Profiling stellar environments of gravitational wave sources

TL;DR

This work demonstrates that detailed information about CBC host environments can be inferred from gravitational waves alone by exploiting phase modulations caused by time derivatives of the center-of-mass line-of-sight velocity. The authors derive phase corrections up to high negative PN orders and connect the derivatives to environment and outer-orbit parameters across SMBH, Bahcall-Wolf NSC, and Plummer GC potentials. Using a Fisher-matrix formalism and emcee sampling, they forecast constraints on parameters such as the SMBH mass M_SMBH, outer-orbit radius R, Plummer/plasma-like scale a_p, and BW exponent α for several detectors (A+, ET, LISA, DECIGO) over multi-year observing runs. They find that, on a single-event basis, it is possible to measure enclosed masses and potential profile slopes with substantial precision out to large radii, enabling insights into CBC formation and evolution in diverse environments, while also quantifying potential biases from model misspecification and showing how Bayesian model selection can mitigate them. Overall, the method provides a pathway to connect CBC proximity to galactic centers with their formation channels, offering a novel, EM-counterpart-independent probe of stellar dynamics in extreme environments.

Abstract

Gravitational waves (GWs) have enabled direct detections of compact binary coalescences (CBCs). However, their poor sky localisation and the typical lack of observable electromagnetic (EM) counterparts make it difficult to confidently identify their hosts, and study the environments that nurture their evolution. In this work, we show that $\textit{detailed}$ information of the host environment (e.g. the mass and steepness of the host potential) can be directly inferred by measuring the kinematic parameters (acceleration and its time-derivatives) of the binary's center of mass using GWs alone, without requiring an EM counterpart. We consider CBCs in various realistic environments such as globular clusters, nuclear star clusters, and active galactic nuclei disks to demonstrate how orbit and environment parameters can be extracted for CBCs detectable by ground- and space-based observatories, including the LIGO detector at A+ sensitivity, Einstein Telescope of the XG network, LISA, and DECIGO, $\textit{on a single-event basis}$. These constraints on host stellar environments promise to shed light on our understanding of how CBCs form, evolve, and merge.

Figures (11)

• Figure 1:
• Figure 2: Circular outer-orbit around an SMBH: The precision of the extraction of parameters over a grid of SMBH masses $M_{\rm SMBH}$ and radii of outer orbit $R$ for a binary system of component masses (from left to right) $5-1.4 \,{\rm M}_{\odot}$ at $\mathrm{100Mpc}$ in A+, $30-30 \,{\rm M}_{\odot}$ at $100\mathrm{Mpc}$ in ET, and $100-100 \,{\rm M}_{\odot}$ at $1\mathrm{Gpc}$ in DECIGO and LISA, with $\cos \vartheta = 0.9$. Top Panels: Relative error in the extraction of mass of SMBHs. The 'solid' lines are the lines of constant $\frac{\Delta M_{\rm SMBH}}{M_{\rm SMBH}}$. Bottom Panel: Relative error in the extraction of radii of the outer orbit. The 'dashed' lines are the lines of constant $\frac{\Delta R}{R}$. The pink regions correspond to the parameter space where either the relative errors are greater than 1 or we do not sample them due to the non-measurability of the kinematic parameters. The blue lines demarcate the regions (lower left) where we do not perform the Fisher matrix analysis, for, in these regions, the assumption $| \Gamma_n (t_{\rm o} - t_{\rm c})^n | \ll 1$ is no longer valid. The arrows point towards the region of increasing precision in the inference of parameters.
• Figure 3: Eccentric outer-orbit around an SMBH: The precision of the extraction of parameters over a grid of SMBH masses $M_{\rm SMBH}$ and semi-major axes $R$ of the outer orbit for a binary system of component masses (from left to right) considered in Figure \ref{['fig: CO_SMBH']} with $e = 0.5$ and $x \approx 1.25$. $x$ was drawn randomly from a uniform distribution between $\frac{1}{1-e}$ and $\frac{1}{1+e}$. Top Panel: Relative error in the extraction of mass of SMBHs. The 'solid' lines are the lines of constant $\frac{\Delta M_{\rm SMBH}}{M_{\rm SMBH}}$. Bottom Panel: Relative error in the extraction of the location of binary in the outer orbit. The 'dotted' lines are the lines of constant $\frac{\Delta x}{x}$. The pink regions, the blue lines, and the arrows have the same meaning as in Figure \ref{['fig: CO_SMBH']}.
• Figure 4: The relative errors in the semi-major axis and eccentricity of the outer orbit over a grid of SMBH masses $M_{\mathrm{SMBH}}$ and semi-major axis $R$ for the binary systems and detector configurations considered in Figure \ref{['fig: EO_SMBH']} of the paper. Top Panels: Relative error in the semi-major axis. The 'solid' lines are the lines of constant $\frac{\Delta R}{R}$. Bottom Panels: Relative error in the eccentricity. The 'dashed' lines are the lines of constant $\frac{\Delta e}{e}$. The pink regions, the blue lines, and the arrows have the same meaning as in Figure \ref{['fig: CO_SMBH']}.
• Figure 5: Effective Keplerian outer-orbit in an NSC with a BW profile: The precision of the extraction of parameters over a grid of MBH masses $M_{\rm MBH}$ and merger distance $r$ for a binary system of component masses (from left to right): $30-30\, {\rm M}_{\odot}$ and $100-100\, {\rm M}_{\odot}$ at $1{\rm Gpc}$ in DECIGO and LISA, respectively, for $\alpha = 1.75$ and $\cos \vartheta = 0.9$ with radial and tangential velocities: $v_r = 250\, {\rm km/s}$ and $r \Omega = 100\, {\rm km/s}$, respectively. Top Panel: Relative error in the mass of the MBH. The 'solid' lines are the lines of constant $\frac{\Delta M_{\rm MBH}}{M_{\rm MBH}}$. Middle Panel: Relative error in the distance of the binary from the NSC's center. The 'dashed' lines are the lines of constant $\frac{\Delta r}{r}$. Bottom Panel: Relative error in the steepness parameter $\alpha$. The 'dotted' lines are the lines of constant $\frac{\Delta \alpha}{\alpha}$. The pink regions, the blue lines, and the arrows have the same meaning as in Figure \ref{['fig: CO_SMBH']}.