JID:PLB AID:30188 /SCO Doctopic: Phenomenology [m5Gv1.3; v 1.134; Prn:15/05/2014; 9:41] P.1(1-6) Physics Letters B ••• (••••) •••–••• 1 Contents lists available at ScienceDirect 66 2 67 3 68 4 Physics Letters B 69 5 70 6 71 7 www.elsevier.com/locate/physletb 72 8 73 9 74 10 75 11 76 12 Solving the tension between high-scale inflation and axion 77 13 isocurvature perturbations 78 14 79 15 Tetsutaro Higaki a, Kwang Sik Jeong b, Fuminobu Takahashi c,d 80 16 81 a 17 Theory Center, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan 82 b 18 Deutsches Elektronen Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany 83 c 19 Department of Physics, Tohoku University, Sendai 980-8578, Japan 84 d Kavli IPMU, TODIAS, University of Tokyo, Kashiwa 277-8583, Japan 20 85 21 86 22 87 a r t i c l e i n f o a b s t r a c t 23 88 24 14 89 Article history: The BICEP2 experiment determined the Hubble parameter during inflation to be about 10 GeV. Such 25 90 Received 19 March 2014 high inflation scale is in tension with the QCD axion dark matter if the Peccei–Quinn (PQ) symmetry Received in revised form 21 April 2014 26 remains broken during and after inflation, because too large axion isocurvature perturbations would be 91 Accepted 8 May 2014 27 generated. The axion isocurvature perturbations can be suppressed if the axion acquires a sufficiently 92 Available online xxxx 28 heavy mass during inflation. We show that this is realized if the PQ symmetry is explicitly broken 93 Editor: J. Hisano 29 down to a discrete symmetry and if the breaking is enhanced during inflation. We also show that, even 94 30 when the PQ symmetry becomes spontaneously broken after inflation, such a temporarily enhanced PQ 95 31 symmetry breaking relaxes the constraint on the axion decay constant. 96 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY license 32 97 3 (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP . 33 98 34 99 35 100 36 1. Introduction mixture of the isocurvature perturbations is tightly constrained by 101 37 the Planck observations [3], which reads 102 38 103 The identity of dark matter is one of the central issues in cos- 0.408 39 fa 104 mology and particle physics. Among various candidates for dark H < 0.87 × 107 GeV (95% CL), (2) 40 inf 11 105 matter, the QCD axion is a plausible and interesting candidate. 10 GeV 41 106 The axion, a, arises as a pseudo-Nambu–Goldstone (pNG) boson 42 neglecting anharmonic effects [4–8]. In particular, a large-field in- 107 in association with the spontaneous breakdown of a global U(1) 43 PQ flation such as chaotic inflation [9] is in conflict with the isocur- 108 Peccei–Quinn (PQ) symmetry [1,2]. If the U(1) symmetry is ex- 44 PQ vature bound. Recently the BICEP2 experiment announced the dis- 109 plicitly broken only by the QCD anomaly, the axion is stabilized at 45 covery of the primordial B-mode polarization [10], which deter- 110 vacuum with a vanishing CP phase, solving the strong CP problem. 46 mines the inflation scale as 111 More important, the dynamical relaxation necessarily induces co- 1 47 2 112 herent oscillations of axions, which contribute to cold dark matter 14 r 48 Hinf 1.0 × 10 GeV , (3) 113 49 (CDM). We focus on the axion CDM which accounts for the total 0.16 114 + 50 dark matter density, throughout this letter. = 0.07 115 r 0.20−0.05 (68% CL), (4) 51 The axion mass receives contributions from the QCD anomaly, 116 where r denotes the tensor-to-scalar ratio.1 After subtracting the 52 − 117 f 1 best available estimate for foreground dust, the allowed range is 53 QCD × −6 a + 118 ma 6 10 eV , (1) modified to r = 0.16 0.06. Therefore one can see from (2) and (3) 54 1012 GeV −0.05 119 55 that there is a clear tension between the inflation scale determined 120 2 56 where fa is the axion decay constant. Because of the light by the BICEP2 and the QCD axion dark matter. 121 57 mass, the axion generically acquires quantum fluctuations of δa 122 58 Hinf/2π during inflation, leading to CDM isocurvature perturba- 123 1 Such large tensor-to-scalar ratio can be explained in various large field inflation; 59 tions. Here Hinf is the Hubble parameter during inflation. The 124 see e.g. [11–24]. The tension with the Planck result can be relaxed in the presence 60 of small modulations in the inflaton potential [25] or hot dark matter/dark radiation. 125 61 2 The isocurvature perturbation bound similarly applies to the so-called axion-like 126 62 E-mail addresses: [email protected] (T. Higaki), [email protected] particles, or general pseudo-Nambu–Goldstone bosons, which are produced by the 127 (K.S. Jeong), [email protected] (F. Takahashi). initial misalignment mechanism and contribute to dark matter. 63 128 64 http://dx.doi.org/10.1016/j.physletb.2014.05.014 129 65 0370-2693/© 2014 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. 130 JID:PLB AID:30188 /SCO Doctopic: Phenomenology [m5Gv1.3; v 1.134; Prn:15/05/2014; 9:41] P.2(1-6) 2 T. Higaki et al. / Physics Letters B ••• (••••) •••–••• 1 There are various known ways to suppress the axion CDM 66 2 isocurvature perturbations. First, if the PQ symmetry is restored 67 3 during inflation (or reheating), there is no axion CDM isocurvature 68 4 perturbations, as the axion appears only when the PQ symme- 69 5 try is spontaneously broken some time after inflation [26,27]. In 70 6 71 this case topological defects such as axionic cosmic strings and 7 72 domain walls are generated, and in particular the domain wall 8 73 number NDW must be unity to avoid the cosmological catastro- 9 74 phe [28]. Second, if the kinetic term coefficient for the phase of the 10 75 PQ scalar was larger during inflation than at present, the quantum 11 76 fluctuations, δa, can be suppressed after inflation. This is possible 12 77 if the radial component of the PQ scalar takes a larger value dur- 13 78 ing inflation [26,29]. The scenario can be implemented easily in a 14 79 supersymmetric (SUSY) theory, as the saxion potential is relatively 15 80 flat, lifted by SUSY breaking effects. Interestingly, a similar effect is 16 81 possible if there is a non-minimal coupling to gravity [30]. Third, 17 82 the axion may acquire a heavy mass during inflation so that its 18 83 quantum fluctuations get suppressed [31]. In Ref. [31] two of the Fig. 1. Constraint on the inflation scale from the axion CDM isocurvature pertur- 19 84 present authors (K.S.J. and F.T.) showed that the QCD interactions bation. The isocurvature constraint (2) is shown by the solid (orange) line, where 20 85 the anharmonic effect is taken into account [8]. The relaxed constraints for Sinf = become strong at an intermediate or high energy scale in the very 16 21 10 GeV, MPl and 15MPl are shown by the dashed, solid, and dot-dashed lines, re- 86 early Universe, if the Higgs field has a sufficiently large expectation spectively. The horizontal band is the BICEP2 result (3). If the axion acquires a heavy 22 3 87 value. mass during inflation, all these constraints disappear as shown in the text. 23 88 In fact, the second solution of Refs. [26,29] is only marginally 24 89 consistent with the BICEP2 result (3), if the field value of the PQ 25 2. Isocurvature constraints on the axion CDM 90 scalar is below the Planck scale. Also the third solution of Ref. [31] 26 91 is consistent with the BICEP2 result only in a corner of the param- 27 The axion, if exists during inflation, acquires quantum fluctu- 92 eter space. Therefore, we need another solution to suppress the 28 ations, δa = Hinf/2π , giving rise to the axion CDM isocurvature 93 axion isocurvature perturbations, as long as we assume that the 29 perturbations. The isocurvature constraint on the axion CDM leads 94 PQ symmetry remains broken during and after inflation. 30 to the upper bound on the Hubble parameter during inflation as 95 31 In this letter, we propose a simple mechanism to suppress the in Eq. (2), which is shown by the solid (red) line Fig. 1. Here the 96 32 axion CDM isocurvature perturbations along the line of the third anharmonic effect is taken into account [8]; the axion CDM isocur- 97 33 solution. Instead of making the QCD interactions strong during vature perturbations get significantly enhanced as the initial field 98 inflation, we introduce a PQ symmetry breaking operator, which 11 34 value approaches the hilltop, as can be seen for fa 10 GeV. 99 35 becomes relevant only during inflation. If the axion acquires a suf- Note that we assume that the axion produced by the initial mis- 100 36 ficiently heavy mass during inflation, the axion CDM isocurvature alignment mechanism accounts for the total CDM density in the 101 37 perturbations practically vanish, evading the isocurvature bound on figure. 102 38 the inflation scale.