Probing globular clusters parameters through gravitational wave lensing with stellar-mass black hole binaries

Technical Summary: Probing Globular Cluster Parameters through Gravitational Wave Lensing with Stellar-Mass Black Hole Binaries

Problem Statement
Globular clusters (GCs) are dense stellar systems that serve as laboratories for studying stellar dynamics and early galaxy formation. A critical parameter for understanding their internal dynamics, formation, and evolution is their mass, typically inferred from the central stellar velocity dispersion ($\sigma_v$). Current methods for determining GC mass rely on electromagnetic (EM) observations, such as velocity dispersion measurements and surface brightness profiles. However, these methods face significant limitations, including line-of-sight projection effects, stellar crowding in dense central regions, shot noise from luminous giant stars, and uncertainties in identifying the true cluster center. These challenges are exacerbated in core-collapsed clusters or systems potentially hosting intermediate-mass black holes (IMBHs). Furthermore, while upcoming wide-field surveys (e.g., LSST, Euclid) will expand GC catalogs with precise positions and distances, measuring internal kinematics like velocity dispersion remains difficult and typically requires dedicated spectroscopic follow-up.

The authors propose a complementary probe: utilizing gravitational waves (GWs) from compact binary coalescences (CBCs) that are lensed by GCs. Unlike strong lensing by galaxies, which produces multiple images, lensing by GCs operates in the wave-optics regime. In this regime, the lens does not create multiple signals but imprints a frequency-dependent amplification on the GW signal. The paper investigates whether these wave-optics signatures can be used to recover the intrinsic properties of the lensing GC, specifically its central velocity dispersion.

Methodology
The study employs a simulation and Bayesian parameter estimation framework to test the recoverability of GC parameters from lensed GW signals.

1. Lens Modeling: GCs are modeled as Singular Isothermal Spheres (SIS). The mass distribution is approximated by a density profile $\rho(r) = \sigma_v^2 / (2\pi G r^2)$, where the velocity dispersion $\sigma_v$ is assumed constant. The effective lensing mass ($M_{SIS}$) is derived as a function of $\sigma_v$ and the lens distance ($D_l$).
2. Signal Simulation: The authors simulate GW signals similar to GW150914 (a stellar-mass black hole binary merger). These signals are lensed using the SIS amplification factor derived from the Fresnel-Kirchhoff diffraction integral. The amplification depends on the dimensionless frequency ($w$) and the impact parameter ($y$).
3. Injection Study: The lensed signals are injected into Gaussian noise generated from the O4 power spectral densities (PSDs) of a three-detector network (LIGO Hanford, LIGO Livingston, and Virgo).
4. Parameter Estimation: The injected signals are recovered using Bayesian inference (via the Bilby package and dynesty nested sampling) with waveform templates that include both source parameters (15 binary parameters using the IMRPhenomXPHM approximant) and lens parameters (SIS lensing mass $M_{SIS}$ and impact parameter $y$).
5. Candidate Identification: From the recovered posterior distributions, a 90% credible sky-localization region is constructed. This region is cross-matched with a catalog of known Milky Way GCs (from [37]) to identify the potential lens.
6. Velocity Dispersion Recovery: Once a candidate GC is identified, its known distance ($D_l$) from the catalog is used to invert the recovered $M_{SIS}$ posterior to estimate the cluster's velocity dispersion ($\sigma_v$) using the relation $M_{SIS} \propto \sigma_v^4 D_l$.

Three specific GCs from the Milky Way catalog were selected for the study based on their distinct central velocity dispersions: Liller-1 ($\sigma_v \approx 19.6$ km s$^{-1}$), NGC 6093 ($\sigma_v \approx 11.1$ km s$^{-1}$), and NGC 2419 ($\sigma_v \approx 5.1$ km s$^{-1}$). For each cluster, 20 injections were performed with varying impact parameters ($y \in [0.001, 5]$).

Key Results

• Recovery of Lensing Mass and Velocity Dispersion: For favorable source-lens alignments (small impact parameters), the injected velocity dispersion values were successfully recovered within the 68% credible intervals. For Liller-1, the median recovered value was 19.7 km s$^{-1}$, closely matching the injected 19.6 km s$^{-1}$.
• Impact of Impact Parameter: The accuracy of the recovery is strongly dependent on the impact parameter $y$. As $y$ increases, the amplification decreases, weakening the lensing imprint on the signal. This leads to poorer sampling of the lensing mass and increased uncertainty in the inferred velocity dispersion. For larger $y$, the recovered median values can shift away from the true value due to noise realizations.
• Sky Localization and Catalog Cross-Matching: The injected cluster was correctly identified within the 90% credible sky-localization region for all tested cases. The method successfully distinguished the lensing cluster from others in the catalog, provided the source was positioned behind the specific target cluster.
• Parameter Degeneracy: The study observed an anti-correlation between the recovered lensing mass ($M_{SIS}$) and the impact parameter ($y$) for $y \sim 1$, consistent with previous findings for point-mass lenses.

Significance and Claims
The paper demonstrates that lensed GWs can serve as a complementary probe for GC dynamics, specifically allowing for the estimation of cluster velocity dispersion and, by extension, mass. This offers a new method to overcome limitations inherent in EM-based mass determination, particularly in dense or core-collapsed regions.

The authors note that the probability of a Milky Way GC acting as a lens for a GW event is low ($\tau \sim 10^{-8}$), highlighting the potential of next-generation detectors to discover such events. The methodology is presented as a tool to:

1. Confidently identify GW events as lensed signals if a Milky Way GC is the lens.
2. Constrain the velocity dispersion and mass of the lensing cluster.
3. Potentially indicate the presence of IMBHs in cluster cores.

The study is modest in its scope, utilizing the SIS model and focusing on Milky Way GCs where distances are well-known. The authors acknowledge that for extragalactic GCs (where catalogs may be incomplete), a degeneracy between lens distance and velocity dispersion would exist, requiring more extensive parameter estimation. They also note that while the current work uses the SIS model, the methodology could be extended to more realistic models (e.g., cored isothermal spheres, King models) in future work. The paper concludes that a global network of at least three GW detectors with comparable sensitivity is essential to enhance the sky localization accuracy required for this technique to be effective.