . (3.28)Thus in the pure trapped regime the value of N compatible with simultaneously solvingthe strong CP problem and accounting for the ensemble of the DM density is N ∼ . (3.29)19his line of constant relic density is highlighted in Fig. 6 ( purple ) for relatively small f a values. Indeed, Eq. (3.26) predicts that the relic density solution in the { m a , f a } planehas to be a line parallel to those defining the canonical QCD and Z N axion bands, becausein all these cases the parametric dependence is m a f a = const . The estimate in Eq. (3.29)disregards anharmonicity corrections and in order to qualitatively account for the latterones the axion DM region will be represented as a band in the figures of Sect. 4. θ tr T QCD > T > T . If the axion velocity at T QCD is large enough, ˙ θ a (cid:29) m a / N , theaxion field has enough kinetic energy to roll many times over the barriers before it startsto oscillate around some minimum, triggering the kinetic misalignment mechanism. Theonset of oscillations is thus delayed until much later than T QCD .In this regime, the WKB (adiabatic) approximation is not valid anymore and the axionenergy density is diluted as ∝ a − due to the expansion of the Universe (as illustrated inFig. 5). This can be shown by noting that ˙ θ a f a is the Noether charge density associatedto the PQ symmetry [56] and therefore it decays as ˙ θ a ∝ a − as long as the axion masscan be neglected. In consequence, a comoving PQ charge q kin can be defined which isa conserved quantity, q kin = ˙ θ a a = const . (3.30)This regime of rapidly decreasing energy density ends up at a temperature T at whichthe kinetic energy becomes of the order of the height of the barrier, and thus the axioncan no longer overcome it,12 ˙ θ a ( T ) f a = 2 m a f a N = ⇒ ˙ θ a, ≡ ˙ θ a ( T ) = 2 m a N . (3.31)At this point the axion will start a second stage of oscillations, see Fig. 5. Enforcing theconservation of q kin from T QCD until T , and using Eq. (3.31), it follows that the scalefactor at the onset of the second stage of oscillations a ≡ a ( T ) can be expressed as q kin = ˙ θ tr a = 2 m a N a = ⇒ a = (cid:32) N ˙ θ tr m a (cid:33) / a QCD . (3.32) T > T . The axion finally starts the second stage of oscillations around one of the N zero-temperature minima at random. Thus, the 1 / N probability of the Z N axion sce-nario [31, 32] to solve the strong CP problem (i.e. the final θ -parameter to correspondingthe CP-conserving point θ = 0) will be maintained when accounting for DM.In order to determine the axion relic density today, the adiabatic approximation isagain appropriate after T . The comoving number of axions N (cid:48) a is constant during this The size of this correction will be different in general for the pure trapped case and the simple ALPregime since in the former H (cid:28) m a , see Appendix B. Equivalently, if the axion kinetic energy at the end of the trapped period ˙ θ f a / m a f a / N . Alternatively, the dilution of the axion velocity can be obtained from the axion EOM in the masslesslimit: ¨ θ a + 3 H ˙ θ a = 0 = ⇒ ˙ θ a a = const. π - π π π θ a θ a = cte st Osc. WKB, N a = cte Kinetic, q a = cte nd Osc. WKB, N a = cteθ a = cte Osc. WKB, N a = cte δθ i = 3 δθ i = 3 − ˙ θ a Delay onset osc. Trapped+Kinetic mis.QCD axion-like mis. − ρ a ∝ a − ∝ a − ∝ a − T .. T QCD T High Temp . Low Temp . − m a ( T ) v s . H T QCD − like . H ( T ) m a ( T ) for Z N axion m a ( T ) for QCD-like axion Figure 5: Axion evolution for the
Trapped+Kinetic misalignment vs. that for acanonical QCD axion, illustrating the strong enhancement of the relic density enhance-ment for the same final zero-temperature mass m a .final period (and different from that during the first oscillation period N a ). Using q kin inEq. (3.32) and the mean axion velocity at T QCD in Eq. (3.23), N (cid:48) a is well approximatedby [56] N (cid:48) a = ρ a, a m a = 2 m a f a a N = ˙ θ tr N f a a . (3.33)Finally the relic density today, ρ a, (cid:12)(cid:12) tr+kin (cid:39) m a a N (cid:48) a , (3.34)21an be written as ρ a, (cid:12)(cid:12) tr+kin = C m a (cid:113) m m QCD a,π √ N (cid:18) √ a a QCD a (cid:19) | θ − π | f a , (3.35)where C denotes a deviation from the analytical –exactly adiabatic and harmonic– esti-mations above, to be computed numerically. For our setup, we find numerically C (cid:39) blue ) with that for aQCD-like axion ( orange ) shows how the delay produced by the kinetic-dominated periodresults in an enhancement of the current axionic relic density, which is even stronger thanin the pure trapped scenario. The period of kinetic misalignment has been shortened inthe figure for illustration purposes.The comparison of the resulting axionic relic density with that for the pure trappedscenario in Eq. (3.26) shows two competing effects. In fact, for the kinetic+trappedscenario: • the dilution factor is smaller, ( √ a a QCD /a ) vs. ( a QCD /a ) ; • the relevant mass scale is larger, (cid:113) m m QCD a,π vs. m a .Hence, it will depend on the specific point of the parameter space whether the ki-netic+trapped or the pure trapped mechanism gives the dominant contribution (seeSection 3.3).The ratio of the predicted axion relic density to the observed DM density can beobtained following same the procedure applied for the simple ALP regime, yielding ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr+kin (cid:39) . (cid:114) m a eV (cid:113) m a m QCD a,π (cid:0) m m (cid:1) / (cid:20) f a GeV (cid:21) | θ − π |N F kin , ( T ) , (3.36)where F kin , ( T ) is a function of the degrees of freedom at play, which can be found inAppendix A together with further details. The relic density depends on the temperaturesof all the different SM copies via the value of m (i.e. the γ k values, see Eqs. (2.3) and(2.8)). In the simplified case in which all the copies of the SM have the same temperature T k (cid:54) =0 ∼ T (cid:48) , Eq. (3.36) can be written –for large N and for the largest possible kineticmisalignment contribution – as ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr+kin , max (cid:39) . (cid:18) f a GeV (cid:19) / z N / N / | θ − π | F kin , ( T ) , (3.37)where F kin , ( T ) is another function of the degrees of freedom at play, see Appendix A.It also follows from the results above that the trapped+kinetic DM solutions behavein the { m a , /f a } plane as ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr+kin ∝ m a f / a , (3.38) Here the maximum axion velocity at the end of the trapped period is considered, i.e. twice the meanaxion velocity in Eq. (3.23).
22n contrast to m a f a = const . for the pure trapped case in Eq. (3.26). The trapped+kinetictrajectories that account for DM are illustrated by double lines in Fig. 6, for differentvalues of the initial misalignment angle θ . We estimate here the minimum kinetic energy necessary in the trapped regime for theaxion to roll over at least O ( N ) maxima after T QCD . The result will be of general interestwhenever f a < f crit a , and of practical relevance mainly for the pure trapped misalignment.Since the trapping forces θ tr (cid:39) π , the corresponding total energy density is alwayslarger than the maximum potential energy (the height of the barrier 2 m a f a / N ). As theaxion rolls over ∼ N different maxima, the energy density is diluted due to the expansionof the Universe, ρ a ( a QCD+ N ) = ρ a, tr (cid:18) a QCD a QCD+ N (cid:19) p , (3.39)where a QCD+ N is the scale factor when the axion has rolled over just N maxima after T QCD , and p parametrizes how strong is the dilution (we find numerically that p ∼ . K min necessary for the axion to be able to roll over just N maxima before oscillating must saturate the condition (cid:18) K min + 2 m a f a N (cid:19) (cid:18) a QCD a QCD+ N (cid:19) p = 2 m a f a N . (3.40)Given the Hubble parameter in a radiation dominated Universe H ( t ) = t and the Fried-mann equation in Eq. (2.4), it follows that the temperature T QCD+ N after overcoming N maxima (corresponding to a time difference of δt ∼ π N /m a , see discussion beforeEq. (3.25)) is T QCD+ N = (cid:20) π g ∗ ( T QCD ) (cid:21) / (cid:115) M Pl t QCD + 2 π N /m a ) (cid:39) T QCD − . π N m a T M Pl , (3.41)where t QCD is the time corresponding to T QCD , and the approximation 2 π N /m a (cid:28) t QCD15 has been used, as in this regime H (cid:28) m a . The corresponding dilution factor reads (cid:18) a QCD a QCD+ N (cid:19) p (cid:39) (cid:18) T QCD+ N T QCD (cid:19) p (cid:39) − . p π N m a T M Pl . (3.42)Substituting it in Eq. (3.40), the minimum kinetic energy required is determined, K min (cid:39) . p π N m a T M Pl f a . (3.43)To evaluate how probable it is to simultaneously solve the strong CP problem and explainDM, K min needs to be compared with the mean kinetic energy generically expected at T QCD . It follows from Eq. (3.23) that K tr ≡ (cid:104) ˙ θ (cid:105) f a = 14 m m QCD a,π (cid:18) a a QCD (cid:19) ( θ − π ) f a . (3.44) Or equivalently 2 π N T (cid:28) m a M Pl .
23n the case in which all mirror copies but the SM have the same temperature T (cid:48) andtaking for example γ (cid:48) = 0 .
2, the latter equation can be rewritten as K tr (cid:39) (cid:32) T ( m QCD a,π ) M (cid:33) / ( θ − π ) f a × κ , (3.45)where κ is a numerical factor that ranges in the interval (4 − N barrier tops whenever m a (cid:46) N π . p (cid:18) T QCD √ M Pl (cid:19) / ( m QCD a,π ) / | θ − π | . (3.46)We compare next the parametric dependence of all DM axion solutions –which also solvethe strong CP problem– discussed up to this point. They are depicted in the { m a , /f a } plane of Fig. 6 and for several values of θ : N = N = N = N = N = N = N = N = N = N = N = N = N = N = Q C D a x i o n SN1987a
S u n nEDM w r o n g m i n i m a ¯ θ Q C D = − − − − − − − − − − − m a [eV] − − − λ [ m ] θ = . / N ( θ − π ) = . θ = . / N θ = . / N θ = . / N . PureTrappedTrapped . + . Kinetic . Simple . ALP f crit a m crit − − − − − − − − − − − − − − − / f a [ G e V − ] Figure 6: Z N axion solutions that solve both the strong CP problem and account forDM. The superimposed single continuous lines indicate either simple ALP solutions orpure trapped ones, while double lines denote trapped+kinetic misalignment regions. Thetenuous oblique lines for various values of N indicate Z N axion solutions to the strongCP problem. Solid colors denote regions experimentally excluded by axion-nEDM [76] orastrophysical data. Further constraints and prospects can be seen in Fig. 11. • In the excluded grey area the strong CP problem is not solved since the conditionin Eq. (3.46) is not satisfied. This precludes any DM solution with N (cid:46)
9, althoughonly for m a (cid:38) m crit (cid:39) × − eV. 24 The stars indicate for each trajectory the point below which f a is large enough( f a > f crit a ) so that the oscillations around the true minimum start after the zerotemperature potential is fully developed. This is the “simple ALP” case with DMsolutions obeying m a f a ∝ const . • Above the stars, double (single) continuous lines indicate trapped+kinetic (puretrapped) trajectories. For large values of f a within the f a < f crit a range, thetrapped+kinetic solution tends to dominate the parameter space, while for small f a the pure trapped mechanism sets the relic density. • The kinetic kick received by a given trapped+kinetic trajectory when its f a valueis smaller than f crit a (but very close to it) can be clearly seen. Above that point, theslope of the trajectory obeys m a f / a = const . , as explained around Eq. (3.38). • The pure trapped regime, that populates the top of the figure for small enough f a ,corresponds to the continuous N ∼
21 line, whose slope obeys m a f a = const . Overall, Fig. 6 shows that good solutions to both the strong CP problem and the natureof DM exist within the Z N axion scenario whenever m a (cid:46) − eV, down to the fuzzyDM region ( m a ∼ − eV). In the range m crit (cid:46) m a (cid:46) − eV the number of worldsmust be N ∼
21. For the lighter axion scenarios, m a (cid:46) m crit , values of N up to ∼
65 arepossible.Finally, note that this analysis has focused only on the axion zero mode as a first step.Within the trapped mechanism the axion field self-interactions may become importantwhenever the axion is not close to the minimum. Non-linearities will induce the produc-tion of higher momentum axion quanta, i.e. axion fragmentation [77]. The consequencesof the fragmentation of the axion field within the trapped misalignment mechanism isleft for a future work.
The trapped mechanism identified in this paper is actually more general than the axionsetup with a non-linearly realized Z N shift symmetry discussed above. Trapped misalign-ment could arise in fact in a large variety of QCD axion or ALP scenarios.In QCD axion models, any additional PQ-breaking source at low energies needs tobe extremely suppressed in order to comply with the nEDM bounds. However, the high-temperature axion potential is largely unconstrained. Indeed, any PQ-breaking sourcewhich is active only at high energies may induce a displacement of the potential minimaat high temperatures. The axion field can then get trapped in its displaced minimumfor a certain period of time, until the low-energy QCD potential develops. This candramatically change the relic density predicted today, due to the trapped misalignmentphase. An enhancement can be expected whenever the transition between the two phasesof the potential is non-adiabatic and takes place at a smaller temperature than theoscillation temperature in absence of trapping. The parametric dependence of the axionrelic density on the { m a , f a } plane is a hallmark of the different misalignment mechanismsdiscussed above. Table 1 summarizes the results presented earlier on (see Eqs. (3.20),(3.26) and (3.38)). It also shows the generic m a dependence of the axion-SM interactioncouplings g aXX ∼ /f a (with X denoting a generic SM field) for each DM production In regimes which are mainly adiabatic, a reduction of the axion relic density may result instead [51]. ρ a, g aXX ∝ /f a Simple ALP √ m a f a m / a Trapped m a f a m a Trapped+Kinetic m a f / a m / a QCD-like m / a f a m / a (3.47)Table 1: Parametric dependence of axion relic density and axion couplings g aXX (for afixed axionic relic density), for the different types of misalignment mechanisms discussedin the text.scenario discussed above. For comparison, the corresponding results for the canonicalQCD axion scenario are shown as well (within the Z N axion there is no regime wherethe QCD-like relic density formula applies other than for N = 1). In Table 1 (and inall figures), for the QCD-like and trapped+kinetic cases the DIGA value α = 8 has beenassumed (see Eq. (3.2)). The differences in slope shown in Table 1 are illustrated qualitatively in Fig. 7. Thecrossing point of all lines but the trapped+kinetic misalignment trajectory correspondsto m crit . The figure shows intuitively how the experimental parameter space is populateddifferently in the trapped mechanisms (blue tones) compared to that of the QCD axion( orange ). For a given m a to which an experiment is sensitive, the value of the signalstrength expected g aXX ∼ /f a is generically larger in the trapped regimes, and thus theexperimental reach is higher. In this section we discuss the implications of the above Z N scenario for axion DM searches,namely those experiments that rely on the hypothesis that the axion sizeably contributesto the DM relic density. More specifically, the Z N axion coupling to photons, nucleons,electrons and to the nEDM will be discussed. The whole mass range from the canonicalQCD axion mass down to the ultra-light QCD axion regime (with masses m a (cid:28) − eV) is considered, including the fuzzy DM regime down to m a ∼ − eV.All the couplings considered but the axion-nEDM coupling are model dependent: theycan be enhanced or suppressed in specific UV completions of the axion paradigm. We willexplore for each of those the parameter space coupling vs. m a . On the right-hand side ofthe figures, though, the naive expectations for f a when assuming O (1) couplings will beindicated as well. This is done for pure illustrative purposes as the relation between f a and axion couplings is model-dependent.In contrast, the value of the axion-nEDM coupling only depends on f a , that is, itonly assumes that the axion solves the strong CP problem. The same model indepen- For arbitrary α , ρ a ∝ m a f (14+3 α ) / (2(4+ α )) a in the trapped+kinetic case, while ρ a ∝ m ( α +2) / ( α +4) a f a inQCD-like misalignment, from which the corresponding m a dependence of g aXX ∝ /f a can be obtained. g aXX ( m a ) for a fixed axionic relicdensity, for the different types of misalignment discussed. For the QCD-like and thepure trapped cases, the DIGA-inspired power-law temperature suppression is assumed,i.e. α = 8. The dashed line reminds that the pure trapped Z N scenario can only takeplace for m a > m crit .dence holds for previous analyses of highly-dense stellar objects and gravitational waveprospects, which in addition did not need to assume an axionic nature of DM. Theseefforts led to strong constraints on the { m a , f a } parameter space [32, 70, 71], which canbe thus directly compared with the prospects for axion-nEDM searches.The figures below depict with solid (translucent) colors the experimentally excludedareas of parameter space (projected sensitivities). The blue tones are reserved exclusivelyfor experiments which do rely on the assumption that axions account sizeably for DM, while the remaining colors indicate searches which are independent of the nature of DM.In case the axion density ρ a provides only a fraction of the total DM relic density ρ DM ,the sensitivity to couplings of axion DM experiments should be rescaled by ( ρ a /ρ DM ) / .Crucially, we have shown in Sect. 3 that in some regions of the { m a , f a } plane the Z N axion can realize DM via the misalignment mechanism and variants thereof, depending onthe possible cosmological histories of the axion field evolution in the early Universe. Theidentified regions will be superimposed on the areas of parameter space experimentallyconstrained/projected by axion DM experiments.In order to compare the experimental panorama with the predictions of a benchmarkaxion model, the figures will also show the expectations for the coupling values within the Z N -KSVZ axion model developed in Ref. [32]. The canonical KSVZ QCD axion solution(i.e. N = 1) is shown as a thick yellow line, embedded into a faded band encompassing themodel dependency of the KSVZ axion [38, 40]. Oblique orange lines will signal insteadthe center of the displaced yellow band that corresponds to solving the strong CP problemwith other values of N , that is, for a Z N reduced-mass axion, see Eq. (1.4).The entire DM relic density can be accounted for within the Z N axion paradigm in27he regions encompassed by the purple band in the figures. These correspond to initialvalues of θ a (from the misalignment mechanism) which range from θ = 3 / N down to θ = 0 . / N . The figures illustrate that in the pure trapped regime the relic densityis independent of the initial misalignment angle. In contrast, for the simple ALP andthe trapped+kinetic mechanisms it does depend on the value of θ (in the latter case,through its dependence on the axion velocity at T QCD ). The effective axion-photon-photon coupling g aγ is defined via the Lagrangian δ L ≡ g aγ aF ˜ F , (4.1)with [78] g aγ = α πf a ( E/N − . , (4.2)where E/N takes model-dependent values ( E and N denote the model-dependent anoma-lous electromagnetic and strong contributions). This coupling is being explored by aplethora of experiments, as illustrated in Fig. 8. For reference, predictions of the bench-mark Z N -KSVZ axion model [32] are depicted for E/N = 0. The conclusion is that, fora large range of values of N , an axion-photon signal can be expected in large portions ofthe parameter space of upcoming axion DM experiments, if the Z N axion accounts forthe entire DM relic density. The axion coupling to nucleons N ( N = p, n ), g aN = C aN m N /f a , (4.3)is defined via the Lagrangian term δ L ≡ C aN ∂ µ a f a N γ µ γ N , (4.4)with [78] C ap = − . . c u − . c d − C a,sea , (4.5) C an = − . . c d − . c u − C a,sea , (4.6)where C a,sea = 0 . c s + 0 . c c + 0 . c b + 0 . c t denotes a “sea-quark”contribution in the nucleon, while in the reference Z N -KSVZ axion model c q = 0 for allquark flavours q . The parameter space of the axion-nucleon coupling vs. axion mass isdisplayed in Fig. 9. It demonstrates that no signals can be expected in the axion DMexperiments foreseen up to now. The band’s width accounts qualitatively for corrections to the analytic solutions obtained in theprevious section, see e.g. comment after Eq. (3.29) and Fig. 6. For instance, in the trapped regime afactor of 2 uncertainty on f a has been applied. = N = N = N = I A X O ADMX
Plasma haloscope − − − − − − − − − − − − m a [eV] CAST
ALPS II A B R A C A D A B R A R B F + U F C A PP ORGANMADMAX K L A S H TOORAD B R A SS D A N C E aLIGO Astrophysics
BabyIAXOBabyIAXO
Neutron stars A D B C α e m . π | g a γ | ∼ f a [ G e V ] − − − − − − − − − − − − | g a γ | [ G e V − ] Figure 8: Axion-photon coupling vs. axion mass. Axion limits readapted from [79] in-clude: laboratory axion experiments and helioscopes ( dark green ) [80–83], axion DMexperiments ( blue ) [84–93] and astrophysical bounds ( green ) [94–98]. Projected sensi-tivities appear in translucent colors. The orange oblique lines represent the theoreticalprediction for solutions to the strong CP problem within the Z N paradigm, for the bench-mark axion-photon couplings E/N = 0 and for different (odd) numbers of SM copies N .In the purple band area the Z N axion can account for the entire DM density, in additionto solve the strong CP problem. The axion coupling to electrons, g ae = C ae m e /f a , (4.7)is defined via the Lagrangian term δ L ≡ C ae ∂ µ a f a eγ µ γ e , (4.8)with [61, 106] C ae = c e + 3 α π (cid:20) EN log (cid:18) f a m e (cid:19) − . (cid:18) GeV m e (cid:19)(cid:21) , (4.9)where in the reference Z N -KSVZ axion model c e = 0 and E/N = 0. The parameter spaceof the axion-electron coupling vs. axion mass is displayed in Fig. 10. It shows that axion-magnon conversion techniques [107, 108] could barely detect a Z N reduced-mass axion29 − − − − − − − − − − − − − − − − − − − − − m a [eV] − − − − − − − − − − − | g a n | f a | Q C D ∼ M P l K S V Z N = N = N = N = N = N = N = N = N = C A S P E r - Z U L F CASPEr-ZULF (projected)
Oldcomagnetometers
Future comagnetometers C A S P E r - w i n d SN1987A
Neutron star cooling
Proton Storage Ring m N C K S V Z a n | g a n | ∼ f a [ G e V ] − − − − − − − ν a [Hz] Figure 9: Axion-neutron coupling vs. axion mass. Axion limits readapted from [79]include: axion DM experiments ( blue ) [99–103] and astrophysical bounds ( green ) [104,105]. Projected sensitivities appear in translucent colors, delimited by dashed lines.The orange oblique lines represent the theoretical prediction for the Z N axion-neutroncouplings, for different (odd) numbers of SM copies N . The purple band encompassesthe region where the Z N axion can account for the entire DM density. The two theoreticalpredictions are represented for the benchmark value c q = 0.which would account for DM. It would be interesting to explore whether experimentswhich probe the axion electron coupling with different techniques, such as QUAX [109],will be able to test this scenario in the future. The axion coupling to the nEDM operator provides a crucial test of the Z N axion setup:it only depends on f a , that is, on the assumption that the axion solves the strong CPproblem, see Eq. (4.11). Unlike the couplings discussed above, it does not depend on thedetails of the UV completion of the axion model. The signal thus depends solely on f a and m a .In fact, standard mechanisms to enhance QCD axion couplings via large Wilson coef-ficients do not modify the size of the nEDM coupling, which is fixed purely by the QCD m a - f a relation. In contrast, the Z N axion setup under discussion is responsible for a“universal” enhancement of all axion couplings (including that to the nEDM operator),for any given m a mass, following the rescaling with N of the modified m a - f a relation inEq. (1.4).Were the relic DM density to be entirely made up of axions, an oscillating nEDM30 − − − − − − − − − m a [eV] − − − − − | g a e | K S V Z LUX (Solar axions) S e m i c o ndu c t o r s Magnons(YIT, NiSP ) Magnons(Scanning)
Red giantsSolar ν White dwarf hint N = N = N = N = N = m e C K S V Z e | g e | ∼ f a [ G e V ] Figure 10: Axion-electron coupling vs. axion mass. Axion limits readapted from [79]include: solar axion experiments ( dark green ) [110, 111], axion DM experiments ( blue )[107, 108] and astrophysical bounds ( green ) [112] and hints (grey band) [113]. Projectedsensitivities appear in translucent colors, delimited by dashed lines. The orange obliquelines represent the theoretical prediction for the Z N axion-electron couplings assuming c e = 0 and E/N = 0, for different (odd) numbers of SM copies N . The purple bandencompasses the region where the Z N axion can account for the entire DM density.signal d n ( t ) could be at reach [76], d n ( t ) = g d (cid:112) ρ DM, local m a cos( m a t ) , (4.10)where g d = C anγ / ( m n f a ) is the coupling of the axion to the nEDM operator, defined viathe Lagrangian term δ L ≡ − i C anγ m n af a nσ µν γ nF µν , (4.11)where C anγ = 0 . e [114] and ρ DM , local ≈ . / cm is the local energy density ofaxion DM. For instance, the axion DM experiment CASPEr-Electric [35, 99, 115] em-ploys nuclear magnetic resonance techniques to search for an oscillating nEDM. Its reachin the { m a , f a } parameter space is displayed in Fig. 11. The reach of other techniques toprobe an oscillating nEDM based on storage rings [116] are also illustrated there, togetherwith present constraints which exclude N (cid:38) Z N axion in a large range of light masses and for 3 ≤ N (cid:46)
65 if such axion Note that the local relic density differs from the mean relic density as measured through the CMBand used in Eq. (3.27). = N = N = N = N = N = N = N = N = N = N = N = N = N = Pulsar Q C D a x i o n SN1987a
S u n nEDM − − − − − − − − − − − m a [eV] − − − λ [ m ] S t o r a g e R i n g C A S P E r P h a s e I C A S P E r P h a s e II C A S P E r P h a s e III
Superradiance − − − − − − − − − − − − − − − / f a [ G e V − ] Figure 11: Model-independent constraints in the { m a , f a } plane, including axion DMexperiments ( blue ) [76, 99, 116] astrophysical ( green ) and finite-density constraints ( or-ange ) [32, 70, 71]. The nEDM bounds have been relaxed by a factor of 3 to take intoaccount the stochastic nature of the axion field, see Ref. [117]. Projected sensitivitiesappear in translucent colors (those relative to finite-density effects can be seen in Fig. 8of [32]). The purple band encompasses the region where the Z N axion can account forthe entire DM density.accounts for DM. In particular, the Z N axion paradigm with reduced mass is –to ourknowledge– the first axion model that could explain a positive signal in CASPEr-ElectricPhase I (and large regions of Phase II), and at the same time solve the strong strong CPproblem. Canonical QCD axion models that exhibit large enough nEDM couplings to bedetectable at these experiments predict automatically too heavy axions and therefore outof their reach. The figure illustrates in addition that future proton storage ring facilitiesmay access the region of interest, albeit only in a small parameter region with large N and θ values.Furthermore, data on highly dense stellar objects share with the oscillating nEDMsignal its model-independence character: they directly provide constraints on the { m a , f a } parameter space. Those data have the added value of not relying on an axionic nature forDM. They have already set strong constraints on the { m a , f a } parameter space [32, 70, 71].For the Z N axion scenario under discussion, they implied a strict limit on the numberof mirror worlds: 3 ≤ N (cid:46)
47 for f a (cid:46) . × GeV. They also provide tantalizingdiscovery prospects for any value of f a and down to N ∼
The very light pseudoscalar DM candidates usually referred to as ultra-light axions, withmass in the range m a ∈ (cid:2) − , − (cid:3) eV, do not customarily address the strong CPproblem. This is because, for the canonical QCD axion, such light masses would implytrans-Planckian decay constants, f a > M Pl . In contrast, the Z N axion scenario is –to ourknowledge– the first axion model of fuzzy DM that can also solve the strong CP problem .Ultra-light axions are typically searched via gravitational interactions, with no refer-ence to their highly suppressed couplings to the SM (for a recent White Paper see [118]).The Z N scenario offers instead an interesting complementarity between strong CP relatedexperiments and purely gravitational probes of fuzzy dark matter.Whenever the ultra-light DM axions obey cosinus-like potentials, they are also subjectto CMB constraints on the matter-power spectrum, due to the presence of anharmonicterms which impact the growth of perturbations (see e.g. Refs. [119, 120]). This impliesa bound around the recombination era on any additional energy density fluctuationsbeyond cold DM, δρ/ρ (cid:46) − , corresponding to a ( z rec ) /f a (cid:46) − [121], with a ( z rec )denoting the axion field amplitude at the recombination redshift z rec . Expanding theaxion potential around the minimum, V ( a ) (cid:39) m a a − λ a a + . . . , the bound translatesinto a condition on the quartic coupling [121] λ a a m a a (cid:12)(cid:12)(cid:12)(cid:12) eq ∼ λ a eV m a (cid:46) − . (4.12)Ref. [121] considered such constraint in the context of the Z N axion model of Ref. [31].We here revisit the latter analysis with the formulae derived in Ref. [32] for the Z N scenario under discussion. For the latter, the potential in the large N limit –Eqs. (1.3)and (1.4)– corresponds to a quartic coupling given by λ a = m a f a / N , (4.13)and the bound in Eq. (4.12) translates into N (cid:46)
85. This constraint does not impact theDM prospects discussed in this paper, as N (cid:38)
65 values are already excluded by eitherfinite density constraint or nEDM data, see Fig. 11.In summary, the Z N scenario under discussion allows us to account for DM and solvethe strong CP problem in a sizeable fraction of the ultra-light axions parameter space: m a ∈ (cid:2) − , − (cid:3) eV. For instance, Fig. 8 shows that a photon-axion signal could beat reach for the upper masses in that range, provided θ takes sizeable values and thekinetic misalignment regime takes place. More importantly, in that entire mass rangea model-independent discovery signal is open for observation at the oscillating nEDMexperiments such as CASPEr-Electric, see Fig. 11. The latter figure also points to theexciting prospects ahead, because the near-future data from high-density stellar systemsand gravitational wave facilities should cover that entire mass region, for any value of θ and down to N ∼
9. 33
Conclusions
This work constitutes a proof-of-concept that an axion lighter –or even much lighter–than the canonical QCD one may both solve the SM strong CP problem and account forthe entire DM density of the Universe. Large regions of the { m a , f a } parameter space tothe left of the of the canonical QCD axion band can accomplish that goal.While the implications of a Z N shift symmetry to solve the strong CP problem (witha 1 / N probability and N degenerate worlds) have been previously analyzed [32], leadingto a lighter-than-usual axion, the question of DM and the cosmological evolution was leftunexplored. We showed here that the evolution of the axion field through the cosmologicalhistory departs drastically from both the standard one and from previously consideredmirror world scenarios.In particular, we identified a novel axion production mechanism which holds whenever f a (cid:46) . × GeV: trapped misalignment , which is a direct consequence of thetemperature dependence of the axion potential. Two distinct stages of oscillations takeplace. At large temperatures the minimum of the finite-temperature potential shifts fromits vacuum value, i.e. θ = 0, to large values, e.g. θ = π , where the axion field gets trappeddown to a temperature T ∼ T QCD . The axion mass is unsuppressed during this trappedperiod and thus of the order of the canonical QCD axion mass. The underlying reason isthat the SM thermal bath explicitly breaks the Z N symmetry, because its temperaturemust be higher than that of the other mirror worlds. This trapped period has a majorcosmological impact: the subsequent onset of oscillations around the true minimum at θ = 0 is delayed as compared with the standard QCD axion scenario. The result is animportant enhancement of the DM relic density. In other words, lower f a values can nowaccount for DM.We have determined the minimum kinetic energy K min required at the end of trappingfor the axion to roll over ∼ N / O ( K min ) in sizeable regions of the parameter space, fuelled by the (much largerthan in vacuum) axion mass at the end of the trapped period. In this pure trapped scenario, the final oscillations start at temperatures smaller but close to T ∼ T QCD .In fact, the axion kinetic energy at the end of trapping is shown to be in generalmuch larger than K min . Trapped misalignment then automatically seeds kinetic misalign-ment [56] between T ∼ T QCD and lower temperatures. The axion rolls for a long time overthe low-temperature potential barriers before final oscillations start at T (cid:28) T QCD , ex-tending further the delay of oscillations around the true minimum ensured by the trappedperiod. In consequence, the trapped+kinetic misalignment mechanism enhances evenmore strongly the DM relic density.Our novel trapped mechanism is more general than the Z N framework consideredhere. It could arise in a large variety of ALP or QCD axion scenarios. For instance, itmay apply to axion theories in which an explicit source of PQ breaking is active only athigh temperatures and the transition to the true vacuum is non-adiabatic. Note also thatin our scenario kinetic misalignment does not rely on the presence of non-renormalizablePQ-breaking operators required in the original formulation [56]. It is instead directlyseeded by trapped misalignment, which is itself a pure temperature effect.For values of the Z N axion scale f a (cid:38) . × GeV, the trapped mechanism doesnot take place, since there is only one stage of oscillations. The T = 0 potential is alreadydeveloped when the Hubble friction is overcome, and the axion oscillates from the start34round the true minimum θ a = 0. The relic density corresponds then to that of a simpleALP regime with constant axion mass, alike to the standard QCD axion scenario.We have determined the current axion relic density stemming from the various mis-alignment mechanisms, analyzing their different dependence on the { m a , f a , N } vari-ables. The ultimate dependence on the arbitrary initial misalignment angle has beendetermined as well for the simple ALP and trapped+kinetic scenarios. For the puretrapped scenario, the relic density turns out to be independent of the initial misalign-ment, which results in a band centered around
N ∼
21 to account for the ensemble of DM.Overall, DM solutions are found within the Z N paradigm for any value of 3 ≤ N (cid:46) f a valuesallowed in the Z N axion paradigm to solve the strong CP problem, all axion-SM couplingsare equally enhanced for a given m a . This increases the testability of the theory in currentand future experiments. In consequence, many axion DM experiments which up to nowonly aimed to target the nature of DM, are simultaneously addressing the SM strongCP problem, provided mirror worlds exist. We have studied the present and projectedexperimental sensitivity to the axion coupling to photons, electrons and nucleons, as afunction of the axion mass and N . It follows that an axion-photon signal is at reachin large portions of the parameter space of upcoming axion DM experiments, while nosuch prospects result for the coupling to nucleons, and only marginally for the couplingto electrons.A different and crucial test is provided by the aG ˜ G coupling (that fixes the value of1 /f a ), which can be entirely traded by an axion-nEDM coupling. The signal has tworemarkable properties, for any given m a : i) in all generality, it does not depend on thedetails of the putative UV completion of the axion model, unlike all other couplingsconsidered; ii) its strength is enhanced in the Z N paradigm, which is impossible in anymodel of the canonical QCD axion. It follows that the Z N paradigm is –to our knowledge–the only true axion theory that could explain a positive signal in CASPEr-Electric phaseI and in a large region of the parameter space in phase II. The reason is that a traditionalQCD axion with an nEDM coupling in the range to be probed by that experiment wouldbe automatically heavier, and therefore outside its reach. Such a signal could insteadaccount for DM and solve the strong CP problem within the Z N scenario. The sameapplies to the Storage Ring projects that aim to detect oscillating EDMs.Furthermore, our results demonstrate a beautiful synergy and complementarity be-tween the expected reach of axion DM experiments and axion experiments which areindependent of the nature of DM. For instance, oscillating nEDM experiments on oneside, and data expected from highly dense stellar objects and gravitational waves on theother, have a wide overlap in their sensitivity reach. Their combination will cover in thenext decades the full range of possible N and m a values, in the mass range from thestandard QCD axion mass down to ∼ − eV, that is, down to the fuzzy DM range.To our knowledge, the Z N axion discussed here is the first model of fuzzy DM which alsosolves the strong CP problem. Acknowledgments
We thank Gonzalo Alonso- ´Alvarez, Gary Centers, Victor Enguita, Yann Gouttenoire, BenjaminGrinstein, Lam Hui, David B. Kaplan, Ryosuke Sato, Geraldine Servant, Philip Sørensen, LucaVisinelli and Neal Weiner for illuminating discussions. The work of L.D.L. is supported by he Marie Sk(cid:32)lodowska-Curie Individual Fellowship grant AXIONRUSH (GA 840791). L.D.L.,P.Q. and A.R. acknowledge support by the Deutsche Forschungsgemeinschaft under Germany’sExcellence Strategy - EXC 2121 Quantum Universe - 390833306. M.B.G. acknowledges supportfrom the “Spanish Agencia Estatal de Investigaci´on” (AEI) and the EU “Fondo Europeo de De-sarrollo Regional” (FEDER) through the projects FPA2016-78645-P and PID2019-108892RB-I00/AEI/10.13039/501100011033. M.B.G. and P. Q. acknowledge support from the EuropeanUnion’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curiegrant agreements 690575 (RISE InvisiblesPlus) and 674896 (ITN ELUSIVES), as well as fromthe Spanish Research Agency (Agencia Estatal de Investigaci´on) through the grant IFT Cen-tro de Excelencia Severo Ochoa SEV-2016-0597. This project has received funding/supportfrom the European Union’s Horizon 2020 research and innovation programme under the MarieSklodowska-Curie grant agreement No 860881-HIDDeN. A Ratios of g ∗ and g s Throughout the paper, different ratios of the number of degrees of freedom in the SM andthe mirror worlds appear. They need to be computed numerically and correspond to O (1)corrections that in large regions of the parameter space can be neglected. This allows thederivation of simple analytical formulas for the different observables, such as the contributionsto N eff , or the axion relic density. In this Appendix we review the definition of these quantitiesand compute them numerically as a function of both the SM and mirror k temperatures fordifferent values of γ k . To this aim, the tabulated effective number of degrees of freedom (for theSM case) as a function of temperature provided in Ref. [122] will be used (see also Fig. 12). − T (GeV) g ( T ) g s ( T ) g ∗ ( T ) Figure 12: Effective number of degrees of freedom.
Ratio of temperatures vs. γ k : b k ( T , T k ) The function b k ( T, T k ) defined in Eq. (2.3), b k ( T, T k ) ≡ (cid:20) g s ( T ) g s ( T k ) (cid:21) / = T k /Tγ k , (A.1) s shown to approach 1 for T k (cid:29) T QCD in the right plot in Fig. 13. Therefore the approximation b k ( T, T k ) ∼ − T ( GeV) . . . . . . . . . b k ( T , T k ) BBN T QCD γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = − − − − T k ( GeV) . . . . . . . . . b k ( T , T k ) BBN T QCD γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = Figure 13: Numerical evaluation of b k ( T, T k ) as a function of the SM temperature T (left)and of the k-th mirror world temperature T k (right), for different values of γ k . Bounds on γ k through N eff : c k ( T , T k ) The ratio c k ( T, T k ) ≡ (cid:20) g ∗ ( T k ) g ∗ ( T ) (cid:21) (cid:20) g s ( T ) g s ( T k ) (cid:21) / , (A.2)which appears in Eq. (2.5), has been depicted in Fig. 14. The left plot shows that its valueis c k ( T = 1 MeV , T k ) (cid:46) . T ∼ c k ( T, T k ) ∼ γ k from Eq. (2.6). − T ( GeV) . . . . . . . . . c k ( T , T k ) BBN T QCD γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = − − − − T k ( GeV) . . . . . . . . . c k ( T , T k ) BBN T QCD γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = Figure 14: Numerical evaluation of c k ( T, T k ) as a function of the SM temperature T (left)and of the k-th mirror world temperature T k (right), for different values of γ k .37 rapped+kinetic misalignment: F kin , ( T , T (cid:48) ) The prediction for the relic density in the kinetic+trapped misalignment scenario in Eq. (3.36)involves the ratio F kin , ( T ) ≡ (cid:32) (cid:113) g ∗ ( T ) g ∗ ( T QCD )3 . (cid:33) . (cid:113) g s ( T ) g s ( T QCD ) , (A.3)which feeds into the final result in Eq. (3.37) (see also Appendix C) through F kin , : F kin , ( T , T (cid:48) ) ≡ F kin , ( T ) × b (cid:48) ( T , T (cid:48) ) / × c (cid:48) ( T BBN , T (cid:48)
BBN ) − / × (3 . /g ∗ ( T )) / = (cid:113) g ∗ ( T ) g ∗ ( T QCD )3 . . (cid:113) g s ( T ) g s ( T QCD ) (cid:34) g s ( T ) g s (cid:0) T (cid:48) (cid:1) (cid:35) / (cid:20) g ∗ ( T BBN ) g ∗ ( T (cid:48) BBN ) (cid:21) / × (cid:20) g s ( T (cid:48) BBN ) g s ( T BBN ) (cid:21) / (cid:18) . g ∗ ( T ) (cid:19) / , (A.4)where T (cid:48) BBN denotes the temperature of all the mirror copies of the SM when the temperatureof the latter is T BBN (cid:39) F kin , can be found in Fig. 15. It shows that its values lie in therange 0 . − .
85, depending on the temperature at the onset of the first stage of oscillations, T , and on the value of γ k . − T ( GeV) . . . . . . . . . F k i n , ( T , T k ) BBN T QCD γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = − − − − T k ( GeV) . . . . . . . . . F k i n , ( T , T k ) BBN T QCD γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = γ k = Figure 15: Numerical evaluation of F kin , ( T , T (cid:48) ) as a function of the SM temperature T (left) and of the k-th mirror world temperature T k (right), for different values of γ k . B Anharmonicity function
The relic density of a scalar field with a harmonic potential can be computed analyticallyusing the adiabatic approximation. However, pseudo Goldstone bosons are in general describedby periodic potentials that only resemble a harmonic potential close to the minimum. If theinitial misalignment angle is large θ (cid:38) π/
2, the onset of oscillations gets delayed leading toan enhancement of the relic density. This enhancement can be parametrized via the so-calledanharmonicity factor f anh . For an ALP-like regime, one can use an empirical formula proposed n Ref. [57] which encodes this deviation from the harmonic result: ρ a, = 12 m a (cid:18) a a (cid:19) θ f a f anh ( θ ) . (B.1)The definition of f anh ( θ ) thus depends on the criteria chosen to define the onset of oscillationsat temperature T . In our case, T correspond to the temperature in which the axion mass termovercomes Hubble friction as given in Eq. (3.14). For the case of a simple ALP regime with − − − − − π − | θ i | f a nh ( θ i ) Numerical resultEmpirical formula
Figure 16: Comparison of the empirical formula in Eq. (B.2) for the anharmonicity factorwith the numerical result in the simple ALP case. constant mass and a cosine potential, and for large misalignment angles π − θ (cid:46) − , ournumerical result is in excellent agreement with the Ref. [57] proposal. However, the latter doesnot apply for small misalignment angles, as it does not have the expected limiting behaviour f anh ( θ ) θ → −−−−→
1. For this reason, we will employ the following anharmonicity prescription: f anh ( θ ) = (cid:104) − log (cid:16) − ( θ /π ) (cid:17)(cid:105) . for π − θ > − , . (cid:104) log (cid:16) . π −| θ | (cid:17) + 4 log (cid:16) log (cid:16) . π −| θ | (cid:17)(cid:17)(cid:105) for π − θ < − , (B.2)where the first expression is inspired by Ref. [123], which obtained an expression for the anhar-monicity factor for the QCD axion (we have adapted it so as to fit our numerical result for theconstant mass ALP); and the second expression is taken from Ref. [57]. C Maximal relic density in the trapped+kinetic regime
The predicted relic density in the trapped+kinetic scenario of Eq. (3.35) can be rewritten as ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr+kin (cid:39) . (cid:114) m a eV (cid:18) m a m QCD a,π m (cid:19) / (cid:32) m QCD a,π m (cid:33) / (cid:18) f a GeV (cid:19) | θ |N F kin , ( T ) , (C.1) here F kin , ( T ) was given in Eq. (A.3). The relic density depends via m on the full temperature-dependent potential in Eq. (3.3) and therefore on the temperatures of all the different worlds(i.e. T and the values of γ k , see Eq. (2.3)). In the simplified case in which all the mirrorcopies of the SM have the same temperature T k (cid:54) =0 = T (cid:48) one can use the potential in Eq. (3.7)to express the factor ( m QCD a,π /m ) / as a function of the parameters of the finite-temperaturepotential, (cid:32) m QCD a,π m (cid:33) / (cid:39) γ (cid:48) b (cid:48) ( T , T (cid:48) ) (cid:113) m QCD a,π M pl T QCD α α +8 (cid:16) . × . (cid:112) g ∗ ( T ) (cid:17) − α α +16 . (C.2)For α = 8 it results in a parametric dependence on the mass scales of the scenario of the form ρ a, ρ DM ∝ (cid:114) m a eV (cid:34) m a (cid:0) m QCD a,π (cid:1) M m T (cid:35) / (cid:18) f a GeV (cid:19) | θ |N , (C.3)from which the parametric dependences shown in Table 1 follow.Finally, the maximal relic density that can be generated by the trapped+kinetic misalign-ment mechanism can be obtained as a function of f a and N , by choosing the value of γ (cid:48) thatsaturates the N eff bound in Eq. (2.6). In the case of the Z N axion mass in Eq. (1.4) we obtain ρ a, ρ DM (cid:12)(cid:12)(cid:12)(cid:12) tr+kin , max (cid:39) . (cid:18) f a GeV (cid:19) / z N / N / | θ | F kin , ( T ) , (C.4)where F kin , ( T ) can be found in Eq. (A.4). Corresponding to γ (cid:48) = 0 . / ( N − / c (cid:48)− / ( T BBN , T (cid:48)
BBN ), with c (cid:48) ≡ c k (cid:54) =0 . eferences [1] R. D. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons,” Phys.Rev. Lett. (1977) 1440–1443. 3[2] R. Peccei and H. R. Quinn, “Constraints Imposed by CP Conservation in the Presenceof Instantons,” Phys. Rev. D (1977) 1791–1797.[3] S. Weinberg, “A New Light Boson?,” Phys. Rev. Lett. (1978) 223–226.[4] F. Wilczek, “Problem of Strong P and T Invariance in the Presence of Instantons,”
Phys. Rev. Lett. (1978) 279–282. 3[5] J. Preskill, M. B. Wise, and F. Wilczek, “Cosmology of the Invisible Axion,” Phys. Lett.B (1983) 127–132. 3, 15[6] L. Abbott and P. Sikivie, “A Cosmological Bound on the Invisible Axion,”
Phys. Lett. B (1983) 133–136.[7] M. Dine and W. Fischler, “The Not So Harmless Axion,”
Phys. Lett. B (1983)137–141. 3, 15[8] J. E. Kim, “Weak Interaction Singlet and Strong CP Invariance,”
Phys. Rev. Lett. (1979) 103. 3[9] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Can Confinement Ensure NaturalCP Invariance of Strong Interactions?,” Nucl. Phys.
B166 (1980) 493–506.[10] A. R. Zhitnitsky, “On Possible Suppression of the Axion Hadron Interactions. (InRussian),”
Sov. J. Nucl. Phys. (1980) 260. [Yad. Fiz.31,497(1980)].[11] M. Dine, W. Fischler, and M. Srednicki, “A Simple Solution to the Strong CP Problemwith a Harmless Axion,” Phys. Lett.
B104 (1981) 199–202.[12] J. E. Kim, “A Composite Invisible Axion,”
Phys. Rev.
D31 (1985) 1733.[13] K. Choi and J. E. Kim, “Dynamical Axion,”
Phys. Rev.
D32 (1985) 1828. 3[14] V. A. Rubakov, “Grand unification and heavy axion,”
JETP Lett. (1997) 621–624, arXiv:hep-ph/9703409 [hep-ph] . 3[15] Z. Berezhiani, L. Gianfagna, and M. Giannotti, “Strong CP problem and mirror world:The Weinberg-Wilczek axion revisited,” Phys. Lett.
B500 (2001) 286–296, arXiv:hep-ph/0009290 [hep-ph] .[16] L. Gianfagna, M. Giannotti, and F. Nesti, “Mirror world, supersymmetric axion andgamma ray bursts,”
JHEP (2004) 044, arXiv:hep-ph/0409185 .[17] S. D. H. Hsu and F. Sannino, “New solutions to the strong CP problem,” Phys. Lett.
B605 (2005) 369–375, arXiv:hep-ph/0408319 [hep-ph] .[18] A. Hook, “Anomalous solutions to the strong CP problem,”
Phys. Rev. Lett. no. 14,(2015) 141801, arXiv:1411.3325 [hep-ph] .[19] H. Fukuda, K. Harigaya, M. Ibe, and T. T. Yanagida, “Model of visible QCD axion,”
Phys. Rev.
D92 no. 1, (2015) 015021, arXiv:1504.06084 [hep-ph] .
20] C.-W. Chiang, H. Fukuda, M. Ibe, and T. T. Yanagida, “750 GeV diphoton resonance ina visible heavy QCD axion model,”
Phys. Rev.
D93 no. 9, (2016) 095016, arXiv:1602.07909 [hep-ph] .[21] S. Dimopoulos, A. Hook, J. Huang, and G. Marques-Tavares, “A Collider ObservableQCD Axion,”
JHEP (2016) 052, arXiv:1606.03097 [hep-ph] .[22] T. Gherghetta, N. Nagata, and M. Shifman, “A Visible QCD Axion from an EnlargedColor Group,” Phys. Rev.
D93 no. 11, (2016) 115010, arXiv:1604.01127 [hep-ph] .[23] A. Kobakhidze, “Heavy axion in asymptotically safe QCD,” arXiv:1607.06552[hep-ph] .[24] P. Agrawal and K. Howe, “Factoring the Strong CP Problem,”
JHEP (2018) 029, arXiv:1710.04213 [hep-ph] .[25] P. Agrawal and K. Howe, “A Flavorful Factoring of the Strong CP Problem,” JHEP (2018) 035, arXiv:1712.05803 [hep-ph] .[26] M. K. Gaillard, M. B. Gavela, R. Houtz, P. Quilez, and R. Del Rey, “Color unifieddynamical axion,” Eur. Phys. J.
C78 no. 11, (2018) 972, arXiv:1805.06465 [hep-ph] .[27] M. A. Buen-Abad and J. Fan, “Dynamical axion misalignment with small instantons,”
JHEP (2019) 161, arXiv:1911.05737 [hep-ph] .[28] A. Hook, S. Kumar, Z. Liu, and R. Sundrum, “High Quality QCD Axion and the LHC,” Phys. Rev. Lett. no. 22, (2020) 221801, arXiv:1911.12364 [hep-ph] .[29] C. Cs´aki, M. Ruhdorfer, and Y. Shirman, “UV Sensitivity of the Axion Mass fromInstantons in Partially Broken Gauge Groups,”
JHEP (2020) 031, arXiv:1912.02197 [hep-ph] .[30] T. Gherghetta and M. D. Nguyen, “A Composite Higgs with a Heavy CompositeAxion,” arXiv:2007.10875 [hep-ph] . 3[31] A. Hook, “Solving the Hierarchy Problem Discretely,” Phys. Rev. Lett. no. 26,(2018) 261802, arXiv:1802.10093 [hep-ph] . 3, 20, 33[32] L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald, “An even lighter QCD axion,”
IFT-UAM/CSIC-20-143 FTUAM-20-21 DESY 21-010 To appear, 2020 . 3, 4, 5, 12, 20,27, 28, 32, 33, 34[33] S. Das and A. Hook, “Non-linearly realized discrete symmetries,”
JHEP (2020) 071, arXiv:2006.10767 [hep-ph] . 3[34] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, “Ultralight scalars as cosmologicaldark matter,” Phys. Rev. D no. 4, (2017) 043541, arXiv:1610.08297[astro-ph.CO] . 4[35] D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, and A. Sushkov, “Proposal fora Cosmic Axion Spin Precession Experiment (CASPEr),” Phys. Rev. X4 no. 2, (2014)021030, arXiv:1306.6089 [hep-ph] . 4, 31[36] D. F. Jackson Kimball et al. , “Overview of the Cosmic Axion Spin PrecessionExperiment (CASPEr),” arXiv:1711.08999 [physics.ins-det] .
37] D. Jackson Kimball et al. , “Overview of the Cosmic Axion Spin Precession Experiment(CASPEr),”
Springer Proc. Phys. (2020) 105–121, arXiv:1711.08999[physics.ins-det] . 4[38] L. Di Luzio, F. Mescia, and E. Nardi, “Redefining the Axion Window,”
Phys. Rev. Lett. no. 3, (2017) 031801, arXiv:1610.07593 [hep-ph] . 4, 27[39] M. Farina, D. Pappadopulo, F. Rompineve, and A. Tesi, “The photo-philic QCD axion,”
JHEP (2017) 095, arXiv:1611.09855 [hep-ph] .[40] L. Di Luzio, F. Mescia, and E. Nardi, “Window for preferred axion models,” Phys. Rev.D no. 7, (2017) 075003, arXiv:1705.05370 [hep-ph] . 27[41] P. Agrawal, J. Fan, M. Reece, and L.-T. Wang, “Experimental Targets for PhotonCouplings of the QCD Axion,” JHEP (2018) 006, arXiv:1709.06085 [hep-ph] .[42] G. Marques-Tavares and M. Teo, “Light axions with large hadronic couplings,” JHEP (2018) 180, arXiv:1803.07575 [hep-ph] .[43] L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, “The landscape of QCD axionmodels,” Phys. Rept. (2020) 1–117, arXiv:2003.01100 [hep-ph] .[44] L. Darm´e, L. Di Luzio, M. Giannotti, and E. Nardi, “Selective enhancement of the QCDaxion couplings,” arXiv:2010.15846 [hep-ph] . 4[45] Z. Berezhiani, “Mirror world and its cosmological consequences,”
Int. J. Mod. Phys. A (2004) 3775–3806, arXiv:hep-ph/0312335 . 4, 5, 9[46] Z. Berezhiani, D. Comelli, and F. L. Villante, “The Early mirror universe: Inflation,baryogenesis, nucleosynthesis and dark matter,” Phys. Lett.
B503 (2001) 362–375, arXiv:hep-ph/0008105 [hep-ph] . 4, 9[47] E. W. Kolb, D. Seckel, and M. S. Turner, “The Shadow World,”
Nature (1985)415–419. 4, 9[48] V. S. Berezinsky and A. Vilenkin, “Ultrahigh-energy neutrinos from hidden sectortopological defects,”
Phys. Rev.
D62 (2000) 083512, arXiv:hep-ph/9908257 [hep-ph] .9[49] G. Dvali, E. Koutsangelas, and F. Kuhnel, “Compact Dark Matter Objects via N DarkSectors,”
Phys. Rev. D (2020) 083533, arXiv:1911.13281 [astro-ph.CO] . 4, 9[50] M. Giannotti, “Mirror world and axion: Relaxing cosmological bounds,”
Int. J. Mod.Phys. A (2005) 2454–2458, arXiv:astro-ph/0504636 . 4[51] S. Nakagawa, F. Takahashi, and M. Yamada, “Trapping Effect for QCD Axion DarkMatter,” arXiv:2012.13592 [hep-ph] . 5, 25[52] R. Alonso and J. Scholtz, “Kick-alignment: matter asymmetry sourced dark matter,” arXiv:2012.14907 [hep-ph] . 5[53] R. T. Co, E. Gonzalez, and K. Harigaya, “Axion Misalignment Driven to the Hilltop,” JHEP (2019) 163, arXiv:1812.11192 [hep-ph] . 5[54] F. Takahashi and W. Yin, “QCD axion on hilltop by a phase shift of π ,” JHEP (2019) 120, arXiv:1908.06071 [hep-ph] .
55] J. Huang, A. Madden, D. Racco, and M. Reig, “Maximal axion misalignment from aminimal model,” arXiv:2006.07379 [hep-ph] . 5[56] R. T. Co, L. J. Hall, and K. Harigaya, “Kinetic Misalignment Mechanism,” arXiv:1910.14152 [hep-ph] . 5, 20, 21, 22, 34[57] A. Arvanitaki, S. Dimopoulos, M. Galanis, L. Lehner, J. O. Thompson, andK. Van Tilburg, “Large-misalignment mechanism for the formation of compact axionstructures: Signatures from the QCD axion to fuzzy dark matter,”
Phys. Rev. D no. 8, (2020) 083014, arXiv:1909.11665 [astro-ph.CO] . 5, 39[58] B. D. Fields, K. A. Olive, T.-H. Yeh, and C. Young, “Big-Bang Nucleosynthesis AfterPlanck,”
JCAP (2020) 010, arXiv:1912.01132 [astro-ph.CO] . 6[59] Planck
Collaboration, Y. Akrami et al. , “Planck 2018 results. I. Overview and thecosmological legacy of Planck,”
Astron. Astrophys. (2020) A1, arXiv:1807.06205[astro-ph.CO] . 6[60]
Planck
Collaboration, N. Aghanim et al. , “Planck 2018 results. VI. Cosmologicalparameters,” arXiv:1807.06209 [astro-ph.CO] . 6, 15[61] S. Chang and K. Choi, “Hadronic axion window and the big bang nucleosynthesis,”
Phys. Lett. B (1993) 51–56, arXiv:hep-ph/9306216 . 6, 29[62] S. Hannestad, A. Mirizzi, and G. Raffelt, “New cosmological mass limit on thermal relicaxions,”
JCAP (2005) 002, arXiv:hep-ph/0504059 .[63] L. Di Luzio, G. Martinelli, and G. Piazza, “Axion hot dark matter bound, reliably,” arXiv:2101.10330 [hep-ph] . 6[64] D. Baumann, “Primordial Cosmology,” PoS
TASI2017 (2018) 009, arXiv:1807.03098[hep-th] . 7[65] Z. Berezhiani, “Through the looking-glass: Alice’s adventures in mirror world,” arXiv:hep-ph/0508233 [hep-ph] . 8[66] R. R. Volkas, A. J. Davies, and G. C. Joshi, “Naturalness of the invisible axion model,”
Phys. Lett.
B215 (1988) 133–138. 8[67] R. Foot, A. Kobakhidze, K. L. McDonald, and R. R. Volkas, “Poincare protection for anatural electroweak scale,”
Phys. Rev. D no. 11, (2014) 115018, arXiv:1310.0223[hep-ph] . 8[68] L. Ubaldi, “Effects of theta on the deuteron binding energy and the triple-alphaprocess,” Phys. Rev.
D81 (2010) 025011, arXiv:0811.1599 [hep-ph] . 8[69] S. Borsanyi et al. , “Calculation of the axion mass based on high-temperature latticequantum chromodynamics,”
Nature no. 7627, (2016) 69–71, arXiv:1606.07494[hep-lat] . 10[70] A. Hook and J. Huang, “Probing axions with neutron star inspirals and other stellarprocesses,”
JHEP (2018) 036, arXiv:1708.08464 [hep-ph] . 12, 27, 32[71] J. Huang, M. C. Johnson, L. Sagunski, M. Sakellariadou, and J. Zhang, “Prospects foraxion searches with Advanced LIGO through binary mergers,” Phys. Rev. D no. 6,(2019) 063013, arXiv:1807.02133 [hep-ph] . 12, 27, 32
72] D. Brzeminski, Z. Chacko, A. Dev, and A. Hook, “A Time-Varying Fine StructureConstant from Naturally Ultralight Dark Matter,” arXiv:2012.02787 [hep-ph] . 12[73] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, “WISPyCold Dark Matter,”
JCAP (2012) 013, arXiv:1201.5902 [hep-ph] . 15[74] G. Alonso- ´Alvarez, T. Hugle, and J. Jaeckel, “Misalignment \ & Co.: (Pseudo-)scalarand vector dark matter with curvature couplings,” JCAP (2020) 014, arXiv:1905.09836 [hep-ph] . 15[75] D. J. E. Marsh, “Axion Cosmology,” Phys. Rept. (2016) 1–79, arXiv:1510.07633[astro-ph.CO] . 15[76] P. W. Graham and S. Rajendran, “New Observables for Direct Detection of Axion DarkMatter,”
Phys. Rev.
D88 (2013) 035023, arXiv:1306.6088 [hep-ph] . 24, 31, 32[77] N. Fonseca, E. Morgante, R. Sato, and G. Servant, “Axion fragmentation,”
JHEP (2020) 010, arXiv:1911.08472 [hep-ph] . 25[78] G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, “The QCD axion,precisely,” JHEP (2016) 034, arXiv:1511.02867 [hep-ph] . 28[79] C. O’Hare, “cajohare/axionlimits: Axionlimits,” July, 2020. https://doi.org/10.5281/zenodo.3932430 . 29, 30, 31[80] R. B¨ahre et al. , “Any light particle search II —Technical Design Report,” JINST (2013) T09001, arXiv:1302.5647 [physics.ins-det] . 29[81] E. Armengaud et al. , “Conceptual Design of the International Axion Observatory(IAXO),” JINST (2014) T05002, arXiv:1401.3233 [physics.ins-det] .[82] CAST
Collaboration, V. Anastassopoulos et al. , “New CAST Limit on theAxion-Photon Interaction,”
Nature Phys. (2017) 584–590, arXiv:1705.02290[hep-ex] .[83] BabyIAXO
Collaboration, A. Abeln et al. , “Conceptual Design of BabyIAXO, theintermediate stage towards the International Axion Observatory,” arXiv:2010.12076[physics.ins-det] . 29[84] C. Hagmann, P. Sikivie, N. Sullivan, and D. Tanner, “Results from a search for cosmicaxions,”
Phys. Rev. D (1990) 1297–1300. 29[85] B. T. McAllister, G. Flower, E. N. Ivanov, M. Goryachev, J. Bourhill, and M. E. Tobar,“The ORGAN Experiment: An axion haloscope above 15 GHz,” Phys. Dark Univ. (2017) 67–72, arXiv:1706.00209 [physics.ins-det] .[86] D. Alesini, D. Babusci, D. Di Gioacchino, C. Gatti, G. Lamanna, and C. Ligi, “TheKLASH Proposal,” arXiv:1707.06010 [physics.ins-det] .[87] HAYSTAC
Collaboration, L. Zhong et al. , “Results from phase 1 of the HAYSTACmicrowave cavity axion experiment,”
Phys. Rev. D no. 9, (2018) 092001, arXiv:1803.03690 [hep-ex] .[88] D. J. Marsh, K.-C. Fong, E. W. Lentz, L. Smejkal, and M. N. Ali, “Proposal to DetectDark Matter using Axionic Topological Antiferromagnets,” Phys. Rev. Lett. no. 12,(2019) 121601, arXiv:1807.08810 [hep-ph] . ADMX
Collaboration, T. Braine et al. , “Extended Search for the Invisible Axion withthe Axion Dark Matter Experiment,”
Phys. Rev. Lett. no. 10, (2020) 101303, arXiv:1910.08638 [hep-ex] .[90] S. Lee, S. Ahn, J. Choi, B. Ko, and Y. Semertzidis, “Axion Dark Matter Search around6.7 µ eV,” Phys. Rev. Lett. no. 10, (2020) 101802, arXiv:2001.05102 [hep-ex] .[91] S. Beurthey et al. , “MADMAX Status Report,” arXiv:2003.10894[physics.ins-det] .[92] D. Alesini et al. , “Search for Invisible Axion Dark Matter of mass m a = 43 µ eV with theQUAX– aγ Experiment,” arXiv:2012.09498 [hep-ex] .[93] M. Lawson, A. J. Millar, M. Pancaldi, E. Vitagliano, and F. Wilczek, “Tunable axionplasma haloscopes,”
Phys. Rev. Lett. no. 14, (2019) 141802, arXiv:1904.11872[hep-ph] . 29[94]
H.E.S.S.
Collaboration, A. Abramowski et al. , “Constraints on axionlike particles withH.E.S.S. from the irregularity of the PKS 2155-304 energy spectrum,”
Phys. Rev. D no. 10, (2013) 102003, arXiv:1311.3148 [astro-ph.HE] . 29[95] A. Ayala, I. Dom´ınguez, M. Giannotti, A. Mirizzi, and O. Straniero, “Revisiting thebound on axion-photon coupling from Globular Clusters,” Phys. Rev. Lett. no. 19,(2014) 191302, arXiv:1406.6053 [astro-ph.SR] .[96] A. Payez, C. Evoli, T. Fischer, M. Giannotti, A. Mirizzi, and A. Ringwald, “Revisitingthe SN1987A gamma-ray limit on ultralight axion-like particles,”
JCAP no. 02,(2015) 006, arXiv:1410.3747 [astro-ph.HE] .[97]
Fermi-LAT
Collaboration, M. Ajello et al. , “Search for Spectral Irregularities due toPhoton–Axionlike-Particle Oscillations with the Fermi Large Area Telescope,”
Phys.Rev. Lett. no. 16, (2016) 161101, arXiv:1603.06978 [astro-ph.HE] .[98] M. D. Marsh, H. R. Russell, A. C. Fabian, B. P. McNamara, P. Nulsen, and C. S.Reynolds, “A New Bound on Axion-Like Particles,”
JCAP (2017) 036, arXiv:1703.07354 [hep-ph] . 29[99] D. F. Jackson Kimball et al. , “Overview of the Cosmic Axion Spin PrecessionExperiment (CASPEr),” Springer Proc. Phys. (2020) 105–121, arXiv:1711.08999[physics.ins-det] . 30, 31, 32[100] T. Wu et al. , “Search for Axionlike Dark Matter with a Liquid-State Nuclear SpinComagnetometer,”
Phys. Rev. Lett. no. 19, (2019) 191302, arXiv:1901.10843[hep-ex] .[101] A. Garcon et al. , “Constraints on bosonic dark matter from ultralow-field nuclearmagnetic resonance,”
Sci. Adv. no. 10, (2019) eaax4539, arXiv:1902.04644[hep-ex] .[102] I. M. Bloch, Y. Hochberg, E. Kuflik, and T. Volansky, “Axion-like Relics: NewConstraints from Old Comagnetometer Data,” JHEP (2020) 167, arXiv:1907.03767[hep-ph] .[103] P. W. Graham, S. Haciomeroglu, D. E. Kaplan, Z. Omarov, S. Rajendran, and Y. K.Semertzidis, “Storage Ring Probes of Dark Matter and Dark Energy,” arXiv:2005.11867 [hep-ph] . 30 Phys. Rev. C no. 3, (2018) 035802, arXiv:1806.07991 [astro-ph.HE] . 30[105] P. Carenza, T. Fischer, M. Giannotti, G. Guo, G. Mart´ınez-Pinedo, and A. Mirizzi,“Improved axion emissivity from a supernova via nucleon-nucleon bremsstrahlung,” JCAP no. 10, (2019) 016, arXiv:1906.11844 [hep-ph] . [Erratum: JCAP 05, E01(2020)]. 30[106] M. Srednicki, “Axion Couplings to Matter. 1. CP Conserving Parts,” Nucl. Phys. B (1985) 689–700. 29[107] S. Chigusa, T. Moroi, and K. Nakayama, “Detecting light boson dark matter throughconversion into a magnon,”
Phys. Rev. D no. 9, (2020) 096013, arXiv:2001.10666[hep-ph] . 29, 31[108] A. Mitridate, T. Trickle, Z. Zhang, and K. M. Zurek, “Detectability of Axion DarkMatter with Phonon Polaritons and Magnons,”
Phys. Rev. D no. 9, (2020) 095005, arXiv:2005.10256 [hep-ph] . 29, 31[109]
QUAX
Collaboration, N. Crescini et al. , “Axion search with a quantum-limitedferromagnetic haloscope,”
Phys. Rev. Lett. no. 17, (2020) 171801, arXiv:2001.08940 [hep-ex] . 30[110] I. M. Bloch, R. Essig, K. Tobioka, T. Volansky, and T.-T. Yu, “Searching for DarkAbsorption with Direct Detection Experiments,”
JHEP (2017) 087, arXiv:1608.02123 [hep-ph] . 31[111] LUX
Collaboration, D. Akerib et al. , “First Searches for Axions and Axionlike Particleswith the LUX Experiment,”
Phys. Rev. Lett. no. 26, (2017) 261301, arXiv:1704.02297 [astro-ph.CO] . 31[112] F. Capozzi and G. Raffelt, “Axion and neutrino bounds improved with new calibrationsof the tip of the red-giant branch using geometric distance determinations,”
Phys. Rev.D no. 8, (2020) 083007, arXiv:2007.03694 [astro-ph.SR] . 31[113] M. Giannotti, I. G. Irastorza, J. Redondo, A. Ringwald, and K. Saikawa, “StellarRecipes for Axion Hunters,”
JCAP (2017) 010, arXiv:1708.02111 [hep-ph] . 31[114] M. Pospelov and A. Ritz, “Theta vacua, QCD sum rules, and the neutron electric dipolemoment,”
Nucl. Phys.
B573 (2000) 177–200, arXiv:hep-ph/9908508 [hep-ph] . 31[115] D. Aybas et al. , “Search for axion-like dark matter using solid-state nuclear magneticresonance,” arXiv:2101.01241 [hep-ex] . 31[116] S. P. Chang, S. Haciomeroglu, O. Kim, S. Lee, S. Park, and Y. K. Semertzidis,“Axionlike dark matter search using the storage ring EDM method,”
Phys. Rev. D no. 8, (2019) 083002, arXiv:1710.05271 [hep-ex] . 31, 32[117] G. P. Centers et al. , “Stochastic fluctuations of bosonic dark matter,” arXiv:1905.13650 [astro-ph.CO] . 32[118] D. Grin, M. A. Amin, V. Gluscevic, R. Hlˇ ozek, D. J. Marsh, V. Poulin,C. Prescod-Weinstein, and T. L. Smith, “Gravitational probes of ultra-light axions,” arXiv:1904.09003 [astro-ph.CO] . 33 Phys. Rev. D no. 2, (2018) 023529, arXiv:1709.07946 [astro-ph.CO] . 33[120] V. Poulin, T. L. Smith, D. Grin, T. Karwal, and M. Kamionkowski, “Cosmologicalimplications of ultralight axionlike fields,” Phys. Rev. D no. 8, (2018) 083525, arXiv:1806.10608 [astro-ph.CO] . 33[121] J. A. Dror and J. M. Leedom, “The Cosmological Tension of Ultralight Axion DarkMatter and its Solutions,” arXiv:2008.02279 [hep-ph] . 33[122] K. Saikawa and S. Shirai, “Primordial gravitational waves, precisely: The role ofthermodynamics in the Standard Model,” JCAP (2018) 035, arXiv:1803.01038[hep-ph] . 36[123] D. Lyth, “Axions and inflation: Sitting in the vacuum,” Phys. Rev. D (1992)3394–3404. 39(1992)3394–3404. 39