Axion-driven hybrid inflation over a barrier
aa r X i v : . [ h e p - ph ] J a n APCTP-Pre2021-001PNUTP-21-A11
Axion-driven hybrid inflation over a barrier
Jinn-Ouk Gong a,b and Kwang Sik Jeong c a Department of Science Education, Ewha Womans University, Seoul 03760, Korea b Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea c Department of Physics, Pusan National University, Busan 46241, Korea
Abstract
We present a scenario where an axion-like field drives inflation until a potential barrier,which keeps a waterfall field at the origin, disappears and a waterfall transition occurs.Such a barrier separates the scale of inflation from that of the waterfall transition. We findthe observed spectrum of the cosmic microwave background indicates that the decay con-stant of the inflaton is well below the Planck scale, with the inflationary Hubble parameterspanning a wide range. Further, our model involves dark matter candidates including theinflaton itself. Also, for a complex waterfall field, we can determine cosmologically thePeccei-Quinn scale associated with the strong CP problem.
Introduction
A very early period of accelerated expansion of the universe, cosmic inflation [1, 2, 3], hasbecome an essential part of the standard cosmological model. Before the onset of the hot bigbang evolution of the universe, inflation provide the necessary initial conditions – otherwiseextremely finely tuned – as confirmed by the observations on the cosmic microwave background(CMB) [4]: the observable universe is homogeneous and isotropic and is spatially flat. Fur-thermore, inflation also explains the origin of the temperature fluctuations of the CMB andinhomogeneous distribution of galaxies on large scales due to the quantum fluctuations duringinflation [5]. The properties of these primordial perturbations have been constrained by decadesof observations, and are consistent with the predictions of inflation [6].To implement inflation, we need a special matter component that behaves similar to thecosmological constant. This is readily realized by a scalar field, the inflaton, which has asufficiently flat potential. Then the motion of the inflaton is damped by the expansion of theuniverse and the energy of the inflaton is dominated by its potential. The corresponding energy-momentum tensor is then close to that of the cosmological constant, driving an inflationaryepoch until this “slow-roll” period does not hold any longer [7, 8]. It is, however, a formidabletask to maintain an unusually flat potential against radiative corrections and quantum gravityeffects [9]: we need to protect the flatness of the potential from any disturbance.A powerful way of protecting the flat inflaton potential is to impose certain symmetries.An axion-like field is an appealing candidate for an inflaton because its mass requires thebreaking of the associated shift symmetry, presumably by non-perturbative effects, naturallymaking it very light. However, the predictions of the minimal axion-driven inflation – naturalinflation [10, 11] – are only marginally consistent with the most recent Planck observations onthe CMB. Additionally, for successful inflation, the axion should have a trans-Planckian decayconstant f & m Pl with m Pl ≡ (8 πG ) − / being the reduced Planck mass [6], which may beoutside the range of validity of an effective field theory description due to quantum gravityeffects.In this article, we present a viable model of inflation driven by an axion-like inflaton φ . Theend of the inflationary phase is triggered not by the violation of the slow-roll conditions but bya waterfall transition like hybrid inflation [12]. The distinctive features of our model are two-fold. First, the waterfall field χ is trapped at the origin during inflation not by a monotonicallyincreasing potential along the χ -direction as in the original hybrid inflation scenario, but bya potential barrier which disappears at a critical value of φ . The barrier separates the scaleof inflation from that of the waterfall phase transition, greatly enlarging the region of themodel parameter space for viable inflation. As we will see, such a setup is natural if the shiftsymmetry associated with φ is broken only non-perturbatively, for instance, by hidden QCDwith quarks coupled to χ . This consequently makes, unlike the original natural inflation, theinflaton decay constant well below the Planck scale so that the effective field theory is trustableagainst unknown quantum gravity effects. The Planck results are accommodated in a broadrange of the inflaton mass and decay constant, but with a certain relationship between them,which opens a possibility to probe our scenario via searches for axion-like particles.Another merit of our scenario is that the inflaton itself can constitute dark matter of theuniverse. The inflationary Hubble scale H inf , which spans a very wide range, may be relatedto the scale of new physics resolving other puzzles of the Standard Model (SM). For the case1f a complex χ , we may identify U(1) χ with the Peccei-Quinn (PQ) symmetry that solves thestrong CP problem [13]. Remarkably, the PQ scale is then determined cosmologically, and thecontribution of the QCD axion to dark matter constrains H inf to be below about 10 GeV.This article is outlined as follows. In Section 2, we describe our model in detail, along withthe parameter space of our interest. We also provide a realization of our model in an ultraviolet(UV) complete theory. In Section 3, we explore the cosmological dynamics of our model, thenbriefly conclude in Section 4.
Our model consists of two scalar fields, the inflaton φ and the waterfall field χ . During inflation, φ rolls down slowly the potential while χ is trapped at the origin by a potential barrier. There,the effective mass squared of χ , µ , is thus positive and is a function of φ . As φ evolves, µ decreases monotonically and vanishes at a critical value of φ , removing the barrier. Theninflation ends almost instantaneously. Here, the barrier brings the separation between thescales for inflation and waterfall phase transition. As will be clear soon, such separation hasan important impact on the dynamics of inflation.Our inflation scenario is successfully realized if φ is an axion-like field with decay constant f . This is because its interactions are well controlled by the shift symmetry, φ → φ + constant,presumably broken only by non-perturbative effects, and the size of its potential terms is finiteand insensitive to f . A dangerous waterfall tadpole can be avoided by imposing a symmetry,for instance, U(1) χ if χ is a complex scalar. Explicitly, we consider the following potential: V ( φ, χ ) = V + µ ( φ ) | χ | − λ | χ | + 1Λ | χ | + U ( φ ) , (1)with λ >
0, where the effective mass squared µ ( φ ) and the inflaton potential U ( φ ) are givenrespectively by ∗ : µ ( φ ) = m − µ cos (cid:18) φf + α (cid:19) , (2) U ( φ ) = M cos (cid:18) φf (cid:19) , (3)where α is a constant phase. Here the positive constant V is fixed by demanding V = 0 at thetrue vacuum, and Λ is the cutoff scale of the theory. The parameter space of our interest is m < µ ≪ M ≪ V . (4)Then the true vacuum appears at χ ∼ √ λ Λ, well below the cutoff scale as long as λ is smallerthan unity, and the vacuum energy density V reads V ∼ λ Λ . (5) ∗ An axion-dependent effective mass squared has been considered first in the model for cosmological relax-ation of the electroweak scale [14], where the Higgs mass is selected cosmologically by the axion evolution. χ -direction for µ ( φ ) >
0, and a barrierseparates them † . It is due to the barrier that inflation can be driven by V which is dependent onΛ but insensitive to µ – the parameter that determines the position and height of the waterfallbarrier.The rate of tunneling over a barrier is proportional to exp( − S E ), where S E is the Euclideanaction of χ evaluated on a bounce solution. Tunneling proceeds dominantly via the Coleman-DeLuccia bounce [18] with S E > S ≡ π / (3 λ ) in the region with µ > H , while through theHawking-Moss instantons [19] with S E = µ /H × S in the opposite region [20]. Here H inf is the Hubble scale during inflation, and we have used the fact that the bounce is insensitiveto the | χ | -term for M ≪ V . For viable inflation, we thus impose the following condition: µ ≫ H . (6)Then χ is heavy enough to be initially fixed at the origin. In addition, the tunneling rate isexponentially suppressed, trapping χ at the origin, until φ reaches a critical point where thebarrier disappears. Bubbles of the true vacuum can be nucleated around the end of inflation,but the U(1) χ phase transition occurs smoothly because soon the barrier disappears.As a simple UV completion of the inflaton potential, we consider a hidden QCD with U(1) χ charged quarks. The terms (2) and (3) are then generated in a controllable way while naturallysatisfying the hierarchies between the mass parameters (4). Vector-like quarks u + u c and d + d c couple to χ through the U(1) χ and the following gauge invariant interactions: m u uu c + yχu c d + y ′ χ ∗ ud c + m d dd c + 116 π φf G µν e G µν , (7)where the hidden confining scale lies in the range m d ≪ Λ h ≪ m u . Here we have taken the fieldbasis where the quark mass parameters are real. Note that the last term above is an anomalousinflaton coupling to hidden gluons, which is the only source of axionic shift symmetry breakingof the inflaton. At low energy scales below m u , heavy quarks u + u c are integrated out to givethe effective mass term (cid:18) yy ′ m u | χ | + m d + δm d (cid:19) dd c , (8)which depends on χ . Here we have included the radiative contribution from a closed loop of χ : δm d = yy ′ π m u log (cid:18) Λ m χ (cid:19) , (9)with m χ being the mass of the radial field component of χ . For small values of χ , d + d c arelighter than Λ h and condensate to form a meson with mass and decay constant around Λ h . Theinflaton mixes with the meson in the presence of the anomalous coupling to hidden gluons, andfinally its effective potential at energy scales below Λ h is obtained by integrating out the heavymeson: ∆ V eff = − (cid:12)(cid:12)(cid:12)(cid:12) yy ′ m u (cid:12)(cid:12)(cid:12)(cid:12) Λ h cos (cid:18) φf + β (cid:19) | χ | + | m d + δm d | Λ h cos (cid:18) φf + β (cid:19) , (10) † One can also consider a case in which λ is negative and there is no potential barrier, for which the situationis qualitatively the same as in hybrid natural inflation [15, 16, 17]. The vacuum energy density is then set notby Λ but by m . As a result, the possible range of M relative to V is more constrained by the Planck results,compared to our model. β = arg( yy ′ /m u ) and β = arg( m d + δm d ). It isclear that the above reduces to (2) and (3) with α = β − β . Also, the hierarchical structure µ ≪ M ≪ V is satisfied naturally if Λ h lies in the range H inf . Λ h ≪ Λ , (11)where we have used that H inf should be lower than Λ h since otherwise instanton effects becomevery weak. On the other hand, m should be smaller than µ because inflation ends when thebarrier disappears. The smallness of m may be accommodated in more speculative idea likesupersymmetry or anthropic selection. The universe undergoes a phase of inflation while the waterfall field is trapped at the origin.During this stage, the inflaton φ evolves down the scalar potential V = V + U ( φ ) = V + M cos (cid:18) φf (cid:19) . (12)Thus, the evolution of φ during inflation is essentially identical to hybrid natural inflation. Theeffective mass squared of the waterfall field (2) crosses zero when φ reaches the critical value: φ c f = cos − (cid:18) m µ (cid:19) − α . (13)Then the sign flip triggers the waterfall phase transition, because there is no potential barrieralong the waterfall field direction, and inflation ends almost instantaneously. Among the modelparameters, m , µ and α affect inflation only through the above combination. Figure 1 showsschematically the inflation and waterfall phases for M ≪ V .At this point, it is very important to note that two crucial ingredients are required in ourmodel, and they make our scenario different from other hybrid inflation scenarios. One isthe axionic shift symmetry associated with the inflaton, which naturally allows the mass scalehierarchies (4). The other is the barrier between two extrema at and off the origin in thewaterfall potential. χ is confined at the origin during inflation by the barrier, which disappearsat φ = φ c . As long as M ≪ V , the evolution along φ changes only the shape of the potentialaround the barrier, with the true vacuum remaining almost the same. It then follows that thevalue of f required for viable inflation can naturally be much lower than the Planck scale.For M ≪ V , inflation is driven by the constant vacuum energy V so that the Hubbleparameter during inflation is given by H ≈ V m . (14)4 .0 0.2 0.4 0.6 0.8 1.0 1.20.00.51.01.5 | |/| | V / V < c = c > c Time | |/| | V / V Figure 1: Schematic display of the inflationary and waterfall phases. The evolution of φ changes the waterfall potential near the origin, as illustrated in the right panel. For φ > φ c ,there is a barrier that traps χ at the origin. The barrier shrinks as φ evolves, and disappearswhen it reaches the critical value, φ = φ c , triggering a waterfall phase. The true vacuum islocated at χ = χ , around which the waterfall potential is rarely affected by the evolution of φ because the variation of the inflaton potential is of order of M , much smaller than V .From the scalar potential (12), the slow-roll parameters are given by, with θ ≡ φ/f , ǫ ≡ m (cid:18) V ′ V (cid:19) ≈ (cid:18) m Pl f (cid:19) (cid:18) M V (cid:19) sin θ , (15) η ≡ m V ′′ V ≈ − (cid:18) m Pl f (cid:19) (cid:18) M V (cid:19) cos θ . (16)Thus, | η | is parametrically much bigger than ǫ , as is usual for natural inflation. The slow-rollconditions, ǫ ≪ | η | ≪
1, are satisfied if the decay constant f satisfies the followingcondition: f & (cid:18) M V (cid:19) / m Pl , (17)but it needs not be above the Planck scale, unlike the original natural inflation that relies onthe existence of a super-Planckian axion decay constant.Let us now examine the cosmological observables. The amplitude of the power spectrumof the curvature perturbation and its spectral index, and the tensor-to-scalar ratio in terms ofthe slow-roll parameters are, according to the most recent CMB observations at the pivot scale k = 0 .
002 Mpc − , constrained as [6] A R = V π m ǫ ∗ ≈ . +0 . − . × − , (18) n R = 1 − ǫ ∗ + 2 η ∗ ≈ . ± . , (19) r = 16 ǫ ∗ < . , (20)5here the subscript ∗ denotes the evaluation at the horizon exit. Since ǫ ≪ | η | , n R is determinedentirely by η . Hence, from (16) and (19), the decay constant is found to have the value f = r − n R cos θ ∗ (cid:18) M V (cid:19) / m Pl ≈ . p cos θ ∗ (cid:18) M V (cid:19) / m Pl , (21)while (20) is translated to the following mild constraint: M V < . − n R cos θ ∗ ≈ . θ ∗ . (22)The number of e -folds N before the onset of the waterfall phase transition can be estimated by N = 1 m Pl Z φφ c dφ ′ √ ǫ ≈ V M (cid:18) fm Pl (cid:19) log (cid:20) tan( θ c / θ/ (cid:21) ≈ .
14 cos θ ∗ log (cid:20) tan( θ c / θ/ (cid:21) , (23)thus we can use interchangeably the field value θ and the gained number of e -folds N from θ .The required number of e -folds is around 60, which fixes the inflaton value at that scale roughly θ ∗ ≈ . (cid:18) θ c (cid:19) − . (cid:18) θ c (cid:19) , (24)neglecting terms of higher order in tan( θ c / e -folds does not need to be very close to the hilltop of the potential.It is interesting to note that the inflaton mass and decay constant are proportional to theinflationary Hubble parameter: combining (18) with (14) and (21), the decay constant and themass of the inflaton are written respectively as f = H inf π (1 − n R ) √ A R tan θ ∗ ≈ . × tan θ ∗ H inf , (25) m φ = s − n R )2 cos θ ∗ H inf ≈ . √ cos θ ∗ H inf , (26)where we have used that the inflaton mass is given by ‡ m φ = M /f for m ≪ M . Figure 2shows the relationship between H inf , m φ and f established by imposing the Planck results. Theinflaton can couple to the SM sector, for instance, to gauge bosons through anomalous couplingsas is naturally expected from its axionic nature. Our scenario thus provides theoretical supportfor experimental searches for axion-like particles in a wide mass range. The rough relation, f ∼ × m φ , indicates that the inflaton should be heavier than about 0 . ‡ If non-perturbatively generated by a hidden QCD, the inflaton mass can be higher than the value given in(26) because hidden quarks get heavier in the true vacuum, increasing the hidden confining scale. (cid:2) (cid:3) (cid:3) (cid:4) (cid:5) (cid:6) (cid:2) (cid:7) (cid:6) (cid:8) (cid:9) (cid:3) (cid:10) (cid:2) (cid:9) (cid:10) (cid:1) (cid:1)(cid:2)(cid:2)(cid:2) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:3)(cid:4) (cid:1)(cid:2) (cid:3)(cid:5) (cid:2)(cid:3)(cid:2)(cid:1)(cid:1)(cid:2)(cid:2)(cid:3)(cid:2)(cid:2)(cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:3)(cid:6) (cid:1)(cid:2) (cid:3)(cid:7) (cid:1)(cid:2) (cid:3)(cid:8) (cid:1) (cid:1)(cid:2)(cid:3) [ (cid:1)(cid:2)(cid:3) ] (cid:1) (cid:2) (cid:3) (cid:4) (cid:5)(cid:6) (cid:7) (cid:8)(cid:9) (cid:10) (cid:3) [ (cid:11) (cid:3) (cid:12) ] Figure 2: Inflaton mass and decay constant compatible with the Planck results on n R and A R .Here we have taken the inflaton value at the horizon exit lying in the range between θ ∗ = 0 . θ ∗ = 1 . r requires H inf < . × GeVas denoted by the vertical dot-dashed line.satisfied with describing briefly the subsequent evolution. After the barrier disappears, thewaterfall field χ soon acquires a tachyonic mass much larger than H inf in magnitude for µ ≫ H . This happens within an e -fold after the inflaton reaches the critical value (13), so χ rolls fast down to the true vacuum. Unlike usual axion-driven inflation where the universe isreheated via an anomalous coupling to photons, the reheating process in our scenario dependsgreatly on the details of the model. Generally speaking, however, depending on the couplings,tachyonic preheating [21] is extremely effective so that the vacuum energy is rapidly transferredto the energy of inhomogeneous oscillations of φ and/or χ [22, 23], subsequently heating up theuniverse to a radiation-dominated regime.After inflation, as χ is relaxed to the one of the degenerate true vacua, U(1) χ symmetry isspontaneously broken. This leads to the formation of cosmic strings [24], which can survive inthe late universe and contribute to the CMB temperature anisotropies, depending on how theassociated Nambu-Goldston boson becomes massive. For instance, for a global U(1) χ , it canobtain a mass non-perturbatively from some confining gauge sector. Then, topological defectsare unstable and collapse by the wall tension if the domain-wall number is equal to unity,or if a small explicit symmetry breaking term is added to lift the vacuum degeneracy [25, 26].Further, the cosmic string loops and the large time-dependent inhomogeneities generated duringtachyonic preheating can act as a source of gravitational waves. The corresponding spectrumof the gravitational waves can span a huge range of frequencies: from O (10 − ) to O (1) hertzfor stable and metastable cosmic strings [27], within the reach of pulsar timing arrays, LIGOand LISA, and from O (1) to O (10 ) hertz for inhomogeneities from tachyonic preheating [28,29, 30]. Such high-frequency gravitational waves are unfortunately beyond the sensitivity ofinterferometric experiments due to the shot noise fluctuations of photons. The gravitationalwaves in relatively low-frequency regime may well be within the reach of future detectors likeadvanced LIGO, Einstein Telescope and Big Bang Observer, which however is possible only forextremely small values of couplings. Explicit construction of such models remains as an open7ossibility. Another distinctive feature of our scenario is the possibility that the inflaton can contributeto dark matter if the inflaton potential arises from a hidden QCD sector as in (7). HavingYukawa interactions with the waterfall field, the hidden quarks have masses increasing with thewaterfall field value. This implies that the confining scale also increases, and thus the hiddenQCD gets stronger after inflation. In the region of large waterfall field values where all thehidden quarks are heavier than the confinement scale, in (2) and (3) we have µ = 0 and M = Λ h , (27)because there are no mesons formed.Let us consider the case that χ is sufficiently large so that the inflaton potential is givenby (27) in the present universe. The inflaton is then stabilized at a CP-conserving minimum,and consequently the inflation sector possesses an accidental Z symmetry, φ → − φ . The Z forbids the inflaton to mix with the waterfall field, and makes it stable if there is no coupling tothe SM sector § . The inflaton starts coherent oscillations around the minimum when the Hubbleparameter H becomes comparable to its mass, i.e. at the temperature fixed by m φ ( T ) = 3 H ( T ) . (28)If T is below Λ h during reheating, oscillations start before reheating ends. Then, the inflatonrelic density from oscillation is roughly estimated byΩ φ h ∼ . θ c (cid:18) T Λ h (cid:19) n (cid:18) f GeV (cid:19) (cid:18) T reh GeV (cid:19) , (29)with T < Λ h , and n = 11 N/ − N ) gauge group. Here T reh is the reheatingtemperature at which the universe becomes completely radiation-dominated, and we have usedthe fact that the scale factor scales as H − / during a matter dominated era. If oscillationsstart after reheating, the relic density can be read off from (29) by replacing T reh with T . Theinflaton can thus account for the observed dark matter in a wide range of f depending on T reh .The inflation sector includes another candidate for dark matter associated with sponta-neously broken U(1) χ . An interesting possibility arising due to a wide allowed range of H inf isto identify U(1) χ with the PQ symmetry so that the phase component of χ becomes the QCDaxion to explain the absence of the CP violation in QCD. This implies that the waterfall andPQ phase transitions are identical. Then, as corresponds to the waterfall field value at the truevacuum, the axion decay constant is cosmologically determined by f a ≈ . × GeV λ / (cid:18) H inf GeV (cid:19) / , (30) § Another way to make the inflaton cosmologically stable is to consider the case with | m d | ≪ | δm d | so that α nearly vanishes, stabilizing φ at a CP-conserving minimum. Such suppression can be achieved, for instance,by invoking supersymmetry to generate δm d from superpotential while m d from K¨ahler potential [31]. GeV to avoid the astrophysical bounds. The axion anomalouscoupling to gluons, which is required to solve the strong CP problem, is generated by addingU(1) χ -charged heavy quarks or extra Higgs doublets as is usually considered ¶ . We also note thatthe domain-wall number should be equal to one since otherwise domain-walls formed during theQCD phase transition overclose the universe. In such a case, axions are produced from coherentoscillations and more efficiently from unstable domain-walls bounded by an axion string. Therelic density is estimated byΩ a h ≈ . × (cid:18) Λ QCD (cid:19) (cid:18) f a GeV (cid:19) . , (31)using the results of the numerical simulation for the axion production [34]. Therefore, theobserved dark matter density indicates low scale inflation with H inf . √ λ × GeV . (32)It would be also interesting to consider other cases where U(1) χ is identified, for instance, withU(1) L associated with the seesaw mechanism generating tiny neutrino masses, or local U(1) B − L to extend the SM. In this article, we have proposed a scenario where natural inflation is implemented. Duringinflation, the waterfall field remains at the origin by a potential barrier, which disappears whenthe inflaton reaches at a critical point, then inflation ends almost instantaneously. The inflatoninteraction responsible for such a barrier can naturally arise if the shift symmetry is brokennon-perturbatively by hidden QCD with quarks coupled to the waterfall field. Interestingly,the Planck results indicate that the inflaton mass and the decay constant are respectively O (0 . × H inf and O (10 − ) × H inf . This raises the possibility of probing our scenario byexperimental searches for axion-like particles if the inflaton couples to the SM sector. It is alsoremarkable that the inflaton can be stable to constitute dark matter if all the hidden quarks getheavier than the confining scale at the true vacuum. Also, for the case of a complex waterfallfield, its phase component can play the role of the QCD axion, contributing to dark matter for H inf below about 10 GeV.
Acknowledgements
This work is supported in part by the National Research Foundation of Korea Grant Numbers2018R1C1B6006061 (KSJ) and 2019R1A2C2085023 (JG). We also acknowledge the Korea-Japan Basic Scientific Cooperation Program supported by the National Research Foundationof Korea and the Japan Society for the Promotion of Science (2020K2A9A2A08000097). JGis further supported in part by the Ewha Womans University Research Grant of 2020 (1-2020-1630-001-1). JG is grateful to the Asia Pacific Center for Theoretical Physics for hospitalitywhile this work was under progress. ¶ For recent reviews, see, for instance, [32, 33]. eferences [1] A. H. Guth, Phys. Rev. D , 347-356 (1981)[2] A. D. Linde, Phys. Lett. B , 389-393 (1982)[3] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. , 1220-1223 (1982)[4] N. Aghanim et al. [Planck], Astron. Astrophys. , A6 (2020) [arXiv:1807.06209 [astro-ph.CO]].[5] V. F. Mukhanov and G. V. Chibisov, JETP Lett. , 532-535 (1981)[6] Y. Akrami et al. [Planck], Astron. Astrophys. , A10 (2020) [arXiv:1807.06211 [astro-ph.CO]].[7] V. Mukhanov, “Physical Foundations of Cosmology,” Cambridge, UK: Univ. Pr. (2005)421 p.[8] S. Weinberg, “Cosmology,” Oxford, UK: Oxford Univ. Pr. (2008) 593 p.[9] D. H. Lyth and A. Riotto, Phys. Rept. , 1-146 (1999) [arXiv:hep-ph/9807278 [hep-ph]].[10] K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. , 3233-3236 (1990)[11] F. C. Adams, J. R. Bond, K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. D ,426-455 (1993) [arXiv:hep-ph/9207245 [hep-ph]].[12] A. D. Linde, Phys. Rev. D , 748-754 (1994) [arXiv:astro-ph/9307002 [astro-ph]].[13] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440-1443 (1977)[14] P. W. Graham, D. E. Kaplan and S. Rajendran, Phys. Rev. Lett. , no.22, 221801(2015) [arXiv:1504.07551 [hep-ph]].[15] G. G. Ross and G. German, Phys. Lett. B , 199-204 (2010) [arXiv:0902.4676 [hep-ph]].[16] G. G. Ross and G. German, Phys. Lett. B , 117-120 (2010) [arXiv:1002.0029 [hep-ph]].[17] G. G. Ross, G. German and J. A. Vazquez, JHEP , 010 (2016) [arXiv:1601.03221 [astro-ph.CO]].[18] S. R. Coleman and F. De Luccia, Phys. Rev. D , 3305 (1980)[19] S. W. Hawking and I. G. Moss, Adv. Ser. Astrophys. Cosmol. , 154-157 (1987)[20] A. Shkerin and S. Sibiryakov, Phys. Lett. B , 257-260 (2015) [arXiv:1503.02586 [hep-ph]].[21] G. N. Felder, J. Garcia-Bellido, P. B. Greene, L. Kofman, A. D. Linde and I. Tkachev,Phys. Rev. Lett. , 011601 (2001) [arXiv:hep-ph/0012142 [hep-ph]].1022] J. Garcia-Bellido and A. D. Linde, Phys. Rev. D , 6075-6088 (1998)[arXiv:hep-ph/9711360 [hep-ph]].[23] E. J. Copeland, S. Pascoli and A. Rajantie, Phys. Rev. D , 103517 (2002)[arXiv:hep-ph/0202031 [hep-ph]].[24] R. Jeannerot, J. Rocher and M. Sakellariadou, Phys. Rev. D , 103514 (2003)[arXiv:hep-ph/0308134 [hep-ph]].[25] G. B. Gelmini, M. Gleiser and E. W. Kolb, Phys. Rev. D , 1558 (1989).[26] S. E. Larsson, S. Sarkar and P. L. White, Phys. Rev. D , 5129 (1997) [hep-ph/9608319].[27] P. Auclair, J. J. Blanco-Pillado, D. G. Figueroa, A. C. Jenkins, M. Lewicki, M. Sakellar-iadou, S. Sanidas, L. Sousa, D. A. Steer and J. M. Wachter, et al. JCAP , 034 (2020)[arXiv:1909.00819 [astro-ph.CO]].[28] J. Garcia-Bellido and D. G. Figueroa, Phys. Rev. Lett. , 061302 (2007)[arXiv:astro-ph/0701014 [astro-ph]].[29] J. Garcia-Bellido, D. G. Figueroa and A. Sastre, Phys. Rev. D , 043517 (2008)[arXiv:0707.0839 [hep-ph]].[30] J. F. Dufaux, G. Felder, L. Kofman and O. Navros, JCAP , 001 (2009) [arXiv:0812.2917[astro-ph]].[31] S. H. Im and K. S. Jeong, Phys. Lett. B , 135044 (2019) [arXiv:1907.07383 [hep-ph]].[32] A. Ringwald, Phys. Dark Univ. , 116 (2012) [arXiv:1210.5081 [hep-ph]].[33] M. Kawasaki and K. Nakayama, Ann. Rev. Nucl. Part. Sci. , 69 (2013) [arXiv:1301.1123[hep-ph]].[34] T. Hiramatsu, M. Kawasaki, K. Saikawa and T. Sekiguchi, JCAP01