Constraints on a mixed model of dark matter particles and primordial black holes from the Galactic 511 keV line
PPrepared for submission to JCAP
Constraints on mixed dark mattermodel of particles and primordial blackholes from the Galactic 511 keV line
Rong-Gen Cai, a,b,c
Xing-Yu Yang, a,b, Yu-Feng Zhou a,b,c a CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, P.O. Box 2735, Beijing 100190, China b School of Physical Sciences, University of Chinese Academy of Sciences, No.19A YuquanRoad, Beijing 100049, China c School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Ad-vanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, ChinaE-mail: [email protected], [email protected], [email protected]
Abstract.
The 511 keV line was first detected in 1970’s but its origin is still unknown. It wasproposed that positrons from dark matter (DM) particles in the halo of galaxy are possiblesources. We consider a mixed DM model consisting of DM particles and primordial black holes(PBHs). With the existence of PBHs, the DM particles may be gravitationally bound to thePBHs and form halo around PBHs with density spikes. These density spikes can enhance theproduction rate of positrons from DM particles, thus they are constrained by the observationsof 511 keV line. We compute the profile of the density spikes, and get the constraints onthe fraction of PBHs in DM f PBH for light dark matter (LDM) scenario and excited darkmatter (XDM) scenario respectively. For LDM with mass in the range of 1MeV ∼ ∼ M − slope in the relative small mass range, and roughly have M / slope in the relativelarge mass range. The most stringent constraint f PBH (cid:46) − appears at the turning pointwhich depends on the mass of XDM. Corresponding author. a r X i v : . [ a s t r o - ph . C O ] J u l ontents In the standard model of cosmology, only 5% of the universe consisting of ordinary baryonicmatter is well-known, and the rest of the universe which consists of nearly 26% dark matter(DM) and nearly 69% dark energy (DE) is barely known [1]. The puzzle of DM dates backto 1930’s [2], and the existence of DM is overwhelming with evidences from various astro-physical observations, such as rotation curves of galaxies, hot gas in clusters, gravitationallensing measurements, galaxy formation, primordial nucleosynthesis and cosmic microwavebackground observations [3, 4]. There are many DM candidates, including astrophysical ob-jects such as primordial black holes (PBHs), and particles beyond the standard model ofparticle physics. Among the particle DM candidates, the most popular are the weakly inter-acting massive particles (WIMPs) and axions, since they have been proposed for some otherreasons in particle physics. Other candidates include sterile neutrinos, light dark matter(LDM), self-interacting dark matter, and many others [3, 4].The PBHs are formed from the gravitational collapse of the overdense regions in the earlyuniverse [5–7], and have attracted considerable attention and have been studied extensively([8, 9], and references therein). Beside being candidate of DM, PBHs can also be the seedsfor galaxy formation [10–13], or the sources of LIGO/VIRGO detection [14, 15]. There are aplenty of scenarios that lead to PBH formation [16, 17], and all of these require a mechanismto generate large overdensities. These overdensities are often of inflationary origin, and willcollapse if they are larger than a certain threshold when reentering the horizon [18–24].The 511 keV line was first detected at the galactic center and was identified as the resultof electron-positron annihilation in 1970’s [25–27]. This line is mostly due to parapositroniumannihilation of thermal or near-thermal positrons [28, 29], and the absence of continuous high-energy spectrum from positron annihilation in flight implies that the initial energy of thesepositrons is less than a few MeV [30], but the origin of these low-energy galactic positronsis still under debate. Several sources have been proposed to explain the low-energy galacticpositrons, such as the β + decay of stellar nucleosynthesis products (e.g. Al, Ti and Ni) [31–33] and LDM [34, 35].However, most of astrophysical sources cannot account for the observed morphology,while DM interactions have the potential to explain the observations because of the DM haloof galaxy [36]. It is possible that the low-energy galactic positrons are produced by directannihilation of LDM ( ∼ few MeV) particles into electron-positron pairs [34], or by the exciteddark matter (XDM) mechanism. In the XDM mechanism, the excited states of heavy DM– 1 –re produced in collisions, and the electron-positron pairs are produced through the decay ofthe excited state into the ground state [37, 38]. The advantage of this mechanism is that theDM mass is relatively unconstrained, and the only requirement is that the splitting betweenthe ground and excited states is less than a few MeV.In principle either PBH or particle can account for the total DM abundance, but there isno evidence that one of them must account for the total abundance, and it seems that a mixedDM model consisting of both PBH and DM particle is more possible. Indeed, such a mixedDM model can lead to many interesting consequences compared with a single componentDM model. Besides, even if PBHs only constitute a small fraction of the DM, they can stillhave significant influence as we will show. Therefore such a mixed DM model needs morestudies, and it will be pretty good if we can make some constraints on such a model throughsome observations.In this paper, we assume that the DM is mixed of PBHs and particles whose interactionscan lead to the productions of electron-positron pairs which can explain the 511 keV lineobservations, i.e., the energy density of DM is given by ρ DM = ρ PBH + ρ χ . (1.1)With the existence of PBHs, the DM particles may be gravitationally bound to the PBHsand form halo around PBHs with density spikes [39]. Since the interaction of DM particlesis related to the particle density, the formation of density spikes can change the interactionand leave some imprints in the 511 keV line observations. Therefore, by analyzing the dataof 511 keV line, one can get constraints on PBHs and DM particles.This paper is organized as follows: we revisit the DM scenario which can explain the511 keV observations in section 2 and compute the density profile of DM particle halo aroundPBHs in section 3, the constraints from the Galactic 511 keV line are shown in section 4,and the last section is devoted to conclusions. Firstly we revisit the scenario that the low-energy galactic positrons which lead to 511 keVline are produced from DM particles in the case that particles account for the total DMabundance. For these positrons produced from DM, its number density is closely connectedto the number density of DM particles. If the positrons are produced by decay of DMparticles, the rate of production is ˙ n e + = n χ Γ d , (2.1)where n e + and n χ are the number densities of positrons and DM particles, respectively, andΓ d is the decay rate of DM particles. But if the positrons are produced from annihilationof DM particles or the decay of excited state into ground state for XDM, then the positronproduction rate is given by ˙ n e + = (cid:104) σv (cid:105) n χ , (2.2)where (cid:104) σv (cid:105) is the thermally averaged cross-section for annihilations or excitations of DMparticles that produce electron-positron pairs.As the experiments show, most positron annihilations are through positronium forma-tion [28, 40, 41], and the positronium fraction f p ≈ .
967 [29]. In this channel, 1 / E = 511keVwhich account for the observations, while the remaining 3 / E < E = 511keV. Thus the total number density of 511 keV photons produced per unittime is ˙ n γ = 2((1 − f p ) + f p /
4) ˙ n e + . (2.3)For any particular model of the Milky Way DM halo, one can compute the intensitydistribution of 511 keV signature as a function of galactic longitude l and latitude b , byintegrating the emissivity ˙ n γ ( r ) along the line of sight (l.o.s.), which is I ( l, b ) = 14 π (cid:90) l . o . s . ˙ n γ ( r ) ds, (2.4)with r = (cid:113) ( s cos b cos l − r (cid:12) ) + s cos b sin l + s sin b, (2.5)where s denotes the distance from the solar system along the line of sight, and r (cid:12) ≈ . (cid:90) I ( l, b ) d Ω . (2.6)Suppose the density profile of Milky Way DM halo is ρ ( r ), the intensity distribution of511 keV signature is I ( l, b ) = 2(1 − f p ) 14 π (cid:90) l . o . s . Γ d ρ ( r ) m χ ds (2.7)for decaying DM, and I ( l, b ) = 2(1 − f p ) 14 π (cid:90) l . o . s . (cid:104) σv (cid:105) ρ ( r ) m χ ds (2.8)for annihilating or excited DM. Noticing that the intensity distribution is I ∼ n χ for decayingDM, while that is I ∼ n χ for annihilating or excited DM, thus the decaying DM will lead toa more spread distribution compared with annihilating or excited DM.Comparing the observation data of 511 keV line with the theoretical predictions, itis found that the decaying DM scenario is disfavored by data, while the annihilating orexcited DM scenario is pretty plausible [43–45]. Using a model-fitting procedure to theINTEGRAL/SPI data, the best fit results are [45] (cid:104) σv (cid:105) ∼ × − ( m χ / GeV) cm s − , (2.9)and Φ ∼ . × − ph cm − s − , (2.10)with the density profile of Milky Way DM halo given by the Einasto profile ρ ( r ) = ρ s exp (cid:18) − (cid:20) α (cid:18) rr s (cid:19) α − (cid:21)(cid:19) , (2.11)where r s ≈ α ≈ .
17 come from the
Via Lactea II simulation [47], and the overalldensity normalization ρ s ≈ . / cm is computed from the local dark matter density [48]. The velocity dependence of this thermally averaged cross-section is neglected for simplicity, which is goodapproximation for MeV DM undergoing pure annihilations [34, 46] and XDM with m χ (cid:38) TeV, and moredetailed study should be done for XDM with m χ (cid:46) TeV [45]. – 3 –
Particle halo around PBHs
There are many experiment constraints on the fraction of PBHs in the DM, it is possible forPBHs with mass in the range 10 − ∼ − M (cid:12) constitute all the DM, but for the stellermass PBHs, the constraints suggest that PBHs subdominant to the rest of DM [9]. Thefraction of PBHs in DM is defined as f PBH ≡ ρ PBH /ρ DM , (3.1)so the corresponding fraction of DM particles is ρ χ = (1 − f PBH ) ρ DM . In this work, weconsider a particles dominating mixed DM model, i.e. f PBH (cid:28) t i , we focus on a particle at position r i with velocity v i . The particle would spend a fraction 2 dt/τ orb of its period at distances between r and r + dr , where τ orb is the period of the particle’s orbital motion around the PBH and dt isthe time it takes for the particle to move from r to r + dr . Given the initial density of DMparticles ρ i ( r i ), at later time t > t i , the density of DM particle halo around PBHs ρ b ( r ) canbe written as the relation ρ b ( r )4 πr dr = (cid:90) πr i dr i ρ i ( r i ) (cid:90) d v i f B ( v i ) 2 dt/drτ orb dr, (3.2)which follows from the Liouville equation and expresses the density conservation law inphase space integrated over the momenta by taking into account the volume transformationin momentum space, where the velocity distribution of DM particles f B ( v i ) is chosen to beMaxwell-Boltzmann distribution. Therefore, we have ρ b ( r ) = 1 r (cid:90) dr i r i ρ i ( r i ) (cid:90) d v i f B ( v i ) 2 τ orb dtdr , (3.3)and more details of the calculation can be found in [39, 49].In figure 1, we show the density profile ρ b of DM particles bound to a PBH as a functionof the rescaled radius r/r g (where r g ≡ GM PBH ) with different values of m χ and M PBH . Fora given m χ , the density profile for the lighter PBH constitutes an envelope to the profile forthe heavier PBH, this is because the maximum rescaled radius that a PBH can gravitationallyaffect, i.e. (3 M PBH / πρ r ) / /r g (where ρ r is the energy density of the radiation-dominateduniverse), is smaller for the heavier PBH. For a given M PBH , ρ b is smaller for the lighter DMparticles. This is because the initial density ρ i ∝ ρ KD (where ρ KD is the energy density ofthe universe when DM particles kinetically decouple from the primordial plasma), and thelight particles kinetically decouple later than the heavy particles, which lead to a smaller ρ i for the lighter DM particles. Due to the symmetry of the orbit, the particle passes the same radius twice, which leads to the factor of2 [39]. – 4 – - - - - - - - - - - - - - - - - Figure 1 . The density profile ρ b of DM particles bound to a PBH with different values of m χ and M PBH . For a given m χ , the density profile for the lighter PBH constitutes an envelope to the profilefor the heavier PBH, these profiles overlap in the small r/r g range and diverge in the large r/r g range. The horizontal dot-dashed lines denote the maximum possible density at present time of theDM particles computed from eq. (4.1). If the DM particles could annihilate, their density will decrease with time, and there will bea maximum possible density at present time which is given by [50] ρ max = m χ (cid:104) σv (cid:105) a t , (4.1)where t ≈ . × s is the age of the universe [1], and (cid:104) σv (cid:105) a is the thermally aver-aged cross-section for annihilations of DM particles. For the LDM scenario , it is obvi-ous that (cid:104) σv (cid:105) a = (cid:104) σv (cid:105) ∼ × − ( m χ / GeV) cm s − , but for the XDM scenario, (cid:104) σv (cid:105) is the thermally averaged cross-section for excitations of DM particles, thus we choose (cid:104) σv (cid:105) a ∼ × − cm s − which matches the observed relic density. Therefore the den-sity profile of DM particles around PBH is ρ χ, PBH ( r ) = min[ ρ max , ρ b ( r )] , (4.2)with critical radius r c satisfying ρ b ( r c /r g ) = ρ max . For the LDM ( ∼ few MeV) annihilating directly to electron-positron pairs, the cross-section obtainedfrom best fit of 511 keV data is too small to give the right relic density. This will not be a problem if thereare additional stronger annihilation channels into invisible particles, for example, dark gauge bosons [46] ordark neutrinos [51]. – 5 –he number of positron produced in the vicinity of a PBH per unit time isΓ PBH = Γ ( i )PBH + Γ ( o )PBH , (4.3)with Γ ( i )PBH = (cid:90) r c πr dr (cid:104) σv (cid:105) m χ ρ = 4 π (cid:104) σv (cid:105) m χ ρ r c , (4.4)Γ ( o )PBH = (cid:90) ∞ r c πr dr (cid:104) σv (cid:105) m χ ρ ( r ) , (4.5)where Γ ( i )PBH and Γ ( i )PBH denote the positron production rate in the inner and outer parts(separated by r c ) of the halo around PBH, respectively.Suppose the energy density of PBHs tracks the density profile of Milky Way DM halo,i.e. ρ PBH ( r ) = f PBH ρ ( r ), the intensity distribution of 511 keV signature from the DMparticles around PBHs is given by I PBH ( l, b ) = 2(1 − f p ) 14 π (cid:90) l . o . s . Γ PBH ρ PBH ( r ) M PBH ds, (4.6)and the total flux of this part isΦ
PBH = (cid:90) I PBH ( l, b ) d Ω = 2(1 − f p ) 14 π Γ PBH M PBH f PBH (cid:90) d Ω (cid:90) l . o . s . ρ ( r ) ds, (4.7)where ρ ( r ) is the Einasto profile given by eq. (2.11).Since the total flux of the 511 keV photons from DM isΦ = Φ PBH + Φ χ , (4.8)where Φ χ is total flux from DM particles which are not bound to PBHs, one can get aconstraint of f PBH from Φ
PBH < Φ ∼ . × − ph cm − s − . (4.9)However, comparing eq. (4.6) with eq. (2.7), we can find that the intensity distribution of511 keV signature from DM particles around PBHs is similar to the one from decaying DM.Recalling that the decaying DM scenario is disfavored by data, which means the positronsfrom DM particles around PBHs must be subordinate, i.e. Φ PBH (cid:28) Φ χ , otherwise themorphology of 511 keV signature will be incompatible with the data. Therefore a morestringent constraint is given byΦ PBH < Φ sens ∼ × − ph cm − s − , (4.10)where Φ sens is the sensitivity of INTEGRAL/SPI [36].We calculate the constraints of f PBH fromΦ ( i )PBH ≡ − f p ) 14 π Γ ( i )PBH M PBH f PBH (cid:90) d Ω (cid:90) l . o . s . ρ ( r ) ds < Φ PBH < Φ sens (4.11)with different m χ for LDM and XDM scenario, which gives f PBH < Φ sens (cid:34) − f p ) 14 π Γ ( i )PBH M PBH (cid:90) d Ω (cid:90) l . o . s . ρ ( r ) ds (cid:35) − . (4.12)– 6 –or the XDM scenario, we calculate the constraint with m χ = 10GeV and m χ = 1TeV,which are the usually used ranges of this scenario. The constraints have M − slope in therelative small mass range, but roughly have M / slope in the relative large mass range, andthe turning point gives the most stringent constraint f PBH (cid:46) − , as shown by the greenand red dot-dashed lines in figure 2. For a given m χ , r c /r g is constant for PBHs with massin the relatively small mass range as shown in the bottom panels of figure 1, which leads toΓ ( i )PBH ∝ M and the M − slope of the constraints.For the LDM scenario, we calculate the constraints with m χ = 10MeV and m χ =100MeV. The constraints with m χ (cid:46) ρ i in eq. (3.3), and therefore leads to a smaller ρ b andsmaller Γ PBH . Since high-energy gamma rays will be produced if the positrons are injectedat even mildly relativistic energy, the positron injection energy is constrained to be (cid:46) m χ (cid:38) f PBH in the LDM scenario are loose.
Figure 2 . Constraints on the fraction of PBHs in DM f PBH from the Galactic 511 keV line withdifferent m χ for LDM and XDM scenario. We also show the constraints from the extragalactic gammaray background (EG γ [52]), galactic 511 keV line from Hawking radiation ( e + [53–55]), gravitationallensing events (HSC [56], EROS [57], OGLE [58]), dynamical effects (SEGUE [59]), and cosmicmicrowave background (CMB [60]). Noticing that these constraints are pretty conservative, which is not only because weuse Φ ( i )PBH instead of Φ PBH , but also because the morphology of 511 keV signature fromobservation data requires Φ
PBH (cid:28) Φ χ . Therefore the more stringent constraint is given byΦ PBH (cid:28) Φ χ , which can be obtained by applying a model-fitting procedure to the observationdata, and will be several orders of magnitude stronger.– 7 –onsidering the velocity dependence of the thermally averaged cross-section (cid:104) σv (cid:105) , weestimate the particle velocity distribution function of the Milky Way DM halo and the spikehalo around PBHs, by using Eddington’s formula [61, 62]. The results show the averageparticle velocity of the spike halo around PBHs is larger than that of the Milky Way DMhalo for particles with m χ (cid:38) The 511 keV line was first detected in 1970’s but its origin is still unknown. The positronsproduced from the direct annihilation of LDM or the decay of the excited state into the groundstate for XDM are the possible origins. Since there is no evidence that DM is composed ofonly one component, we consider a mixed DM model of particles and PBHs. With theexistence of PBHs, the DM particles may be gravitationally bound to the PBHs and formhalo around PBHs with density spikes. These density spikes can enhance the production rateof positrons from DM particles, thus they can be constrained by the observations of 511 keVline. In this work, we compute the profile of the density spikes, and get the constraints onthe fraction of PBHs in DM. For XDM with mass in the range of 10GeV ∼ M − slope in the relative small mass range, and roughly have M / slopein the relative large mass range. The most stringent constraint f PBH (cid:46) − appears atthe turning point which depends on the mass of XDM particle. For LDM with mass inthe range of 1MeV ∼ Acknowledgments
We thank Dr. S.J. Wang for useful discussions. RGC and XYY are supported in part bythe National Natural Science Foundation of China Grants No. 11947302, No. 11991052,No. 11690022, No. 11821505 and No. 11851302, and by the Strategic Priority ResearchProgram of CAS Grant No. XDB23030100, and by the Key Research Program of FrontierSciences of CAS. YFZ is partly supported by the National Key R&D Program of China No.2017YFA0402204 and by the National Natural Science Foundation of China (NSFC) No.11825506, No. 11821505, No. U1738209, No. 11851303 and No. 11947302.
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