Effects of dark dipole radiation on eccentric supermassive black hole binary inspirals

TL;DR

This work tackles the final-parsec problem for supermassive black hole binaries by exploring a dark-sector scenario in which eccentric binaries source light scalar or vector fields, generating dipole radiation that can hasten mergers. The authors derive general flux formulas for massive scalar and vector dipole radiation from localized periodic sources in flat spacetime and apply them to eccentric Keplerian binaries, embedding these fluxes into the adiabatic evolution of the orbit. A simplified stochastic gravitational-wave background model is then constructed and confronted with current pulsar timing array data via Bayesian analysis, revealing a mild preference for nonzero dipole strength and limited sensitivity to the boson mass $m$ in the range explored ($m\lesssim 10^{-25}$ eV). The results show that while dark-dipole radiation can augment the inspiral for certain parameters, it does not universally resolve the final-parsec problem; nonetheless, its imprint on the SGWB offers a novel probe for dark-sector physics with upcoming PTA data at lower frequencies.

Abstract

The final-parsec problem has long posed a central challenge in understanding the merger of supermassive black hole binaries. In this paper, we investigate a scenario in which a dark scalar or vector field is sourced by eccentric binaries, leading to accelerated mergers through additional dipole radiation, and thereby extending the range of masses for which the binary merges within a Hubble time. The radiation fluxes from an eccentric charged Keplerian binary are derived using general results for localized periodic sources in flat spacetime. We find that dipole radiation, although insufficient to fully resolve the final-parsec problem, can alter the low-frequency spectrum of the stochastic gravitational-wave background from supermassive black hole binary inspirals. We construct a simplified model for the spectrum and perform a Bayesian analysis using the current pulsar timing array data.

Figures (6)

• Figure 1: Left: Eccentricity evolution with semimajor axis, starting from $e = 0.9$ and $a = 1\,\mathrm{pc}$. Right: Comparison between $P_\text{gw}$ (dashed lines) and $P_\text{dip}$ (solid lines) during the inspiral in the scalar case, with $\alpha=0$. The energy flux is expressed in units of solar luminosity $L_\odot$. For both panels, we choose $M = 10^{10}\,M_{\odot}$ and $q = 0.1$.
• Figure 2: Left: SMBHB merger time as a function of $e_0$ and $M$ for $m_\phi=10^{-27}\,\mathrm{eV}$. The gray plane marks the Hubble time, and regions with merger times exceeding $20\,\mathrm{Gyr}$ are truncated. Right: $e_0(M)$ curves corresponding to $t_\text{merger}=t_\text{Hubble}$. The parameter space of successful mergers expands as $\gamma^2$ increases. For both panels, we choose $q=0.1$.
• Figure 3: Characteristic strain spectrum for a single source population of SMBHBs with $M = 10^{10}\,M_{\odot}$, $q = 0.1$ and $z = 1$. Panels (a)–(d) illustrate the effects of varying initial eccentricity $e_0$, dipole strength $\gamma$, boson mass $m$ and coupling strength $\alpha$. The total merger rate $N$ is chosen such that $h_c=9.685\times 10^{-15} (f/\mathrm{yr}^{-1})^{-2/3}$ for $e_0=0$ in the vacuum case (black solid line).
• Figure 4: Left: Characteristic strain spectrum for a representative population of SMBHBs using posterior-median parameters, overlaid with PTA data. The black line corresponds to the vacuum case with $e_0=0$ and $h_c(f) \approx 1.793\times10^{-15}\,\bigl(f/\mathrm{yr}^{-1}\bigr)^{-2/3}$. Right: Posterior distributions of the model parameters for $m=10^{-27}\,\mathrm{eV}$. Results for the scalar field are shown in blue, with posterior medians and $1\sigma$ intervals: $\gamma^2 = 0.46^{+0.34}_{-0.29}$, $e_0 = 0.50^{+0.31}_{-0.31}$, $\psi_0 = -2.39^{+0.12}_{-0.14}$. Results for the vector field are shown in orange, with posterior median and $1\sigma$ intervals: $\gamma^2 = 0.37^{+0.37}_{-0.25}$, $e_0 = 0.52^{+0.30}_{-0.32}$, $\psi_0 = -2.29^{+0.15}_{-0.18}$.
• Figure 5: Posterior distributions in the scalar case (left) and the vector case (right). Results for $m=10^{-27}\,\text{eV}$ ($10^{-25}\,\text{eV}$) are shown in blue (orange).