Jhep08(2019)147

Home , Inflaton

JHEP08(2019)147 Springer July 8, 2019 : August 4, 2019 August 5, 2019 August 27, 2019 : : : Received Revised Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP08(2019)147 b,c [email protected] , . and Lorenzo Ubaldi 3 a,c 1907.00984 The Authors. c Cosmology of Theories beyond the SM, Eﬀective Field Theories

A new mechanism for producing axion dark matter is proposed. By invoking , [email protected] SISSA and INFN SezioneVia di Bonomea Trieste, 265, 34136 Trieste,Institute Italy for Fundamental PhysicsVia of Beirut the 2, Universe, 34014 Trieste, Italy E-mail: International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy b c a Open Access Article funded by SCOAP and higher. Keywords: ArXiv ePrint: tions of the axionthe and cosmological inflaton fields perturbations, play reheatingkinetic various the cosmological mixing universe, roles, also and including relates servingmechanism generating the tames as dark axions dark matter that matter. lifetime wouldup with otherwise new The overdominate the windows the in reheating the universe, temperature. axion and parameter thus The space, opens including decay constants at the GUT scale Abstract: low-scale inflation and a kinetic mixingthe between axion the axion is and driven thecan to inflaton, give it a rise is to field shown the point that observed dark slightly matter displaced abundance. from In the this potential framework, different minimum, combina- which Takeshi Kobayashi Inflaxion dark matter JHEP08(2019)147 ] gives us confidence that 1 10 ]. The simplest models of inflation involve a 4 – 1 – 3 19 – 19 2 3 6 5 19 7 5 13 20 18 ]. 1 6 16 ]. While (very) low-scale inflation remains as a possibility, the improved obser- 5 B.1 Generic cosmologicalB.2 background Slow-roll inflation 3.3 Reheating and dark matter stability 2.2 General axions 3.1 Low-scale inflation 3.2 End of inflation 2.1 Axions coupled to a strong sector however there are alsoBBN baryogenesis [ models thatvational limit can on operate primordial atscale gravitational scales waves from has not above, tightened much now themonomial above disfavoring terms bound [ simple on the high-scale inflation inflation models composed of single of exponentially accelerated expansionscalar field [ slowly rolling downtook a flat place, potential. nor We the knoware temperature neither fairly at at certain what which scale thatBang the the the Nucleosynthesis process plasma (BBN) reheat to was successfully temperature reheated takegenerating must place. at the be Most the matter-antimatter of above end asymmetry the the require known of mechanisms MeV even it. for higher scale reheating We for temperatures, the Big The remarkable agreement between(CMB) measurements and of the the Lambdawe cosmic Cold understand microwave well Dark background several Matter aspectssupports (ΛCDM) of the model the picture [ early that history of the the Hot universe. Big The Bang existing data cosmology started with inflation, a period 1 Introduction B Misalignment of stabilized axion C QCD inflaxion 4 Parameter space 5 Conclusions A Diagonal basis 3 Inflaxion mechanism Contents 1 Introduction 2 Inflaton-axion mixing JHEP08(2019)147 (1.1) ] and . An 17 – < π reason to 15 ? eV has been a 22 π < θ − forest [ a priori − 10 α ∼ σ . 2 m / ]. Axions can have a wide 1 eV. This also implies that if 12 – is the misalignment angle which 22 eV 10 − ? ; considering the axion potential σ θ 22 f 10 m ? − ], as well as axion-like fields that can θ 9 ∼ 10 – = σ 7 2 ? m σ ] for reviews). However it was recently pointed f – 2 – GeV ? 14 , θ problem [ 17 is sufficiently smaller than unity and/or the mass 12 ? 10 θ , the angle lies within the range CP πf × 1 . + 2 0 is the decay constant, and ] (see e.g. [ σ GeV and the mass ∼ f ]. In the vacuum misalignment scenario, the relevant dynamics 13 σ 17 → Ω 18 10 σ eV. An ultralight axion dark matter with ∼ 1 makes up the observed dark matter abundance if, for instance, the 22 f − , is smaller than the axion mass, so that the axion possesses a potential 10 | ∼ inf ? H θ . | σ GeV, the axion would overdominate the universe and give rise to the so-called is the axion mass, m σ 17 m 10 In this paper we explore the possibility that the axion couples to the inflaton via a The most well-studied mechanism that gives rise to the axion displacement is the Another feature essential to our understanding of the universe is the existence of cold & and the inflatonforbid being such the a mixing relevantis as degrees it to of respects demonstrate freedom, the thataxion axion there this dark shift is kinetic symmetry. matter. mixing no Theof gives We main inflation, find goal rise of the to this most a work interesting new scenario mechanism to for producing be when the Hubble scale galaxy rotation curves [ for the axion field takes placethe after equation inflation of is motion over, and of thebe the abundance neglected. single is axion derived field, by solving assuming that couplings to other fieldskinetic can mixing term. From the point of view of an effective field theory, with the axion cosmological moduli problem unless satisfies subject of active investigation over the yearsstructures as of it the leaves distinct universe signatures [ inout the small-scale that these are in tension with observational data on the Lyman- parametrizes the initial fieldto displacement have as a periodicity axion with decay constant is f the inflationary epoch, the vacuum misalignment gives rise to a present-day abundance of Here vacuum misalignment scenario, in whichthe the inflationary symmetry epoch, breaking happens andobservable before thus universe. or inflation The during renders axion stays the frozenthan at axion this the field initial field Hubble homogeneous value while within expansionand its the mass rate behaves is of as smaller cold the dark early matter. universe, For then an eventually axion-like starts field to whose oscillate mass stays constant since they are embedded. Two(i) properties that of they axions areglobal are U(1) pseudo of symmetry, particular Nambu-Goldstone and as importance Bosonsof such in (pNGBs) the obey this axion a of study: field shift a symmetry, frommatter (ii) its spontaneously in that potential the broken an minimum later initial gives universe. displacement rise to a relic abundance of axion dark dark matter. Its actualposed. nature Among remains them, axions a provideQCD mystery, a axion but viable which several and solves candidates well the studiedarise have strong possibility; been for these pro- instance include the fromrange of string masses theory and compactifications couplings to [ matter, depending on the theoretical framework in which JHEP08(2019)147 . 5 -folds. e ]. 27 , 26 window. However . f ; here, with the aim 4 inflaxion . In such a case the axion already has QCD -folds or more, which could pose a challenge Λ e < 20 – 3 – inf we explain axion misalignment and slow-roll inflation , we are led to consider low-scale models of inflation. < H we comment on the implications for the QCD axion. σ σ B C m In our scenario, on the other hand, we consider even lower m 1. In turn, this opens up the large 1 where we present the basic construction of our mechanism. . The field evolution will be dominated by purely classical 2 σ | ? θ | < m inf H we describe how it can produce dark matter and give rise to a consistent 3 ] the authors considered the QCD axion, without any direct coupling to the we introduce a field basis that diagonalizes the kinetic and mass terms of the A 25 ], but to our knowledge not in the context of inflation. Our inflaton can also be , 23 24 – 19 Our mechanism can be applied to generic axions. We discuss both the QCD axion and Kinetic mixings among multiple axions have beenThe considered axion, in being a previous pNGB, is litera- expected to be light. As in this work we focus on infla- For other interesting ideas on axion dark matter with low-scale inflation, see e.g. [ 1 2.1 Axions coupledLet to us a start strong by considering sector of an such axion that a is field, coupled to we some focus strong on sector. the As an QCD example axion. Then our mechanism is described by the Some calculational details and possibilitiesIn with appendix the QCD axioninflaton-axion are system. left to In the appendix appendices. in the diagonal basis. In appendix 2 Inflaton-axion mixing Then in section cosmological history. Weof explore concentrating the on parameterwe the space mainly in salient focus section features onevolution. axion-like of particles Then we the whose conclude mass with dynamics a stays discussion while constant of directions throughout avoiding for the further complications, cosmic research in section for inflationary model building. inflation scales of dynamics, and there will be no strict requirement onaxion-like the particles in number section of inflationary Given a long enoughconfiguration period which of results inflation, in the aically two probabilistic competing peaked distribution effects at for lead the values for to misalignment this an angle, mechanism typ- equilibrium toextraordinarily be long predictive, period it of is inflation of important 10 to reach equilibrium; this requires an tionary Hubble scales lower than In refs. [ inflaton, and studied the regime a potential during inflation,its but minimum, is but not its oscillating; quantum classically fluctuations the tend field to slowly keep rolls it toward displaced from the minimum. searches. In the diagonalcombination basis, of the the oscillating inflaton and field axion. that serves Hence, as we dubture dark [ it matter the is aaxion-like, linear but this is not a requirement necessary to the end of our proposed mechanism. the axion would simply beits subject potential to where damped its oscillations, energymatter would density abundance. relax vanishes, and to We thus the willits would minimum show minimum not of that contribute during the to inflation, kineticnew the thus windows mixing dark of providing keeps the the a parameter axion source space displaced for of from axion misalignment. dark matter, This relevant opens to up some experimental that noticeably affects its dynamics during inflation. In this case, without a kinetic mixing JHEP08(2019)147 , φ (2.7) (2.5) (2.6) (2.1) (2.2) (2.3) (2.4) as the . (The φ σ, . 200 MeV, ν CP ≈ φ∂ Σ µ 2 and ∂ α QCD µν P − α 1 α g . , which was manifest √ ) has a shape capable − ] σ φ ) ( − 28 QCD , which is not violated by φ ( V , . Λ Φ πf Σ V 2 − πkF, V is also parity-odd and that the ) gives the potential and mass α + 2 2 T Σ F φ σ φ 2 − ν − α 2.6 α 1 Φ → for φ∂ ν − α √ µ ∂ cos σ 1 ∂ Φ . − − √ µ 1 µν ) and ( ∂ 1 g = 1 2 conserving points µν 2.5 = Φ 2 g GeV − F the gluon field strength. We refer to 1 2 F min 12 < α < – 4 – is derivatively coupled to another real field ) CP Φ µν A . The shift symmetry of − 1 T , φ ) in ( 10 ˜ σ G ( V Aµν ˜ Σ G − , then Σ is a standard canonical axion with a decay 2 Σ G 2 G σ, f V Aµν m ( α eV Σ F , its mass at the minimum is [ G πkF, 6 similarly obtains an effective potential, but with a peri- 1 2 − f σ → − ) = 2 s π 1 2 σ ) g = 2 QCD 10 , are at the 2 s π √ 32 ,T g )Σ fixed. If we assume that V ,F × 2 32 = (Σ min = 0, then the global minima of the full scalar potential at zero α 6 ) as a symmetry under a shift of both Σ and Φ that keeps the of the kinetic mixing is restricted to lie within the range U σ Σ + Σ + . The coupling to the gluons yields an effective potential for Σ − ν ≈ ) basis, f σ 2.4 α ∂ min 1 ν 2 Σ which can roughly be approximated by a cosine, φ Σ = 0) + √ α µ m σ, φ σ∂ ∂ µ conservation is not necessary for the dark matter production mechanism − α/ ,T πF ∂ ( ) as µν 1 g φ (Σ preserves parity, then our theory in the vacuum respects µν − √ ( CP 1 2 g U V V 1 2 − = ; rewriting as (Σ − = F is the axion decay constant, πf = g f ), is realized in ( g − L . − Now, let us consider the effects of The kinetic terms can be diagonalized by redefining the fields, for instance, as L √ 2.1 σ √ assumption of we describe later,minimum but of is usefultemperature, for illustrating the nature of the vacuum.) Setting the for in ( combination Φ minimum of Going back to theodicity ( 2 with periodicity 2 The potential vanishes at temperatures muchwhile higher than at the temperatures QCD below scale Λ Λ Ignoring for a moment theconstant potential for which the Lagrangian takes the form odd under parity, respects athe discrete shift kinetic symmetry mixing term.(otherwise By one requiring of the the eigenvalues physicaldimensionless of degrees coupling the of freedom kinetic would matrix be to non-dynamical be or positive a ghost), the where inflaton, since we willof be inducing supposing cosmic that inflation its as effective well potential as providing a graceful exit from it. The axion field, following simple model where the axion JHEP08(2019)147 (2.8) (3.1) (3.2) (3.3) — the . ]), and σ Ψ] 30 , 29 and σ, φ, [ via the kinetic c φ L σ + σ ν φ∂ is the Hubble rate, and spacetime-dependent and µ ∂ φ σ. 2 σ µν a/a σ, 2 σ -conserving minima. The same α g α m = ˙ m . − , ) which do not violate the axion’s CP − ) 2 3 H ) + φ ) x may or may not depend on the tem- . However in an expanding universe, ( φ d φ to other matter fields which we have σ/f ( ( 2 to be a pNGB of some global U(1) sym- 0 V 0 σ ) min φ t V σ − m ( a αV − φ ) cos( to be a constant, and ignored the matter ν φ + and ( σ rendering the field 2 and Φ ) = ) = φ∂ I ˙ σ – 5 – ˙ m µ φ σ -derivative, dt V ∂ t H H − min µν g = ) as ) produces dark matter, and gives rise to a consistent + 3 + 3 1 2 2 ¨ φ σ 2.1 − 2.8 ds )( )(¨ 2 2 2 σ α α 2 σ − − m 1 2 (1 (1 − σ ν σ∂ . µ ; this in turn generates an effective linear potential for ∂ φ /dφ represents the couplings of µν ) is sufficiently flat, the Hubble friction can keep the field Φ in slow-roll motion g φ c . An overdot represents a ( 1 2 c L V − L dV an integer. In Minkowski spacetime, displacing the fields from a minimum results = k g ) = We will see in the following sections that different combinations of We should also remark that from an effective field theory point of view, there can φ − L ( 0 √ Here we havecoupling supposed the axionV mass Let us now show howcosmological the history. model We ( consider a flat FRW universe with metric in which the homogeneous equations of motion of the fields read inflaxions — will play various cosmologicalcosmological roles perturbations, such a as reheaton a for canonical inflaton heating producing up the the universe,3 and dark matter. Inflaxion mechanism The mass term ofaround the one axion of is the understoodperature. minima, to and arise the from expanding axioncollectively denoted the mass as periodic Ψ; potential its details will later be discussed. to any strong sector. Hencemetry hereafter that we consider is explicitly brokenanalyze (for a instance by model quantum of gravitational the effects form [ However even ifparameter they regions were different from not, what our we show mechanism2.2 in of the following axion discussions. General production axions In can the following, work, we will but also be in discussing axion-like fields which are not necessarily coupled effect is seen in thethus original sourcing basis ( mixing. We discuss these dynamics in detail in section further be potential couplingsdiscrete of shift the symmetry. form Throughout this paper we assume such terms to be negligible. in coupled oscillations. Theof fields’ values the averaged oscillations over a stillgiven period correspond that longer to than Σ thewhile timescale it approaches aminima, minimum. the axion When Σ Φ would also is get effectively displaced frozen from at its a point away from the with JHEP08(2019)147 for , are (3.6) (3.7) (3.4) (3.5) B φ ) during inflation, 3.6 , . 1 ˙ ) can thus be rephrased as the requirement φ φ 2 σ (

m V ]. Hence also in this case the axion approaches αH , 3 . ˙ φ 32 Λ σ , Hφ 2 31 2 σ ' − . A more detailed derivation of this solution, 0 H m < m < f, ) ) are negligible, then one can diagonalize the ﬁelds such 3 V – 6 – φ φ 2 σ ( inf ( − π inf 0

2 V m H H ) that the field displacements during inflation take ' − However, with the kinetic mixing with the inflaton, ' αV ˙ φ 3

) where the inﬂaton settles down to after the end of 3.6 φ ' H ( σ for a QCD axion). Then our requirements translate into αφ σ V

Our mechanism is insensitive to the detailed form of the 2 QCD ) is assumed to give rise to a sufficiently long period of cosmic φ ( ) to zero to obtain the quasi-static solution, V 3.2 . Because the axion settles down to the field value ( B = 0. It is seen from ( ). φ represents the typical value of the Hubble rate during inflation. 3.6 , and in cases where the axion potential arises only below some energy scale, we f inf H Let us set the minimum of Furthermore, we assume the inflationary scale to be even lower than the axion mass, If the third and higher order derivatives of When the axion mass is not much larger but only comparable to the Hubble scale, the axion would not 3 2 that they do not interact withthat each other. one Our assumption of for detailed the discussions. diagonal fields possesses a flatoscillate enough but instead potential rapidly roll tothe towards drive the solution minimum inflation. ( [ See appendix inflation as hierarchical values, together with the verificationgiven of in appendix the slow-roll approximationthe for initial the axion non-canonical misalignmentLagrangian for parameters, the unlike post-inflationary the evolution case is for uniquely the fixed vacuum misalignment by scenario. the Upon moving to the farthe right slow-roll hand conditions and side, used we 3 have assumed the inflaton potential to satisfy This forces the axiontually to be undergo stabilized a at damped thethe oscillation minimum. axion’s along effective its potential potential,the minimum and is left thus hand shifted even- side from of the ( origin. This is seen by setting where exists during inflation.constant We identify thedenote symmetry this breaking by Λ scale (whicha with is bound Λ the on the axion de decay Sitter temperature during inflation, The inflaton potential inflation, produce curvature perturbations in gooda agreement with graceful observation, and exit provide frominflaton inflation. potential, however, wesuch assume that the the energy U(1) scale symmetry of is inflation broken to before be inflation, and sufficiently also low that the axion potential 3.1 Low-scale inflation JHEP08(2019)147 (3.9) (3.8) at the )), the φ m B.10 ]. 1. Then since 33 φ is satisfied (cf. ( 2 . | H . However the mass 2 p 2 σ end φ | , 2M 2

m 2 . end 1 2 ), the field values of , end σ α 2 φ 2 σ 2 − H m m α φ 3.7 . − 2 φ p i 2 σ 2 1 m ϕ − m α ν m 6 1 2 p ∂ i C CαM ' ' ϕ – 8 – 2 φ µ ' ∂ RH 2 RH m = µν tot DM g ), the expressions at the leading order in ρ ) as ρ 1 2 undergoes harmonic oscillations about the origin, and σ, ϕ − 3.9 3.10 2 DM end + ϕ RH , m 2 σ ϕ RH , α φ m using ( X ' ' and ) and the contribution from =DM i DM 3.9 during inflation. This can break down towards the end of inflation, DM 2 DM ϕ = DM | ϕ m ϕ g σ to start oscillating with this initial amplitude at the end of inflation, 2, and the total density of the universe / − L √ , on the other hand, is much heavier (note that from ( DM ) we have displayed each of the coefficients of 2 DM | | ), and this field will reheat the universe via its decay. ϕ ϕ RH αφ 2 RH ϕ 3.12 | 2 DM m is a model-dependent dimensionless parameter that is typically larger than unity m C ' We defer the discussions on decay to the next section, and here we estimate the energy In order to analyze the field dynamics in the post-inflationary epoch, let us redefine 2 DM DM Considering the ratio between itsρ energy density which isobtained initially as dominated by the potential energy, Here due to the factor 2 in ( hierarchy depending on the details ofof the inflation exit the from inflation. inequalityfield is However displacement let still of us marginally suppose satisfied, that at i.e., the end the same mass asindicates. the axion, andm will serve as a dark matter candidate,densities as of its the subscript oscillating fields. As we saw in ( where in ( Each of the fields thus behaves as pressureless matter in the post-inflationary epoch. Here, under the mass hierarchya ( simple form of The exact expressions for the diagonal fields and their masses are given in appendix the axion, i.e., Let us thus impose this condition hereafter. the fields such that the kinetic and mass terms in the Lagrangian are diagonalized, a dark matter candidate, the inflaton mass after inflation needs to be larger than that of JHEP08(2019)147 , . ). p . tot DM M ρ ρ 3.15 3 . For inf GeV, , and (3.16) σ ' H √ 2 15 , which 3 m 2 µ/ 10 10 α RH φ (red). Field (red), with ρ ∼ ≈ × − m φ 1 αφ αφ ' since both m √ 6 = 4 / 5 DM = 0 denoting when φ µ inf ρ ) together with the ) has become approxi- and t is suppressed by the m H 3.3 inf 3.16 ∼ = 0 is set to the end of DM t H , i.e., ) is (blue), and t ϕ , with 10 (blue), and σ σ RH . ≈ σ 1.0 ϕ 3.16 with a frequency ) and ( σ 7 π/m ), and m 3.2 is of order 10. The non-diagonal φ/µ − C DM e (orange), ϕ 2 + 0.5 ), with DM ϕ φ/µ φ/µ e (orange curve), − . The horizontal axis shows the dimensionless σ φ 1 sech( – 9 – 0.0 . One clearly sees through this example that the + − /m 2 1 inf µ αφ 2 φ H ∝ end . Time is in units of 2 3 φ ) m H = φ p 10 ( /m can be defined at all times, one should keep in mind that this forms 0.5 V only after inflation when the potential ( ≈ ) = DM σ - αM end φ ϕ φ ( + RH H m p ϕ V . One sees that the density fraction of DM αφ 1. These conditions are enough to specify the field dynamics in ) whose numerical coefficient 5 5 3 φ component. becomes unity. The parameters are chosen as (i.e. the oscillation period of αM 10 15 20 15 10 = - − a - - 2 σ gives 3.14 | RH DM ), and thus the universe is dominated by . The inflation scale for the potential ( ∝ µ α ϕ ϕ | α ˙ π/m H/H 3.10 further carry an oscillatory mode with frequency − and φ , and φ σ starts oscillating at around the end of inflation m m redshift as . Field evolution at the end of inflation for 2 and = 1. Shown is the case of DM 2 σ RH ϕ ρ The actual field evolutions at around the end of inflation can be seen in figure = 10 This potential is adopted as a toy model for describing theAlthough late the stage field of inflation and the subsequent Depending on the inflaton potential at around the end of inflation, the fields may exchange energy 6 7 5 ˙ φ H/H through a parametric resonance; then the dark matter fraction might beoscillatory modified from phase, the hence estimate we of do ( on not worry scales whether that it have produces exited observationally viable the curvature Hubble perturbations horizona long before diagonal the basis endmately together of quadratic. inflation. with an initial amplitude ( fields corresponds to the inflation when m terms of the normalizedvalues quantities; of in particular,hence the our plot choice of isfield independent of the explicit Shown are time evolutions of the field values normalized by time in units of 2 mass hierarchy ( this plot we numericallyFriedmann solved equation, the for equations a of case motion where ( the inflaton potential is of the form − This density ratio staysand constant while the fields oscillate harmonically, Figure 2 values are normalized by JHEP08(2019)147 . ) φ < f 3.10 | (3.18) (3.17) (3.19) (3.20) σ | . This is )), giving DM 3.12 ϕ . . ψ, 5 obtains through gives RH and ) to ( ϕ 2 φ ¯ 2 ψiγ DM RH A.8 φ α ϕ /m with other matter fields ϕ 2 σ 1 − σ φff m with a periodicity of order 1 g √ σ (after inflation) + and + φ µν for both 2 σ ˜ F DM 8 m ϕ µν plane (from ( 2 φ 2 σ σ φF - 2 φ m m φ α m < f, 4 φγγ | ' G + to decay and start the Hot Big Bang cosmology, – 10 – −→ , φ DM end µν RH ˜ ϕ F | ϕ RH µν ϕ 2 σF α ), which at the leading order in − α 4 σγγ 1 G . 3.12 √ 2 ) − . This allows φ Ψ] = DM (during inflation) /m DM σ ϕ )) would continue to be placed close to 0 after inflation, carrying only a ϕ m σ, φ, 2 σ [ ' A.7 c m σ L than in ( can couple to other matter fields through shift-symmetric operators. Together being long-lived. For this scenario, it is essential that the condition ( ) does not hold and the axion mass remains larger, then the heavier diagonal | RH H σ 00 ϕ ϕ ]. V | DM 3.10 ’s coupling terms, they provide decay channels 23 ϕ , φ For concreteness, let us for example study the following matter couplings: Before ending this section we should remark that, although we have been treating If ( As we will show in the next section, the couplings of 21 The phenomenology of kinetically mixed axions interacting with other matter fields was also studied in 8 e.g. [ Here one particularly sees thatis the suppressed couplings by the ( dark matter field 3.3 Reheating andThe dark axion matter stability with easily seen from inverting ( and impose this as a constraintthat on the the model axion parameters. may Violation rotateresulting of along this random condition its implies axion periodic misalignmentexplain potential multiple the angle times dark at during matter the inflation;following abundance, discussions. the end however, of we inflation will not may consider also this be possibility able in to the the decay constant. Theminima, quadratic and thus approximation we is have been validLet implicitly in assuming us the the rewrite axion vicinity this displacement to of condition be at the of the potential end of inflation in a more conservative form of field (i.e. tiny fraction of the totalthe energy lighter density. field, Reheating which would is thencould complete typically survive with also until the today the decay to longer-lived of serve one, as and dark thus matter. neitherthe of axion the potential fields as a quadratic, it is actually periodic in Consequently the diagonal basismost of rotates the in energy the density in the post-inflation universe to the heavier diagonal field. provide decay channels for bothlarger for diagonal fields, andwhile induce decay rates thatholds are typically so that theinflation, mass i.e., hierarchy between the inflaton and the axion flips at the end of JHEP08(2019)147 ) φ RH DM ϕ ϕ 3.22 and (3.22) (3.23) (3.21) , with σ ). Here, are (here /M computed ψ ˜ F 3.13 , m , RH 3 φ , φ σF 3 φ m m m π is generically more is a Dirac fermion, π 2 φγγ using ( π 2 φff 8 to be a pseudoscalar 64 would quickly decay 2 σγγ g ψ G φ 64 DM 2 φ G 2 / / ϕ m 2 RH 3 5 / ) ) ϕ 5 2 2 ) 1 α 1 α 2 2 and α α − − σ is suppressed compared to that is tiny, − (1 m surviving until today, while (1 α (1 DM ' ' ϕ DM ) is suppressed by the mass hierarchy ' ) ¯ ϕ f ) γγ f RH γγ ϕ → → → would contain the suppression RH RH and RH ϕ ϕ – 11 – ϕ φff Γ( Γ( g DM Γ( ϕ . It should be noted that for the channels ( ) are , the decay rate of 5 , , φ ) . The decay rates to fermions, if they are much lighter φ φ 4 φ 4 5 σ 7 σ , φ 7 m ) 3 σ m m m m becomes comparable to the Hubble rate. If Γ φ m /m σ π π /m σ π RH 2 2 φγγ φff 8 m σ . On the other hand, the two-photon decay rates induced by 2 σγγ g 64 ( m G 3 64 2 ) G m ). One can also envisage a situation where the coupled fermions 2 ( α φ α ∝ ' ∝ ' ψ 3.19 ' ) /m ) σ ) ¯ m RH f are coupling constants. Here we have considered γγ RH m Γ f γγ ( Γ , which allows for the possibility of / / → φff → ∝ → RH g DM DM ϕ , DM is the field strength of either photons or hidden photons, DM RH DM ϕ γγ ) induced by the inflaton ϕ ) Γ ϕ µν σ not only due to kinematics, but further because of the suppression of the effective Γ( / ( Γ( F φ Γ( are 3.23 DM G Thus we have seen through the examples that, unless The decay rates of the dark matter and reheaton fields to photons via RH ˜ some cutoff scale. If instead we reheated into a hidden sector, then we should address F ϕ stable than quickly decaying and reheatingwhen the the universe. total decay The rateas time Γ above of exceeds reheating the can Hubble be rate estimated at as the end of inflation, then with ratio Γ and ( of coupling as seen inare ( lighter than thewould inflaton be but kinematically heavier forbidden. than the axion, in which case the decay of than both scalars ( with ratio Γ where we have writtenthe the ratio diagonal between masses theas in decay Γ terms rates of of φF we neglect the possibility thatcancel the each contributions other) to the photon couplings through M how to subsequently reheatis the beyond the visible scope one.sake of of this That keeping paper. the requires argumentpreheating, more Here as which model concise we could and be only building, clear very considerthe which efficient as QCD perturbative and possible. inflaxion. reheating, would Another We for be leave option of the such is particular a to interest study in study to the future case work. of and so that its kineticthe visible mixing sector term above withfields the to the BBN the axion temperature, visible photons conserves thethe and/or parity. obvious SM the choice Standard gauge As is Model symmetry, (SM) to we the fermions. couple want coupling In the to this scalar reheat case, given where JHEP08(2019)147 ) = B (3.28) (3.29) (3.30) (3.31) (3.26) (3.27) (3.24) (3.25) DM 0 ρ , and that φ ]. In order m 1 in appendix H ϕ eV [ , 33 2 − / 1 10 continues to redshift as × does not necessary mean that 0 ] reh 1 DM H H end 35 ϕ , . , ≈ . 34 end end 0 2 φ 2 σ > H 4 H H reh m m RH 2 T > H ≤ . ) α 3 2 , T reh RH RH − , ) T . 0 ( T 10 Λ ∗ ( g ∗ ∼ s < f. 4 MeV < 2 for Γ for Γ < H g , which is a function of the temperature as 30 π 2 0 s & reh – 12 – reh reh s π . 45 s DM = T T 2 end Γ reh RH ) yields the ratio of the dark matter density to the 2 0 RH 2 T 2 reh H Γ reh = H ( H s ). H 3.25 2 p 2 φ 2 σ = 3.6 M m m 3 2 reh α 6 H 2 ) and ( C ' 3.15 Thus the Hubble rate at reheating can be estimated as DM 9 Ω opens the radiation-dominated era, and assuming the decay products to , we require its lifetime to be longer than the age of the universe, RH ) using the entropy density ϕ DM after reheating. Assuming the entropy of the universe to be conserved after reh ϕ 3 /s ) form a diagonal basis only after inflation. Hence Γ − 0 s a ( RH ∝ , ϕ As for One should remember that the inflaton mass during inflation was much smaller than 9 DM DM DM reh ϕ ( inflaton fluctuations shouldcould have decay decayed during during inflation; inflation.realization this of would the The further quasi-static stabilized damp solution the field ( field’s (i.e. oscillation and thus lead to a very quick Combining these with ( total density in the present universe, Under these simplifying assumptions,ρ the energy density of reheating, the dark matterρ densities today and upon reheating can be related as Moreover, in cases wherereheating the temperature axion to mass be arises also only lower than below this, some scale Λ, we assume the where the Hubble constantto in compute its the present-day currentthe abundance, universe reheating let is temperature. using make scale some Firstly, simplifying so we assumptions thatinflationary assume about epoch, the it symmetry not stays broken to and exceed the the axion symmetry exists break- throughout the post- which in turn sets a lower bound on Γ This temperature is required from BBN to be at least of [ The decay of quickly thermalize, the initial temperature upon reheating is computed via after inflation ends. JHEP08(2019)147 ) ), 3.29 (4.1) (4.2) (4.3) (4.4) 3.18 ), ( 3.10 4 in order to ), ( , which combined 3.5 n > reh ]. Although some of ), ( H 1 is extremely close to unity, ) from above as | 3.4 ≥ 3 [ α . | 0 3.31 RH upon reheating do not affect to make up the dark matter ≈ ∗ ) s . ( 10. We also note that the detailed DM g DM 4 n ϕ − 2 ∼ n . ) and then it might be possible to have dark n C 1 n . ) follows Γ 0 3.31 0 φ σ reh reh H to be long-lived imposes constraints on the 0 H H m m reh H 3.24 ) where the dependencies on other parameters are H = 3, the inflaxion cannot provide all of the H < DM ) and find that, unless 2 4.2 n ∼ ϕ − – 13 – . 3.21 RH 10 DM RH Γ DM φ σ Γ ) translates into . Γ ) should satisfy Ω Γ . From ( m m )) bounds the abundance ( 4.1 σ ) are violated, we may still get dark matter. Violation DM ), the axion mass temporarily disappears upon reheating 3.31 2.2 ). Furthermore, for Ω and 3.29 3.29 3.29 φ 0, then ( 1 (cf. ( < alone, which gives ) and ( n > 2 ˜ α F ), and ( to quickly decay while 1, this bound shows that the power must satisfy 3.28 σF 3.28 RH alone does not allow inflaxion dark matter. However we should also note that if ( A similar story holds for a derivative coupling of the axion to fermions of ϕ ˜ F 0 10 ), ( σF /H ) yields 3.27 reh ) implies there would be a second symmetry breaking after reheating; this would lead H 3.27 ), ( Requiring Before turning to an analysis of the inflaxion parameter space, let us mention that even Instead of using the simplifying assumption ( 3.28 10 3.26 dark matter. dropped, one can alsothe use coupling the fullis expression not ( satisfied, then thematter abundance with is axionic no couplings longer only. given by ( Since reproduce the observed dark matterphoton abundance. coupling One particularly sees that with the axion- with a positive power This combined with Here, if the ratio of the decay rates is determined by the mass ratio as required for our mechanism tothat provide satisfies dark all matter, these below we conditions. explore the parameter space sort of matter couplingswith of ( The conditions we have imposed on( the inflaxion parameters areof ( the universe,the its conditions abundance were ( introduced only to simplify the computation and thus not necessarily to generation of topologicalthe defects other which hand may by eventually violating emituntil ( inflaxion the dark cosmic matter. temperaturewhether On cools the observed down dark again matter to abundance can Λ; also it be4 would achieved in be this interesting Parameter case. to space analyze values of the numberthe of order-of-magnitude relativistic estimate. degrees of freedom if the conditions ( of ( where in the far right hand side we have substituted JHEP08(2019)147 ˜ F as as (4.5) (4.6) (4.7) φF plane. is fixed is written 3(a) φ 3(a) reh 3, and its . T end m 0 - H σ ≈ m ) excluding the is approximately DM 4.7 σ m in the eV). For the chosen set σ / is governed by this axion φ , ) taking different shapes in m ( µν DM ). The other parameters are ψ, 4.7 ˜ 10 5 F ϕ 9 . The inflaton mass µν 3 ¯ ψiγ − φ φF , 4 φff φγγ = 10 g G ) setting the lower bound of the displayed , both of the inflaton and axion couplings end + φff + the decay of g DM 3.26 µν > H were allowed to decay only through the axion’s ϕ µν ˜ F set equal to the electromagnetic fine structure ˜ – 14 – reh F as T µν γ RH DM µν , then the decay of the dark matter into fermions α Γ = 10, and the coupling constant in figure ϕ σF σF RH 3(b) C γ πf γ α πf 8 α decays predominantly through the inflaton’s couplings in ) which excludes the green regions in the plots, dark mat- 2, 8 , if the fermion mass lies between those of the dark matter / , with < m = is the same as the inflationary energy scale up to a factor f 3.5 by the inflaton couplings. Different inflaton couplings open = RH c ψ c of the inflaxion dark matter field, and since we are supposing ϕ L = 1 = 1. On the other hand, the inflaton’s couplings such as m reh 3(b) L reh 2 T n α T (see also discussions in Footnote DM = 5) fulfill the requirement. < m n , and in figure ( end 1 DM . − H GeV, ψ to make up the entire dark matter abundance, i.e. Ω ∗ as 5 ) excluding the red regions, instantaneous reheating ( m g 17 = we show the parameter space for an axion-like ) would be saturated if DM GeV ¯ instantaneously decays at the end of inflation and reheats the universe, ψiγ The allowed windows are shown in white. We note that 3.27 ϕ 3 which gives 3(b) φ reh 10 137. To simplify the analysis, we have further imposed = 10 3.27 − 11 σ H / RH µ f ϕ ψ ∂ . coupling; one sees that at higher = 10 5 The conditions on the decay rates obviously depend on the details of the matter cou- In figure γ inf -range. ˜ In the plots we have ignored the time variation of the Hubble rate during inflation, thus F = 7) and µ H 11 φγγ reh n ¯ different windows inside the triangularline. area In between particular the inand green figure region reheaton and as the red dashed as the allowed windows. Forcan the dark provide matter the maincondition decay ( channel. TheσF red dashed linescoupling, in while the at plots lower show where the the same as theinstantaneous mass reheating, that depends on plings. As can bethe seen two plots, from the the reheaton blue regions excluded by ( of parameters, the conditionsstabilization during that inflation most ( stronglyter restrict stability the ( parameterblue space regions, are as well axion asT the BBN constraint ( rendering fixed to G by requiring value is shown by the black solid contours labeled with log tional to the decayconstant 1 constant so that and in figure where the axion’s coupling to the SM/hidden photons is taken to be inversely propor- ( Here we assumed the axionignored mass the to conditions be that constant involve throughout Λ. the cosmic The evolution matter and couplings thus are chosen in figure ψγ JHEP08(2019)147 , 1 −

φff

0 DM >H Γ region , could GeV 10

reh inf and −1 10

T −

GeV

�� 5 m ]) G = 10 channel. ψ 5 −10 eV ˜ [ F φγγ GeV would require � is increased, and as ¯ G σ ψiγ �� 0 17 σF φ f and reheating tempera- = 10 m + ( σ ˜ F 10 m = 10 φγγ 5 G f σF

log

- inf

); given a vacuum misalignment (b)

1.1

� RH 10

17 Γ

]) [ ( = 10 8 6 4 2 0 2

- 10 reh f GeV log T . Black solid contours show the inﬂaton mass 3 −

– 15 –

0 DM >H Γ = 10 GeV, the axion coupling is very small and hard to eV it could almost be within reach of the upcoming 10

17 inf φff −7 g

38]. The inﬂaton coupling,

σ 23] experimental constraints on such couplings compare

inf

�� Γ 5 m 37, eV, which on the other hand is in tension with the small- ]) f σ ˜ eV F −22 [ m � σ φF 10 �� 0 m + ∼ ( ˜ 3. With F σ 10 σF m /eV). Red dashed lines show the edge of the dark matter stability exclusion 5 φ log - m /2. In the left plot the inﬂaton is coupled to photons with (a) ( 1, to make up the dark matter abundance with 10 36] and forecasted [ =1 � |∼ 10 α -

θ . Parameter space of axion-like ﬁelds in terms of its mass

| . Regions where the inﬂaxion makes up the dark matter is shown in white. The colored

]) [ ( 2 0 2 8 6 4

reh

- 10 reh log GeV T T If it is the SM photon that the axion and inflaton are coupled to, it is natural to ask how With the conventional vacuum misalignment scenario, axions with masses and decay values. We also note that the allowed windows become larger as is taken closer to unity. | σ α ABRACADABRA experiment [ is set by BBN. The axion decay constant is fixed to ture | constants as shown hereextremely tiny would values. overdominate This the can beangle universe, readily of checked unless from ( the initial angle takes constant to regions are excluded from the requirementster of stability axion (red), stabilization during and inflation instantaneous (green), reheating dark (blue). mat- The lower end of the displayed and in the right plot to fermions with are increased, the regions sandwichedm between the blue and red shift or expand to smaller would be kinetically forbiddenextended and to hence the the red right dashed edge line. of the In allowed both window plots, would as be the coupling constants Figure 3 an ultralight mass in terms of log test, but in the mass region region if the dark matter inflaxion were allowed to decay only through the the existing [ scale observations of thedark universe. matter However abundance withnumber can the of be inflaxion conditions we achieved mechanism, have with the imposed observed much only for larger the axion purpose masses, of simplifying even theto under analysis. those a in our figure JHEP08(2019)147 , ) ], 40 4.5 3(a) obtained DM ϕ ) requires the reheating 3.29 ] and the CAST helioscope [ )). ) was adopted only to simplify the 39 3.19 3.29 (cf. ( ]. The effective coupling for φγγ 36 GeV can make up dark matter without having G – 16 – 17 10 ∼ , which would be experimentally excluded. On the other . However, since ( f C φγγ . However within the allowed window shown in figure G RH GeV with the reheaton decaying into the SM photons as in ( is the QCD axion, the condition ( 9 ϕ σ 10 & f An important feature of the inflaxion mechanism is that the kinetic mixing, together In the plots we have considered an axion-like field whose mass stays constant during the to different combinations ofgenerate the the cosmological inflaton perturbations, reheat and thekinetic the universe, and mixing axion serve also as fields, dark induces matter. i.e.both inflaxion The the the dark axion matter inflaxions, couplings and to to thetemperature interact inflaton each of with couplings. the other This universe in particles with particular through the relates the lifetime reheating of inflaxion dark matter. Moreover, the dark axions with similarly large decayinflaxion constants. mechanism is The uniquely dark determined matterfor by the abundance the Lagrangian produced vacuum parameters, by misalignment. unlike the isocurvature the perturbations, case Also it escapes due from conventional to cosmological constraints the on axions. fact thatwith inflaxion the dark rolling matter of the is inflaton, free rotates of the diagonalized basis in the field space. This leads in the conventional vacuum misalignment scenario.particles We with showed in decay particular constants that axion-like of extremely tiny initial misalignment angles,scale or structure ultralight of masses the that universe. wouldof spoil Although possibilities in the this small- for paper we the did QCD not investigate axion, the full our range mechanism can in principle also work for QCD axion and inflaton, and anThe inflaton axion mass during after inflation inflationminimum, that is which is driven enables larger to than the adark the axion field matter axion combined abundance. mass. point with slightly This theto displaced opens serve inflaton up from as to dark new the matter parameter make potential with windows up masses for and the the decay observed constants axion, that enabling would lead it to overproduction 5 Conclusions We have proposed akey dynamical ingredients production of mechanism for the axion construction dark are matter. low-scale inflation, The three a kinetic mixing between the also cause tension withradiation. observations, As including a the prooflow-scale of BBN reheating concept, and in we appendix CMB present the constraintsanalysis, parameter it on space would extra for be QCDwe important inflaxions leave to with for explore future the work. possibilities beyond this restriction, which temperature to be belowobservationally viable the cosmological QCD scenario. scale,in (The regions which same in might is the poseinflaxion true window plots a at for close challenge axion-like to for fieldsrequires the lying realizing a lower very an large edges.) coupling hand, Under a this reheaton decaying constraint, into to non-SM particles open at a such QCD low reheating temperatures could depending on the massit of is out ofthrough the reach inflaton of is current much smaller experiments than [ cosmic evolution. When saturate the upper bounds from globular cluster stars [ JHEP08(2019)147 ]). 49 – ]. An (which 47 46 – GB G 44 σ , given that there is a background ˜ F σF ]), or 43 – 17 – ]. Moreover, if the duration of inflation is minimal 42 ]. Furthermore, parts of the axion window that our mechanism opens 41 ]. The inflaxion mechanism could also provide a new cosmological picture of 22 We stress that some of the requirements for the inflaxions discussed in this paper, The basic idea of the mechanism is to dilute away the axion density with low-scale Veronica Guidetti, Ann Nelson, and Giovanni Villadoro for helpful discussions. Acknowledgments We thank Toyokazu Sekiguchi for usefultian comments Cespedes, on Michele a Cicoli, draft, Djuna and Croon, Enrico Gilly Elor, Barausse, Diptimoy Sebas- Ghosh, Marco Gorghetto, compactifications via non-perturbativefashion corrections to to [ thethe Kählerpotential axiverse, in in which aexcept most for similar axions those coupled settle to down theThis inflaton to picture and the thence may vacuum get offer excitedWe during a to leave low-scale make these way up inflation, topics to the for dark avoid future matter. the investigation. cosmological moduli problem in the axiverse. Another possibility we didbe not interesting study to is systematically aparameter study non-perturbative space the decay capable of range of explaining of thedirection the possibilities for reheaton. dark further matter to study It of fully is will our toThe explore universe. realize inflaton-axion the inflaxions Another kinetic within axion important an mixing ultraviolet-complete can framework. arise, for instance, in the low-energy limit of string such as the axion potentialsimplifying the to analyses. stay In constant particular duringtemperature. for reheating, the However QCD were by axion, lifting introduced this such merelywithin implied simplifying the a for conditions, very framework, low other such reheating scenarios as can a also temporal arise vanishing of the axion potential upon reheating. also construct variants with,can e.g., also a allow axions shift-symmetric to Gauss-Bonnet induceof baryogenesis coupling [ helical electromagneticalternative fields possibility which would can besome in to hidden principle gauge generate interactions be an that become additional produced strong small as only potential, during in inflation possibly (see [ from e.g. [ a non-negligible amplitude, thenfeatures its in mixing the with curvature the perturbation inflaton spectrum could on give observable rise scales. inflation, to oscillatory while slightly excitinginflaxion achieves the this through axion a derivative from coupling with its the slow-rolling post-inflationary inflaton. vacuum One may state. The even relax the recentof tensions the in universe the [ up measurements can of be the probedthat matter by the fluctuation upcoming matter amplitude experiments couplings suchparity-violating the signatures as diagonal in ABRACADABRA. fields the pick sky Weand up [ also thus through at remark the the time axion when could the give CMB scales rise exit to the horizon the axion is still oscillating with even if the axion decay constantpossibility is of large. producing These smoking-gun facts signaturesfrom about for the other distinguishing inflaxion the axion couplings inflaxion production offer mechanism scenarioscosmic the such strings. as For the instance, vacuumclose misalignment the to or case the emission where age from inflaxion of dark the matter universe decays can with give a rise lifetime to a number of observable signals, and may matter lifetime can be rather short — not much longer than the age of the universe — JHEP08(2019)147 to ˜ φ (A.4) (A.5) (A.6) (A.7) (A.8) (A.1) (A.2) (A.3) , ). Here and Ψ] 2.8 σ σ, φ, [ are guaranteed c L , ϕ 3 + . ) themselves can also take 3 ∗ 2 ) φ ˜ β , φ ∗ up to quadratic order, 2 φ − ∗ α ˜ φ φ , − φ and ( 00 ∗ σ 2 φ + 4 σ ˜ ( φ O 2 m αV ) α + 2 2 β − O − + ) ∗ 2 σ σ 2 − 00 ) are real, hence ) ∗ φ 0 . . ∗ V 00 0 (1 ∗ 00 2 V ∗ − 2 ∗ X ( A.3 , V V α | V p φ 00 ( ∗ 2 + 2 − 00 + / in ( V ∗ α ∗ X 1 1 ∗ 2 V V ˜ φ − φ β 1 2 − p 2 − 1 1 | | − 2 − β – 18 – ' 2 σ 2 β and 1 φ ) + 2 i , etc., and we note in particular that the effective = α H ∗ m ∗ ϕ | + σ φ = φ = ) around some field value 2 2 i ) is rewritten in terms of these fields as + = 12 φ ˜ φ m − m φ ( 2 | 0, however the discussion in this appendix also applies 2.8 X 1 2 φ V ˜ X φ, ϕ ( 0. Below we ignore any temperature dependence of the 0 > ∗ − /dφ + V i ) 2 σ < = φ ϕ m ,X + ( ν ασ 2 σ 2 σ 00 ∗ ∂ ∗ m i ϕ V ' dV V m ϕ , one obtains L µ = 2 σ = ϕ ∂ ) = 0 ∗ 2 can be negative. φ µν V /m ) can be simplified. By expanding each of the coefficients of ( β can be either positive or negative. (In the main text the axion mass g 00 2 ∗ V ), 1 2 V 00 ∗ m ∗ A.3 − φ V ( V = = X i ∗ = V g In the presence of a hierarchy between the masses such that We compute the leading order expressions for the coefficients, since − L 12 √ leading order in hierarchical field values that depend on their masses. the expression ( with the mass squareds given by Here, note that One can check that theto coefficients be of real fields. The Lagrangian ( where The kinetic and potentialfollowing field terms redefinitions, up to quadratic order can then be diagonalized by the where mass squared squared is considered toto be tachyonic masses, i.e. axion mass.) We can complete the square by the shift In this appendix we diagonalizewe expand the the kinetic effective and potential mass terms in the Lagrangian ( A Diagonal basis JHEP08(2019)147 ). A.8 (B.5) (B.2) (B.3) (B.4) (B.1) (A.9) , respectively. φ can similarly be | , denote respectively and 00 ) ∗ φ terms in the potential L V H ( also possesses a super- 3 ϕ 00 ϕ ) H ∗ V using the diagonal basis , φ ϕ ) ) ' | | σ and − L ), the effective potentials in L 2 σ ) φ ϕ L ϕ . L ( ( m ( | ϕ 2 L 00 B.2 ϕ ) basis, this corresponds to O . ( U ϕ U 00 2 2 L , 2 p α σ, φ 2 σ ) m M φ 1 2 − m . 2 σ ( | 0 ,U 1 00 = ) ∗ m + Ψ] are neglected. φ V 2 ' ( αV | 00 0 ) ∗ 0 , ∗ 2 H to be a slow-rolling inflaton. Let us combine V 2 V ' , η ) into V σ, φ, 2 [ ( 2 L ˜ H φ ' c , whose masses at the leading order are ϕ – 19 – ), and suppose the axion mass to be larger than ) 00 L = 0. In the ( − A.5 ). ∗ ) undergoes a damped oscillation about its origin, 2 σ ) , m L 00 ϕ ∗ L ∗ 00 L H ∗ ϕ m 2 σ 3.1 H V ϕ αV ( ϕ ϕ V 0 ( ϕ ( 0 m ˜ φ, V ' ' U is displaced from its origin. ( U ) = are understood to be in terms of σ 2 L L σ ,U ↔ ϕ m 2 ) V p ), one sees that the heavier field ( ) 2 φ ) agree as 2 σ ( U M . Hence φ 2.2 ( 2 V and = V H σ, m ' . In the following discussions, the U ) A L ϕ 2 H ( ) and L U m ϕ ( ) and the matter coupling U A.1 ) combined with ( as well as the Hubble rate, φ A.9 We consider a flat FRW universe ( The expressions in the opposite hierarchical regime where the derivatives of Hence the values of the slow-roll parameters defined in the diagonal basis Then one can checkthe that two once bases the axion is stabilized as ( B.2 Slow-roll inflation We now consider the lighterits diagonal field mass term and the constant offset in ( and thus eventually gets stabilized at which shows that the axion field The approximate expressions forFrom ( the diagonal fieldsHubble mass, for i.e. such a case are given in ( Let us analyzeintroduced the in appendix field displacementexpansion of ( a stabilized axion that of obtained by exchanging ( B Misalignment of stabilizedB.1 axion Generic cosmological background the lighter and heavier of the two fields up to overall signs which do not affect the physics. Here JHEP08(2019)147 (B.6) (B.7) (B.8) (B.9) (C.1) (B.10) becomes larger GeV, the axion GeV, the axion | 00 ]. 12 12 V 55 | 10 10 – as the time of 53 ]. However in this case φ f f . 52 , ) . L ) 50 ϕ . . | φ ( 0 ( ] 00 6 ∗ 0 / V U 7 . 50 V 2 continues to be stabilized at 0, this which renders the axion abundance | H ' − H . 2 2 π p 1 makes up the observed dark matter ' − 2 ). When these parameters are smaller ϕ . H L ˙ ˙ GeV M ˙ φ φ f H φ ϕ changes from positive to negative, corre- ( 2 ' p 2 σ | ∼ 12 H ? H V a ) can also be written as 2 σ 3 ? = 2 m θ 3 M αH 10 θ | GeV. However for m 2 2 3 L B.2 , 12 , ] symmetry breaking happens before or during ) ' 2 ? ϕ – 20 – ) 7 θ φ ) 10 L ' − ( φ ϕ × ]). On the other hand for ( V σ ( ) = ˙ ∼ 1 L V . U 51 ' f ) becomes ϕ 0 ( 2 = 1. Given that ' B.1 ∼ U 2 2 H ) due to anharmonic effects [ 2 σ p H Ω 2 p M ) becomes important or the inflaton mass ˙ C.1 3 H/H M 3 − A.1 follows the slow-roll attractor, L ϕ 1, the mass hierarchy ( 13 ), the end of inflation can also be described in terms of | towards the end of inflation. η | 2 σ B.4 take the same forms as those for a canonical single-field inflaton. Moreover, during 1 (“anthropic axion window” [ m The end of inflation, i.e. when the sign of ¨ φ With 13 2 ? could make up the entirelarger dark than matter the only expression if the ( axion isocurvature perturbationsaxion would dark also matter is be severely enhanced constrained and by thus CMB measurements such [ anharmonic An axion with anabundance initial if misalignment the angle decaywould constant overdominate the is universe unless theθ misalignment angle is tuned to lie near zero, i.e. Before turning to theaxion dark QCD matter. inflaxion, Ifthe we the inflationary briefly epoch Peccei-Quinn review [ and thethe is QCD conventional not axion story restored arising afterwards, of from then a QCD the vacuum misalignment present-day is abundance [ of the potential expansion ( than C QCD inflaxion Using ( This relation, however, can receive corrections if for instance the higher order terms in sponds to the time when happens when the inflaton potential and kinetic energies are related by Hence we see thatfor with the axion stabilized,slow-roll the inflation, slow-roll the conditions axion displacement and ( approximations than unity, One can further check that from these approximations follow agree with those defined similarly in terms of JHEP08(2019)147 1 > , and σ W m N , mass f 60], or introduce /2, and the inflaton =1 ) is merely introduced α 3.29 � -

inf 17

8 0 >H – 21 – - log ΓDM log . Black solid contours show the inﬂaton mass in terms =1[ 3 − W 10 14 N 7 × - � -

] [

0.15 0.10 0.05 0.20

φff reh GeV T g . To simplify the analyses, reh T symmetry so that all vacua are gauge equivalent [ W N Z 4, the reheaton mass is much lighter than eV. Under this condition, one /eV). can in principle be met for the QCD axion. In the parameter region we will φ . Parameter space of the QCD axion in terms of its decay constant 4 m 61–64]. In these cases, axions are radiated from the axionic domain walls and ( 10 region is set by BBN. The inflaton-axion mixing constant is fixed to The inflaxion mechanism for the QCD axion is subject to conditions involving the If, on the other hand, the symmetry breaks after inflation or is restored upon reheating, reh reheating temperature is coupled to fermions with then axionic domain walls wouldcan be form made and unstable overdominate by the settingso-called the universe. domain number The of wall degenerate walls vacua number) however of to the axion potential (the with such extremely low reheating temperatures,as the a axion window proof shown here ofmake should up be concept the considered (see dark matter the isof main shown dark in text matter white. for stability TheT colored (red) discussions). regions and are Regions instantaneous excluded from reheating where the (blue). the requirements QCD The lower inflaxion end can of the displayed Figure 4 of log and gauge the QCD scale. In particular, the condition ( further breaking of thevacua [ axion’s shift symmetrycosmic strings to and generate thus can anaxion in dark energy principle matter make produced bias up is the between still dark uncertain; the matter. see [ However the amount of reheating temperature to be belowobservationally the viable QCD cosmology scale, with the whichto poses inflaxions. a simplify Although challenge the ( for analyses realizingwe and an not demonstrate a as necessary asection condition proof for the of inflaxion conceptshow framework, that in below all figure the conditions listed at the beginning of JHEP08(2019)147 ; ). 3 − φff and 2.6 ) be- G 10 in the f ]. in the Phys. × , 4.6 68 , DM φff g ϕ = 5 67 φff and g as shown in ( , . RH -range is set by the f ϕ eff GeV. We should also reh N T 16 10 3; the black solid contours . from the simplifying condi- 0 ∼ = 10, required instantaneous GeV, which would be experi- 9 ≈ f ) excluding the red region, and C 10 QCD ), then a very large coupling DM 3.27 & is a function of 4.5 GeV, which includes the region close f 2 and σ / 19 m 10 = 1 × α in terms of the axion decay constant 1 4 – 22 – ]. However the detailed value of this model-dependent factor < f < 28 is decreased, the position of the allowed window shifts GeV φff ]. as well as the inflaton and fermions with ), which permits any use, distribution and reproduction in g . The axion mass 17 ]. , and the lower end of the displayed 14 10 reh ) excluding the blue region. The white region denotes the al- reh eV). Moreover, we have considered couplings as in ( T SPIRE × / T Planck 2018 results. VI. Cosmological parameters 4.7 φ SPIRE IN [ ) are the dark matter stability ( IN m [ ]). We leave a detailed study of the QCD inflaxion cosmology to ( ), while the upper end is set to Λ A New Type of Isotropic Cosmological Models Without Singularity CC-BY 4.0 10 ], the non-SM fermion is assumed to have a time-varying mass, arising 3.29 70 1 , values, hence larger axion and inflaton masses; the value of This article is distributed under the terms of the Creative Commons 3.26 f 69 (1980) 99 ), and fixed the inflaton mass such that Ω collaboration, ) and ( 4.7 B 91 ). We adopted the parameter values 3.26 3.29 Lett. Planck arXiv:1807.06209 If the inflaton is coupled instead to photons as in ( The parameter space is shown in figure A.A. Starobinsky, For the QCD axion-photon coupling, there is a further multiplicative factor of order unity which is set 14 [2] [1] by the electromagnetic and colordoes anomalies not [ affect the analysis here since the decay channels are dominated by the inflaton’s coupling. References decay (see e.g. [ future work. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. to the right edge of the plot, is disfavoredis by required black in hole order superradiance tomentally limits open excluded. [ a However QCD this axion statementand window assumes the at a picture perturbative may decay differ of when the one reheaton, takes into account the possibility of non-perturbative instantaneous reheating ( lowed window. As the coupling towards smaller plot is chosen such thatremark that the the allowed range window of is 3 centered at are labeled with log tween the axion and photons, the latter interaction dominantly determinesdisplayed parameter the range. decay rates The conditions ofbesides that both ( most strongly restrict the parameter space the reheating temperature We used a linear scaleBBN for constraint ( tion ( reheating ( very light non-SM fermions, whichsector couple above to the SM BBN particleson temperature. extra and radiation Here, thus [ in thermalize order thefor to visible instance avoid from the a observationalnon-relativistic constraints after coupling reheating to so a that rolling they do scalar not field, contribute which to ∆ quickly renders the fermion example of the accompanying reheating scenario would be that the reheaton decays into JHEP08(2019)147 ] 119 (2000) Phys. D 96 ] Mon. , 85 , Phys. Rev. , Constraining String Phys. Rev. Mesons , ]. B Phys. Rev. hep-th/0605206 (2007) 733 constraints on arXiv:1807.06211 , [ Phys. Rev. Lett. , (2018) 083027 ]. , α ]. 79 SPIRE IN arXiv:1408.3936 Mon. Not. Roy. Astron. ]. Phys. Rev. Lett. [ ]. [ , , D 98 SPIRE SPIRE First constraints on fuzzy Lyman- IN (2006) 051 IN ][ SPIRE forest ][ SPIRE 06 IN α IN ][ ]. (1978) 223 ][ (2014) 004 Phys. Rev. Rev. Mod. Phys. 40 , ]. 12 JHEP , Ultralight scalars as cosmological dark , SPIRE ]. ]. IN ][ SPIRE Galactic rotation curves versus ultralight dark JCAP ]. ]. IN , arXiv:0905.4720 Hidden axion dark matter decaying through mixing SPIRE SPIRE The type IIB string axiverse and its low-energy ][ [ arXiv:1610.08297 – 23 – Invariance in the Presence of Instantons IN IN [ hep-ph/9406308 [ arXiv:1206.0819 [ SPIRE SPIRE Cold and fuzzy dark matter T Baryogenesis and Dark Matter from Axion dissipation through the mixing of Goldstone [ [ IN IN Phys. Rev. Lett. ]. ][ ][ , and P Flux compactification ]. ]. CP Conservation in the Presence of Instantons SPIRE ]. (1981) 467 (1981) 347 forest data and hydrodynamical simulations ]. (2010) 123530 IN (1994) 213 (2012) 146 arXiv:1810.00880 Axions In String Theory (2017) 043541 α [ ][ SPIRE Planck 2018 results. X. Constraints on inflation SPIRE 195 10 arXiv:1703.09126 SPIRE IN D 23 IN D 81 SPIRE [ IN [ ][ IN B 336 D 95 [ ][ JHEP arXiv:1703.04683 arXiv:1708.00015 , [ [ A New Light Boson? Problem of Strong The Inflationary Universe: A Possible Solution to the Horizon and Flatness (2019) 035031 Phys. Rev. Phys. Rev. (2017) 4606 collaboration, , First Order Phase Transition of a Vacuum and Expansion of the Universe ]. ]. ]. (1977) 1440 (1978) 279 , Phys. Rev. Phys. Lett. , , astro-ph/0003365 38 40 D 99 471 [ SPIRE SPIRE SPIRE IN arXiv:1805.00122 hep-th/0610102 IN IN phenomenology with QCD axion and[ the 3.5 keV X-ray line matter: Implications of the[ soliton-host halo relation bosons the mass of lightSoc. bosonic dark matter using SDSS Lyman- ultralight scalar dark matter:(2017) Implications 123514 for the early and late universe matter dark matter from Lyman- (2017) 031302 [ Axiverse 1158 Lett. [ Planck [ Lett. Not. Roy. Astron. Soc. Problems Rev. F. Wilczek, R.D. Peccei and H.R. Quinn, S. Weinberg, A.H. Guth, G. Elor, M. Escudero and A. Nelson, K. Sato, M. Cicoli, M. Goodsell and A. Ringwald, T. Higaki, N. Kitajima and F. Takahashi, N. Bar, D. Blas, K. Blum and S. Sibiryakov, K.S. Babu, S.M. Barr and D. Seckel, E. Armengaud, N. Palanque-Delabrouille, C. D.J.E. Yèche, Marsh and J. Baur, T. Kobayashi, R. Murgia, A. De Simone, M. V. Irˇsiˇcand Viel, L. Hui, J.P. Ostriker, S. Tremaine and E. Witten, Viel,V. Irˇsiˇc,M. M.G. Haehnelt, J.S. Bolton and G.D. Becker, A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J.W. March-Russell, Hu, R. Barkana and A. Gruzinov, P. Svrˇcekand E. Witten, M.R. Douglas and S. Kachru, [9] [6] [7] [8] [4] [5] [3] [20] [21] [18] [19] [16] [17] [14] [15] [12] [13] [10] [11] JHEP08(2019)147 ] , 05 D , , 13 (2018) Phys. ]. , (2004) (2019) JHEP Phys. , , ]. 02 D 98 Phys. Rev. SPIRE D 70 , ]. IN (2018) 035017 SPIRE Nature Phys. ][ JHEP , IN astro-ph/0002127 , [ SPIRE ][ IN D 98 ]. Revisiting the bound on (2014) 191302 Phys. Rev. ][ , Phys. Rev. , The QCD axion, precisely 113 Spectator field models in light SPIRE ]. IN ][ Phys. Rev. (2000) 023506 , ]. arXiv:1602.01086 SPIRE [ Gravity and global symmetries arXiv:1303.6255 IN The 3.5 keV Line from Stringy Axions ]. ]. ]. [ D 62 ][ ]. SPIRE Cosmological Constraints on Rapid Roll IN arXiv:1901.10652 Experimental Targets for Photon Couplings of ]. Phys. Rev. Lett. [ ][ , SPIRE SPIRE SPIRE SPIRE IN IN IN MeV scale reheating temperature and IN ][ ][ ][ ]. ]. (2013) 042 – 24 – SPIRE ][ Axion Misalignment Driven to the Bottom Review of Particle Physics (2016) 141801 Phys. Rev. arXiv:1709.06085 QCD axion window and low-scale inflation IN , [ ]. 10 The ALP miracle: unified inflaton and dark matter ][ Broadband and Resonant Approaches to Axion Dark 117 SPIRE SPIRE ]. IN IN (2019) 052012 arXiv:0905.1752 Rapid roll Inflation with Conformal Coupling SPIRE Stochastic axion scenario ][ ][ [ JCAP ]. IN , ]. SPIRE ][ (2018) 006 Wormholes and masses for Goldstone bosons arXiv:1805.08763 D 99 IN collaboration, [ 02 ][ SPIRE hep-th/9502069 SPIRE arXiv:1702.03284 arXiv:1511.02867 arXiv:1707.02987 New CAST Limit on the Axion-Photon Interaction [ IN [ [ [ IN (2009) 023 ][ Design and implementation of the ABRACADABRA-10 cm axion dark arXiv:0709.1952 ][ Phys. Rev. Lett. [ 09 JHEP , ]. , Phys. Rev. What is the lowest possible reheating temperature? , arXiv:1705.02290 arXiv:1812.11186 (1995) 912 (2018) 015042 [ [ (2017) 044 (2016) 034 (2017) 192 SPIRE JCAP ]. , astro-ph/0403291 IN collaboration, 05 01 10 [ [ D 52 D 98 arXiv:1706.07415 [ (2008) 043519 SPIRE arXiv:1406.6053 IN arXiv:1805.07362 axion-photon coupling from Globular[ Clusters CAST (2017) 584 030001 Matter Detection matter search thermalization of neutrino background [ 043506 Particle Data Group Inflation of spectral index after Planck 136 77 (2019) 162 JHEP Rev. Rev. JCAP JHEP the QCD Axion [ A. Ayala, I. Dom´ınguez,M. Giannotti, A. Mirizzi and O. Straniero, Y. Kahn, B.R. Safdi and J. Thaler, J.L. Ouellet et al., S. Hannestad, T. Kobayashi, S. Mukohyama and B.A. Powell, T. Kobayashi, F. Takahashi, T. Takahashi and M. Yamaguchi, M. Kawasaki, K. Kohri and N. Sugiyama, R. Alonso and A. Urbano, L. Kofman and S. Mukohyama, G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G.R. Villadoro, Kallosh, A.D. Linde, D.A. Linde and L. Susskind, F. Takahashi, W. Yin and A.H. Guth, R. Daido, F. Takahashi and W. Yin, R.T. Co, E. Gonzalez and K. Harigaya, P. Agrawal, J. Fan, M. Reece and L.-T. Wang, P.W. Graham and A. Scherlis, M. Cicoli, V.A. Diaz, V. Guidetti and M. Rummel, [39] [40] [37] [38] [35] [36] [32] [33] [34] [30] [31] [28] [29] [25] [26] [27] [23] [24] [22] JHEP08(2019)147 , ]. , ]. B ]. Phys. , SPIRE (1986) IN SPIRE [ ]. ]. (1982) IN SPIRE (2005) 009 IN 48 D 33 Phys. Lett. ][ 07 (2008) 005 , SPIRE SPIRE ]. ]. (1992) 3394 IN IN 09 ][ ]. ][ (1974) 3 [ JCAP SPIRE SPIRE 67 , D 45 ]. IN (2006) 018 IN JCAP ]. Phys. Rev. ]. SPIRE ][ hep-ph/9505253 , , ][ IN , 10 [ Phys. Rev. Lett. SPIRE Vector dark matter production , SPIRE SPIRE IN arXiv:1304.0922 [ IN IN Decaying dark matter and the ][ Phys. Rev. JCAP ][ ][ , , ]. astro-ph/0511774 arXiv:1704.01122 ]. [ [ Dimensionless constants, cosmology and Cosmological Consequences of the (1982) 1867 ]. Isocurvature Constraints and Anharmonic (2013) 032 Zh. Eksp. Teor. Fiz. SPIRE 48 astro-ph/9812088 , Geometric Baryogenesis from Shift Symmetry arXiv:1810.07208 IN SPIRE [ Cosmological signature of new parity violating [ 09 SPIRE ][ IN IN ][ Can slow roll inflation induce relevant helical [ – 25 – arXiv:1005.5322 [ (2006) 023505 (2017) 011801 Accidental Peccei-Quinn symmetry protected to JCAP arXiv:1505.05511 arXiv:1612.04824 , Update of axion CDM energy [ [ (2019) 015 (1999) 1506 119 D 73 Strongly broken Peccei-Quinn symmetry in the early Universe Cosmic Strings and Domain Walls in Models with Goldstone Phys. Rev. Lett. 04 (1992) 189 ]. 83 Constraining the inflationary energy scale from axion , ]. Axionic domain wall production during inflation (2011) 037 ]. N-flationary magnetic fields Is there a Peccei-Quinn phase transition? Anharmonic evolution of the cosmic axion density spectrum (2015) 067 ]. astro-ph/9405028 03 SPIRE JCAP [ B 283 SPIRE arXiv:1507.06387 , IN 09 (2017) 131101 SPIRE [ IN Phys. Rev. ][ IN ][ , SPIRE ][ JCAP 118 IN Phys. Rev. Lett. , [ , JCAP ]. Cosmic and Local Mass Density of Invisible Axions , Phys. Rev. Lett. ]. Removing the cosmological bound on the axion scale 8 , Axions and inflation: Sitting in the vacuum (1994) 7690 Phys. Lett. σ Of Axions, Domain Walls and the Early Universe (2015) 010 , ]. SPIRE 10 SPIRE IN D 50 IN (1990) 353 [ [ SPIRE arXiv:0806.0497 IN hep-ph/0405256 astro-ph/0606534 cosmology arbitrary order Spontaneous Breakdown of Discrete Symmetry 1156 246 Rev. Effects on QCD Axion Dark Matter and PseudoGoldstone Bosons [ [ JCAP 889 other dark matters at the end of inflation [ [ magnetic fields? tension in interactions Phys. Rev. Lett. L. Di Luzio, E. Nardi and L. Ubaldi, Ya. B. Zeldovich, I. Yu. Kobzarev and L.B. Okun, P. Sikivie, A.D. Linde and D.H. Lyth, D.H. Lyth and E.D. Stewart, T. Kobayashi, R. Kurematsu and F. Takahashi, A. Vilenkin and A.E. Everett, K.J. Bae, J.-H. Huh and J.E. Kim, D.H. Lyth, K. Strobl and T.J. Weiler, M.S. Turner, M. Tegmark, A. Aguirre, M. Rees and F. Wilczek, G.R. Dvali, M. Dine and A. Anisimov, F. Takahashi and M. Yamada, M.M. Anber and L. Sorbo, R. Durrer, L. Hollenstein and R.K. Jain, M. Bastero-Gil, J. Santiago, L. Ubaldi and R. Vega-Morales, A. Lue, L.-M. Wang and M. Kamionkowski, A. De Simone, T. Kobayashi and S. Liberati, K. Enqvist, S. Nadathur, T. Sekiguchi and T. Takahashi, [60] [61] [57] [58] [59] [55] [56] [52] [53] [54] [50] [51] [47] [48] [49] [44] [45] [46] [42] [43] [41] JHEP08(2019)147 , , ]. , ]. Phys. , SPIRE IN SPIRE (1996) 4237 IN ][ ][ D 53 ]. ]. Long-term dynamics of SPIRE Phys. Rev. ]. IN , arXiv:1411.2263 Gauge-preheating and the end of [ ]. ][ SPIRE arXiv:1806.05566 IN [ SPIRE ][ IN ]. SPIRE ][ IN ][ Discovering the QCD Axion with Black Holes SPIRE (2015) 084011 IN Axions from Strings: the Attractive Solution Cosmology of Biased Discrete Symmetry Breaking ][ – 26 – Evading the cosmological domain wall problem (2018) 091E01 ]. Biased domain walls arXiv:1502.06506 D 91 [ Lattice formulation of axion inflation. Application to 2018 arXiv:1812.03132 SPIRE [ hep-ph/9608319 IN arXiv:1706.05097 [ [ [ PTEP (2015) 034 Phys. Rev. , , 12 arXiv:1806.04677 (2019) 002 [ 06 (1997) 5129 (1989) 1558 JCAP Stochastic and resolvable gravitational waves from ultralight bosons (2017) 131101 , JCAP D 55 D 39 119 (2018) 151 , ]. 07 SPIRE IN [ preheating and Gravitational Waves Rev. Lett. axion inflation JHEP cosmological axion strings Phys. Rev. Phys. Rev. J.R.C. Cuissa and D.G. Figueroa, R. Brito et al., P. Adshead, J.T. Giblin, T.R. Scully and E.I. Sfakianakis, M. Gorghetto, E. Hardy and G. Villadoro, M. Kawasaki, T. Sekiguchi, M. Yamaguchi and J. Yokoyama, A. Arvanitaki, M. Baryakhtar and X. Huang, D. Coulson, Z. Lalak and B.A. Ovrut, S.E. Larsson, S. Sarkar and P.L. White, G.B. Gelmini, M. Gleiser and E.W. Kolb, [70] [68] [69] [65] [66] [67] [63] [64] [62]