Decaying Dark Matter in Halos of Primordial Black Holes

We investigate photon signatures of general decaying dark-matter particles in halos of primordial black holes. We derive the halo-profile density and the total decay rate for these combined dark-matter scenarios. For the case of axion-like particles of masses below $\mathcal{O}(10)$ keV, we find strong bounds on the decay constant which are several orders of magnitude stronger than the strongest existing bounds, for all halo masses above $\mathcal{O}(10^{-5})$ solar masses. Using future X-ray measurements, it will be possible to push these bounds on such combined dark-matter scenarios even further.

Introduction -In the standard model of cosmology, the energy density of the Universe consists of approximately 25 % in the form of a pressureless, nearly perfect fluid of non-relativistic objects, so-called (cold) dark matter. A large number of potential dark-matter candidates have been proposed so far. The perhaps most well-studied class is constituted by hypothetical particles which only weakly interact with the other standardmodel particles. Amongst this variety, there are so-called WIMPs (cf. [1]), sterile neutrinos (see Ref. [2] for an early discussion on their role as dark-matter components and Refs. [3][4][5][6]), axions [7][8][9][10], and axion-like particles (ALPs) [11]. The latter, whose characteristics we will use in this work, constitutes a class of pseudo Nambu-Goldstone bosons which are coupled to photons. For photons, their mass and decay constant are related, while this is generally not the case for ALPs.
Besides microscopic candidates like ALPs, dark matter might also be constituted by macroscopic objects such as primordial black holes (PBHs) [12,13]. 1 These are black holes which have been produced in the very early Universe. The interest in PBH constituting parts of the dark matter [17] has been revived recently [18][19][20][21][22][23][24][25], in particular through the gravitational-wave discovery of black-hole binary mergers [26,27]. The possible PBH formation mechanisms are very diverse and there is a large number of scenarios, which lead to their formation. All of these have in common that they require some mechanism to generate large overdensities.
Even though most of the emphasis in dark-matter research has been focused on one-component scenarios, models with more than one component have been investigated, including mixed types of both microscopic as well as macroscopic nature. On the one hand, a small fraction of PBHs could provide seeds for super-massive black holes in the galactic centres [28]. On the other hand, in view of the fact that it appears difficult, although not impossible, to have the entire dark matter in the form of PBHs or UCMHs (cf. Ref. [20] including a summary of relevant constraints), the class of particle dark matter provides a vital supplementary and major candidate.
In all of those combined scenarios, the particles will be gravitationally bound to the PBHs. This could lead to strong decay [29] and/or annihilation signatures [30,31].
Halos -As mentioned above, in a combined darkmatter scenario consisting of a large fraction of particles and a small fraction of PBHs, the former will be gravitationally bound to the latter. For WIMPs, this has been studied by Eroshenko [30] and the authors [31]. However, this formation mechanism, which happens in the radiation-dominated epoch, is not specific to any particular WIMPs model. In fact, the essential ingredients are the mass and the velocity distribution of the particles. Hence, we will generalize the results to investigate general halo formation and follow Ref. [31], wherein the technical details can be found. Figure 1 presents the halo-profile density as a function of radius r (in units of the Schwarzschild radius r s = 2.95 · 10 3 km) for accreted particles of mass m ∈ {0.1, 10 3 , 10 5 , 10 7 } keV around a PBH of a solar mass, assuming a Maxwellian velocity profile. As expected, lighter particles lead to a more extended halo. Outside of the halo's core, its profile follows ρ(r) ∝ m 2/5 r −3/2 .
Decay -For decay signatures, distinct from annihilations, and unless the halo is extremely close to the telescope, it is practically point-like, and hence its total mass M matters rather than its concrete density profile. Given an individual decay rate, the total decay rate is readily obtained using Γ total = N Γ, with N = M/m being the number of particles within the halo. For ALPs (see arXiv:1811.05810v1 [astro-ph.CO] 14 Nov 2018 Sec. 111 of Ref. [32] for a recent review), we may write where G aγγ is the decay constant. For the QCD axion, Eq. (1) simply becomes For sterile neutrinos, a similar expression holds (see Ref. [29]). Figure 2 shows the total decay rate Γ total aγγ as a function of the halo mass M for different values of the particle mass m, where we assumed for illustrational purpose G aγγ = G QCD aγγ . Furthermore, it holds that Γ total aγγ ∝ M m 5 .
Constraints -In Ref. [29], we proposed and investigated a scenario in which the dark matter is constituted by halos of sterile neutrinos around PBHs. Therein, we studied the possibility that in a certain observational time frame with a certain probability one of those compact objects propagates at a given minimum distance near the telescope. Through the halo's nearness, its radiation may dominate the photon flux from other sources onto the telescope. If the halo's minimum distance is small enough, its signature will be detected.
It is now tempting to generalise this set-up. Applied to ALPs, we derive new limits on the maximally-allowed decay constant G aγγ by focussing on X-ray data. Now, Refs. [3,4,33] provide bounds on G aγγ as well as instrumental sensitivity flux limits. Specifically, the data used come from observations of the Large Magellanic Cloud with XMM-Newton (XMM obs ID: 0127720201) [33] (see Ref. [34] for instrumental characteristics). We compare the observed X-ray fluxes to those originating from the decays of the ALP halos using the methodology of Ref. [29], described in the previous paragraph. Utilising the relation from the total decay rate to the decay constant [see Eq. (1)], we obtain constraints for the latter. Concretely, for a selection of halos with different halo masses (M ∈ {10 −5 , 10 −1 , 10 3 } M ), we compare their X-ray fluxes to the one received by the telescope assuming no halos. Furthermore, we suppose that the local dark-matter density is spatially homogeneous and takes a value of 0.3 GeV cm −3 . This determines the average distance d between two halos: The velocity distribution of the halos is assumed to be Maxwellian. As described above, subject to this distribution with a certain probability P , a halo will move near the telescope and shed photons onto it. As we showed in Ref. [29], P is approximately given by where θ ∈ [0, π] and φ ∈ [0, 2π) are the opening angles of a detector. Above, r Φ is the distance from the detector such that a certain flux Φ A through its effective area A is observed; it is given by [29] r For a given observational time, it is then easy to determine the X-ray flux of the halos and compare it to that of the background. By virtue of Eq. (1), this can thus be used to constrain G aγγ . As a function of the ALP mass m a , our results are depicted in Fig. 3. In this figure, it can be observed that an increase of M leads to a decrease of the constraint line of G aγγ . Also, smaller values of the X-ray energy lead to stronger constraints. The physical reason for this is that, for a fixed M , the number N a = M/m a of ALPs within the halo is increasing with decreasing m a , whereas the background X-ray flux is given and fixed, the single halo moving in the vicinity of the telescope contains more decaying particles the smaller their mass is. This extra factor of 1/m a is responsible for the increased detection prospects towards smaller mass.
In Fig. 3, we observe that for m a below O(10) keV, we obtain bounds on G aγγ which are several orders of magnitude stronger than the strongest existing bounds, for all M above O(10 −5 ) solar masses. In particular, for 0.5 keV ≤ m a ≤ 100 keV, we find that the following values of G aγγ will, at least, be excluded The obtained bounds in this paper can be regarded a conservative, as we have not included ALP-to-photon conversion due to magnetic fields. The latter can occur in two distinct situations: internally, for instance for charged and rotating black holes, and externally, for instance from galactic magnetic fields. Either case only increases the photo emission from the halo objects, and hence strengthens the bounds. It would be interesting to investigate these instances in the future, but it is beyond the scope of the present paper.
Conclusions -We have investigated decay signatures from a two-component dark-matter scenario in which most of the dark matter is constituted by axion-like particles (ALPs) complemented by primordial black holes (PBHs). We have studied how the former accrete around the latter and calculated the halo profile (shown in Fig. 1). Then, we have studied the decay signatures (visualised in Fig. 2) from which we have derived bounds on the decay constant (depicted in Fig. 3). We have found that this combined scenario leads to detection prospects which, for small ALP masses less than or equal to O(10) keV and for halos heavier than 10 −5 M , are far better than the pure ALP scenario.