Probing Boson Clouds with Supermassive Black Hole Binaries

TL;DR

This work extends the framework of boson-cloud resonances around rotating black holes to supermassive black hole binaries by incorporating astrophysical evolution histories and orbital backreaction. It shows that environmental energy-loss channels prior to the gravitational-wave-dominated stage can sustain or deplete the boson cloud, notably affecting hyperfine transitions and potentially producing a floating-orbit phase. The study analyzes both bound-state transitions and ionization to unbound states, deriving how ionization power and cloud depletion modify orbital evolution and gravitational-wave signals. It demonstrates that multi-messenger observations—electromagnetic measurements of orbital-period decay and gravitational-wave observations across a broad frequency range—offer complementary pathways to detect ionization effects and constrain boson properties, with outcomes strongly dependent on the total mass $M$, mass ratio $q$, and gravitational fine-structure constant $\alpha = GM\mu$ and the SMBHBs’ evolutionary histories.

Abstract

Rotating black holes can generate boson clouds via superradiance when the boson's Compton wavelength is comparable to the black hole's size. In binary systems, these clouds can produce distinctive observational imprints. Recent studies accounting for nonlinearities induced by orbital backreaction suggest that if the binary forms at a large separation, resonance transitions can significantly deplete the cloud, minimizing later observational consequences except for very specific orbital inclinations. In this paper, we extend this framework to supermassive black hole binaries (SMBHBs), considering the influence of their astrophysical evolutionary histories. We find that, before entering the gravitational wave (GW) radiation stage, the additional energy loss channels can accelerate orbital evolution. This acceleration makes hyperfine resonant transitions inefficient, allowing a sufficient portion of the cloud to remain for later direct observations. We further discuss the ionization effects and cloud depletion occurring at this stage. Based on these theoretical insights, we explore how multi-messenger observations for SMBHBs can be utilized to detect the ionization effects of boson clouds by examining changes in the orbital period decay rate via electromagnetic measurements and variations in GW strain over a wide frequency band. Our findings reveal a complex dependence on the binary's total mass, mass ratio, and boson mass, emphasizing the significant role of astrophysical evolution histories in detecting boson clouds within binaries.

Figures (5)

• Figure 1: Constraints for the initial state $|a\rangle=\left|211\right>$ hyperfine transition on the plane of the gravitational fine structure constant $\alpha$ and the inclination angle $\iota$ for SMBHBs with a mass ratio $q\approx 0.1$, shown for $M=10^6M_{\odot}$ (left) and $M=10^{10} M_{\odot}$ (right), and $\Delta m=-1$ (upper) and $\Delta m=-2$ (lower). The shaded gray region represents the parameter space where the transition is non-adiabatic, with the boundary values determined by Eq. (\ref{['eq: iota_crit2']}) for $\alpha_{\rm gr}\gtrsim\alpha\gtrsim \alpha_{\rm NA}$ and Eq. (\ref{['eq: iota_crit1']}) for $\alpha\gtrsim \alpha_{\rm gr}$, respectively. In the adiabatic transition regime, the colored lines represent contours of the initial state population at resonance breaking, $\left| c_{a,1} \right|^2$ in Eq. (\ref{['eq:Gammabreaking1']}). When $\alpha\gtrsim \alpha_{\rm GW}$, the cloud depletion due to its GW emission in Eq. \ref{['eq:Mc_GW_decay']} becomes significant. The vertical gray lines mark various critical values of $\alpha$, as listed in Tab. \ref{['tab:alpha']}.
• Figure 2: The form factors $\mathcal{F}_{\rm ion}$ in Eq. (\ref{['eq:Fion']}) and $\mathcal{F}_{\rm DF}$ in Eq. (\ref{['eq:FormDF']}) as functions of $x$ for a counter-rotating orbit with $\iota=\pi$. The inset provides a comparison on a linear scale.
• Figure 3: Numerical results of ionization-to-GW power ratio $\mathcal{R}$ in Eq. (\ref{['eq:Rx']}) as a function of $x$, taking into account cloud depletion, for the BH mass $M=10^6M_{\odot}$ (upper left), $M=10^{8}M_{\odot}$ (upper right), and $M=10^{10}M_{\odot}$ (lower), respectively. Different colored lines represent various benchmark values of $\alpha$, chosen according to Tab. \ref{['tab:alpha']}. Solid lines correspond to the mass ratio $q=0.1$, while dashed lines correspond to $q=0.01$.
• Figure 4: Orbital period decay rate $|\dot{T}|$ as a function of $\bar{r}$ (or $T$) and $\alpha$, accounting for cloud depletion in Eq. (\ref{['eq:Tdot']}) whenever $\mathcal{R}\gg1$, for the BH mass $M=10^6M_\odot$ (upper left), $M=10^8 M_{\odot}$ (upper right) and $M=10^{10} M_{\odot}$ (below). In all panels, different colored lines represent contours of different $|\dot{T}|$ values. Solid lines correspond to the mass ratio $q=0.1$ and dashed lines to $q=0.01$. The gray curves indicate values of $x= \alpha^2\bar{r}$. The horizontal gray dashed lines denote $\alpha_{\rm inst}, \alpha_{\rm NA}, \alpha_{\rm GW}$ and $\alpha_{\rm gr}$, from bottom to top for different masses, as shown in Tab. \ref{['tab:alpha']}.
• Figure 5: GW cycle number $\mathcal{N}(f)$ (left) and characteristic strain $h_c$ (right) for a single SMBHB during the inspiral stage, accounting for ionization effects. Top, middle, and bottom panels correspond to BH masses $M=10^6M_{\odot}$, $M=10^{8}M_{\odot}$ and $M=10^{10}M_{\odot}$, respectively, with the total observation time $T_{\rm obs}=10$ yr. The inspiral stage ends at the inner-most stable circular orbit, i.e. $\bar{r}=6$ for a Schwarzschild BH. Different colored lines represent various benchmark values of $\alpha$. Solid and dashed lines denote binaries with $q=0.1$ and $q=0.01$, respectively. Black dotted lines (left) show the product $f T_{\rm obs}$. Gray lines (right) indicate sensitivity curves for various experiments, including LISA L3 design sensitivity LISA:2017pwj, projections for $\mu$Hz GW detection with asteroid test-mass Fedderke:2021kuy and with fast radio burst (FRB) timing assuming 1000 FRB events detected per year Lu:2024yuo, and the NANOGrav (11 yr) results NANOGrav:2017wvv. All sources are shown at a redshift of $z=0.1$, or equivalently, a luminosity distance of $d_L\approx 10^{22}\,\mathrm{km}$.