We show that supersymmetric axion models breaking the PQ symmetry by the
interplay of non-renormalizable supersymmetric terms and soft supersymmetry
breaking terms provide a natural framework not only for generating the axion
scale from soft supersymmetry breaking scale $m_{3/2}$ but also for enhancing
it during inflation by factor of order $\sqrt{H_I/m_{3/2}}$ where $H_I \simeq
10^{14}$ GeV according to the recent BICEP2 result. In this scheme, the PQ
symmetry can stay broken throughout the whole history of the Universe if the
reheat temperature is below $10^{10}$ GeV, or $m_{3/2}$ when the PQ fields
couple strongly to thermal (Standard Model) particles. It is also shown that
parametric resonance during preheating is not effective enough to induce
non-thermal PQ symmetry restoration. As a consequence, axion models with the
QCD anomaly $N_{DW}>1$ can be made free from the domain wall problem while the
axion isocurvature perturbation is suppressed sufficiently for the axion scale
during inflation larger than about $M_P (\Omega_a h^2/0.12)^{1/2} ( F_a/10^{12}
\mbox{GeV})^{0.6}$ GeV.Comment: 5 page

Chun, Eung Jin

Physics Letters B

English

arXiv

AbstractWe show that supersymmetric axion models breaking the PQ symmetry by the interplay of non-renormalizable supersymmetric terms and soft supersymmetry breaking terms provide a natural framework not only for generating the axion scale from soft supersymmetry breaking scale m3/2 but also for enhancing it during inflation by factor of order HI/m3/2 where HI≃1014 GeV according to the recent BICEP2 result. In this scheme, the PQ symmetry can stay broken throughout the whole history of the Universe if the reheat temperature is below 1010 GeV, or m3/2 when the PQ fields couple strongly to thermal (Standard Model) particles. It is also shown that parametric resonance during preheating is not effective enough to induce non-thermal PQ symmetry restoration. As a consequence, axion models with the QCD anomaly NDW>1 can be made free from the domain wall problem while the axion isocurvature perturbation is suppressed sufficiently for the axion scale during inflation larger than about MP(Ωah2/0.12)1/2(Fa/1012 GeV)0.6 GeV

Elsevier - Publisher Connector 

Axion dark matter with high-scale inflation 

Physics Letters B 735 (2014) 164–167Contents lists available at ScienceDirectPhysics Letters Bwww.elsevier.com/locate/physletbAxion dark matter with high-scale inﬂationEung Jin ChunKorea Institute for Advanced Study, Seoul 130-722, Republic of Koreaa r t i c l e i n f o a b s t r a c tArticle history:Received 30 April 2014Accepted 6 June 2014Available online 13 June 2014Editor: A. RingwaldWe show that supersymmetric axion models breaking the PQ symmetry by the interplay of non-renormalizable supersymmetric terms and soft supersymmetry breaking terms provide a natural framework not only for generating the axion scale from soft supersymmetry breaking scale m3/2 but also for enhancing it during inﬂation by factor of order √HI/m3/2 where HI  1014 GeV according to the recent BICEP2 result. In this scheme, the PQ symmetry can stay broken throughout the whole history of the Universe if the reheat temperature is below 1010 GeV, or m3/2 when the PQ ﬁelds couple strongly to thermal (Standard Model) particles. It is also shown that parametric resonance during preheating is not effective enough to induce non-thermal PQ symmetry restoration. As a consequence, axion models with the QCD anomaly NDW > 1 can be made free from the domain wall problem while the axion isocurvature perturbation is suppressed suﬃciently for the axion scale during inﬂation larger than about MP (Ωah2/0.12)1/2(Fa/1012 GeV)0.6 GeV.© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.1. IntroductionThe recent measurement of CMB B-mode polarizations by BICEP2 points to large primordial tensor perturbations with the tensor-to-scalar ratio r = O(0.1) [1]. This reveals a high-scale in-ﬂation with the Hubble parameter,HI  1014 GeV(r0.16)1/2(1)which corresponds to the inﬂation energy scale of V 1/4I ≡(3H2I M2P )1/4  2 × 1016(r/0.16)1/4 GeV. Furthermore, the chaotic inﬂation with a quadratic potential [2] seems to ﬁt nicely other CMB observables measured by Planck [3]. If conﬁrmed, it has pro-found implications for the axion dark matter [4–7]. In this work, we present a natural framework where the PQ symmetry is never restored throughout the whole history of the Universe, and the PQ symmetry breaking scale during inﬂation is much larger than the present one so that the domain wall problem occurring in axion models with the QCD anomaly NDW > 1 can be avoided, and the axion isocurvature density perturbation can be suﬃciently sup-pressed.2. Axion and strong CP problemThe axion was introduced to resolve the strong CP problem in a dynamical way (for a recent review, see, [8,9]). Being a pseudo-Goldstone boson of a QCD-anomalous U (1) symmetry (PQ symme-http://dx.doi.org/10.1016/j.physletb.2014.06.0170370-2693/© 2014 The Author. Published by Elsevier B.V. This is an open access article SCOAP3.try) the axion couples to the CP-odd QCD ﬁeld strength term;LQCD = aFag2s32π2Gaμν G˜μνa (2)where Fa is the axion decay constant and G˜μνa = 12εμνρσ Gaρσ . That is, the QCD θ angle is replaced by a dynamical ﬁeld a: θ ≡ a/Fa . After the QCD phase transition, instanton effect generates the ax-ion potential,V (a) =m2a F 2a[1− cos aFa](3)where the axion mass is given by ma =√Z/(1 + Z)( fπmπ/Fa)with the up and down quark mass ratio Z =mu/md , and mπ ( fπ ) is the pion mass (decay constant). The axion potential sets the vac-uum expectation value 〈a〉 = 0 ensuring no θ contribution to the neutron electric dipole moment.3. Axion cold dark matterThe occurrence of the axion potential (3) also plays an impor-tant role in cosmic axion production. If the PQ symmetry is broken after inﬂation, there appear three sources of axion production: co-herent oscillation from initial misalignment by θi , axionic string formation upon the PQ symmetry breaking, and domain wall pro-duction during the QCD phase transition. Summing these up one gets [6,9]under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by E.J. Chun / Physics Letters B 735 (2014) 164–167 165Ωah2 ≈ 0.18〈θ2i 〉αt.d.(Fa1012 GeV)1.19(Λ400 MeV)(4)where 〈θ2i 〉 ≈ 2π2/3 is a value averaged over possible ranges of the initial misalignment angle θi which is not uniform in our Hub-ble volume, and αt.d. takes into account contributions from strings and domain walls. One then ﬁnds that the axion cold dark matter (CDM) satisﬁes Ωah2  0.12 for Fa ≈ 1.4 ×1011α−0.84t.d. GeV (see [6]and references therein for different estimates of αt.d.).Note that the θ angle deﬁned in (2) by θ ≡ a/Fa has a period-icity θ ≡ θ + 2πNDW where NDW is the QCD anomaly, and thus there appear distinct NDW θ vacua: 0, 2π, · · · , 2π(NDW − 1) in the axion potential (3). It is then required to have NDW = 1 in order not to produce stable string-domain wall networks which overcloses the Universe. The KSVZ model with one pair of heavy quarks has NDW = 1 while the original DFSZ model has NDW = 6[8]. Let us remark that various hybrid models can be constructed to obtain NDW = 1.If the PQ symmetry is broken before or during inﬂation, axionic strings formed during the PQ phase transition are eﬃciently di-luted away and no domain wall can form. Thus, any axion model with NDW > 1 is allowed, and the axion dark matter can be pro-duced from the coherent oscillation as well as from (massless) axion ﬂuctuations during inﬂation, δa ≈ HI/2π . This leads toΩah2 ≈ 0.18[θ2i +(HI2π F I)2]( Fa1012 GeV)1.19(Λ400 MeV).(5)Here a uniform initial misalignment angle θi is taken as our Uni-verse might be expanded from a small patch with given θi during inﬂation, and the axion decay constant during inﬂation F I is as-sumed to be different from the present value Fa and thus could suppress the ﬂuctuation contribution to evade the problem with axion isocurvature perturbation [10,11].The axion ﬂuctuations produced during inﬂation become isocur-vature density perturbations of the axion dark matter after the axion acquires mass at the QCD phase transition. The recent Planck measurements constrain the isocurvature power spectrum Pa of the axion CDM compared to the scalar power spectrum PR [3]:Pa ≡ 4ξ2 (HI/2π)2(F Iθi)2 + (HI/2π)2  0.04PR (6)where ξ ≡ Ωa/ΩCDM , and PR  2.2 × 10−9. This essentially rules out the axion as a CDM candidate [5,6] if F I = Fa . In order to see how much the severe constraint from the isocurvature perturba-tions can be relaxed for F I  Fa , let us ﬁrst consider two regions separately: (i) F Iθi < HI/2π , and (ii) F Iθi > HI/2π . In the case (i) having Pa ≈ 4ξ2, Eqs. (5) and (6) require ξ < 4.7 × 10−6 together withF I  9× 1015(Fa1012 GeV)0.6GeV, andθi  0.0018(1012 GeVFa)0.6. (7)In the case (ii), one ﬁnds the requirements:F I  4.2× 1018√ξ(Fa1012 GeV)0.6GeV, andθi ≈ 0.82√ξ(1012 GeVFa)0.6. (8)Thus, the axion CDM with ξ = 1 can be obtained for Fa ∼ 1012 GeVand θi ∼ 1 if F I ∼ MP is allowed.4. PQ symmetry and supersymmetry breakingThe axion decay constant changing with time is a generic fea-ture as far as there is no particular reason to forbid inﬂaton cou-pling to the PQ ﬁelds. A simple and systematic framework can be obtained in the context of supersymmetry which has to be broken by the inﬂaton potential of order V I during inﬂation [12]. Further-more, the PQ symmetry breaking scale can be correlated to the supersymmetry breaking scale in a ﬂat potential model with the superpotential [13,14]:WPQ = λ Pn+2QMnP(9)where n is a positive integer, and the U (1)P Q charge 1 and −(n + 2) are assigned to P and Q , respectively. Recall that the dynamical generation of the axion scale from soft supersymme-try breaking terms can also resolve the μ problem by introducing a Higgs bilinear operator WKN = hPn+1HuHd/MnP [15] which en-sures μ ∼m3/2 and realizes a supersymmetric version of the DFSZ axion model having NDW = 6. The potential domain wall problem is evaded if the PQ symmetry is broken before/during inﬂation and remains broken during the whole history of the Universe as will be discussed below.In order to deliver the essential features, let us consider a sim-pliﬁed PQ superpotential together with the quadratic potential for chaotic inﬂation1:W =mχχ2 + λn + 3φn+3MnP(10)where χ is the inﬂaton superﬁeld and φ is a representative of the PQ superﬁelds. As is well-known [12], a non-renormalizable coupling allowed in the Kahler potentialδK ∼ 1M2Pχ †χφ†φ (11)delivers the supersymmetry breaking effect by the inﬂaton to the PQ sector leading to the scalar potential V = Vsoft + VSUSY :Vsoft =(CmH2 +m2φ)|φ|2 + [ (CAH + A)λn + 3φn+3MnP+ h.c.],VSUSY = |λ|2 |φ|2n+4M2nP(12)where Cm and CA are constants of order one, and mφ and A are the usual soft terms of order m3/2. Minimization of the above scalar potential gives rise to the vacuum expectation value 〈φ〉 =φ0 as follows [17]:φ0 ∼⎧⎪⎨⎪⎩(HIMnP )1n+1 ∼ F I during inﬂation,(Min[H,mφ]MnP )1n+1 after inﬂation,(mφMnP )1n+1 ∼ Fa after reheating.(13)One can see that the PQ symmetry remains broken during the whole history of the Universe if the reheating occurs at the Hub-ble parameter smaller than mφ , which sets an upper limit on the reheat temperature:TR  1010(mφ200 GeV)1/2GeV. (14)1 Note that a special form of the Kahler potential is required to guarantee the quadratic term in the scalar potential: V I =m2|χ |2 [16].166 E.J. Chun / Physics Letters B 735 (2014) 164–167To maintain the initial PQ symmetry breaking vacuum stays con-nected to the present one. The reheat temperature needs to be even lower if a PQ ﬁeld has a sizable coupling to any ﬁelds in thermal equilibrium and thus obtains a thermal mass. For instance, the PQ ﬁeld may couple to a right-handed neutrino, W ∼ yφNNto realize the seesaw mechanism [13,14], or to a heavy quark, W ∼ yP Q Q c in the KSVZ model. These couplings generate ther-mal mass-squared of order δm2φ ∼ y2T 2R which restores the PQ symmetry if δm2φ > m2φ . To avoid this, we put a rough condition ofTR mφy. (15)Another important issue concerning symmetry non-restoration is parametric resonance which can leads to huge production of bosonic ﬂuctuations in the process of preheating during the in-ﬂaton oscillation period, and thus non-thermal symmetry restora-tion [18,19]. If this happens, subsequent symmetry breaking brings back the topological defect production and thus the results in (4)should be applied for NDW = 1 while ruling out axion models with NDW > 1. This is a generic feature if the PQ ﬁeld has a di-rect coupling to the inﬂaton: δL ∼ g2|χ |2|φ|2. Even without such a coupling, the PQ symmetry can be restored if there is a large initial value φi during the inﬂation oscillation period before the reheat-ing, and PQ symmetry non-restoration is ensured if Fa  10−4φi[20]. In supersymmetric theories, there appear couplings between the inﬂaton and PQ ﬁelds through the supersymmetry breaking effect (12) although direct couplings in the superpotential are for-bidden. In this case, parametric resonance is never effective as we will see in the following. To simplify the analysis, let us take n = 2and ignore the A terms (mφ  A and Cm  CA ), and assume neg-ative mass-squared terms2:V = −(CmH2 +m2φ)|φ|2 + λ2M2P |φ|6. (16)When the mass term is constant (during inﬂation and H mφ ), the homogeneous ﬁeld value φ0 = 〈φ〉 is set by φ0 =√CmHIMP /√3λ during inﬂation, and φ0 =√mφMP /√3λ after reheating for H  mφ . During the period of inﬂaton oscillation before reheating φ0 follows the equation of motionφ¨0 + 3Hφ˙0 + 3λ2M2Pφ50 − CmH2φ0 = 0 (17)ignoring the m2φ term. It is solved byφ0 = A√HMP√3λ(18)where H = 2/3t and the coeﬃcient A is determined to be A =( 49Cm + 14 )1/4 which is slightly different from the ones during inﬂa-tion and after reheating. Treating properly the transition between the inﬂation (reheating) and oscillation periods, one would be able to ﬁnd a smooth transition of A(t) connecting the different val-ues during inﬂation, inﬂaton oscillation, and radiation-dominated periods. For our analysis, we will take a simple approximation of an abrupt change of periods and use the solution (18) to ﬁnd the evolution of ﬂuctuations of the PQ ﬁeld: φ = φ0 + δφ, during the oscillation period. The linearized equation of motion for the ﬂuc-tuations in Fourier space is2 For a detailed analysis with general soft terms for the original superpotential (9), see Ref. [21].δ¨φk + 3H δ˙φk −(k2a2+ C˜mH2)δφk = 0 (19)where C˜m = 119 Cm + 54 . Note that the source term does not contain the oscillating term of the inﬂaton contrary to the usual parametric resonance case with a direct coupling between the inﬂation and PQ ﬁelds in non-supersymmetric models [18–20]. Therefore, no para-metric resonance can occur and it should remain true even after considering the transition effect properly. One can indeed ﬁnd that an asymptotic behavior of solutions to Eq. (19) is given byδφk ∼ sin(kt3/2)k4/5t4/3(20)which is an oscillating and decaying function showing no reso-nance effect.Let us ﬁnally see how large Fa  1012 GeV with F I  MP in (8)can be allowed in our framework. As we have F I/Fa =√CmHI/mφindependently of n, the ﬁrst relation of (8) translates toCm  3ξ(mφ102 GeV)(1013 GeVFa)0.8(21)and Fa ∼ 1013 GeV requires a small Yukawa coupling of λ ∼ 10−6for n = 2. Note that one may have Fa  1012 GeV allowing trans-Planckian PQ ﬁeld values during inﬂation, F I  MP , which, how-ever, does not lead to a dangerous effect in the potential as it comes with a small Yukawa λ.5. ConclusionThe QCD axion is a well-motivated hypothetical particle which is a pseudo-Goldstone boson of the PQ symmetry solving the strong CP problem and is a good candidate for the cold dark mat-ter. In generic axion models with the QCD anomaly NDW > 1, the PQ symmetry needs to be broken before/during inﬂation and never restored after inﬂation in order to prohibit production of stable domain walls. However, the isocurvature perturbation pro-duced during inﬂation rules out axions as a dark matter candidate if the PQ symmetry breaking scale F I during inﬂation is the same as the present one Fa . In general, these two symmetry breaking scales need not to coincide, for instance, if couplings between in-ﬂation and PQ ﬁelds are allowed. Supersymmetry provides a nat-ural way to realize such a situation. Generic order-one couplings between inﬂaton and PQ ﬁelds allowed in the Kahler potential lead to mediation of the inﬂaton supersymmetry breaking effect to the PQ sector. A smooth connection of such an early super-symmetry breaking to the present one can be realized if the PQ symmetry breaking is induced by soft supersymmetry breaking terms as presented in this work. As a consequence, the strin-gent constraint from isocurvature perturvations can be avoided if F I  MP (Ωah2/0.12)1/2(Fa/1012 GeV)0.6.For this to happen, the reheat temperature has to be low enough; TR  1010 GeV in order to keep the Hubble parameter below the soft supersymmetry breaking scale m3/2. If PQ ﬁelds couple strongly to thermal particles through a coupling y, it is required to have TR m3/2/y in order to forbid the induced ther-mal masses which can restore the PQ symmetry. There could also be non-thermal symmetry restoration due to parametric resonance during the inﬂaton oscillation period. Having the inﬂaton coupling to the PQ ﬁelds only through the Hubble parameter, the inﬂaton supersymmetry breaking effect, parametric resonance is shown to be ineffective in our framework.6. Note addedThe author thanks Kiwoon Choi for sharing their results [22]which has some overlap consistent with ours.E.J. Chun / Physics Letters B 735 (2014) 164–167 167References[1] P.A.R. Ade, et al., BICEP2 Collaboration, arXiv:1403.3985 [astro-ph.CO].[2] A.D. Linde, Phys. Lett. B 129 (1983) 177.[3] P.A.R. Ade, et al., Planck Collaboration, arXiv:1303.5082 [astro-ph.CO].[4] T. Higaki, K.S. Jeong, F. Takahashi, arXiv:1403.4186 [hep-ph].[5] D.J.E. Marsh, D. Grin, R. Hlozek, P.G. Ferreira, arXiv:1403.4216 [astro-ph.CO].[6] L. Visinelli, P. Gondolo, arXiv:1403.4594 [hep-ph].[7] A.G. Dias, A.C.B. Machado, C.C. Nishi, A. Ringwald, P. Vaudrevange, arXiv:1403.5760 [hep-ph].[8] J.E. 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Eung Jin Chun

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Axion dark matter with high-scale inflation

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Axion Dark Matter with High-Scale Inflation

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