Hawking Radiation of Nonrelativistic Scalars: Applications to Pion and Axion Production

TL;DR

This paper addresses the need for accurate modeling of Hawking radiation from asteroid-mass Primordial Black Holes when massive scalar particles such as pions and axion-like particles are produced non-relativistically. By re-deriving and numerically solving the massive-scalar greybody factors in Schwarzschild geometry, the authors provide a non-relativistic emission rate that differs significantly from massless or simplified treatments, and they incorporate these results into a full gamma-ray spectrum that includes primary photons, decay photons from pions/ALPs, and FSR. They demonstrate, via applications to pion and ALP scenarios, that previous approximations can bias PBH parameter inference with upcoming MeV gamma-ray data (e.g., AMEGO-X), and they show that the non-relativistic corrections are potentially detectable in GC observations. The work also discusses potential suppression for composite hadrons and provides open-source code to compute the spectra (HoRNS), highlighting the importance of precise Hawking radiation predictions for multi-messenger PBH studies and future searches for PBHs via gamma rays and gravitational waves.

Abstract

In studying secondary gamma-ray emissions from Primordial Black Holes (PBHs), the production of scalar particles like pions and axion-like particles (ALPs) via Hawking radiation is crucial. While previous analyses assumed relativistic production, asteroid-mass PBHs, relevant to upcoming experiments like AMEGO-X, likely produce pions and ALPs non-relativistically when their masses exceed 10 MeV. To account for mass dependence in Hawking radiation, we revisit the greybody factors for massive scalars from Schwarzschild black holes, revealing significant mass corrections to particle production rates compared to the projected AMEGO-X sensitivity. We highlight the importance of considering non-relativistic $π^0$ production in interpreting PBH gamma-ray signals, essential for determining PBH properties. Additionally, we comment on the potential suppression of pion production due to form factor effects when producing extended objects via Hawking radiation. We also provide an example code for calculating the Hawking radiation spectrum of massive scalar particles.

Figures (8)

• Figure 1: The effective potential $V_{\rm eff}$ normalized by $r_s^2$ for different particle masses and angular momenta in Schwarzschild spacetime. As $r_s\to \infty$, the potential approaches $r_s^2 m^2$. The solid lines of $r_s m=0.34$ may represent the potential for $\pi^0$ with $m=135$ MeV around the Schwarzschild black hole of $10^{14.5}$ g. Note that the larger the angular momentum, the higher the peak of the potential, and hence the lower the transition rate (with the same energy). Therefore, s-wave dominates the radiation in the non-relativistic limit.
• Figure 2: Upper panel: The transition rate $|T_{\omega l}|^2$ of massless and massive scalar particles with different angular momenta. Lower panel: The greybody factor $\Gamma_{\omega}=\sum_{l}(2l+1)|T_{\omega l}|^2$ of massless and massive scalars. Note that there is a hard cutoff at $\omega=m$ ($r_s\omega=r_s m=0.34$) for the massive particle. As $\omega$ increases, the modes of higher $l$ are excited, hence the bumps in the lower panel. Note that $r_s\omega=\omega/4\pi T_H$ for Schwarzschild black holes.
• Figure 3: The normalized cross section $\sigma/27\pi G^2M^2$ of massless (dashed lines) and massive scalars (solid lines). In the low-energy limit, the cross section of the massive scalar blows up, and the massless approaches the black hole area $16\pi G^2M^2$. In the high-energy limit, the massive and the massless cross sections both approach the geometrical optics limit $27\pi G^2M^2$.
• Figure 4: The particle production rate (log-plot) measured at infinity for massless and massive scalars. The production rates of different modes of different $l$ are compared. The dark blue line shows the total production rate when summing over all contributions of angular momentum, and there is a cut-off at $\omega=m$ ($r_s\omega=r_s m=0.34$) for the massive particle. The production rate of massless particles peaks at $r_s \omega\approx 0.2$, which in the simple case of Schwarzschild black holes means $\omega/T_H=4\pi\times0.2\approx 2.5$.
• Figure 5: The ratio of the production rates of $\pi^0$ calculated by different methods with $M=10^{14.5}g$ and $r_s m=0.34$. The blue curve is the ratio $\left(\frac{dN}{dtd\omega}\right)_{\rm massless}/\left(\frac{dN}{dtd\omega}\right)_{\rm NR\space calculation}=\sigma_{\rm massless}/v^2\sigma_{\rm massive}$, namely the deviation of approximating the massive greybody factor by the massless. The green curve shows $\left(\frac{dN}{dtd\omega}\right)_{\rm Massless\times v^2}/\left(\frac{dN}{dtd\omega}\right)_{\rm NR \space calculation}=\sigma_{\rm massless}/\sigma_{\rm massive}$, which indicates the validity of the approximation Eq. \ref{['eq:vsqmassless']}. In the high-energy limit, all curves converge to 1 as indicated in Fig. \ref{['Fig.sigma']} that $\sigma_{\rm massless} \to \sigma_{\rm massive}$ in the high-energy limit and meanwhile $v^2=1-m^2/\omega^2\to 1$. However, they differ at the non-relativistic limit, where the massless calculation gives roughly an extra $20\%$ of the photon numbers under the current parameters. The additional $v^2$ factor in the Massless$\times v^2$ method suppresses the production rate in the non-relativistic limit.