The background gas humming and multi-messenger transients of stalled supermassive black hole binaries

Abstract

We establish the multi-messenger mechanics of episodic mass transfer in supermassive black hole binaries stalled within circumbinary discs. Utilizing continuous wavelet transforms, we isolate localized gas clumps at the cavity edge and track their evolution. By regularizing the forced fluid equations at Lindblad resonances via the inhomogeneous Airy differential equation, we bypass linear singularities to extract the finite wave amplitudes that trigger non-linear shock formation. These shocks produce bounded accretion bursts. We model the time-domain thermal luminosity, deriving an analytical power spectral density that forms a harmonic cascade. The superposition of the accretion streams generates a spectral beat frequency, providing an exact mathematical extraction of the binary mass ratio. The radiative cooling of shock-accelerated electrons produces a multi-wavelength spectral energy distribution from a synchrotron radio continuum to an inverse-Compton gamma-ray tail. We identify a relativistic signature: a discontinuous, high-frequency gravitational wave sideband termed the ``background gas humming''. This emission arises from the highly asymmetric, transient fluid geometry of the accretion shocks. Evaluating the asymptotic properties of the Airy regularization, we show that this humming manifests as a sequence of discrete high-frequency bursts with temporal quiescence gaps that systematically compress as the cavity shrinks. We show that the instantaneous mass of the gas actively trapped within the cavity violently amplifies prior to decoupling, culminating in a terminal burst near 4.0 mHz that serves as a multi-messenger precursor to the final vacuum inspiral.

Figures (7)

• Figure 1: The 2D surface density profile of the circumbinary disc normalized to the unperturbed far-field value $\Sigma_{\text{ref}}$. The spatial coordinates are scaled to the binary separation $a$. We use a logarithmic color map to resolve the density depletion within the central cavity alongside the dense outer rim. The continuous gravitational torque truncates the disc at $r_{\text{cav}} = 2a$ (dashed line), forcing the density to plummet to a numerical floor of $10^{-4}$ inside $r=a$. The $m=1$ fluid instability creates a pronounced mass pile-up along the cavity rim at $\theta = 0$, reaching a peak normalized density of roughly $0.94$, whereas the antipodal side only attains a maximum density of $0.10$. This localized, coherent overdensity serves as the primary mass reservoir stripped by the supermassive black holes (white circles) during their orbital approach.
• Figure 2: Transient mass accretion streams dictated by the orbital beat frequency. The primary black hole (solid blue) and secondary black hole (dashed orange) alternate in sweeping past the slower-moving $m=1$ mass reservoir. Because the relative beat period evaluates to $1.55 T_{\text{orb}}$, the combined total accretion rate (grey solid line) yields a deterministic, perfectly periodic sequence of mass injections arriving exactly every $0.77 T_{\text{orb}}$. The transient mass transfer events dominate the continuous flow, pulsing from a residual cavity baseline of $0.5$ arbitrary units up to a maximum transient peak of precisely $25.5$. The bounded burst width ($t_{\text{visc}} = 0.08 T_{\text{orb}}$) governs the rapid viscous draining of the minidiscs, establishing the deep quiescence gaps that physically separate the discrete flares.
• Figure 3: Airy regularization of the transient density wave excited at the minidisc Lindblad resonance. The horizontal axis represents the scaled radial distance $\xi$ from the resonance turning point at $\xi=0$. The filled grey curve denotes the localized forcing function $S(\xi)$ generated by the impacting beat-frequency accretion stream. The standard WKB amplitude envelope (dashed red line) unphysically diverges to infinity at the turning point due to the vanishing radial wavenumber. The rigorous solution to the inhomogeneous Airy equation (solid blue line) resolves this singularity. The forcing naturally excites a bounded, finite wave that decays exponentially in the evanescent zone ($\xi > 0$) and propagates leftward as an oscillatory wave train ($\xi < 0$). This finite initial amplitude serves as a precursor to non-linear shock steepening.
• Figure 4: Analytical electromagnetic signatures of the episodic accretion shocks. Left: The Power Spectral Density (PSD) of the transient light curve. Driven by the beat frequency ($\nu_{\text{beat}} \approx 0.646 \nu_{\text{orb}}$), the flares Fourier transform into a broad harmonic cascade. The equal-mass symmetry ($q=1.0$, solid blue stems) perfectly suppresses the odd harmonics, yielding a fundamental observable frequency of $1.29 \nu_{\text{orb}}$. An asymmetric mass ratio ($q=0.5$, red circles) breaks this symmetry, allowing the odd harmonics to emerge. The high frequencies are suppressed by a Gaussian envelope corresponding to the $t_{\text{visc}}$ duration of the shocks. Right: The theoretical multi-wavelength spectral energy distribution ($\nu F_\nu$) during a single burst. Strong-shock Fermi acceleration yields an extended synchrotron radio continuum ($p=2$). The thermal minidisc provides an ultraviolet peak, which serves as the seed population for inverse-Compton scattering, resulting in a dominant, high-energy X-ray/gamma-ray tail.
• Figure 5: Analytical timescale crossing defining the decoupling moment. The horizontal axis represents the evolutionary time in millions of years, while the vertical axis denotes the normalized structural timescales. The viscous timescale of the gas $t_{\text{visc}}$ (blue line) decays gradually as $a^{3/2}$. Conversely, the relativistic gravitational wave decay timescale $t_{\text{gw}}$ (red line) collapses precipitously as $a^4$. Their exact mathematical intersection defines the decoupling threshold at precisely $t_{\text{dec}} = 8.0$ Myr, where the binary outpaces the external circumbinary disc.