Constraining primordial black holes as dark matter using AMS-02 data

Primordial black holes (PBHs) are the plausible candidates for the cosmological dark matter. Theoretically, PBHs with masses $M_{\rm PBH}$ in the range of $4\times10^{14}\sim 10^{17}\,{\rm g}$ can emit sub-GeV electrons and positrons through Hawking radiation. Some of these particles could undergo diffusive reacceleration during propagation in the Milky Way, potentially reaching energies up to the GeV level observed by AMS-02. In this work, we utilize AMS-02 data to constrain the PBH abundance $f_{\rm PBH}$ by employing the reacceleration mechanism. Under the assumption of a monochromatic PBH mass distribution, our findings reveal that the limit is stricter than that derived from Voyager 1 data. This difference is particularly pronounced when $M_{\rm PBH}\lesssim10^{15}\,{\rm g}$, exceeding an order of magnitude. The constraints are even more robust in a more realistic scenario involving a log-normal mass distribution of PBHs. Moreover, we explore the impact of varying propagation parameters and solar modulation potential within reasonable ranges, and find that such variations have minimal effects on the final results.

Our investigation specifically focuses on the electrons and positrons emitted from PBHs, aiming to impose constraints on PBH abundance f PBH at the mass range 4 × 10 14 g ≲ M PBH ≲ 10 17 g.In contrast to photons and neutrinos, the energy of these cosmic ray (CR) electrons/positrons is subject to environmental influences.Previous studies have primarily concentrated on energy loss resulting from interactions with the interstellar medium and radiation during propagation.However, the impact of random shocks in interstellar space, capable of reaccelerating low-energy CR particles, has gained attention.This reacceleration mechanism has been extensively discussed and has demonstrated its ability to self-consistently fit both the proton and Boronto-Carbon ratio [48][49][50][51].Furthermore, this investigation presents a novel opportunity for exploring electrons and positrons emitted from PBHs with energies even lower than the detectability threshold of the AMS-02 experiment, i.e., the lower energy CRs undergo a process of reacceleration during propagation, expanding the observable energy range to a higher value.
In our study, we employ a meticulous calculation method incorporating the positron fraction data [52] and the combined electron and positron data [53] from AMS-02 to impose limits on PBH abundance.This allows us to constrain f PBH with increased precision compared to using the Voyager 1 data of all-electrons [54,55], as the AMS-02 data offers higher accuracy.Our approach involves leveraging more accurate numerical calculation tools and incorporating updated propagation parameters [56].This refined methodology aims to provide a more comprehensive and precise assessment of the f PBH constraints based on the latest observational data.We employ the LikeDM code [57] to calculate the flux of electrons and positrons produced by PBHs, and get new limits on PBH abundance using the AMS-02 data.
This paper is organized as follows.In Sec. 2, we briefly review the production of positrons and electrons through Hawking radiation.In Sec. 3, the propagation process of positrons and electrons is described.Sec. 4 presents the results, followed by a conclusion in Sec. 5.

Electrons and positrons from evaporating PBHs
Black holes, including PBHs, can emit particles through a phenomenon known as Hawking radiation.The emitted radiation is characterized as thermal radiation, exhibiting a temperature that is inversely proportional to the mass of the black hole.The non-rotating PBH temperature T H is described as [21,22] where M PBH is the PBH mass.The corresponding primary emission spectrum of emitted electrons and positrons can be expressed as [58] where Γ e is the electron absorption probability, approximately modeled as at high energies for GM PBH E/(ℏc 3 ) ≫ 1 [58].In this work, we only consider the non-rotating PBHs with primary electron/positron emission.For one thing, the spin of PBHs introduces a slight modification to the injected energy spectrum [31], and this alteration tends to smooth out as the propagation.For another, the contribution of secondary electrons and positrons resulting from the decay of unstable particles is found to be negligible, constituting only 1% of the total flux.This calculation has been performed using the publicly available code BlackHawk [59,60].
In general, the Hawking radiation of PBHs provides an electron/positron source, described as where g(M PBH ) represents the mass distribution of PBHs normalized to ρ ⊙ , and ρ PBH (r) denotes the total PBH mass density as a function of the distance from the Milky Way center r.It is assumed that ρ PBH (r) traces the DM density ρ DM (r), and can be expressed as ρ PBH (r) = f PBH ρ DM (r), with f PBH being the PBH abundance.Under the hypothesis of a common mass for all PBHs (i.e., a monochromatic mass distribution), Eq. ( 1) is reduced to In this work, we also concentrate on log-normal distribution, defined as [61] g where µ is the mass for which the density is maximal, and σ is the width.

The propagation of evaporated electrons and positrons
The Hawking radiation from PBHs serves as an additional source of electrons and positrons, and these particles undergo diffusive propagation within the Milky Way as part of CRs.We first calculate the primary emission spectra by using the BlackHawk [59,60].Subsequently, we integrate this information as the CR source into the propagation calculation code.Numerical tools, such as GALPROP [62] and DRAGON [63], have been developed for CR propagation calculations.In this work, we employ the public code LikeDM [57] to calculate the propagation process.For a specified source distribution, the LikeDM code employs the Green's function method, relying on numerical tables obtained from GALPROP, to calculate the propagation process.This approach has been verified to provide a good approximation to the GALPROP output while being significantly more efficient.The propagation is assumed to occur within a diffusion reacceleration framework, with the determination of propagation parameters relying on the boron-to-carbon ratio data and the diffuse γ-ray emission observed by Fermi-LAT [64].It is worth noting that the chosen parameters have been updated compared to Ref. [30].The main propagation parameters are shown in Tab. 1, including the diffusion coefficient D xx , the characteristic halo height z h and the Alfvenic speed v A describing the reacceleration effect.Additionally, we apply the simple force-field approximation [65] with a broad range of modulation potential to describe the solar modulation.The PBH density is modeled using the Navarro-Frenk-White (NFW) profile [66] ρ NFW = ρ s r s where r s = 20 kpc and ρ s = 0.26 GeV cm −3 represent the scale radius and scale density [67], respectively.Assuming all DM in the Milky Way consists of PBHs with a monochromatic mass distribution, we present the electron and positron spectra after propagation in Fig. 1.The fluxes Φ e + +e − are obtained with Prop.6 in Tab. 1 and a solar modulation potential of 0.6 GeV.Due to the reacceleration, the energies of the ejected electron/positron at sub-GeV scale are boosted to the GeV range, aligning with the range covered by the AMS-02 data [52,53].Therefore, the AMS-02 data can be employed to constrain the fraction of PBHs.As shown in Fig. 1, the flux Φ e + +e − decreases rapidly with the PBH mass M PBH .
The astrophysical CR background encompasses conventional primary electrons, such as those originating from supernova remnants, as well as secondary electrons and positrons generated through inelastic collisions between CR nuclei and the interstellar medium.As we seek spectral features that stand out from the "smooth" background, it is reasonable to assume that the majority of the observational data can be well-fitted by the background.Following Ref. [57], we utilize the empirical model, which includes the primary electrons, secondary electrons/positrons, and the electron/positron excess from the extra source: , Therefore, the total background energy spectrum of electrons plus positrons Φ bkg, e ± is Φ bkg, e ± = φ e − + 1.6φ e + + 2φ s .
where the factor 1.6 accounts for the asymmetry of electrons and positrons generated in pp collisions [69].The best-fit parameters can be found in Tab.III of Ref. [57].In general, we directly fit the CR data with the model described above, focusing solely on the numerical shape without calculating the propagation of the background.In contrast, for the PBH source, we carefully calculate the propagation.When incorporating the contribution of the PBH source into the model, we optimize the fitting results by introducing the adjustment factors α i E β i , with i = e − , e + , and s correspond to φ e − , φ e + , and φ s , respectively.This adjustment enables us to derive more conservative constraints.

Results
In this work, we constrain the PBH abundance f PBH through maximum likelihood fitting by utilizing data from the AMS-02 positron fraction [52] and total electron plus positron Fig. 2 Constraints on PBHs abundance f PBH as a function of the PBH mass M PBH , obtained in this study using AMS-02 data (solid black line) and Voyager 1 data (solid red line).The results from Ref. [30] using Voyager 1 data are also incorporated (light blue and pink dashed lines with background, and dot-dashed lines without background).We adopt Prop.6 in Tab. 1, and set the solar modulation potential to 0.6 GV and 0 GV for the case with AMS-02 and Voyager 1 respectively.Fig. 2(a) assumes a monochromatic mass distribution, while Fig. 2(b) considers a log-normal mass distribution.
To begin with, we adopt Prop.6 from Tab. 1 with a halo height z h = 15 kpc and set the solar modulation potential to 0.6 GV.The constraint on f PBH for monochromatic mass distribution of PBHs is depicted as the solid black line in Fig. 2(a).For comparison, we also derive the limit from Voyager 1 data [54,55] with Prop.6 (solid red line).Given that the Voyager 1 spacecraft has already crossed the heliopause threshold, the solar modulation potential is naturally set to 0 GV.Additionally, constraints from Ref. [30] based on the Voyager 1 data (light blue and pink lines) are incorporated.To maintain consistency with Ref. [30], we fit the Voyager 1 data with a power-law form energy spectrum as Φ Vbkg, e ± = 593.97E−1.31 (with χ 2 dof = 7.73/9), omitting the adjustment factors.In contrast to the methodology employed in Ref. [30], our method refined by improving numerical calculation tools and incorporating the latest propagation parameters.Fig. 2(a) clearly illustrates the stronger constraint derived from the AMS-02 data compared to Voyager 1, particularly by more than an order of magnitude when M PBH ≲ 10 15 g.This superiority is largely attributed to the remarkably small errors in the AMS-02 measurements.Fig. 2(b) shows the constraints on f PBH for log-normal mass distribution PBHs portrayed by Eq. (2).We set the central value of the lognormal distribution µ ≲ 10 17 g, and cut at 4 × 10 14 g as PBHs with lower masses have already evaporated.Various values of the width σ are taken into account, and the constraints notably strengthen with increasing σ .For σ = 0, the constraint is consistent with that of monochromatic mass distribution (solid black lines in Figs.2(a) and 2(b)).
For a more comprehensive analysis, with the assumption of monochromatic mass distribution, we investigate the constraints with different sets of propagation parameters and solar modulation potentials.The Props. 1 to 6 outlined in Tab. 1 are used to limit the PBH abundance, as illustrated in Fig. 3(a).Here, the solar modulation potential is fixed at 0.6 GV.It can be seen that these different propagation parameters behave similarly when constraining the PBH abundance.Additionally, in accordance with Prop.6, various cases of solar modulation are also considered in Fig. 3(b), with modulation potentials set at 0.4 GV, 0.5 GV, and 0.6 GV, respectively.The limit on f PBH gradually weakens with modulation potential as more electrons and positrons are shielded by the solar magnetic field.Overall, different choices of propagation parameters or solar modulation within a reasonable range do not significantly impact the results as depicted in Figs. 3.

Conclusion
PBHs with M PBH ≲ 10 16 g are anticipated to inject sub-GeV electrons and positrons into the environment through Hawking radiation.Hypothesizing a diffusion plus reacceleration model of propagation, we have computed the fluxes for different PBH masses.The results reveal that part of sub-GeV electrons and positrons can be accelerated to the GeV level, falling within the observational range of AMS-02.In this work, we utilized the flux of these GeV electrons and positrons to constrain the PBH abundance f PBH in the Milky Way using AMS-02 data.Employing Prop.6 in Tab. 1 and  a solar modulation potential for 0.6 GV, we have explored two cases involving monochromatic and log-normal mass distributions of PBHs.Our results limit f PBH ≪ 0.1 for M PBH ≲ 10 16 g, thus largely ruling out the possibility of PBHs within this mass range as a significant contributor to DM.We have also presented the constraints on f PBH with the remaining five sets of propagation parameters in Tab. 1, demonstrating that different parameter choices do not significantly affect the results.Furthermore, while lower solar modulation potentials strengthen the limit on f PBH , their impact is marginal within reasonable parameter space.

Fig. 1
Fig.1The fluxes Φ e + +e − originating from the evaporation of PBHs with M PBH = 10 15 g and 10 16 g, considering Prop.6 in Tab. 1 alongside a solar modulation potential of 0.6 GeV.The AMS-02 measurements are also presented for comparison[68].

Fig. 3
Fig. 3 Constraints on PBHs abundance f PBH as a function of the PBH mass M PBH , with the monochromatic mass distribution assumption.Fig. 3(a) adopts six groups of propagation parameters given in Tab. 1, with a solar modulation potential of 0.6 GV.Fig. 3(b) utilizes Prop.6 with three different solar modulation potentials.