Solving the Tension between High-Scale Inflation and Axion Isocurvature Perturbations
aa r X i v : . [ h e p - ph ] A ug KEK-TH-1711, DESY-14-033, TU-958, IPMU14-0060
Solving the Tension between High-Scale Inflationand Axion Isocurvature Perturbations
Tetsutaro Higaki a,⋆ , Kwang Sik Jeong b, ∗ , Fuminobu Takahashi c, d † a Theory Center, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan b Deutsches Elektronen Synchrotron DESY,Notkestrasse 85, 22607 Hamburg, Germany c Department of Physics, Tohoku University, Sendai 980-8578, Japan d Kavli IPMU, TODIAS, University of Tokyo, Kashiwa 277-8583, Japan
Abstract
The BICEP2 experiment determined the Hubble parameter during inflation to be about10 GeV. Such high inflation scale is in tension with the QCD axion dark matter if the Peccei-Quinn (PQ) symmetry remains broken during and after inflation, because too large axion isocur-vature perturbations would be generated. The axion isocurvature perturbations can be sup-pressed if the axion acquires a sufficiently heavy mass during inflation. We show that this isrealized if the PQ symmetry is explicitly broken down to a discrete symmetry and if the breakingis enhanced during inflation. We also show that, even when the PQ symmetry becomes sponta-neously broken after inflation, such a temporarily enhanced PQ symmetry breaking relaxes theconstraint on the axion decay constant.
PACS numbers: ⋆ email: [email protected] ∗ email: [email protected] † email: [email protected] . INTRODUCTION The identity of dark matter is one of the central issues in cosmology and particlephysics. Among various candidates for dark matter, the QCD axion is a plausible andinteresting candidate. The axion, a , arises as a pseudo-Nambu-Goldstone (pNG) bosonin association with the spontaneous breakdown of a global U(1) PQ Peccei-Quinn (PQ)symmetry [1, 2]. If the U(1) PQ symmetry is explicitly broken only by the QCD anomaly,the axion is stabilized at vacuum with a vanishing CP phase, solving the strong CP prob-lem. More important, the dynamical relaxation necessarily induces coherent oscillationsof axions, which contribute to cold dark matter (CDM). We focus on the axion CDMwhich accounts for the total dark matter density, throughout this letter.The axion mass receives contributions from the QCD anomaly, m QCD a ≃ × − eV (cid:18) f a GeV (cid:19) − , (1)where f a is the axion decay constant. Because of the light mass, the axion genericallyacquires quantum fluctuations of δa ≃ H inf / π during inflation, leading to CDM isocur-vature perturbations. Here H inf is the Hubble parameter during inflation. The mixtureof the isocurvature perturbations is tightly constrained by the Planck observations [3],which reads H inf < . × GeV (cid:18) f a GeV (cid:19) . (95% CL) , (2)neglecting anharmonic effects [4–8]. In particular, a large-field inflation such as chaoticinflation [9] is in conflict with the isocurvature bound. Recently the BICEP2 experimentannounced the discovery of the primordial B-mode polarization [10], which determinesthe inflation scale as H inf ≃ . × GeV (cid:16) r . (cid:17) , (3) r = 0 . +0 . − . (68%CL) , (4)where r denotes the tensor-to-scalar ratio. After subtracting the best available estimate Such large tensor-to-scalar ratio can be explained in various large field inflation; see e.g. [11–24]. The r = 0 . +0 . − . . Therefore one can seefrom (2) and (3) that there is a clear tension between the inflation scale determined bythe BICEP2 and the QCD axion dark matter. There are various known ways to suppress the axion CDM isocurvature perturbations.First, if the PQ symmetry is restored during inflation (or reheating), there is no ax-ion CDM isocurvature perturbations, as the axion appears only when the PQ symmetryis spontaneously broken some time after inflation [26, 27]. In this case topological de-fects such as axionic cosmic strings and domain walls are generated, and in particularthe domain wall number N DW must be unity to avoid the cosmological catastrophe [28].Second, if the kinetic term coefficient for the phase of the PQ scalar was larger duringinflation than at present, the quantum fluctuations, δa , can be suppressed after inflation.This is possible if the radial component of the PQ scalar takes a larger value during infla-tion [26, 29]. The scenario can be implemented easily in a supersymmetric (SUSY) theory,as the saxion potential is relatively flat, lifted by SUSY breaking effects. Interestingly,a similar effect is possible if there is a non-minimal coupling to gravity [30]. Third, theaxion may acquire a heavy mass during inflation so that its quantum fluctuations getsuppressed [31]. In Ref. [31] two of the present authors (KSJ and FT) showed that theQCD interactions become strong at an intermediate or high energy scale in the very earlyUniverse, if the Higgs field has a sufficiently large expectation value. In fact, the second solution of Refs. [26, 29] is only marginally consistent with theBICEP2 result (3), if the field value of the PQ scalar is below the Planck scale. Also thethird solution of Ref. [31] is consistent with the BICEP2 result only in a corner of theparameter space. Therefore, we need another solution to suppress the axion isocurvatureperturbations, as long as we assume that the PQ symmetry remains broken during and tension with the Planck result can be relaxed in the presence of small modulations in the inflatonpotential [25] or hot dark matter/dark radiation. The isocurvature perturbation bound similarly applies to the so called axion-like particles, or generalpseudo Nambu-Goldstone bosons, which are produced by the initial misalignment mechanism andcontribute to dark matter. The idea of heavy QCD axions during inflation was considered in Refs. [32–34] to suppress the axionabundance, not the isocurvature perturbations. f a & GeV and also N DW = 1. 4 I. ISOCURVATURE CONSTRAINTS ON THE AXION CDM
The axion, if exists during inflation, acquires quantum fluctuations, δa = H inf / π ,giving rise to the axion CDM isocurvature perturbations. The isocurvature constraint onthe axion CDM leads to the upper bound on the Hubble parameter during inflation as inEq. (2), which is shown by the solid (red) line Fig. 1. Here the anharmonic effect is takeninto account [8]; the axion CDM isocurvature perturbations get significantly enhanced asthe initial field value approaches the hilltop, as can be seen for f a . GeV. Note thatwe assume that the axion produced by the initial misalignment mechanism accounts forthe total CDM density in the figure.Let us here briefly study how much the second solution mentioned in the Introductioncan relax the isocurvature bounds on the Hubble parameter. To see this, we write thekinetic term of the PQ scalar, S , whose expectation value determines the axion decayconstant, as: L K = ζ ∂S † ∂S ⊃ ζ | S | ( ∂θ ) , (5)where we define S = | S | e iθ , and ζ ( >
0) parametrizes a possible deviation from the canoni-cally normalization. In general, ζ may depend on S , the inflaton or other fields: ζ = ζ (Φ),where Φ denotes such fields collectively. During inflation, the canonically normalized ax-ion, a = h ζ | S |i inf θ , acquires quantum fluctuations of order H inf at the horizon exit,namely, δθ = H inf π h ζ | S |i inf . (6)Note that the quantum fluctuations, δθ , at super-horizon scales remain constant, even if ζ | S | evolves in time during and after inflation. Thus, the quantum fluctuations of thecanonically normalized axion in the low energy is given by δa = h ζ | S |i h ζ | S |i inf H inf π , (7)where the subindices 0 and inf denote that the variables are estimated in the low energyand during inflation, respectively. For ζ = 1, if | S | takes a large VEV during inflation and5ettles down at a smaller field value in the low energy, the axion quantum fluctuationsare suppressed by a factor of h S i inf /f a . Alternatively, we may assume that ζ depends onthe inflation field φ . If the inflaton takes a large field value ζ may be enhanced duringinflation, suppressing the axion quantum fluctuations, similarly.In Fig. 1, we show the upper bounds on H inf for h S i inf = 10 GeV, M P l , and 15 M P l ,which are relaxed with respect to the case of h S i inf = f a . We can see however that super-Planckian values of h S i inf are necessary to resolve the tension between the BICEP2 resultand the axion CDM. There is an interesting possibility that the PQ scalar is the inflaton.Then it takes super-Planckian values of order O (10) M P l during inflation, the axion decayconstant f a between 10 GeV and 10 GeV will be allowed. In the next section, we will show that the tension between the BICEP2 result andthe axion CDM can be resolved without invoking super-Planckian field values for the PQscalar.
III. SUPPRESSING ISOCURVATURE PERTURBATIONS BY EXPLICIT PQBREAKINGA. Basic idea
Now we give a basic idea to suppress the axion isocurvature perturbations. The es-sential ingredient for our mechanism is the PQ symmetry breaking enhanced only duringinflation. To illustrate our idea, let us consider a simple set-up, L = 12 ( ∂φ ) + ∂S † ∂S − V ( φ, S ) (8) We note that the one cannot take h S i inf arbitrarily large, as it might affect the inflaton dynamics.In non-SUSY case, we need h S i . H M P l , whereas, in SUSY case, we need h S i inf . M P l , sinceotherwise the energy density stored in the PQ scalar significantly affects the inflaton dynamics. Here we assume that the kinetic term is not significantly modified at large field values. In the case ofthe running kinetic inflation [16–18], the kinetic term is modified, but the suppression factor is basicallysame, as we shall see discuss later. f a GeV / H G e V / i n f BICEP2
FIG. 1: Constraint on the inflation scale from the axion CDM isocurvature perturbation. Theisocurvature constraint (2) is shown by the solid (orange) line, where the anharmonic effect istaken into account [8]. The relaxed constraints for h S i inf = 10 GeV , M
P l and 15 M P l are shownby the dashed, solid, and dot-dashed lines, respectively. The horizontal band is the BICEP2result (3). If the axion acquires a heavy mass during inflation, all these constraints disappearas shown in the text. with V ( φ, S ) = 12 m φ + λ (cid:0) | S | − f (cid:1) + δV P QV , (9)where φ is the inflaton, m is the inflaton mass, λ is the quartic coupling of S , f determinesthe VEV of | S | in the low energy, and δV P Q represents the explicit PQ symmetry breakingterms. The BICEP2 result (3) is consistent with the chaotic inflation model with aquadratic potential, but the extension to other inflation models is straightforward.Let us first consider the following PQ symmetry breaking term, δV P QV = kS N + h . c ., (10)7hich breaks the U(1) PQ down to a Z N subgroup. Here and in what follows we will takethe Planck scale M P l ≃ . × GeV to be unity unless otherwise noted. Suppose that | S | takes a VEV much larger than f during inflation. This is possible if S is coupled tothe inflation sector and acquires a large tachyonic mass. Then, for sufficiently large N ,the axion mass receives a sizable contributions, δm a ∼ k | S | N − , during inflation. If δm a & H , (11)the axion quantum fluctuations are significantly suppressed at super-horizon scales. Theadditional contribution to the axion mass becomes negligibly small when S settles downat | S | = f after inflation. Note that N must be very large in order not to spoil theaxion solution to the strong CP problem. For instance, N >
14 is required for f a =10 GeV [35].Alternatively, we may consider the following interaction, δV P QV = kφS N ′ + h . c ., (12)which depends on the inflaton φ . As the inflaton takes a super-Planckian field value, theabove PQ symmetry breaking operator is enhanced only during inflation, giving rise toa heavy mass for the axion. This enhancement is efficient especially in the large-fieldinflation models. If the inflaton is stabilized at the origin, this PQ symmetry breakingoperator vanishes in the present vacuum. Therefore, in this case, there is no tight lowerbound on N ′ .Thus, one can indeed realize PQ symmetry breaking which becomes relevant onlyduring inflation, by making use of the dynamics of either the PQ scalar or the inflatonfield. We will study the dynamics of the PQ scalar during inflation in detail in the nextsubsection. This is indeed the case if we impose a Z symmetry on both φ and S , for N ′ being an odd integer. . A case in which the PQ symmetry is spontaneously broken during inflation Now let us study our scenario in detail. The enhancement of the PQ symmetry breakingcan be realized if during inflation the saxion acquires a VEV much larger than in thepresent vacuum, or if the symmetry breaking operators depend on the inflaton field value.The former possibility requires a relatively flat potential for the saxion. To this end weconsider a SUSY axion models where the saxion has a flat potential lifted only by SUSYbreaking effects and non-renormalizable terms. We however emphasize here that SUSYis not an essential ingredient for our solution.The superpotential for the PQ sector can be divided into two parts, W = W PQ + W PQV , (13)where W PQ includes the interactions invariant under global U(1) PQ , whereas W PQV breaksU(1) PQ down to its subgroup Z N . To be concrete let us consider W PQV = λS N , (14)where S is a PQ superfield developing a vacuum expectation value. We do not imposeU(1) R symmetry and assume that λ is of order unity. In general, the coefficient of thePQ symmetry breaking operator can depend on other fields, such as the inflaton. Weshall consider this possibility later.The point is that the global U(1) PQ symmetry can be broken to, e.g., a discrete sym-metry by quantum gravity effects. For instance, in string theory, anomalous U(1)s whichcome from gauge symmetries can be broken to discrete symmetries [36] by stringy instan-tons, which do not exist in the usual gauge theory like QCD [37, 38].In order to suppress the axion CDM isocurvature perturbations, the saxion must bestabilized at a large field value during inflation so that the axion mass receives a largecontribution from the PQ-violating operator (14). On the other hand, after inflation ends, This is for simplicity; in the presence of U(1) R symmetry, the PQ breaking effect can be suppressedin a certain set-up, and we would need a slightly more involved model to have a sufficiently large PQbreaking.
9n order not to spoil the axion solution to the strong CP problem, the saxion should settledown at a smaller field value where the axion mass receives a negligible contribution fromthe PQ violating operator (14). For this, the saxion potential needs to be significantlymodified during inflation by its coupling to the inflaton.Let us first study the condition for the saxion to have a minimum at a large field value.The relevant terms in the scalar potential of S can be parameterized by V = m S | S | − ( A λ λS N + h . c . ) + λ N | S | N − , (15)where soft SUSY breaking terms are included. Note that, during inflation, m S includesthe Hubble-induced mass induced by the coupling to the inflaton. In the present vacuum,the typical scale for the soft SUSY breaking terms is given by the gravitino mass, m / .Note that some of soft SUSY breaking masses can be loop-suppressed depending on themediation scheme. One can find that, if the soft terms satisfy the condition | A λ | > N − m S , (16)there appears a minimum at large S where the terms in (15) dominate the potential. Forinstance, if the mass of S is tachyonic, m S <
0, the above condition is satisfied. Evenif the soft mass is non-tachyonic, sufficiently large A -term creates a minimum at a largefield value.Let us examine the scalar potential at large S during inflation. The interactions withthe inflaton field generically induce soft terms as q | m S | ∼ | A λ | ∼ H inf , (17)for the Hubble parameter H inf larger than m / . If the condition (16) is satisfied duringinflation, the potential develops a minimum at | S | ∼ (cid:18) λ H inf M P l (cid:19) / ( N − M P l . (18) We have omitted interactions with other PQ scalars assuming that arg( S ) is one of the main compo-nents of the axion after inflation, and is the dominant component during inflation. S is tachyonic, i.e. the Hubble-induced mass term isnegative. At the local minimum the axion acquires a mass around H inf , m a ∼ λA λ | S | N − ∼ H , (19)which highly suppresses axion quantum fluctuations at super-horizon scales. Thereforethe discrete PQ symmetry provides a simple mechanism to suppress isocurvature pertur-bations while protecting the approximate global PQ symmetry against (more) harmfulquantum corrections.For the suppression mechanism to be viable, it is crucial to make sure that S settlesdown to the true vacuum after the inflation is over. The potential may develop a localminimum at | S | ∼ m / ( N − / due to the A -term after inflation, and then S could be trappedthere. This would spoil the axion solution to the strong CP problem as the axion acquiresa too large mass, in general. The simplest solution to this issue is that the PQ breakingoperator depends on the inflaton field value, and it disappears after inflation. Then, thePQ scalar will settle down at the true vacuum after inflation without being trapped in thewrong one. In the case that the PQ breaking term does not depend on the inflaton, weneed to consider the dynamics of the PQ scalar after inflation. To avoid the trapping inthe wrong vacuum, there should be a period during which the condition (16) is violatedso that S is pushed toward the true vacuum. We do not, however, want the PQ symmetryto be restored, since then the isocurvature perturbations would be erased irrespective ofthe above mechanism.Let us study a concrete axion model to see that the above conditions can be indeedrealized. We consider the axion model with W PQ = κX ( S ˜ S − f ) , (20)where S and ˜ S have the same PQ charge with an opposite sign, and X is a PQ singlet.In the low energy, S and ˜ S are stabilized at h S i ∼ h ˜ S i ∼ f if their soft masses arecomparable. Thus, the value of f is close to the axion decay constant, f a . For H inf < f ,the scalar potential generated by W PQV develops a local minimum at large S along the F -flat direction S ˜ S = f , if the Hubble-induced mass for S is tachyonic or small enough11ompared to the A -term so that the condition (16) is satisfied. The axion then obtains amass around H inf for A λ ∼ H inf at the minimum. After inflation the Hubble-induced massfor S can be positive, m S ∼ H , where H denotes the Hubble parameter. Then the PQscalars are stabilized at S ∼ ˜ S ∼ f in the low energy. Note that the U(1) PQ symmetrycan remain broken even if S acquires a positive Hubble-induced mass after inflation. Letus explain why this is the case. For H inf > f , the PQ symmetry can be broken duringinflation if the Hubble-induced mass of S is negative. After inflation, however, the PQsymmetry would be restored if both S and ˜ S acquire a positive Hubble mass. On theother hand, the PQ symmetry remains spontaneously broken if ˜ S has a negative Hubblemass after inflation so that ˜ S takes a large VEV. After the Hubble parameter becomessufficiently small, these PQ scalars are stabilized along the F-flat direction.It is noted that the saxion starts to oscillate around the true vacuum with a large initialamplitude after the false vacuum is destabilized. The saxion oscillation can dominate overthe energy density of the universe after the reheating, if the saxion is sufficiently long-lived.We have nothing new to add to the saxion cosmology studied extensively in the literatures[40–43]. The decay of such saxion reheats the universe again, and the axion producedfrom the decay behaves as the dark radiation whereas the SM radiation is produced bythe decay into the Higgs and/or the gauge bosons [44–47]. Interestingly, the presence ofdark radiation can solve the tension between the BICEP2 and the Planck results [48].Lastly let us examine how large N should be in order not to spoil the axion solutionto the strong CP problem. In general, the PQ breaking term contributes to an additionalCP phase. The contribution becomes sufficiently small when the axion mass is dominated The change of the Hubble-induced mass is possible if the Hubble-induced mass depends on the inflatonfield values in the large-field inflation. Also this is possible if the inflation ends with the waterfall fieldas in the hybrid inflation, since the Hubble-induced mass can be generated from different couplings inthe K¨ahler potential after inflation. We assume that ˜ S does not participate in the explicit PQ-symmetry breaking. This may be realizedif the PQ symmetry breaking is accompanied with another PQ scalar S ′ like S ′ M S N . Then theholomorphic nature of the superpotential can forbid or suppress the PQ symmetry breaking due to S ′ .
10 11 12 13 14 15 165101520253035 N Too large strong CP phase
FIG. 2: The lower bound on N as a function of f a . We have set m / = 100TeV, λ = 1 and A λ = m / . by the QCD instanton contribution,∆ m a ∼ A λ λN f N − a < − ( m QCD a ) , (21)where m QCD a is given by Eq. (1), and ∆ m a represents the contribution of the PQ breakingterm to the axion mass in the true vacuum. This condition puts a lower bound on N .We show it in Fig. 2 for the case with λ ∼ A λ ∼ m / . For instance, N >
12 isnecessary for f a = 10 GeV. If PQ symmetry breaking terms are present only duringinflation, there is no such lower bound.
C. A case in which the PQ symmetry is spontaneously broken after inflation
If the PQ symmetry is restored during and/or after inflation, and if it becomes sponta-neously broken after inflation, there is no isocurvature constraint. In this case, however,13xionic cosmic strings are produced at the phase transition, whereas domain walls aregenerated at the QCD phase transition. The domain wall number N DW must be unityto avoid the overclosure of the Universe; in this case, the axions are copiously producedby domain wall annihilations, and the right amount of axion dark matter is generated for f a ≈ GeV [28].If there is an additional PQ symmetry breaking term enhanced during inflation, we canactually relax the above constraint on the domain wall number and the decay constant.To simplify our argument, let us assume that the explicit PQ symmetry breaking isproportional to S , and that the coefficient depends on another scalar field ψ . The PQsymmetry breaking can be temporarily enhanced, if ψ takes a large VEV until some timeafter the PQ symmetry breaking, and it settles at the origin in the low energy. The pointis that, after the PQ symmetry breaking, the explicit breaking term induces domain wallsattached to the axionic strings. Then, if there is only unique vacua, i.e., the domain wallnumber associated with the PQ symmetry breaking is unity, strings and domain wallsdisappear when the domain wall energy exceeds that of axionic strings. (Note that thedomain wall number N DW for the QCD anomaly has nothing to do with the domainwall number associated with the extra PQ symmetry breaking.) When domain walls andcosmic strings annihilate, axions are produced. Such axions will be diluted by cosmicexpansion and/or decay into the SM particles as their mass remains heavy until theexplicit PQ symmetry breaking becomes suppressed. Interestingly, the axion field valueis set to be a value which minimizes the PQ symmetry breaking after the domain wallannihilation. Thus, the large decay constant as well as N DW > IV. DISCUSSION AND CONCLUSIONS
We have seen that, if the PQ scalar takes super-Planckian values during inflation,there is an allowed region for f a where the BICEP2 results become consistent with theaxion CDM. Let us here briefly discuss one inflation model with the PQ scalar identified14ith the inflaton. In non-SUSY case, the PQ scalar can be stabilized by the balancebetween the negative mass and the quartic coupling as in Eq. (9). Then, it is possibleto realize the quadratic chaotic inflation model with the PQ scalar, if the kinetic term ofthe PQ scalar is significantly modified at large field values, based on the running kineticinflation [16–18]. For instance, we can consider L = | ∂S | + ξ ( ∂ | S | ) − λ (cid:0) | S | − f (cid:1) , (22)where ξ ≫ λ is the quarticcoupling. The large value of ξ can be understood by imposing a shift symmetry on | S | : | S | → | S | + C , where C is a real constant. Then the ξ -term respects the symmetry,while the other terms explicitly break the shift symmetry. Note that the shift symmetryis consistent with the PQ symmetry, as it is the radial component of S that transformsunder the shift symmetry. Let us denote the radial component of S as σ = | S | . At largefield values σ ≫ / √ ξ , the canonically normalized field is ˆ σ ∼ √ ξσ , and the scalarpotential becomes the quadratic one in terms of σ . The quadratic chaotic inflation canbe realized by the PQ scalar. In this case, h ˆ S i inf is of order 15 M P l , and one can see fromFig.1 that there is a region between f a ≃ GeV and 10 GeV where the isocurvatureconstraint becomes consistent with the BICEP2 result. Thus, one interesting way to evadethe isocurvature bound is to identify the PQ scalar with the inflaton.In this letter, we have proposed a simple mechanism in which the axion acquires aheavy mass during inflation, leading to the suppression of the axion CDM isocurvatureperturbations. The point is that the U(1)
P Q symmetry is explicitly broken down to itsdiscrete subgroup, Z N . If the PQ-breaking operators are significant during inflation, theaxion can acquire a sufficiently heavy mass, suppressing the isocurvature perturbations.There are two ways to accomplish this. One is that the PQ-breaking operators dependon the inflaton field value. If the inflaton takes a large-field value during inflation, thePQ-breaking operator becomes significant, while it becomes much less prominent if the In other words, the ordinary kinetic term and the potential term are relatively suppressed as theyexplicitly break the shift symmetry. N reads N >
12 for f a = 10 GeV and the gravitino mass oforder 100 TeV. We have discussed a concrete PQ scalar stabilization to show that it ispossible that the saxion is fixed at a large field value during inflation, while it settles downat the true minimum located at a smaller field value, without restoring the PQ symmetry.Toward a UV completion of our scenario based on the string theory, we may considernon-perturbative effects, W ⊃ ( φ − φ ) n e ±A , or K ⊃ ( φ † − φ † ) m e ±A + h . c ., (23)where the exponentials ∝ e ±A break a U (1) PQ symmetry down to a discrete one. Here A is a (linear combination of) string theoretic axion multiplet, and φ is the inflaton. If theinflaton φ develops non-zero expectation value during the inflation (and φ = φ in thetrue vacuum), such an axion obtains a large mass, which may suppress the isocurvatureperturbations. Similar terms may also help the overshooting problem of moduli duringinflation, producing the high potential barrier against decompactification [49] , evenwithout the coupling to the inflaton in the superpotential [51].In nature there may be other kinds of pseudo Nambu-Goldstone bosons such as axion-like particles, and the isocurvature perturbation bound can be similarly applied to them.The tension between the isocurvature perturbations and the high-scale inflation can besolved by our mechanism. That it to say, we can add a symmetry breaking operator whichbecomes relevant only during inflation. We can do so by either introducing an inflatonfield charged under the symmetry or by assuming that the radial component significantlyevolves during and after inflation. In particular, our mechanism can relax the isocurvature The so-called Kallosh-Linde model can give a large mass to the relevant axion multiplet: W ⊃ ( φ − φ ) n ( W + Ae ± a A + Be ± b A ). See also [50]. . Note added:
Isocurvature constraints on the QCD axion and axion like particles wereconsidered also in Refs. [55, 56] soon after the BICEP2 announcement.
Note added 2:
After submission of this paper we noticed that some of our argumentshas an overlap with Ref. [57] where it was pointed out that the axion isocurvature per-turbations can be suppressed if the PQ symmetry is badly broken during inflation andthe alignment of the axion by the PQ symmetry breaking can eliminate domain walls.
Acknowledgment
This work was supported by Grant-in-Aid for Scientific Research on Innovative Ar-eas (No.24111702, No. 21111006, and No.23104008) [FT], Scientific Research (A) (No.22244030 and No.21244033) [FT], and JSPS Grant-in-Aid for Young Scientists (B) (No.24740135 [FT] and No. 25800169 [TH]), and Inoue Foundation for Science [FT]. Thiswork was also supported by World Premier International Center Initiative (WPI Pro-gram), MEXT, Japan [FT]. [1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440 (1977); Phys. Rev. D , 1791(1977).[2] For a review, see J. E. Kim, Phys. Rept. , 1 (1987); H. Y. Cheng, Phys. Rept. , 1(1988); J. E. Kim and G. Carosi, Rev. Mod. Phys. , 557 (2010); A. Ringwald, Phys.Dark Univ. (2012) 116; M. Kawasaki and K. Nakayama, arXiv:1301.1123 [hep-ph].[3] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5082 [astro-ph.CO].[4] M. S. Turner, Phys. Rev. D , 889 (1986).[5] D. H. Lyth, Phys. Rev. D , 3394 (1992).[6] K. J. Bae, J. -H. Huh and J. E. Kim, JCAP , 005 (2008) [arXiv:0806.0497 [hep-ph]].
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