Cluster X-ray line at $3.5\,{\rm keV}$ from axion-like dark matter

The recently reported X-ray line signal at $E_\gamma \simeq 3.5\, {\rm keV}$ from a stacked spectrum of various galaxy clusters and the Andromeda galaxy may be originating from a decaying dark matter particle of the mass $2 E_\gamma$. A light axion-like scalar is suggested as a natural candidate for dark matter and its production mechanisms are closely examined. We show that the right amount of axion relic density with the preferred parameters, $m_a \simeq 7 \,{\rm keV}$ and $f_a \simeq 4\times 10^{14}\, {\rm GeV}$, can be naturally obtainable from the decay of inflaton. If the axions were produced from the saxion decay, it could not have constituted the total relic density due to the bound from structure formation. Nonetheless, the saxion decay is an interesting possibility, because the $3.5\, {\rm keV}$ line and dark radiation can be addressed simultaneously, being consistent with the Planck data. Small misalignment angles of the axion, ranging between $\theta_a\sim 10^{-4} -10^{-1}$ depending on the reheating temperature, can also be the source of axion production. The model with axion misalignment can satisfy the constraints for structure formation and iso-curvature perturbation.


Introduction
It has been recently reported by two groups [1,2] that there exists an unidentified line in a stacked X-ray spectrum of galaxy clusters and Andromeda galaxy at an energy, E γ 3.5 keV, after the EPIC data of XMM-Newton observations [3] were analyzed. As a result, the measured line energy and flux are consistent with monochromatic photon(s) due to a decaying dark matter(DM), having the mass of m DM ∼ 7 keV and the lifetime of τ DM ∼ 10 28 sec. It is intriguing that the X-ray spectrum of galaxy clusters at different redshifts shows a consistent line signal for dark matter. For a further confirmation of the observed X-ray line, however, we need more objects and a better spectral resolution as in Astro-H mission [4]. It has been shown that the observed X-ray line signal can be explained well by the decay of sterile neutrino that appears in a minimal extension of the Standard Model explaining observed small neutrino masses [1,2,5]. Nonetheless, it would be worthwhile to investigate alternative dark matter models for the X-ray line [6].
We consider an axion-like dark matter for explaining the 3.5 keV X-ray line † . Axionlike particles are ubiquitous in string compactifications and QCD axion models, etc, as they appear as pseudo-Goldstone bosons at low energies after the breakdown of accidental global symmetries [7]. When the axion-like scalar with 7 keV decays into a pair of photons due to anomaly interactions, it can accommodate the observed X-ray line consistently with the axion decay constant, f a 4 × 10 14 GeV. Furthermore, we discuss the mechanisms for producing the correct relic density of axion-like dark matter by either the axion misalignment or the non-thermal production from the decay of the inflaton or the scalar partner of axion, so called saxion. Depending on the temperature at which axion or saxion starts to oscillate coherently and on whether saxion decays after or before reheating after inflation, we divide our discussion into different scenarios of the axion production and impose on each case the bounds coming from structure formation and iso-curvature perturbations.
The paper is organized as follows. We begin with a description of the model for axionlike dark matter explaining the X-ray line and discuss the cosmological bounds involved with the non-thermal production of axions and the axion misalignment angle. Then, various scenarios of axion production will be discussed in the cases of inflaton/saxion decays and axion misalignment. Then, conclusions will be drawn.

A keV scale axion for the X-ray line
The recently reported X-ray line at about 3.5 keV may be explained by a 7-keV dark matter decaying two photons. Axion-like scalar is a very good candidate for such a light dark matter. In this section, assuming that the X-ray line spectrum stems from a 7-keV axion dark matter, we discuss the necessary properties of the axion and possible cosmological constraints on that.

Axion properties
We introduce an axion-like particle as a pseudo-Goldstone scalar associated with a broken anomalous U (1) symmetry. After the symmetry-breaking, the model-independent effective Lagrangian for axion a and saxion s (the radial component of the symmetry breaking field) can be expressed as where α em is the fine structure constant of electromagnetic interaction, f a is the axion coupling constant, and F µν andF µν are the field-strength tensor and its dual for electromagnetic field, respectively. We note that c 1,2,3 are dimensionless parameters of order one and we have included an extra charged fermion f that is responsible for the generation of anomalies. Here, the saxion interactions are motivated by a supersymmetric axion model [10] where the axion chiral multiplet A with A| θ=0 = s+ia appears in the superspace interactions as d 2 θAW α W α and d 2 θ m f e c 3 A/fa Φ L Φ R where W α is the field strength superfield and Φ L,R are matter chiral multiplets containing the extra charged fermion. We can add the Lagrangian for inflation by L I .
For a keV-scale axion, the first term of the second line in Eq.
(1) provides the main decay channel with a rate given by ( where we set c 1 = 1 for simplicity. This axion decay can be a possible origin of the recently reported X-ray spectrum if the axion saturates the dark matter relic density and has the following properties [1,2] m a = 7.1 keV , τ a = 1.14 × 10 28 sec → Γ a = 5.73 × 10 −53 GeV where τ a is the life-time of axion. This implies that the axion coupling constant should be f a 4 × 10 14 GeV m a 7 keV (4)

Cosmological constraints
Our keV-scale axion can be constrained by astrophysics and cosmology, namely, structure formation or iso-curvature perturbation of dark matter, depending on how it is produced.

Structure formation
The keV axion may be a decay product of the inflaton and/or saxion. Suppose that the mother particle, denoted as X, decays to two axions, each of which carries the energy of when the axion mass is ignored. Then, in order not to destruct large scale structures, axion should be non-relativistic around the epoch when a Hubble patch contains mass energy corresponding to the galactic-sized halo (corresponding to T ∼ T * ≡ 300 eV) (see for example [11,12]). This implies that where the initial momentum of axion is p a,i E a,i for E a,i m a , g * S (T ) is the relativistic degrees of freedom at a temperature T , and T i is the background temperature when axion is produced from the decay of the mother particle X. Here, we used g * S (T * ) = 3.9 and g * S (T i ) = 200 for obtaining the lower bound of T i .

Isocurvature perturbation
If axion dark matter is produced mainly from the coherent oscillation caused by axion misalignment, it is subject to the bound on dark matter iso-curvature perturbation. The recent Planck data combined with WMAP polarization data leads to a constraint on the fraction of iso-curvature perturbation by [13], at 95% CL, where P R 2.2 × 10 −9 and P S are the power spectrum of curvature and isocurvature perturbations, respectively. The iso-curvature perturbation of axion dark matter can be expressed as [14] where a osc is the initial oscillation amplitude of axion caused by misalignment, H I is the expansion rate during inflation, and we assumed axion dark matter from the misalignment saturates the observed relic density of dark matter. As described in the next section, for a osc f a , Ω a ∝ a 2 osc , hence one finds

Scenarios of axion production
Axion-like scalars can be produced from decays of heavy particles or coherent oscillations.
In this section, we discuss how we can obtain a right amount of keV-scale axion while satisfying various constraints given in the previous section. In particular, we consider the axion production from inflaton/saxion decay and axion misalignment in both cases of high and low reheating temperature after primordial inflation. In order to match the relic density of dark matter to the observed one, Ω a = 0.268 [13], we quote the necessary axion abundance at present as

Inflaton decay
In inflation scenarios where inflation is responsible for the density perturbation of the present universe, inflation should decay mainly to SM particles. It can also partially decay to axions we are considering now. In the sudden decay approximation, the axion abundance from such a partial decay of inflaton is given by where Br(I → aa) and Br(I → SM) are respectively the branching fraction of inflaton (I) to axions and to SM particles, and T R and m I are the reheating temperature and mass of inflaton, respectively. We assumed Br(I → SM) 1 in the far right side of Eq. (11). Compared to Eqs. (6) and (10), we find that a right amount of axion relic density can be obtained while satisfying the constraint from structure formation, provided that

Saxion decay
The saxion, the radial component of the complex field containing axion, can play a crucial role in axion production, since it can decay into a pair of axions via the axion kinetic term, In this case, the decay rate of saxion to a pair of axions is given by In the presence of an extra heavy charged fermion coupled with saxion via a Yukawa coupling in the following form, there is an additional contribution to the saxion decay rate, where in the second line, use is made of the Yukawa interaction from the effective Lagrangian in Eq. (1). Then, for Γ s→aa Γ s→f f , the branching fraction of saxion decaying to a pair of axions is given by Thus, for |c 3 | 7, we can obtain a small branching fraction, Br(s → aa) ∼ 10 −2 , as required for structure formation in a later discussion.
In the early universe, saxion might be at the broken phase with H I m s , but could undergo a coherent oscillation as H m s . Then, the saxion decay might be the main source of axion production, although structure formation constrains the branching fraction to axions, similarly to the case of inflaton decay. In the following argument, for simplicity, we express the full decay width of saxion as and we will use the sudden decay approximation for saxion decay.

High T R
If inflaton decays before saxion starts its oscillation, from Eq. (18) and with g * (T i ) = 200, we find that the constraint from structure formation (Eq. (21) In order to get a right relic density for axion from the saxion decay by and Eq. (10), the required saxion abundance should be If saxion is produced via coherent oscillation while universe is dominated by radiation, we obtain Y s (t i ) = Y s (t s,osc ) where Y s (t s,osc ) is the initial abundance of saxion in coherent oscillation, given by Here, we assumed that the initial saxion misalignment is almost the same as the axion decay constant f a . As shown in the left plot of Fig. 1 Therefore, the allowed region with Br(s → aa) O(10 −3 ) in the left panel of Fig. 1 tends to be saxion-dominated. In such a case, the abundance of axion from the decay of saxion is given by where T R,s is the late-time reheating temperature after the decay of saxion, and we assumed that saxion decays mainly to SM particles. Using Eq.
Note that in the case of saxion domination, the constraint from structure formation still applies by T R,s 6 × 10 −3 m s . As shown in the right panel of Fig. 1 where Y a with Eq. (28) is depicted as a function of m s and Br(s → aa), the constraint from structure formation, shown in red-dashed line, limits the parameter space so that a right amount of axion dark matter can be obtained only for Br(s → aa) 10 −2 .

Low T R
Inflaton decay might be delayed to a time after axion production, i.e., Γ I < Γ s which requires m s > π 2 90 g * (T R ) × 64π where we used with g * (T R ) = 200. In chis case, the constraint from structure formation reads which results in where g * S (T * ) = 3.91 and g * S (T R ) = 200 were used. If saxion starts its coherent oscillation as H m s , the abundance of axion when inflaton decays is given as Hence, we find that, once Eq. (32) is satisfied, axion abundance coming from the decay of saxion turns out to be too small if inflaton decays after axion production.

Axion misalignment
The keV-scale mass of axion is far below the typical mass scale of inflation. In addition, the mass of axion is generated by the anomaly only after the associated symmetry is broken. Hence, if the symmetry were broken after inflation, a typical axion misalignment would be of order of the axion decay constant. On the other hand, if the symmetry were broken before or during inflation, the amount of misalignment can be much smaller than the axion decay constant. In the following argument, we assume the latter case to allow a wide range of misaligned axion field values.

High T R
The energy density of axion at the onset of oscillation can be expressed as where a osc is the initial oscillation amplitude, and a osc f a was assumed. We assume that inflaton decays before axion starts its oscillation. Then, the present abundance of misalignment axion is where we used g * (T osc ) = 200, and the upper-bound saturates the relic density.
A remark is in order here. Considering primordial inflation, one notice that such an intermediate scale misalignment with the symmetry-breaking scale larger than a osc by several orders of magnitude requires that the modulus associated with the axion should be in the broken phase during inflation so as for a Hubble patch to be occupied by a particular value of a osc . In addition, as already shown in Eq. (9), for a osc 10 10 GeV, the expansion rate of the primordial inflation should be less than of order of O(10 6 ) GeV in order not to produce too much iso-curvature perturbation caused by perturbations of a osc .
If there is a coherent oscillation of saxion, and the saxion decay is delayed to a very late time, saxion may dominate the universe at some point. Comparing the Hubble scale evaluated at saxion domination to the axion mass as we find that saxion domination can take place after the onset of axion oscillation for m s 10 10 GeV. In this case, the axion abundance from the misalignment is given by where T R,s is the late-time reheating temperature from the decay of saxion, and we assumed that saxion decays mainly to SM particles. Hence, using Eqs.

Low T R
Inflaton might decay at a very late time, and axion might start its oscillation when the universe is still dominated by inflaton. In this case, the axion abundance is given by where T R is the reheating temperature of inflaton. Hence a osc is upper-bounded as Note that, since T R 10 MeV, saxion had to be in the broken phase before or during inflation, otherwise cold axion would be over-produced.

Conclusion
We proposed a simple model for keV-scale axion dark 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 in the decay of a heavy particle or from the coherent oscillation of axion caused by misalignment. Depending on the axion production mechanisms and the amount of dark matter relic density, we showed how the keV-sale axion model is constrained by structure formation and iso-curvature perturbation.
We found that axions produced from the inflaton decay can saturate the observed relic density of dark matter while satisfying the constraint from structure formation, provided that the reheating temperature is higher than the inflaton mass by two orders of magnitude and the branching fraction of the inflaton decay to axions is less than about 0.01. In the case of saxion decay, we considered the case that inflaton decays first, and saxion, which in the broken phase, then starts its coherent oscillation. In this case, saxion mass should be about 10 6 GeV for the branching fraction of saxion into axions of about 10 −3 . The smaller the branching fraction is, the smaller the saxion mass is required. If the saxion of coherent oscillation dominates the universe at a later time, a wider range of parameter space is allowed. On the other hand, if inflaton decays at a late time after the decay of saxion, the axion relic density cannot saturate the observed dark matter relic density, without disturbing the structure formation due to relativistic axions.
In the case of axion misalignment, if inflaton decays before axion starts its oscillation, the misalignment angle θ a should be about 10 −4 to saturate the relic density. On the other hand, if the reheating temperature of primordial inflation is about 10 MeV, it is possible to have a natural value of θ a ∼ 0.1. In saxion domination, the misalignment axion constrains the parameter space more such that some allowed region of axions produced in the case of saxion decay is removed, although the limitation depends on the misalignment angle.