Black hole scalar sirens in the Milky Way

Abstract

Hypothetical light scalar particles trigger the superradiant instability around spinning black holes (BHs), causing clouds of scalars to grow around the BH. In the presence of sufficiently strong particle self-interactions (characterized by the decay constant $f$), scalars are ejected from BH orbits, resulting in coherent, non-relativistic emissions that continuously carry away the BH's angular momentum. Parameters exist for which cloud growth is much faster, and scalar depletion is much slower, than the age of the Galaxy. This defines a distinct class of astrophysical sources of scalars, which we call \emph{BH scalar sirens} -- BHs that persistently emit scalars effectively forever. We compute the scalar background from the expected population of $N_\text{BH}\sim 10^{8}$ isolated stellar-mass BHs in the Milky Way, which are sirens for scalars in the mass range $10^{-13}$--$10^{-11}\,$eV and $f\lesssim 10^{14}$--$10^{9}\,$GeV. This provides a detection target independent of early-universe scalar production or cosmological initial conditions. The generated observable signals are up to two orders-of-magnitude larger than those expected from a misaligned cosmic scalar in this mass range. The energy spectrum of emitted scalars is distinctly broader and at higher velocities (up to $\sim 10^{-1}c$) than that of virialized dark matter, and encodes the mass and spin distributions of the BH population. While stellar-mass Milky Way BHs are our primary target, our framework extends to supermassive, intermediate-mass and light BHs. Given the difficulty of directly observing populations of isolated BHs, scalar emissions offer a novel probe of these otherwise invisible objects, highlighting the potential for joint discovery between scalars and BHs, and broadly motivating searches for scalars over many orders-of-magnitude in mass.

Figures (13)

• Figure 1: Amplitude at the Sun-Earth location of the two local scalar observables most relevant to direct experimental detection: the dimensionless angle $\theta \equiv \Phi/f$ (top), and the scalar wind $f^{-1}\hat{\bm{J}}_\parallel\cdot \bm \nabla \Phi$ (anti)parallel to the line-of-sight to the Galactic Center (bottom), for a population ($N_\text{BH} = 10^8$) of stellar-mass BH scalar sirens in the Milky Way, assumed to be exponentially distributed with characteristic spread $R_s=3\, \mathrm{kpc}$ from the Galactic Center. Due to the increased velocity of the scalars, the wind from the MW stellar-mass BH siren population is up to two orders of magnitude larger than that of a scalar relic produced through a pre-inflationary misalignment mechanism, \ref{['eq:misalignment']}, with initial misalignment angle $\theta_i \approx 1$. We assume a characteristic BH mass scale $M_s=9.5\,\mathrm{M}_\odot$ and a unitless BH spin parameters $a_\star$ distributed according to a power law with "pitch" $\beta$ (Eq. \ref{['eq:spindistro']}). The solid red line assumes $\beta=0.65$, the best fit to existing spin measurements of stellar-mass BHs Draghis:2023vzj; the translucent red region spans from $\beta=0$ to $\beta=0.9$.
• Figure 2: Schematic depiction of the "autoionization" of the gravitational atom and the flow of particles (blue arrows) and of the component of angular momentum parallel to the BH spin axis (orange arrows) in a steady-state BH scalar siren. The dashed box corresponds to the SR cloud at quasi-equilibrium. One particle (light blue) can be thought of as "climbing out" of the BH-cloud system in a three-step process assisted by "auxiliary" cloud particles (dark blue), carrying $3\hbar$ units of angular momentum with them. Once the process reaches equilibrium, the cloud is (very nearly) unchanged at the end and the process repeats. While we heuristically represent the emission of a "classical" point particle here, the emission is really a continuous wave process. The "phase information" (which the heuristic picture above fails to capture) carried by every particle is such that the emission takes place coherently over the volume of the cloud.
• Figure 3: Schematic depiction of the radiated scalar field (light blue) in the non-rotating rest frame of a spinning BH scalar siren (center), with some generic offset from the Galactic Center (GC; red cross). The angular momentum vector of the BH points out of the page. Effects of external gravitational potentials (such as that of the galaxy) on the propagating wave are neglected. As per \ref{['eq:radiation']}, when viewed from along the spin axis, points of equal phase are outwardly traveling spirals: $\text{Re}[e^{-i(\omega_\phi^\text{BH} t-\mu v_\phi^\text{BH} r-3\varphi)}] = \cos(\omega_\phi^\text{BH} t-v_\phi^\text{BH} r-3\varphi)$. In the distant radiation zone, the wavefronts appear as outgoing shells, \ref{['eq:radiation']}. An observer traveling in this field measures the scalar values, \ref{['eq:observed_field']}, and the local gradient, \ref{['eq:gradient']}.
• Figure 4: Left: Spacetime diagram for the observation of non-relativistic scalar radiation from the worldlines of BH scalar sirens (strong self-interactions, small $f$) in the Galactic Center and disk. Right: Corresponding diagram for massless gravitational waves emitted by superradiant clouds in the regime of negligible self-interactions (large $f$). The distance to the Galactic Center (GC) is approximately $8\,\mathrm{kpc}$ or $0.03\, \mathrm{Myr}$. Dotted worldline segments denote inert periods; emission begins after a delay set by the superradiance timescale. In the weak-interaction regime, individual BHs may emit GWs only for a finite time that can be short compared to the age of the MW, so only sources emitting at the retarded time $t=t_\text{today}-r_\text{BH}^\text{o}/c$ contribute to the signal. In contrast, in the scalar-siren regime the emission of massive scalars is effectively eternal after cloud growth. Because scalar propagation is timelike, the relevant lookback time \ref{['eq:lookback time']} satisfies $t_\text{lb}(\alpha) = r_\text{o}^\text{G}/v_\phi> r_\text{o}^\text{G}/c$ is parametrically controlled by $\alpha$, yet remains short compared to the Galactic age. Consequently, the scalar signal can be estimated directly from the total number of BH sirens today, with greatly reduced sensitivity to the detailed formation history of Galactic black holes. The two regimes are mutually exclusive in the sense that angular-momentum extraction is dominated either by GW emission (negligible self-interactions) or by scalar radiation (strong self-interactions). Finally, unlike massless gravitational waves, massive scalar radiation may remain gravitationally bound to the MW and accumulate over time, although this cannot be achieved for stellar-mass sirens. (\ref{['sec:low_alpha_bound']}).
• Figure 5: In the Galactic frame, with one of its arms represented as a blue line, we compute the sum total scalar emission of the set of $\sim 10^8$ BHs near the Galactic Center. The power spectral density, \ref{['eq:power_spectral_density']}, is obtained from the incoherent sum of coherent emitters. It can be recast in the form of a kinetic theory calculation of the radiative power emanating from a "gas" of BHs at the Galactic Center, with mass and (continuous, classical) spin internal degrees of freedoms. Because both the observer and the BHs are bound to the Galaxy, their velocity is small relative to that of the emitted scalars of suitably large mass $\mu$. Position vectors are in red, velocity vectors are in blue-green.