The DFSZ axion in the CMB
TThe DFSZ axion in the CMB
Ricardo Z. Ferreira ∗ Nordita, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, SE-106 91 Stockholm, SwedenandInstitut de F´ısica d’Altes Energies (IFAE) and TheBarcelona Institute of Science and Technology (BIST),Campus UAB, 08193 Bellaterra, Barcelona
Alessio Notari † Departament de F´ısica Qu`antica i Astrofis´ıca & Institut de Ci`encies del Cosmos (ICCUB),Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain
Fabrizio Rompineve ‡ Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA (Dated: December 14, 2020)We perform for the first time a dedicated analysis of cosmological constraints on DFSZ QCD axionmodels. Such constructions are especially interesting in light of the recent Xenon-1T excess and ofhints from stellar cooling. In DFSZ models, for m a (cid:38) . m a ≤ . c aπ ismaximal. Constraints on m a , instead, can be significantly relaxed when c aπ is small. In particular,we point out that in the so-called DFSZ-II model, where the axion coupling to leptons does notvanish simultaneously with c aπ , production via muons gives m a ≤ . m a can be fully lifted. We then combine cosmological data withrecent hints of a DFSZ axion coupled to electrons from the Xenon-1T experiment, finding in thiscase that the axion mass is constrained to be in the window 0 .
07 eV (cid:46) m a (cid:46) . .
3) eV for theDFSZ-I (DFSZ-II) model. A similar analysis with stellar cooling hints gives 3 meV (cid:46) m a (cid:46) . σ for a DFSZ-I axion. I. INTRODUCTION
The QCD axion [1, 2] is arguably among the bestmotivated hypothetical addition to the Standard Model(SM) of particles physics. By coupling to QCD, itprovides an elegant solution to the CP problem ofstrong interactions, while its oscillations could explainthe observed dark matter in the Universe. However,its experimental detection is inevitably very challenging,due to its very weak interactions with the SM. It isthus of crucial importance to investigate complementaryprobes of the QCD axion, such as cosmological andastrophysical observations, and to combine them withcurrent laboratory searches.The possibility to detect imprints of the QCDaxion in the Cosmic Microwave Background (CMB) isparticularly exciting. Indeed, through the weak butunavoidable interactions with the SM, QCD axion quantacan be thermally produced above [3–6], across [7] and/or ∗ [email protected] † [email protected] ‡ [email protected] below [8–11] the ElectroWeak Phase Transition (EWPT). Depending on the axion mass, this can lead to apopulation of axions that remain relativistic up to theepoch of recombination, in contrast to the non-relativisticpopulation generated by the misalignment mechanismand by the decay of topological defects (see [12–14] forrecent numerical estimates of this latter contribution).Such a thermally produced population would then actas dark radiation and eventually as warm dark matter.The CMB is sensitive to the fraction of energy densityin these components, therefore constraints on the QCDaxion mass and couplings can be in principle derivedwith CMB observations. In particular, the amount ofdark radiation at recombination is commonly expressedin terms of the effective number of extra neutrino speciesbeyond the three SM ones, ∆ N eff , with current CMBand BAO data imposing ∆ N eff < .
296 [15] at 95%C.L. (the addition of Pantheon and SH ES supernovaemeasurements favors a non-vanishing ∆ N eff at slightlymore than 2 σ level [16, 17] ∆ N eff = 0 . +0 . − . ). Theabundance of a relic species that decouples from the earlyUniverse plasma at a temperature T D is suppressed bythe total number of degrees of freedom of the plasma atdecoupling, and thus it becomes larger if T D decreases. a r X i v : . [ h e p - ph ] D ec Therefore, the largest relic axion abundance arises whenthermalization occurs below the QCD phase transition(QCD PT), through scattering of the lightest particles:leptons [3, 11] and pions [18, 19]. For this to be the case,the axion mass m a should be larger than about ∼ .
05 eVand correspondingly the axion decay constant f , that isthe scale suppressing the axion couplings to SM fields,should be smaller than about 10 GeV. By using CMBdata and focusing on pion scattering, strong constraintson the axion mass have already been derived in one classof axion models, the KSVZ scenario [20, 21], where theQCD axion coupling to SM fermions is negligible, since itvanishes at tree level (see [18, 19] for the original works,and [22–25] for recent updates using Planck 2013, 2015and 2018 data, respectively).However, in an important alternative class of axionmodels, the DFSZ scenario [26, 27], the QCD axioncouples to SM fermions at tree level, thanks to thepresence of an extra Higgs doublet. This opens upnew possibilities: axion production from pion scatteringscan be either enhanced or suppressed (see also [28,29]) compared to the KSVZ scenario, and productionvia leptons can also become important, both featuresdepending on the details of the UV construction. Thefirst aim of this paper is thus to pin down the range ofDFSZ axion masses and couplings which are compatiblewith current cosmological datasets and to providepredictions for next generation CMB experiments [30].To our knowledge, such an analysis has not beenconsistently performed to date. Our interest in cosmological probes of the DFSZ axionis further motivated by some recent laboratory andastrophysical observations, which may be interpreted ashints of the existence of a QCD axion with a significantcoupling to electrons (although these observations cannotbe simultaneously addressed). First, on the experimentalfront, the Xenon-1T (X1T) collaboration has recentlyreported an excess of electron recoil events [31]. Thiscan indeed be explained (with a 3 . σ significance) bysolar axions interacting with electrons with a coupling g ae , if f /c e ≡ m a /g ae (cid:39) × GeV or equivalently m a (cid:39) . /c e eV. Since in the KSVZ model c e is loop-suppressed, the range of axion masses that can explainthe X1T excess has been excluded by cosmological datalong ago [19]. The situation is very different in the DFSZscenario, where c e (cid:46) / σ ) at a DFSZ axion Recent works [24, 25] fixed the axion-pion coupling according tothe KSVZ scenario (and did not discuss production via leptons).Therefore, such bounds on the QCD axion mass cannot beapplied to DFSZ scenarios (especially when correlating with theXenon-1T excess). with m a ∼ /c e meV, which can be probed by CMB datawhen c e (cid:28) m a > .
02 eV is also known to be constrained bythe neutrino signal from SN 1987A (see [38–40] for theoriginal works, [41, 42] for the latest updates and [43]for a critical take). A weaker but possibly more robustconstraint comes from horizontal branch (HB) stars,whose cooling rates generically impose m a (cid:46) . N eff . In Section IV wediscuss the details of the data analysis and present theMCMC results. Finally, in Section V we conclude bypresenting the main conclusions of our work for DFSZaxions. Additional details on scattering rates and theBoltzmann equation are provided in the Appendix. II. AXION INTERACTIONS
The main focus of this paper is the QCD axion, a , whose general Lagrangian above the scale of QCDconfinement and below the EW scale is: L a = 12 ( ∂ µ a ) + ∂ µ a f (cid:88) ψ c ψ j µψ + α s π af G ˜ G + L aγ − ¯ q L M q q R + h.c. , (1)where j µψ = ¯ ψγ µ γ ¯ ψ is the axial current for a genericStandard Model (SM) fermion ψ (quark or lepton), L aγ = g aγγ aF ˜ F / M q is the diagonal quark mass matrix. The continuousshift symmetry of the axion field is only broken by theterm proportional to G ˜ G in eq. (1). Alternatively, toremove the axion coupling to gluons we may performa chiral rotation of the quark fields, R a = e i a f Q a withtr Q a = 1. It is convenient to define Q a = M − q / tr( M − q ),to avoid tree-level mixing with pions [47]. This rotationwill affect the second and fourth term in eq. (1) andinduce an axion dependence in the quark mass term: L a = 12 ( ∂ µ a ) + ∂ µ a f (cid:88) ψ c ψ j µψ + L aγ − ¯ q L M q R + h.c. , (2)where M = R a M q R a . The coefficients in (2) are givenby c q = c q − Q a and the axion-photon Lagrangian isthe same as above but with g aγγ replaced by g aγγ = g aγγ − N c α/ (2 πf )Tr( Q a Q ), where N c is the number ofcolors, α the fine-structure constant and Q is the electriccharge matrix of the quarks.As it stands, the Lagrangian (1) requires a UVcompletion around the scale f . This is usually achievedby introducing a complex scalar field Φ, with a U (1)Peccei-Quinn (PQ) symmetry broken at the scale f [48, 49]. The axion then arises as the phase of thisfield and the coupling to SM gluons can be generatedaccording to two main constructions. In the KSVZmodel [20, 21], Φ is only coupled at tree-level to heavydark fermions in the UV, therefore c ψ = 0. In the DFSZmodel [26, 27], Φ couples to an extended Higgs sectorwith two doublets H u and H d . Thus a coupling of theaxion to SM quarks is automatically present and one has: KSVZ : c u = c d = 0 , (3) DFSZ : c u = 13 cos ( β ) , c d = 13 sin ( β ) , (4)where the notation u and d stands for a universal couplingto up-type and down-type quarks of any generation andtan β ≡ v u /v d , where v u,d are the VEVs of H u and H d .Below the scale of QCD confinement, stronginteractions generate a periodic potential V ( a ) for theaxion, which thereby acquires a mass: m a = √ z z f π m π f (cid:39) . (cid:18) GeV f (cid:19) eV . (5)Here z = m u /m d (cid:39) . +0 . − . , f π (cid:39) . m π (cid:39) .
98 MeV [50]. Because of its interactions, the QCD axion field canbe thermally produced in the early Universe. This leadsto a population of relic axions which, depending onthe axion mass, acts as a dark radiation (DR) or awarm dark matter component. Assuming relic axionsdecouple, their present energy density in the relativistic Throughout this work we neglect uncertainties on the valuesof f π and m π , since they do not affect our results. We usethe uncertainties on z as given in [50]. Smaller uncertaintieshave been recently obtained from lattice simulations: z =0 . limit would be ρ DR = π / g a T a , and their abundanceis usually parameterized in terms of the effective numberof neutrino ( ν ) species, as (see e.g. [52])∆ N eff ≡ ρ DR ρ ν = g a g ν (cid:18) T a T ν (cid:19) (cid:39) . g a (cid:18) . g ∗ s ( T dec ) (cid:19) / , (6)where T ν is the neutrino temperature, g i is the number ofinternal degrees of freedom in a given species (times 7/8for a fermion), so that g a = 1 for an axion, while g ∗ s is thetotal number of entropy degrees of freedom in the plasma,and T dec is the temperature at which the axion decouplesfrom the primordial plasma. The smaller is T dec thelarger is the contribution to ∆ N eff . For this reason, inthis work we will focus only on axion interactions whichcan lead to the lowest decoupling temperatures, aroundor below the QCD PT (see instead [3–6, 8–11, 53] fordecoupling above the QCD PT), thus to the largest valueof ∆ N eff . These are the interactions with pions andleptons, which we now separately introduce. A. Interactions with pions
Below the confinement scale the axion couples topions through the Lagrangian: L aπ = c aπ f π ∂ µ af (cid:2) ∂ µ π π + π − − π (cid:0) ∂ µ π + π − − π + ∂ µ π − (cid:1)(cid:3) , (7)where (see e.g. [54]) c aπ = −
13 ( c u − c d ) = − (cid:18) c u − c d − − z z (cid:19) . (8)Depending on the PQ breaking model and taking theuncertainties on z into account [50], we have: KSVZ : c aπ (cid:39) . +0 . − . , (9) DFSZ : c aπ (cid:39) . +0 . − . −
19 cos(2 β ) . (10)The values of c aπ as a function of sin β are shown in Fig. 1for the two models. We have included constraints onsin β , from tree-level unitarity of fermion scattering [46,54], i.e. tan β ∈ [0 . , β isnot visible in the plot, since it corresponds to β (cid:39) π/ β , the axion-pion coupling in the DFSZ model can be significantlysuppressed compared to its value in the KSVZ modelwhen sin β is small, and can actually even vanish when Interactions with nucleons are also present but do not lead to asignificant thermal production of axions, due to the small densityof nucleons in the thermal plasma.
Figure 1. The axion-pion coupling c aπ for the KSVZ(red) and DFSZ (blue) models. The bands corresponds tovarying the ratio m u /m d within the 1 σ (dark) and 2 σ (light)uncertainty [50]. The band bordered by the dashed linescorresponds instead to the central value and 2 σ uncertainty ofRef. [51]. The gray band is the region excluded by unitaritybounds [54]. the 2 σ uncertainty on z from [50] is taken into account.Using instead the most recent lattice results [51] yields c aπ (cid:38) .
014 at 2 σ . On the other hand, when β ≈ π/ π π ± → aπ ± and π + π − → aπ . The thermally averaged rates forthese processes were first computed by [18] and updatedby [19], which foundΓ ππ → aπ = A (cid:18) c aπ f π f (cid:19) T h ( x π ) , (11)where x π ≡ m π /T , A = 0 .
215 and h ( x ) is anumerical function normalized to h (0) = 1 that decaysexponentially below T = m π (see Appendix A). The rateabove leads to thermal decoupling below the QCD phasetransition for m a (cid:38) . c aπ (cid:38) .
1. For c aπ (cid:28) .
1, (11) predicts a decoupling temperature abovethe QCD phase transition for m a (cid:46) N eff below and report detailed calculations of thedecoupling temperature in Appendix A.To summarize, the dependence on sin β implies thatthermal production of axions from pion scattering in theDFSZ model can be significantly suppressed or enhancedcompared to the KSVZ model. This leads to importantdifferences with respect to previous work, which assumedKSVZ scattering rates (see e.g. [23–25]). B. Interactions with leptons
We now turn to the axion couplings to leptons ( (cid:96) ). Inthe DFSZ scenario these couplings are universal and ariseat tree level. However, two submodels exist, dependingon whether the coupling to leptons is identified with thecoupling to down or up type quarks:
DFSZ-I : c (cid:96) = c d (12) DFSZ-II : c (cid:96) = − c u , (13)where c u,d are given in (3). By means of (8), one canrelate the axion-lepton couplings above to the couplingto pions. It is then easy to see that in the DFSZ-I case c (cid:96) is small when c aπ is small, whereas in the DFSZ-II c (cid:96) increases as c aπ decreases.In the DFSZ model the axion can then be produced attree level through scatterings involving a photon: (cid:96) ± γ → (cid:96) ± a, (cid:96) + (cid:96) − → γa . The total thermally averaged rate isgiven by [11]Γ a(cid:96) → (cid:96)γ = B c (cid:96) (cid:18) m (cid:96) f (cid:19) T g ( x (cid:96) ) , (14)where x (cid:96) ≡ m (cid:96) /T , B ≈ . · − and g (cid:96) is a functionnormalized to g (cid:96) (0) = 1 that decays exponentially below T = m (cid:96) (see Appendix A). We thus haveΓ a(cid:96) → (cid:96)γ Γ ππ → aπ = (cid:18) c (cid:96) c aπ (cid:19) r (cid:96) (cid:16) m (cid:96) T , m π T (cid:17) . (15)The function r (cid:96) is shown in Fig. 2 for (cid:96) = τ, µ . Notethat r τ (cid:28) r µ and r µ (cid:28) T (cid:46) m π / r µ (cid:38)
1. However, this is relevant only when the axiondoes not thermalize at higher temperatures due to pionscatterings, which only occurs for f (cid:46) GeV (see Fig. 8in Sec. A). This region is strongly disfavored by stellarcooling (see e.g. [54]) and by laboratory searches [31, 55].Thus, we conclude that production from leptons isnegligible compared to production from pions for therange of f of interest, unless c (cid:96) (cid:29) c aπ . This latterpossibility occurs in the DFSZ-II case, when sin β (cid:28) c aπ →
0. At largerdecoupling temperatures (thus smaller values of ∆ N eff according to (6)) the contribution from taus can also berelevant. In the DFSZ-I case, axion production from pionscatterings dominates over the production from leptons,as long as f (cid:38) GeV.
C. Laboratory and Astrophysical Hints
Some recent laboratory and astrophysical observationsmay favor QCD axion models with significant axion-electron interactions, such as the DFSZ model. Here weconsider: the X1T excess in electron-recoil events [31]
Figure 2. Normalized ratio of axion production rates fromlepton and pion scatterings as a function of x = m π /T ,according to (15) (see Appendix A for details). and observations of faster-than-expected cooling ratesof white dwarfs (see e.g. [32, 33]) and red giant stars(see [34–36]). Both observations can be interpreted interms of an axion particle, with suggested coupling toelectrons , typically written in terms of g ae ≡ m e c e /f ,given (at 1 σ ) by Xenon-1T [31]: g ae (cid:39) . +0 . − . × − , (16) Star Cooling [56]: g ae (cid:39) . +0 . − . × − . (17)These hints are mutually incompatible and furtherobservations exist (see e.g. [54] for a review), whichconstrain the interpretation of both the aforementionedobservations in terms of an axion particle. For thisreason, we will perform separate analyses in this work,both with and without taking the hints above intoaccount. In the KSVZ model the axion does not couple toleptons at tree-level, therefore m a (cid:38) m a (cid:38)
46 eV)is required to explain the stellar (X1T) hints. Suchlarge axion masses in the KSVZ model have already beenexcluded by cosmological data [19, 23] and independentlyby astrophysical observations (see e.g. [54] for a review).For this reason, we focus on the DFSZ axion scenarioin this work. The region of axion masses and tan β compatible with either the stellar cooling or the X1Thints within the unitarity bounds on tan β are shown inFig. 3. In the X1T case, it is in principle possible that the excess is drivenby axion-photon and/or axion-nucleon couplings, in addition tothe axion-electron coupling. This possibility can be realized onlyin the KSVZ and in the DFSZ-II model, in the region allowed byunitarity. We discard it here, since it would require m a (cid:38) The X1T 1- σ confidence region was obtained by averaging the90%C.L. regions in each of the first two plots in figure 8 of [31], inthe limit of vanishing photon and nucleon coupling, respectively. III. CONTRIBUTION TO ∆ N eff The abundance of thermally produced axions affectsthe epoch of recombination and is thus probed by CMBobservations. This is conventionally quantified by theparameter ∆ N eff (6), which can be accurately computedas (see e.g. [11]) ∆ N eff (cid:39) .
85 ( Y ∞ a ) / , (18)where Y a ≡ n a /s is the axion abundance, n a the axion number density and s the total entropydensity. The superscript ∞ means that the quantityis evaluated at asymptotically small temperatures, sinceafter decoupling Y a is a conserved quantity. This way,one also takes into account axion production even whenthe axion is not thermalized. The axion abundance Y a can be computed by solving the Boltzmann equation sHx dY a dx = (cid:18) − d ln g (cid:63),s d ln x (cid:19) n eq a (cid:88) i Γ i (cid:18) − Y a Y eq a (cid:19) , (19)where x = m π /T , n eq a = ζ (3) T /π , Y eq a = n eq /s arethe equilibrium values for the axion number density andabundance, and (cid:80) i Γ i sums over the thermally averagedscattering rates of the different production channels. Dueto the effects of the QCD phase transition, the behaviorof g ∗ s has to be extracted from lattice computations (weuse the most recent results from [57]).In this work we focus on the contribution to ∆ N eff from pion (11), muon and tau (14) scatterings. Theseare the relevant processes for thermalization around orbelow the QCD phase transition, thus leading to a valueof ∆ N eff which can be probed by current data. We solvethe Boltzmann equation (19) from T i = 100 m π to T f =1 MeV, setting the initial axion abundance to zero. Thecontribution from pions relies on the thermally averagedpion scattering rate (11) which ceases to be reliable attemperatures larger than roughly 160 MeV (see e.g. thebehavior of g (cid:63),s in [57]), the precise value requiring adedicated analysis beyond the scope of this paper. Fordefiniteness, in this work we use the latest determinationof the QCD critical temperature from lattice calculations,i.e. T QCD (cid:39)
158 MeV [58], and include pions in theBoltzmann equation only below this temperature (seeAppendix B for a discussion on how our results areaffected by a different choice of this cutoff temperature).On the other hand, when T dec > T QCD , our computationcan only provide a lower bound on ∆ N eff , since it is inprinciple possible that a sizable contribution arises fromscatterings with pions, or heavier mesons and baryons,during the QCD phase transition. Nonetheless, an upper See instead [3–7] for thermalization above and across the EW PTand [8–11] for thermalization below the EW PT but excludingpions (see also [53] for the possible correlations with the X1Texcess).
CMB-S4
Planck
Planck
DFSZ-II
Figure 3. Parameter region compatible with Xenon-1T excess (curvy gray) and stellar hints (hatched blue) for the DFSZ-I(left) and DFSZ-II (right) axion. Regions where ∆ N eff is large enough to be probed at 2 σ by Planck18 and CMB-S4 are shadedin dark and light gray, respectively. The regions to the right of the dot-dashed blue (green, purple) line are disfavored by whitedwarf (horizontal branch stars, supernovae 1987A) observations [42, 44, 56]. bound on such a contribution can be estimated assuming T dec = T QCD , since decoupling at higher temperatureswould give a smaller ∆ N eff according to (6). Regardingthe contribution from leptons, it is only relevant for theDFSZ-II model at small values of tan β , as discussed inSec. II B.Contours of ∆ N eff in the parameter space of the DFSZmodel are shown in Fig. 3, obtained by also fixing z = 0 .
47 and T QCD (cid:39)
158 MeV (see Appendix Bfor further results). We focus on values that can beconstrained at 2 σ by current (Planck) and future (CMB-S4 [30]) CMB datasets. We find that the existence ofa DFSZ axion with m a (cid:38) . σ significance; this is true for both theDFSZ-I and DFSZ-II submodels, when tan β (cid:38)
1. Atsmaller values of tan β , we see that ∆ N eff is suppressed,as expected from our discussion in Secs. II A and II B,and we expect constraints on the axion mass to besignificantly relaxed (indeed, when taking uncertaintieson z into account, axion production can be significantlyshut off). In the DFSZ-II submodel, axion productionfrom lepton scattering remains relevant even at smallvalues of tan β , as discussed in Sec. II B, thus reducing thepossibility to relax constraints. When comparing withthe constraints coming from astrophysical observations,we see that current cosmological data is competitive with the horizontal branch stars bound [44] and it can be moreconstraining than white dwarf cooling observations [56]in the DFSZ-II case for large c aπ .Fig. 3 also shows that current hints for a DFSZaxion can be independently investigated by means ofcosmological data. This is especially true for the DFSZ-II scenario, where both the stellar cooling and X1T hintslead to observable values of ∆ N eff for m (cid:38) . N eff . Interestingly,most of this region can be probed at 2 σ by CMB-S4. IV. DATASETS AND RESULTS
We now turn to a quantitative assessment of theDFSZ axion scenario in light of the latest cosmologicaldatasets. To this aim, we use the Boltzmann code
CLASS [59, 60], which we have modified to include thethermally produced QCD axion as an extra light speciesbeyond the SM neutrinos, in complete analogy withthe case of an extra neutrino species which is alreadyincluded in
CLASS . In particular, we have implementedthe following changes:1 We used a Bose-Einstein distribution with onedegree of freedom to describe the thermallyproduced axions, rather than the Fermi-Diracdistribution which is implemented in
CLASS forextra neutrino species.2 We assigned a temperature to the extra species,given by T a = T ν (cid:0) ∆ N eff (cid:1) / , as dictated by (6).With these modifications, we obtain a cosmologicalmodel with six ΛCDM parameters plus two additionalparameters: c aπ (or alternatively sin β ) and m a . We thenuse Monte Python [61, 62], in its version 3 . .
2, to performa Markov chain Monte Carlo analysis of this model. Wemodel neutrinos, using the standard treatment of thePlanck collaboration, as two massless and one massivespecies with m ν = 0 .
06 eV and T ν = 0 . Weobtain constraints on m a by means of the following twocombinations of cosmological datasets:a) Planck 18 + BAO : Planck 2018 high- (cid:96) and low- (cid:96)
TT, TE, EE and lensing data [63], plus BAOmeasurements from 6dFGS at z = 0 .
106 [64], fromthe MGS galaxy sample of SDSS at z = 0 .
15 [65],and from the CMASS and LOWZ galaxy samplesof BOSS DR12 at z = 0 .
38, 0 .
51, and 0 .
61 [66].b)
Planck 18 + BAO + Pantheon + SH ES :same as above plus SH ES 2019 measurementof the present day Hubble rate H = 74 . ± .
42 km/s/Mpc [67], and the Pantheon supernovaedataset [68].We first focus on general constraints, obtained usingonly the datasets a) and b) above, which in ourfigures are referred to as “w/o SN + H ” and “w/SN + H ” respectively. Note that we improve theanalysis compared to previous related work [19, 22–24]by solving the Boltzmann equation to obtain the axionabundance rather than using approximate estimations ofthe decoupling temperature.As discussed in Sec. II B, in the DFSZ-I model thermalaxion production is dominated by pion scatteringsindependently of the value of the axion-pion coupling.In this case, the contribution to ∆ N eff has a simpledependence on c aπ , and thus we choose the latter asan independent parameter in our MonteCarlo analysis,together with m a (our results then only depend weaklyon z through the axion mass (5), therefore we fix z = 0 . m a and c aπ in Fig. 4. We use a logarithmic prior on c aπ :ln c aπ = [ − , − .
27] to encompass both small and largevalues of c aπ , see Fig. 1.One can appreciate that the cosmological bound on m a becomes stronger as c aπ increases whereas constraintsare significantly relaxed as c aπ decreases. In order to We checked that adding (cid:80) m ν as a free parameter in our runsdoes not significantly affect the upper bounds on m a presentedhere. Figure 4. Constraints on the axion coupling to pions c aπ andon the axion mass m a when axion production via leptons canbe neglected. This is the case for the the DFSZ-I model at anyvalue of c aπ , whereas in the DFSZ-II model the constraintsshown in the figure are reliable only for c aπ (cid:38) O (0 . derive precise bounds, we perform a dedicated analysis oftwo representative values of c aπ = 0 . , . c aπ (see Fig. 1). The corresponding upper boundson m a at 95% C.L. are reported in Table I for our twodatasets. Smaller values of c aπ are possible at 2 σ whenusing the uncertainty on z from [50] (see Fig. 1) and as c aπ → m a is fully removed.A separate analysis is required for the DFSZ-II model,since thermal production from leptons is relevant in thiscase for small values of c aπ , as discussed in Sec. II B.By including both leptons and pion scattering rates inthe Boltzmann equation and implementing the results in CLASS as described above, we obtain the upper boundreported in Table I for the same representative valuesof c aπ . As expected, when c aπ is small, c aπ = 0 . c aπ is O (0 .
1) the contribution fromleptons is negligible and the bound is the same as in theDFSZ-I model.We now turn to the results obtained by combiningcosmological datasets with either laboratory orastrophysical hints, which we implement by including aGaussian likelihood on g ae in our runs, according to (16)and (17). In these cases, thermal axion production fromleptons is always negligible compared to pions. Forconvenience, we trade the parameter c aπ for sin β (ortan β ) and impose unitarity constraints on the latterparameters by choosing flat priors on sin β = [0 . , c aπ → z varies within its experimental uncertainties, we nowkeep z as an extra parameter, which we constrain by Figure 5. Constraints on the axion mass and the angle β between the two Higgs doublets in the DFSZ model, obtained bycombining cosmological data Planck18+BAO (+Pantheon+SH ES) with the Xenon-1T hint (16). Left: DFSZ-I model, Right:DFSZ-II model. Dark (light) shaded contours correspond to 1(2) σ regions.DFSZ-I Planck 18+BAO (+SN+ H ) c aπ = 0 . m a ≤ .
20 (0 .
29) eV c aπ = 0 . m a ≤ .
84 (0 .
82) eVDFSZ-II Planck 18+BAO (+SN+ H ) c aπ = 0 . m a ≤ .
20 (0 .
29) eV c aπ = 0 . m a ≤ .
60 (0 .
61) eVTable I. Constraints on m a for the DFSZ-I and DFSZ-IImodels at 95% C.L., for two representative choices of axioncoupling to pions. including a Gaussian likelihood according to [50].We start by considering the X1T hint (16). We showthe combined constraints on the fundamental parameters m a and sin β of the DFSZ-I and DFSZ-II models inFig. 5. In the DFSZ-I case the constraints on m a aresignificantly relaxed at small values of sin β , as expectedfrom Fig. 3. In particular, we find m a (cid:46) . β takes the minimal value allowed by the unitarityconstraint, sin β = 0 .
24, with the Planck18+BAO+X1Tdataset. The parameter sin β is, on the other hand,unconstrained. This is in contrast with the DFSZ-II case,where the constraint on m a is only slightly relaxed atlarge values of sin β . In particular, we find m a (cid:46) .
27 eVfor sin β close to its largest allowed value, which wefind to be sin β (cid:46) . N eff increases rapidly when the mass increases, whereasin the DFSZ-I case the amount of relic axions in theX1T band is almost constant and the main cosmologicaleffect is the fact that the axion behaves as a warmdark matter component. When using the dataset b) we find essentially no difference for the DFSZ-I case,whereas the constraint is further relaxed in the DFSZ-IIcase. The SH ES measurement of H , being strongly intension with the other datasets, prefers a larger ∆ N eff ,thereby allowing for slightly larger axion masses. Wenow move to the stellar hints. We focus on the DFSZ-II case, since in the DFSZ-I scenario thermal axionproduction is small in the stellar hint band (see left plotin Fig. 3). In this particular analysis, it is convenientto trade sin β for tan β to facilitate convergence andimpose a logarithmic prior log tan β = [ − . , .
23] fromunitarity. We show the combined constraints on thefundamental parameters m a and tan β of the DFSZ-IImodel in Fig. 6 (left). As expected from the discussionabove, we find strong constraints, m a (cid:46) .
21 eV whenlog tan β is close to its largest allowed value, which wefind to be log tan β (cid:46) .
85, and a significant relaxationof these bounds when SH ES is included. We also showthe combined constraints on ∆ N eff and m a : as expectedfrom Fig. 3 (right), a significant amount of relic axionscan be produced in the stellar hint band, which thendrives strong constraints on the axion mass.To summarize, we find that current cosmological dataimpose strong constraints on the axion mass m a in theDFSZ-II scenario, which allow to rule out a significantfraction of the parameter space in which the QCD axioncan explain the X1T excess or the stellar cooling hints.On the other hand, we find that the DFSZ-I modelis significantly less constrained by current cosmological Figure 6. Left: Constraints on the axion mass and the angle β between the two Higgs doublets in the DFSZ-II model. Right:Constraints on the axion mass and the amount of relic axions ∆ N eff . Both figures are obtained by combining cosmological dataPlanck18 + BAO (+Pantheon+SH ES) with the stellar cooling hint (17). Dark (light) shaded contours correspond to 1(2) σ regions. data. Next generation of CMB experiments will beable to further improve the bounds on the axion mass.In particular, they will be able to probe most of theparameter space where the DFSZ-I model can explainthe X1T excess (see Fig. 3). V. CONCLUSIONS
A sizeable thermal axion relic abundance can affectthe CMB in a way similar to massive neutrinos. In thiswork, we used the latest cosmological data to constrainthe DFSZ scenario, when axions are produced throughscatterings of pions and leptons. This occurs for m a (cid:38) . c aπ , is not fixed andcan be enhanced or suppressed, depending on the vevsof two Higgs doublets which characterize this class ofrealizations of the Peccei-Quinn mechanism. Therefore,cosmological constraints on the axion mass stronglydepend on such coupling. In particular, by solving theBoltzmann equation, implementing the thermal axion ina Boltzmann code and performing a MCMC analysis, wefound that: on the one hand, for maximal axion-pioncouplings the axion mass is constrained to m a ≤ . c aπ is reduced. Forinstance, when c aπ is suppressed by a factor of 10 fromits maximal value, we find m a ≤ .
60 (0 .
84) eV at 95%C.L. in the DFSZ-II (DFSZ-I) case and for even smallervalues of c aπ only the DFSZ-II model can be constrainedby CMB observations.Our approach is particularly well motivated in lightof recent hints of a DFSZ-like QCD axion coupling toelectrons, both from Xenon-1T experiment and fromobservations of stellar coolings. In particular, the formeris in tension with astrophysical constraints and it isthus crucial to understand the extent to which currentand future cosmological data can shed light on theviability of axion models to explain such signals. Inthe DFSZ model, the couplings to leptons is universaland related to the coupling to pions, thus the hintscan potentially have direct implications for the CMB.We performed a combined analysis and found thatCMB data already restricts the mass range in whichthe DFSZ axion addresses the Xenon-1T excess: thisis particularly so for the DFSZ-II scenario, where wedetermined 0 .
07 eV (cid:46) m a (cid:46) . m a (cid:46) . σ most ofthe parameter space of the DFSZ-I model, if the latter isto address the X1T excess. In the case of stellar hints,0the reported axion-electron coupling is smaller and so wewere only able to constrain the interpretation within theDFSZ-II model to the mass range 3 meV (cid:46) m a (cid:46) . m (cid:38) . . σ significance. However, it is in principle possible thata further significant contribution to ∆ N eff arises fromscatterings across the QCD PT (pions and/or heaviermesons). This would then extend the mass range thatcan be probed by CMB-S4 at 2 σ , possibly down tomasses which are closer to the SN 1987A bound, andthus provide the opportunity to firmly establish thelower bound on the QCD axion mass. Extending thecomputation of ∆ N eff across the QCD PT is thus a veryinteresting task for future work.Furthermore, we have restricted our attention to theminimal particle content of QCD axion models. However,the latter may in general couple to extra fields, belongingfor instance to a dark sector. This possibility mayin fact be particularly well motivated in the DFSZscenario, where solving the domain wall problem requiresadditional contributions to the axion potential (see [69]for a solution with a light dark sector). In this case,scatterings of dark sector fields may also lead to a largervalue of ∆ N eff . We leave these interesting questions forfuture work.Finally, let us mention that the outlook onexperimental searches for a DFSZ axion is promising:in particular the mass range considered in this workcan be probed via the axion coupling to photons, bye.g. the IAXO helioscope [70] (and/or to a lesser extentby its scaled down version BabyIAXO [71]), and viathe axion-electron coupling by next generations electronrecoil detectors such as PandaX-4T [72], LZ [73] andXENONnT [74]. ACKNOWLEDGMENTS
We thank Luca di Luzio and Giovanni Villadoro foruseful discussions. We acknowledge use of Tufts HPCresearch cluster. The work of A.N. is supported bythe grants FPA2016-76005-C2-2-P, PID2019-108122GB-C32, ”Unit of Excellence Mar´ıa de Maeztu 2020-2023” ofICCUB (CEX2019-000918-M), AGAUR2017-SGR-754.RZF acknowledges support by the Spanish MinistryMEC under grant FPA 2017-88915-P and the SeveroOchoa excellence program of MINECO (SEV-2016-0588). IFAE is partially funded by the CERCA programof the Generalitat de Catalunya. The work of FR issupported in part by National Science Foundation GrantNo. PHY-2013953.
Figure 7. The numerical functions h ( x ) (from [19]) and g ( x ).. Appendix A: Scattering rates
For completeness, here we summarize the results onthe thermally averaged rates which have been used inthis paper.
1. Pions
The thermally averaged rates for axion productionfrom pion scatterings is given by [18, 19]:Γ ππ → aπ = A (cid:18) c aπ f π f (cid:19) T h ( x π ) , (A1)where x π = m π /T , A = 0 .
215 and h ( x ) is a numericalfunction normalized to h (0) = 1, shown in Fig. 7.The decoupling temperature can then be estimated bysetting Γ ππ → aπ ( T π dec ) = H ( T π dec ). The result is shownin Fig. 8 for three representative choices of c aπ . Thelargest value of c aπ in Fig. 8 corresponds to the maximalcentral value of c aπ in Fig. 1. Above the QCD phasetransition, the thermal production of axions from pionscannot be reliably computed according to (11). Forthis reason, we have shaded the region above T = 158MeV, which corresponds to the most recent value of theQCD critical temperature obtained by means of latticecalculations [58]. We comment on the implications ofthe QCD phase transition on our computation of ∆ N eff below.
2. Leptons
The axion-lepton interaction opens up three differentchannels of axion production: (cid:96) ± γ → (cid:96) ± a, (cid:96) + (cid:96) − → γa .The total thermally averaged production rate is given bythe sum over the three channels and is given by (see [11]for details) Γ a(cid:96) → (cid:96)γ = B c (cid:96) (cid:18) m (cid:96) f (cid:19) T g ( x (cid:96) ) , (A2)1 Figure 8. The axion decoupling temperature T π dec as afunction of the axion mass m for the processes ππ → aπ ,according to the condition Γ ππ → aπ ( T π dec ) = H ( T π dec ) and therate (A1). The solid (dashed, dot-dashed) curves correspondto the choice c aπ = 0 .
225 (0 . , . T QCD = 158 MeV[58]. where B = 1 . × − , x (cid:96) = m (cid:96) /T and g ( x (cid:96) ) is anumerical function normalized to g (0) = 1, shown inFig. 7. Appendix B: Solution of the Boltzmann equation
Here we provide more details about the computation of∆ N eff from the Boltzmann equation. We have solved (19)including the thermally averaged pion (11) and leptonscattering rates (14) and computed ∆ N eff by meansof (18). The solution to the Boltzmann equation dependsonly on two parameters: c aπ and m a . It is also sometimesconvenient to trade c aπ for sin β (or tan β ), according to (8). We include pions only below the QCD criticaltemperature, which we take to be 158 MeV based on thelatest lattice calculations [58].We show in Figs. 9 (10) the result of our calculationin the DFSZ-I (DFSZ-II) scenario for two representativechoices c aπ = 0 .
225 (left) , . N eff for large c aπ . Instead, forsmall values of c aπ , production from muons (and to alesser extent from taus) is relevant only in the DFSZ-IImodel, as shown in Fig. 10 (right) thus yielding a largervalue of ∆ N eff than in the DFSZ-I model for the samevalues of c aπ and m a .In both cases, the dependence of ∆ N eff on c aπ is wellreproduced by the function∆ N eff, fit (cid:39) Aα / (cid:16) B α . /C (cid:17) C . (B1)where α ≡ ( m a / eV)( c aπ / . N eff ∝ α / ∝ f − / for small masses can be obtainedanalytically [11]. When production from pions isdominant (DFSZ-I and DFSZ-II at large coupling), wefind A ≈ . , B ≈ .
96 and C = − .
95 to givea good fit, shown as a red curve in Fig. 10 (left).When leptons and pions are both relevant, the values of
A, B, C depend on the precise value of c aπ . For c aπ =0 . A ≈ . , B ≈ . C ≈ − . CLASS .The shaded regions in Figs. 9 and 10 correspond tovarying the temperature below which we include pionscatterings in the Boltzmann equation, between 158 MeV(lower curve) and 200 MeV (upper curve). Finally, thegreen line corresponds to the value of ∆ N eff if the axiondecouples at T = 158 MeV. Below the line, the axion doesnot thermalize below the QCD PT and so there could beadditional contributions from pion scatterings at highertemperatures which need to be computed using differentmethods. In this regime we also neglect contributionsfrom axion couplings to heavier quarks which could givea signal up to ∆ N eff (cid:39) .
05 when decoupling happenswhile in weakly coupled regimes ( T (cid:38) [1] S. Weinberg, A New Light Boson? , Phys. Rev. Lett. (1978) 223–226.[2] F. Wilczek, Problem of Strong P and T Invariance inthe Presence of Instantons , Phys. Rev. Lett. (1978)279–282.[3] M. S. Turner, Thermal Production of Not SO InvisibleAxions in the Early Universe , Phys. Rev. Lett. (1987) 2489. [Erratum: Phys. Rev. Lett.60,1101(1988)].[4] E. Masso, F. Rota, and G. Zsembinszki, On axion thermalization in the early universe , Phys. Rev. D (2002) 023004, [ hep-ph/0203221 ].[5] P. Graf and F. D. Steffen, Thermal axion production inthe primordial quark-gluon plasma , Phys. Rev.
D83 (2011) 075011, [ arXiv:1008.4528 ].[6] A. Salvio, A. Strumia, and W. Xue,
Thermal axionproduction , JCAP (2014) 011, [ arXiv:1310.6982 ].[7] F. Arias-Aragon, F. D’Eramo, R. Z. Ferreira, L. Merlo,and A. Notari, Production of Thermal Axions across the Figure 9. The thermal axion abundance, obtained by solving numerically the Boltzmann equation including production frompion, muon and tau scatterings, in the DFSZ-I model. We show two representative examples with c aπ = 0 .
225 and c aπ = 0 . N eff if the axion decouples at T = 158 MeV.Figure 10. The thermal axion abundance, obtained by solving numerically the Boltzmann equation including productionfrom pion, muon and tau scatterings, in the DFSZ-II model. We show two representative examples with c aπ = 0 .
225 and c aπ = 0 . CLASS . Solutions obtainedby considering each component separately are also shown as dashed lines in the right plot (see legends). In the left plot thecontributions from leptons are not visible. The green line is the value of ∆ N eff if the axion decouples at T = 158 MeV. ElectroWeak Phase Transition , arXiv:2012.04736 .[8] C. Brust, D. E. Kaplan, and M. T. Walters, New LightSpecies and the CMB , JHEP (2013) 058,[ arXiv:1303.5379 ].[9] D. Baumann, D. Green, and B. Wallisch, New Targetfor Cosmic Axion Searches , Phys. Rev. Lett. (2016), no. 17 171301, [ arXiv:1604.08614 ].[10] R. Z. Ferreira and A. Notari,
Observable Windows forthe QCD Axion Through the Number of RelativisticSpecies , Phys. Rev. Lett. (2018), no. 19 191301,[ arXiv:1801.06090 ].[11] F. D’Eramo, R. Z. Ferreira, A. Notari, and J. L. Bernal,
Hot Axions and the H tension , JCAP (2018) 014,[ arXiv:1808.07430 ].[12] M. Gorghetto, E. Hardy, and G. Villadoro, Axions fromStrings: the Attractive Solution , JHEP (2018) 151, [ arXiv:1806.04677 ].[13] M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, andJ. Urrestilla, Scaling Density of Axion Strings , Phys.Rev. Lett. (2020), no. 2 021301,[ arXiv:1908.03522 ].[14] M. Gorghetto, E. Hardy, and G. Villadoro,
More Axionsfrom Strings , arXiv:2007.04990 .[15] Planck
Collaboration, N. Aghanim et. al.,
Planck 2018results. VI. Cosmological parameters , Astron.Astrophys. (2020) A6, [ arXiv:1807.06209 ].[16] G. Ballesteros, A. Notari, and F. Rompineve,
The H tension: ∆ G N vs. ∆ N eff , arXiv:2004.05049 .[17] M. Gonzalez, M. P. Hertzberg, and F. Rompineve, Ultralight Scalar Decay and the Hubble Tension , JCAP (2020) 028, [ arXiv:2006.13959 ].[18] S. Chang and K. Choi, Hadronic axion window and the big bang nucleosynthesis , Phys. Lett. B (1993)51–56, [ hep-ph/9306216 ].[19] S. Hannestad, A. Mirizzi, and G. Raffelt, Newcosmological mass limit on thermal relic axions , JCAP (2005) 002, [ hep-ph/0504059 ].[20] J. E. Kim, Weak Interaction Singlet and Strong CPInvariance , Phys. Rev. Lett. (1979) 103.[21] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can Confinement Ensure Natural CP Invariance ofStrong Interactions? , Nucl. Phys.
B166 (1980) 493–506.[22] M. Archidiacono, S. Hannestad, A. Mirizzi, G. Raffelt,and Y. Y. Wong,
Axion hot dark matter bounds afterPlanck , JCAP (2013) 020, [ arXiv:1307.0615 ].[23] E. Di Valentino, E. Giusarma, M. Lattanzi, O. Mena,A. Melchiorri, and J. Silk, Cosmological Axion andneutrino mass constraints from Planck 2015temperature and polarization data , Phys. Lett. B (2016) 182–185, [ arXiv:1507.08665 ].[24] M. Millea,
New cosmological bounds on axions in theXENON1T window , arXiv:2007.05659 .[25] W. Giar`e, E. Di Valentino, A. Melchiorri, and O. Mena, New cosmological bounds on hot relics: Axions & Neutrinos , arXiv:2011.14704 .[26] M. Dine, W. FisCHLer, and M. Srednicki, A SimpleSolution to the Strong CP Problem with a HarmlessAxion , Phys. Lett.
B104 (1981) 199–202.[27] A. R. Zhitnitsky,
On Possible Suppression of the AxionHadron Interactions. (In Russian) , Sov. J. Nucl. Phys. (1980) 260. [Yad. Fiz.31,497(1980)].[28] L. Di Luzio, F. Mescia, E. Nardi, P. Panci, andR. Ziegler, Astrophobic Axions , Phys. Rev. Lett. (2018), no. 26 261803, [ arXiv:1712.04940 ].[29] D. S. M. Alves and N. Weiner,
A viable QCD axion inthe MeV mass range , JHEP (2018) 092,[ arXiv:1710.03764 ].[30] CMB-S4
Collaboration, K. N. Abazajian et. al.,
CMB-S4 Science Book, First Edition , arXiv:1610.02743 .[31] XENON
Collaboration, E. Aprile et. al.,
Observationof Excess Electronic Recoil Events in XENON1T , arXiv:2006.09721 .[32] J. Isern, E. Garcia-Berro, S. Torres, R. Cojocaru, andS. Catalan, Axions and the luminosity function of whitedwarfs: the thin and thick discs, and the halo , Mon.Not. Roy. Astron. Soc. (2018), no. 2 2569–2575,[ arXiv:1805.00135 ].[33] A. H. C´orsico, L. G. Althaus, M. M. Miller Bertolami,and S. Kepler,
Pulsating white dwarfs: new insights ,Astron. Astrophys. Rev. (2019), no. 1 7,[ arXiv:1907.00115 ].[34] N. Viaux, M. Catelan, P. B. Stetson, G. Raffelt,J. Redondo, A. A. R. Valcarce, and A. Weiss, Neutrinoand axion bounds from the globular cluster M5 (NGC5904) , Phys. Rev. Lett. (2013) 231301,[ arXiv:1311.1669 ].[35] O. Straniero, I. Dominguez, M. Giannotti, andA. Mirizzi,
Axion-electron coupling from the RGB tip ofGlobular Clusters , in13th Patras Workshop on Axions, WIMPs and WISPs,pp. 172–176, 2018. arXiv:1802.10357 .[36] O. Straniero, C. Pallanca, E. Dalessandro,I. Dominguez, F. Ferraro, M. Giannotti, A. Mirizzi, andL. Piersanti,
The RGB tip of galactic globular clustersand the revision of the bound of the axion-electron coupling , arXiv:2010.03833 .[37] L. Di Luzio, M. Fedele, M. Giannotti, F. Mescia, andE. Nardi, Solar axions cannot explain the XENON1Texcess , Phys. Rev. Lett. (2020), no. 13 131804,[ arXiv:2006.12487 ].[38] M. S. Turner,
Axions from SN 1987a , Phys. Rev. Lett. (1988) 1797.[39] G. Raffelt and D. Seckel, Bounds on Exotic ParticleInteractions from SN 1987a , Phys. Rev. Lett. (1988)1793.[40] A. Burrows, M. S. Turner, and R. Brinkmann, Axionsand SN 1987a , Phys. Rev. D (1989) 1020.[41] J. H. Chang, R. Essig, and S. D. McDermott, Supernova 1987A Constraints on Sub-GeV DarkSectors, Millicharged Particles, the QCD Axion, and anAxion-like Particle , JHEP (2018) 051,[ arXiv:1803.00993 ].[42] P. Carenza, T. Fischer, M. Giannotti, G. Guo,G. Mart´ınez-Pinedo, and A. Mirizzi, Improved axionemissivity from a supernova via nucleon-nucleonbremsstrahlung , JCAP (2019), no. 10 016,[ arXiv:1906.11844 ]. [Erratum: JCAP 05, E01 (2020)].[43] N. Bar, K. Blum, and G. D’Amico, Is there a supernovabound on axions? , Phys. Rev. D (2020), no. 12123025, [ arXiv:1907.05020 ].[44] A. Ayala, I. Dom´ınguez, M. Giannotti, A. Mirizzi, andO. Straniero,
Revisiting the bound on axion-photoncoupling from Globular Clusters , Phys. Rev. Lett. (2014), no. 19 191302, [ arXiv:1406.6053 ].[45] I. M. Bloch, A. Caputo, R. Essig, D. Redigolo,M. Sholapurkar, and T. Volansky,
Exploring NewPhysics with O(keV) Electron Recoils in DirectDetection Experiments , arXiv:2006.14521 .[46] F. Bj¨orkeroth, L. Di Luzio, F. Mescia, E. Nardi,P. Panci, and R. Ziegler, Axion-electron decoupling innucleophobic axion models , Phys. Rev. D (2020),no. 3 035027, [ arXiv:1907.06575 ].[47] H. Georgi, D. B. Kaplan, and L. Randall,
Manifestingthe Invisible Axion at Low-energies , Phys. Lett. B (1986) 73–78.[48] R. D. Peccei and H. R. Quinn,
CP Conservation in thePresence of Instantons , Phys. Rev. Lett. (1977)1440–1443.[49] R. D. Peccei and H. R. Quinn, Constraints Imposed byCP Conservation in the Presence of Instantons , Phys.Rev.
D16 (1977) 1791–1797.[50]
Particle Data Group
Collaboration, P. Zyla et. al.,
Review of Particle Physics , PTEP (2020), no. 8083C01.[51] M. Gorghetto and G. Villadoro,
TopologicalSusceptibility and QCD Axion Mass: QED and NNLOcorrections , JHEP (2019) 033, [ arXiv:1812.01008 ].[52] D. Baumann, Primordial Cosmology , PoS
TASI2017 (2018) 009, [ arXiv:1807.03098 ].[53] F. Arias-Aragon, F. D’Eramo, R. Z. Ferreira, L. Merlo,and A. Notari,
Cosmic Imprints of XENON1T Axions , arXiv:2007.06579 .[54] L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, The landscape of QCD axion models , Phys. Rept. (2020) 1–117, [ arXiv:2003.01100 ].[55]
LUX
Collaboration, D. Akerib et. al.,
First Searchesfor Axions and Axionlike Particles with the LUXExperiment , Phys. Rev. Lett. (2017), no. 26261301, [ arXiv:1704.02297 ]. [56] M. Giannotti, I. G. Irastorza, J. Redondo, A. Ringwald,and K. Saikawa, Stellar Recipes for Axion Hunters ,JCAP (2017) 010, [ arXiv:1708.02111 ].[57] S. Borsanyi et. al., Calculation of the axion mass basedon high-temperature lattice quantum chromodynamics ,Nature (2016), no. 7627 69–71,[ arXiv:1606.07494 ].[58] S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D.Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K.Szabo,
QCD Crossover at Finite Chemical Potentialfrom Lattice Simulations , Phys. Rev. Lett. (2020),no. 5 052001, [ arXiv:2002.02821 ].[59] J. Lesgourgues,
The Cosmic Linear Anisotropy SolvingSystem (CLASS) I: Overview , arXiv:1104.2932 .[60] D. Blas, J. Lesgourgues, and T. Tram, The CosmicLinear Anisotropy Solving System (CLASS) II:Approximation schemes , JCAP (2011) 034,[ arXiv:1104.2933 ].[61] B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, Conservative Constraints on Early Cosmology: anillustration of the Monte Python cosmological parameterinference code , JCAP (2013) 001,[ arXiv:1210.7183 ].[62] T. Brinckmann and J. Lesgourgues, MontePython 3:boosted MCMC sampler and other features , Phys. DarkUniv. (2019) 100260, [ arXiv:1804.07261 ].[63] Planck
Collaboration, N. Aghanim et. al.,
Planck 2018results. V. CMB power spectra and likelihoods , Astron.Astrophys. (2020) A5, [ arXiv:1907.12875 ].[64] F. Beutler, C. Blake, M. Colless, D. Jones,L. Staveley-Smith, L. Campbell, Q. Parker,W. Saunders, and F. Watson,
The 6dF Galaxy Survey:Baryon Acoustic Oscillations and the Local HubbleConstant , Mon. Not. Roy. Astron. Soc. (2011)3017–3032, [ arXiv:1106.3366 ].[65] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden, and M. Manera,
The clustering of the SDSSDR7 main Galaxy sample – I. A 4 per cent distance measure at z = 0 .
15, Mon. Not. Roy. Astron. Soc. (2015), no. 1 835–847, [ arXiv:1409.3242 ].[66]
BOSS
Collaboration, S. Alam et. al.,
The clustering ofgalaxies in the completed SDSS-III Baryon OscillationSpectroscopic Survey: cosmological analysis of the DR12galaxy sample , Mon. Not. Roy. Astron. Soc. (2017),no. 3 2617–2652, [ arXiv:1607.03155 ].[67] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, andD. Scolnic,
Large Magellanic Cloud Cepheid StandardsProvide a 1% Foundation for the Determination of theHubble Constant and Stronger Evidence for Physicsbeyond Λ CDM , Astrophys. J. (2019), no. 1 85,[ arXiv:1903.07603 ].[68] D. Scolnic et. al.,
The Complete Light-curve Sample ofSpectroscopically Confirmed SNe Ia fromPan-STARRS1 and Cosmological Constraints from theCombined Pantheon Sample , Astrophys. J. (2018),no. 2 101, [ arXiv:1710.00845 ].[69] F. Ferrer, E. Masso, G. Panico, O. Pujolas, andF. Rompineve,
Primordial Black Holes from the QCDaxion , Phys. Rev. Lett. (2019), no. 10 101301,[ arXiv:1807.01707 ].[70] I. Irastorza et. al.,
Towards a new generation axionhelioscope , JCAP (2011) 013, [ arXiv:1103.5334 ].[71] BabyIAXO
Collaboration, A. Abeln et. al.,
Conceptual Design of BabyIAXO, the intermediate stagetowards the International Axion Observatory , arXiv:2010.12076 .[72] PandaX
Collaboration, H. Zhang et. al.,
Dark matterdirect search sensitivity of the PandaX-4T experiment ,Sci. China Phys. Mech. Astron. (2019), no. 3 31011,[ arXiv:1806.02229 ].[73] LZ Collaboration, D. Akerib et. al.,
The LUX-ZEPLIN(LZ) Experiment , Nucl. Instrum. Meth. A (2020)163047, [ arXiv:1910.09124 ].[74]
XENON
Collaboration, E. Aprile et. al.,
ProjectedWIMP Sensitivity of the XENONnT Dark MatterExperiment , JCAP (2020) 031, [ arXiv:2007.08796arXiv:2007.08796