Eur. Phys. J. C (2014) 74:3062 DOI 10.1140/epjc/s10052-014-3062-5

Regular Article - Theoretical Physics

Cluster X-ray line at 3.5 keV from -like dark matter

Hyun Min Lee1, Seong Chan Park2, Wan-Il Park3,a 1 Department of Physics, Chung-Ang University, Seoul 156-756, Korea 2 Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea 3 School of Physics, KIAS, Seoul 130-722, Korea

Received: 10 July 2014 / Accepted: 31 August 2014 / Published online: 20 September 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract The recently reported X-ray line signal at Eγ interesting to investigate the possible DM candidates with 3.5 keV from a stacked spectrum of various galaxy clusters existing data. One obvious candidate is the sterile and the Andromeda galaxy may be originating from a decay- in models for the neutrino masses [1,2,4] but there could be ing dark matter of the mass 2Eγ . A light axion-like alternative DM explanations [5,6]. scalar is suggested as a natural candidate for dark matter and In this paper, we propose an axion-like particle as the its production mechanisms are closely examined. We show source of the X-ray line at 3.5 keV.1 Axion-like (or that the right amount of axion relic density with the pre- simply ) are ubiquitous in various extensions of the 14 ferred parameters, ma 7 keV and fa 4 × 10 GeV, SM including stringy models with various compactification can be naturally obtainable from the decay of inflaton. If the schemes where pseudo-Goldstone appear in the low axions were produced from the saxion decay, it could not energy after the breakdown of accidental global symmetries have constituted the total relic density due to the bound from [7]. The mass spectra of the axion-like particles can span a structure formation. Nonetheless, the saxion decay is an inter- wide range, covering the keV scale. We may identify one of esting possibility, because the 3.5 keV line and dark radiation them as the axion-like particle explaining the X-ray line.2 The can be addressed simultaneously, being consistent with the axion-like particle could be produced in the early universe Planck data. Small misalignment angles of the axion, rang- by various mechanisms and account for the observed DM −4 −1 ing between θa ∼ 10 Ð10 depending on the reheating amount: e.g. the axion misalignment, the decay of the radial temperature, can also be the source of axion production. The partner of the axion-like DM, dubbed the saxion, and also model with axion misalignment can satisfy the constraints the decay of the inflaton. We show that an axion-like DM, for structure formation and iso-curvature perturbation. produced preferably in the decay of the inflaton, can provide a good fit to the observed X-ray line by its decay into a pair of through anomaly interactions with the decay 1 Introduction 14 constant, fa 4 × 10 GeV. The rest of the paper is organized as follows. In Sect. 2, It has been recently reported by two groups that there exists a we describe our model for an axion-like DM and discuss line signal at 3.5 keV in a stacked X-ray spectrum of galaxy the cosmological bounds from structure formation and iso- clusters and the Andromeda galaxy [1,2]. Since no source curvature perturbation in Sect. 3. In Sect. 4, three scenarios of of X-ray lines, such as atomic transitions, is known at this axion production and relevant observational bounds on them energy, the observed line may suggest the existence of a new are discussed. Finally, conclusions are drawn in Sect. 5. source. It would be tantalizing to notice that the line is con- sistent with the monochromatic signal due to a decay- 2 The model with a 7 keV axion for the X-ray line ing dark matter (DM) with the mass, mDM 7 keV, and the τ 28 lifetime, DM 10 s[1,2]. Even though a further confir- We introduce an axion-like particle as a pseudo-Goldstone mation of the line by independent and refined observations, associated with a broken anomalous U(1) symmetry. e.g. Astro-H mission [3], is required, it would be timely and 1 We note that there have appeared related papers on the keV axion H. M. Lee, S. C. Park and W.-I. Park are contributed equally. dark matter [8,9] while we were finalizing our work. 2 The axion-like particles, in general, do not play the role of the QCD a e-mail: [email protected] axion. 123 3062 Page 2 of 8 Eur. Phys. J. C (2014) 74:3062

After the symmetry breaking, the model-independent effec- • Structure formation tive Lagrangian for the axion a and the saxion s (the radial The keV axion may be a decay product of the inflaton component of the symmetry-breaking field) can be expressed and/or the saxion. Suppose that a mother particle, denoted as as X, decays to two axions, each of which carries the energy of L = 1(∂ )2 + 1(∂ )2 + s (∂ )2 − 1 2 2 μs μa μa ms s 2 2 2 fa 2 α = / −1 2 2−1 μν+c1 em ( ˜ μν − μν) Ea,i m X 2(5) maa Fμν F aFμν F sFμν F 2 4 8π fa c2 ¯ μ 5 ¯ / μ ¯ + (∂μa) f γ γ f + i f Dμγ f − m f f f when the axion mass is ignored. The axion of our inter- fa est is expected to be out of thermal equilibrium at tem- c3(s+ia)/fa ¯ −(m f e fL f R + h.c.) + LI (1) peratures well below GUT scale [9]. Hence the momen- tum of the axion is simply red-shifted once it is pro- where αem is the fine structure constant of electromagnetic duced from the decay of X. Then, in order not to destruct interaction, fa is the axion coupling constant, and Fμν and ˜ large scale structures, the axion should be non-relativistic Fμν are the field-strength tensor and its dual for electromag- around the epoch when a Hubble patch contains energy netic field, respectively. We note that c1,2,3 are dimension- less parameters of order one and we have included an extra corresponding to the galactic-sized halo (corresponding ∼ ≡ charged f that is responsible for the generation of to T T∗ 300 eV) (see for example [12,13]). α anomalies. For example, in a supersymmetric axion model More precisely, the most recent analysis of Lyman- for- est data shows that the comoving free-streaming length of [10,11], the axion chiral multiplet A with A|θ=0 = s + ia 2 α DM at the matter-radiation equality is constrained to be appears in the superspace interactions as d θ AW Wα and 2 c A/f ¯ at 95 % CL [14] d θ m f e 3 a where Wα is the field strength super- field, and and ¯ are matter chiral multiplets containing λ <λc ≈ . the extra charged fermion. But the results in our work do not fs fs 0 12 Mpc (6) depend on the presence of supersymmetry. We can add the L Lagrangian for inflation, I . λc where we introduced fs representing the observational For a keV-scale axion, the first term of the second line in bound on the free-streaming length. Including the previ- Eq. (1) provides the main decay channel with a rate given by ous various analyses of Lyman-α forest data [15] and the α2 3 emma phase space densities derived from the dwarf galaxies of →γγ = (2) a π 3 2 the Milky way [16Ð18], leads to the bound on the free- 64 fa streaming length ranging between 0.12 Mpc  λc  where we set c = 1 for simplicity. This axion decay can be a fs 1 0.60 Mpc. The comoving free-streaming length of the possible origin of the recently reported X-ray line at 3.5keV, axion produced from the decay of a mother particle X if the axion saturates the dark matter relic density and has the is computed as following properties [1,2]: 28    m = 7.1keV,τ= 1.14 × 10 s teq v tnr teq v a a λ ≡ adt = dt + adt = . × −53 fs or a 5 73 10 GeV (3) τ R(t) τ R(t) t R(t) X   X nr  1/2 where τ is the lifetime of the axion. This implies that the 2tnr τX 1 teq a = 1 − + ln axion coupling constant should be Rnr tnr 2 tnr        m 3/2 1/2 1/4 14 a 1 H m X /2 Teq fa 4 × 10 GeV . (4) ≈ 0 7keV H0 X  ma T0 For notational convenience in the later section, we use f , ≡     / a 0 1 m 2 T 3 2 4 × 1014 GeV. × 1 + ln X a 0 (7) 2 H0 m X /2 Teq

3 Cosmological constraints where va is the velocity of the axion, R(t) is the scale fac- tor, tnr is the time when the axion becomes non-relativistic, Our keV-scale axion can be constrained by astrophysics and τX ( X ) is the lifetime (decay rate) of X, related to the τ = (π 2 / )−1/2 / 2 and cosmology, namely, the structure formation or the iso- decay temperature TX by X g∗ 90 MP TX . 1/2 curvature perturbation of dark matter, depending on how the In the second line, we used R(t) = Req(t/teq) for 1/2 axion is produced. t < teq, and va(t) = (tnr/t) for tnr  t  teq. 123 Eur. Phys. J. C (2014) 74:3062 Page 3 of 8 3062

The constraint Eq. (6) is now interpreted as we can obtain a right amount of the keV-scale axion while satisfying various constraints given in the previous section.    /   T 0.12 Mpc 200 1 4 7keV In particular, we consider the axion production from the infla- X  0.2 (8) λc ( ) ton/saxion decay and the axion misalignment in both cases of m X fs g∗ TX ma high and low reheating temperatures after primordial infla- where {···}in the last line of Eq. (7) was approximated tion. In order to match the relic density of dark matter to to {···}=8.27. the observed one, a = 0.268 [19], we quote the necessary • Iso-curvature perturbation axion abundance at present as   Axions in our scenario can be produced by either the decay −5 7keV of inflaton/saxion or the axion misalignment. In the case Ya 6.9 × 10 . (12) m of saxion decay, the iso-curvature perturbation of the sax- a ion is potentially problematic. However, it could be sup- 4.1 Inflaton decay pressed since the saxion could have a Hubble scale mass during inflation. So, we consider only the case of axion In inflation scenarios where the inflaton is responsible for misalignment in our discussion. the density perturbation of the present universe, the inflaton The recent Planck data combined with WMAP polariza- should decay mainly to SM particles. It can also partially tion data leads to a constraint on the fraction of the iso- decay to axions we are considering now (see for example curvature perturbation by [19] Ref. [21] for producing dark matter from the decay of infla- P ton). In the sudden decay approximation, the axion abun- S < 0.041, (9) dance from such a partial decay of the inflaton is given by PR   3 TR BrI →aa TR = . → −9 Ya 0 75 BrI aa (13) at 95 % CL, where PR 2.2 × 10 and PS are the 4 m I BrI →SM m I power spectra of curvature and iso-curvature perturba- where BrI →aa and BrI →SM are respectively the branching tions, respectively. The iso-curvature perturbation of the fractions of inflaton (I ) to axions and to SM particles, and TR axion dark matter can be expressed as [20] and m I are the reheating temperature and mass of inflaton,   → 2 respectively. We assumed BrI SM 1 in the far right side of ∂ ln H 4 P = r a I (10) Eq. (13). Comparing Eq. (13)toEq.(8) and (12), we find that S ∂θ π I osc 2 fa a right amount of the axion relic density can be obtained while satisfying the constraint from structure formation, provided where r is the fractional contribution of axion DM to the that observed relic density of DM, θ is the initial misalign-    osc λc ( ) 1/4 ment angle,3 and H is the expansion rate during inflation, −4 fs g∗ TR I BrI →aa  4.6 × 10 . (14) I 0.12 Mpc 200 and fa is the axion coupling constant during inflation. As θ  ∝ θ 2 described in the next section, for osc 1, a osc, hence one finds 4.2 Saxion decay    1.2 × 106 GeV θ f I The saxion, the radial component of the complex field con- H  osc a . I −4 (11) r 10 fa taining the axion, can play a crucial role in the axion pro- duction, since it can decay into a pair of axions via the axion I kinetic term, Note that fa can be much larger than fa at zero tempera- ture. In addition, θosc can be O(1) if the axion relic density 1 s L ⊃ (∂a)2 . (15) can be diluted by some amount due to, for example, a late- 2 fa time entropy production. In this case, the decay rate of the saxion to a pair of axions is given by 3 4 Scenarios of axion production 1 ms → = . (16) s aa π 2 64 fa Axion-like scalars can be produced from decays of heavy par- ticles or coherent oscillations. In this section, we discuss how 4 We assumed that the momentum of inflaton was red-shifted and neg- ligible relative to its mass when it decayed. Otherwise, the dependence 3 We assume the misalignment angle is constant until the commence- on the momentum of inflaton would have had to be included. We thank ment of axion coherent oscillation. Anupam Mazumdar for pointing this out. 123 3062 Page 4 of 8 Eur. Phys. J. C (2014) 74:3062

In the presence of an extra heavy charged fermion coupled where ts is the time when saxion decays. If the saxion is with the saxion via a Yukawa coupling in the following form: produced via coherent oscillation at ts,osc and decays at ts while the universe is dominated by radiation, we obtain L ⊃−λs ff¯ , (17) Ys(ts) = Ys(ts,osc) where Ys(ts,osc) is the initial abundance there is an additional contribution to the saxion decay rate, of the saxion in coherent oscillation, given by

3/2 3/2 2 2 2 2 2     λ 4m f c3m f ms 4m f 2 1/2 ¯ = m 1− = 1 − fa MPl s→ ff π s 2 π 2 2 Y (t , ) ∼ 8 ms 8 fa ms s s osc MPl ms     where in the far right-hand side, the Yukawa interaction 2 10 1/2 −4 fa 10 GeV from the effective Lagrangian in Eq. (1) was used. Then, 4.3 × 10 . (23) f , m  a 0 s for s→aa s→ ff¯ , the branching fraction of the saxion decaying to a pair of axions is given by Here, we assumed that the initial saxion misalignment is 2 ms 1 almost the same as the axion decay constant fa. Then, Br → > . (18) s aa 2 2 2 from Eqs. (12), (22), and (23), one finds the condition for 8c3m f 2c3 a right amount of the axion dark matter, Thus, for |c3|=O(1Ð10) which may be a natural expecta- −2 tion, we obtain Brs→aa = O(10 Ð0.1) that may match to ∼ HTR,rd Brs→aa Brs→aa all the requirements in some region of parameter space, as      2 1/2 discussed in the following sections. ≡ . × −2 7keV fa,0 ms . 8 0 10 10 In the early universe, the saxion might be at the broken ma fa 10 GeV phase with HI  ms, but it could undergo a coherent oscil- (24) lation as H  ms. Then, the saxion decay might be the main source of axion production, although structure forma- Comparing Eqs. (21) and (24), we find that there could = λc tion constrains the branching fraction to axions, similarly to exist a viable parameter space for fa fa,0 if fs is the case of inflaton decay. In the following argument, for pushed up to the value over 0.54 Mpc, corresponding to simplicity, we express the full decay width of the saxion as about 1 keV mass of thermal warm dark matter. How- ever, such a large free-streaming length is in a strong ten- = → /Br → (19) s s aa s aa sion with structure formation, being excluded by the most and we will use the sudden decay approximation for saxion recent analysis of Lyman-α forest data at the 9σ CL [14]. decay. We now take a smaller fa, for which the axion does not saturate the observed relic density DM but the observed

• High TR flux of the X-ray line can be still obtained. Even in this If the saxion decays while the universe is dominated by case, since the abundance of the axion is proportional to 2 radiation, the temperature of the universe right after the fa , the photon flux caused by the decay of the axion does decay of the saxion is given by not depend on fa. Therefore, the required Brs→aa is given by   π 2 −1/4  Ts = g∗(Ts) s→aa MPl/Brs→aa, (20) ∼ HTR,3.5 90 Brs→aa Brs→aa     1/2 ≡ . × −2 7keV ms , 8 0 10 10 (25) where Eq. (19) was used in the right-hand side. We find ma 10 GeV that the constraint from structure formation (Eq. (8)) is

now translated to which does not depend on fa. Note that, if the abundance   of the axion is much smaller than the observed relic den- λc 2  HTR,fs ≡ . × −3 fs sity of DM, the constraint on the free-streaming length of Brs→aa Brs→aa 3 98 10 0.12 Mpc the axion is irrelevant as long as the axion DM is non-       2 2 relativistic around the epoch of CMB decoupling. Partic- × ms fa,0 ma . 10 (21) 10 GeV fa 7keV ularly, if the energy density of axion is about or less than O(10) % of active neutrino species, the axion can play the The abundance of axions produced from the decay of the role of dark radiation and only the correct photon spectrum v2  O( . ) v saxion is given by would require a 0 1 with a being the velocity of axion at present. However, it turns out that the require- Ya = 2Brs→aaYs(ts), (22) ment does not generate any new stronger constraint on 123 Eur. Phys. J. C (2014) 74:3062 Page 5 of 8 3062

Brs→aa.TheBrs→aa required to have a fractional axion However, comparing to Eq. (21), we find that the allowed DM, r ≡ a/ CDM is region of parameter space is opened only if

      HTR,rd λc 2 4 2 Brs→aa ∼ r × Br → . (26) 4 fs fa ma s aa ms  10 GeV . 0.12 Mpc fa,0 7keV (33) The energy contribution of axion dark radiation is given by In this case, one finds    ρ ρ 4/3       a CDM r 8 11 CDM λc 4 2 4 = r = 3 + r (27) HTR,fs  × −9 fs fa ma . ρν r ρν 7 4 r Brs→aa 4 10 0.12 Mpc fa,0 7keV (34) ρ /ρ ≤  bnd ≡ . and it is constrained to be a ν Neff 0 26 [22]. Hence one finds Now the abundance of the axion from the decay of the saxion is given by    / −1   8 11 4 3 ≤ + r  bnd T r 3 Neff 3 s 7 4 Ya Brs→aa CDM 2 m s     bnd 1/2   / − N − Br → m 1 2 f , = 1.16 × 10 5 eff . (28) 7.6 × 10 5 × s aa s a 0 . −2 6 0 26 10 10 GeV fa (35) Therefore, in order to have a sizable axion dark radiation matching to observation, one needs where we assumed that the saxion decays mainly to SM particles, and used Eq. (20) in the second line. A right    − amount of the flux for the 3.5keVX-ray line is obtained 4/3 1 HTR,DR 8 11 if Brs→aa ∼ Br → ≡ 3 + s aa 7 4   −5 Ya 6.9 × 10 (7keV/ma) × r  bnd HTR,rd. = . (36) Neff Brs→aa (29) f 2 f 2 CDM a a,0

Equating Eqs. (25) and (29), we find that the amount of Combined with Eq. (35), the above equation leads to hinted dark radiation and the 3.5keVX-ray line can be     ( ) 1/2 4 explained simultaneously if = SD,3.5 ≡ . g∗ Ts 10 GeV Brs→aa Brs→aa 13 3 200 ms       1/2 6 2 f − N fa 7keV a 3.4 × 10 3 eff (30) × . (37) , fa,0 0.26 fa 0 ma

HTR,3.5 Using Eq. (33), one finds and Brs→aa = Brs→aa .     If the decay of the saxion is delayed, the saxion may start ( ) 1/2 . 2 ∼ SD,3.5 g∗ Ts 0 12 Mpc to dominate the universe when H HSD with Br → > 13.3 s aa 200 λc     fs   2 4 4 × fa 7keV , ∼ fa (38) HSD ms (31) fa,0 ma MPl which contradicts Eq. (34). Therefore, it is not possible > and HSD s, in other words, to get a right amount of photon flux while satisfying the     constraint from structure formation in this case of saxion 2 4 1 ms MPl domination. Brs→aa > . (32) • 64π fa fa Low TR 123 3062 Page 6 of 8 Eur. Phys. J. C (2014) 74:3062

Inflaton decay might be delayed to a time after axion pro- dance coming from the decay of the saxion would have duction, i.e., I < s, which requires turned out to be too small.   π 2 −1/2 LTR 4.3 Axion misalignment Brs→aa < Br → ≡ s→aa/ I = g∗(T ) s aa 90 R     2 2 The keV-scale mass of the axion is far below the typical 1 ms MPl ms × . (39) expansion rate of inflation. In addition, the mass of the axion 64π M f T Pl a R is generated by the anomaly only after the associated symme- In this case, the free-streaming length is given by try is broken. Hence, if the symmetry were broken after infla- tion, a typical axion misalignment would have been of order        2/3 1/6 1/4 of the axion decay constant. On the other hand, if the symme- 1 H0 I m X /2 Teq λfs ≈ try were broken before or during inflation, the amount of mis- H0 s H0 ma T0         alignment could have been much smaller than the axion decay 4/3 1/3 2 3/2 × + 1 s H0 ma T0 constant. In the following argument, we assume the latter case 1 ln / 2 H0 I m X 2 Teq to allow a wide range of misaligned axion field values. (40)

• High TR and the constraint from structure formation (Eq. (6)) is The energy density of the axion at the onset of oscillation interpreted as can be expressed as

 LTR,fs 1 Brs→aa Brs→aa ρ = 2θ 2 2     a,osc ma osc fa (43) λc 3/2 3/2 2 ≡ fs H0 − 0.12 Mpc 5.33 × 10 38 GeV θ θ        where osc is the initial misalignment angle, and osc 1 1/2 3/2 3/8 × I ma T0 is assumed. We assume that the inflaton decayed before H m /2 T the axion started its oscillation. Then, the present abun- 0  s eq   /   / dance of the misaligned axion is 1 4 3 H 1 3 × 1 + ln s 0 2 H √  − /     0 I π 2 1 4 2 1/2 3 2 fa MPl     −3/2 Ya = g∗(T ) θ . 2 3/2 8 90 osc osc M m × ma T0 LTR Pl a Brs→aa m X /2 Teq (44)   /   λc 3 2 m 3/2 = . × −12 fs a This can be consistent with the observed relic density if 1 5 10 .   0 12 Mpc 7keV   /    fa,0 1 2 1/2 × LTR ; − fa,0 7keV Brs→aa (41) θ  2 × 10 4 (45) fa osc fa ma in the last line we used an approximation, {···}= 8.26. where we used g∗(T ) = 200, and the upper-bound sat-  osc If the saxion starts its coherent oscillation as H ms,for urates the relic density. fa around of larger than intermediate scale which may A remark is in order here. Considering the primordial appear naturally in theories beyond the inflation, one notice that θosc  1 requires that the modu- (for example, SUSY or string theories), the abundance of lus associated with the axion should be in the broken phase the axion when the inflaton decays is given as during inflation so as for a Hubble patch to be occupied     by a particular value θ . In addition, as already shown 2 osc 1 f T −4 a R in Eq. (11), for θosc  10 , the expansion rate of the Ya(TR) Brs→aa 2 MPl ms primordial inflation (HI ) should be less than of order   /   / 6 − m 3 2 m 1 2 of O(10 ) GeV in order not to produce too much iso-  2.5 × 10 22 a s 7keV 1010 GeV curvature perturbation caused by perturbations of θosc. (42) The recent data of BICEP2 hinted that the Hubble scale 14 during inflation is HI ∼ 10 GeV [23]. If it is con- whereinthelastlineweusedEq.(41). We find that if the firmed, axion cannot be the main component of DM, inflaton decayed after axion production, the axion abun- unless fa  MPl. However, it may be still possible to 123 Eur. Phys. J. C (2014) 74:3062 Page 7 of 8 3062

obtain a right amount of the X-ray line flux for fa  fa,0 axions produced from the saxion decay would be in a strong while satisfying the constraint from iso-curvature pertur- tension with the bounds from the Lyman-α forest data. The bation. For example, the photon flux from the decay of tension can be removed if axions produced from the sax- θ 2 the misaligned axion is proportional to osc and does not ion decay is of a subdominant contribution of the dark mat- depend on fa as long as the initial abundance of the axion ter relic density at present. This, in turn, requires a smaller is given by Eq. (43). Since r in Eq. (11) is proportional to axion decay constant to keep the observed 3.5keVX-ray 2  fa , one can take fa fa,0 to push up the bound of HI line the same. In this case, it is interesting to notice that the to O(1014) GeV hinted by BICEP2 data, while keeping axions can play the role of dark radiation simultaneously, −4 θosc ∼ 2 × 10 . being compatible with Planck data. If the saxion of coherent • Low TR oscillation dominated the universe at a later time, the con- The inflaton might decay at a very late time, and the axion straint from structure formation would have made it difficult might start its oscillation when the universe was still dom- to accommodate the 3.5keVX-ray line signal consistently. inated by the inflaton. In this case, the axion abundance is Also, if the inflaton decayed at a late time after the decay of given by the saxion, the axion relic density could have not saturated   the observed dark matter relic density, without disturbing the 2 structure formation due to relativistic axions. = 1 TR θ 2 fa Ya osc (46) 8 ma MPl We also discussed the axion misalignment for the axion production. In this case, if the inflaton decayed before the where TR is the reheating temperature of the inflaton. axion started its oscillation, the misalignment angle θosc −4 Hence, compared to Eq. (12), θosc is upper-bounded as should be about 10 to saturate the relic density. On the    other hand, if the reheating temperature of primordial infla- 1/2 θ  . fa,0 1GeV . tion is about 10 MeV, it is possible to have a natural value of osc 0 4 (47) θ ∼ . fa TR osc 0 1. In the case of saxion domination, the parameter space of the misaligned axion is more constrained, due to the Note that, since TR  10 MeV, θosc ∼ O(0.1−1) is axions produced from the saxion decay. allowed and the saxion can be in the symmetric phase during inflation although we then may have to worry about Acknowledgments This research is supported in part by Basic Sci- the domain wall problem. The iso-curvature perturbation ence Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology bound on HI in Eq. (11) may be satisfied marginally, being (2013R1A1A2007919) (HML), (2011-0010294), (2011-0029758), and compatible with BICEP2 data, within some errors, for (2013R1A1A2064120) (SCP) and (2012R1A2A1A01006053) (WIP). θ ∼ O( ) I ∼ O( 16−17) HML is also supported by the Chung-Ang University Research Grants osc 1 and fa 10 GeV. in 2014.

Open Access This article is distributed under the terms of the Creative 5 Conclusion Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. We proposed a simple model for the keV-scale axion dark Funded by SCOAP3 /LicenseVersionCCBY4.0. matter whose decay product into monochromatic photons can be the source of the recently reported X-ray spectrum at about 3.5 keV. Such a light axion can be produced from the decay of a heavy particle or from the coherent oscillation of the References axion caused by misalignment. We showed how the keV-sale axion model is constrained by structure formation and iso- 1. E. Bulbul, M. Markevitch, A. Foster, R.K. Smith, M. Loewen- curvature perturbation, depending on the axion production stein, S.W. Randall, Astrophys. J. 789, 13 (2014). arXiv:1402.2301 [astro-ph.CO] mechanisms and the amount of dark matter relic density. 2. A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, J. Franse, We found that axions produced from the inflaton decay arXiv:1402.4119 [astro-ph.CO] can saturate the observed relic density of dark matter and 3. T. Takahashi, K. Mitsuda, R. Kelley, H. Aharonian, F. Aarts, H. satisfy the constraint from structure formation, provided that Akamatsu,F.Akimoto,S.Allenetal.,arXiv:1210.4378 [astro- ph.IM] the reheating temperature was not smaller than the inflaton 4. H. Ishida, K.S. Jeong, F. Takahashi, Phys. Lett. B 732, 196 (2014). mass by an order of magnitude and the branching fraction of arXiv:1402.5837 [hep-ph] the inflaton decay to axions was less than about O(10−4).In 5. D.P. Finkbeiner, N. Weiner, arXiv:1402.6671 [hep-ph] the case of saxion decay, on the other hand, if the saxion was 6. J.-C. Park, S.C. Park, K. Kong, Phys. Lett. B 733, 217 (2014). arXiv:1403.1536 [hep-ph] coherently produced in a broken phase and decayed during 7. J.E. Kim, G. Carosi, Rev. Mod. Phys. 82, 557 (2010). the epoch of radiation domination after inflaton decay, the arXiv:0807.3125 [hep-ph] 123 3062 Page 8 of 8 Eur. Phys. J. C (2014) 74:3062

8. T. Higaki, K.S. Jeong, F. Takahashi, Phys. Lett. B 733, 25 (2014). 16. A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, JCAP 0903, 005 arXiv:1402.6965 [hep-ph] (2009). arXiv:0808.3902 [hep-ph] 9. J. Jaeckel, J. Redondo, A. Ringwald, Phys. Rev. D 89, 103511 17. D. Gorbunov, A. Khmelnitsky, V. Rubakov, JCAP 0810, 041 (2014). arXiv:1402.7335 [hep-ph] (2008). arXiv:0808.3910 [hep-ph] 10. J.E. Kim, H.P. Nilles, Phys. Lett. B 138, 150 (1984) 18. S. Horiuchi, P.J. Humphrey, J. Onorbe, K.N. Abazajian, M. 11. T. Higaki, R. Kitano, Phys. Rev. D 86, 075027 (2012). Kaplinghat, S. Garrison-Kimmel, Phys. Rev. D 89, 025017 (2014). arXiv:1104.0170 [hep-ph] arXiv:1311.0282 [astro-ph.CO] 12. M. Viel, J. Lesgourgues, M.G. Haehnelt, S. Matarrese, A. Riotto, 19. P.A.R. Ade et al. [Planck Collaboration], arXiv:1303.5082 [astro- Phys. Rev. D 71, 063534 (2005). astro-ph/0501562 ph.CO] 13. R.S. de Souza, A. Mesinger, A. Ferrara, Z. Haiman, R. Perna, 20. T. Kobayashi, R. Kurematsu, F. Takahashi, JCAP 1309, 032 (2013). N. Yoshida, Mon. Not. R. Astron. Soc. 432, 3218 (2013). arXiv:1304.0922 [hep-ph] arXiv:1303.5060 [astro-ph.CO] 21. P.S. Bhupal Dev, A. Mazumdar, S. Qutub, Physics 2, 26 (2012). 14. M. Viel, G.D. Becker, J.S. Bolton, M.G. Haehnelt, Phys. Rev. D arXiv:1311.5297 [hep-ph] 88(4), 043502 (2013). arXiv:1306.2314 [astro-ph.CO] 22. P.A.R. Ade et al. [Planck Collaboration], Astron. Astrophys. 15. K. Markovic, M. Viel, Lyman-α forestandcosmicweaklensingina (2014). arXiv:1303.5076 [astro-ph.CO] warm dark matter universe. Publications of the Astronomical Soci- 23. P.A.R. Ade et al. [BICEP2 Collaboration], Phys. Rev. Lett. 112, ety of Australia, vol. 31, e006, pp 1Ð20 (2014). arXiv:1311.5223 241101 (2014). arXiv:1403.3985 [astro-ph.CO] [astro-ph.CO]

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