Bayesian Analysis of Gravitational Wave Microlensing Effects from Galactic Double White Dwarfs

1. Problem Statement

The paper addresses the challenge of detecting gravitational wave (GW) microlensing effects induced by compact objects within the Milky Way (such as black holes, globular clusters, and dark matter subhalos) on signals from Double White Dwarf (DWD) systems.

• Context: DWDs are primary targets for future space-based detectors like Taiji (operating in the milliHertz frequency band). Due to their abundance, they offer a high-statistics population for probing lensing.
• Challenge: While ground-based detectors (LIGO/Virgo) have detected GWs, no lensed events have been confirmed. Detecting microlensing in the Taiji band requires distinguishing subtle wave-optics signatures (diffraction/interference fringes) from unlensed signals amidst detector noise.
• Goal: To evaluate the detectability of these microlensing effects, estimate the parameters of the lens system, and determine the conditions under which lensed and unlensed models can be statistically distinguished using Bayesian inference.

2. Methodology

The authors employ a rigorous Bayesian inference framework combined with advanced waveform modeling.

• Physical Model:

  • Source: Monochromatic GW signals from DWDs, modeled with standard polarization and phase evolution.
  • Lens: Modeled as Point Mass Lenses (PML). This simplifies the analysis while representing a diverse population of Galactic objects.
  • Lensing Effect: The amplification factor $F(f, y)$ is calculated using wave optics formalism, incorporating the Einstein radius, source frequency, and impact parameter. The impact parameter $y(t)$ is time-dependent due to the relative transverse velocity ($v_{eff}$) between the source and the lens.
  • Parameters Investigated:
    • Lens Mass ($M_L$): $[10, 10^6] M_\odot$
    • Effective Velocity ($v_{eff}$): $[50, 500]$ km/s
    • Initial Separation ($L$): $[R_E, 3R_E]$ (where $R_E$ is the Einstein radius).

• Data Simulation:

  • Simulated 4-year observation data streams for the Taiji detector.
  • Utilized second-generation Time Delay Interferometry (TDI) channels ($A2$ and $E2$) to ensure noise orthogonality.
  • Fast-Slow Decomposition: To handle the computational cost of Bayesian inference over long durations, the authors employed an accelerated waveform calculation method. This separates the rapidly varying GW phase (fast part) from the slowly varying lensing amplification and TDI response (slow part), allowing for sparse grid sampling and efficient Fourier transforms.

• Statistical Framework:

  • Parameter Estimation: Used the Bilby framework with the nessai nested sampling algorithm to estimate posterior distributions for source and lens parameters.
  • Model Selection: Compared two hypotheses:
    1. Lensed Model ($M_L$): Includes lens parameters ($M_L, v_{eff}, L$).
    2. Unlensed Model ($M_U$): Standard DWD signal without lensing.
  • Metric: The Bayes Factor ($\ln B_{L/U} = \ln Z_L - \ln Z_U$) was used to quantify the evidence for the lensed hypothesis over the unlensed one.

3. Key Contributions

1. Accelerated Waveform Generation: Extended the fast-slow decomposition method to include lensed DWD signals and second-generation TDI channels, significantly improving computational efficiency for Bayesian analysis.
2. Systematic Parameter Space Exploration: Conducted a controlled variable analysis across a wide range of lens masses, velocities, and initial separations to map the detectability landscape for Taiji.
3. Quantification of Degeneracies: Identified and characterized the strong degeneracy between the lens mass ($M_L$), the effective velocity ($v_{eff}$), and the source amplitude ($A$), which complicates independent parameter estimation.
4. Detection Thresholds: Established specific physical thresholds (in terms of mass and impact parameter) required to distinguish lensed events from unlensed noise.

4. Key Results

• Impact of Lens Mass ($M_L$):
  • $M_L \geq 10^6 M_\odot$: Lensing effects are strong and clearly distinguishable from unlensed signals (High Bayes Factor).
  • $M_L = 10^5 M_\odot$: The lensed and unlensed models are statistically indistinguishable under current priors. The Bayes factor is insufficient to confirm lensing.
• Impact of Initial Separation ($L$):
  • $L = R_E$ to $2R_E$: Lensing signatures are detectable.
  • $L = 3R_E$: Even for massive lenses ($10^6 M_\odot$), if the impact parameter is large ($3R_E$), the lensed and unlensed models become indistinguishable.
• Impact of Effective Velocity ($v_{eff}$):
  • Variations in $v_{eff}$ (within $50-500$ km/s) have a negligible effect on the distinguishability of the models. While the Bayes factor decreases slightly as velocity decreases, the models remain distinguishable within the tested range. The waveform shape is dominated by $M_L$ and $L$, not $v_{eff}$.
• Parameter Degeneracies and Biases:
  • A strong tripartite degeneracy exists between $M_L$, $v_{eff}$, and the source amplitude $A$.
  • Bias in Unlensed Fits: When lensed data is analyzed using an unlensed model, the inferred amplitude $A$ is overestimated (to compensate for the missing magnification), and the initial phase $\phi_0$ exhibits significant distortion.
  • Constrained Parameters: While $M_L$ and $v_{eff}$ are poorly constrained individually due to degeneracy, the combination of parameters that produces the amplification is well-constrained.

5. Significance and Conclusion

• Feasibility for Taiji: The study confirms that the Taiji mission has the capability to detect GW microlensing from Galactic DWDs, but only for specific configurations: Massive lenses ($>10^5 M_\odot$) and close alignments ($L < 3R_E$).
• Astrophysical Implications:
  • Successful detection would provide direct probes of black hole formation and the population of globular clusters.
  • It offers a unique method to test the Cold Dark Matter paradigm by probing dark matter subhalos on small scales.
• Future Directions: The authors note that current results rely on idealized single-point mass lenses. Future work should consider multiple dynamic lenses and joint electromagnetic observations to break the $M_L-A-v_{eff}$ degeneracy and improve parameter inference.

In summary, this paper provides a critical roadmap for the Taiji collaboration, defining the physical limits of GW microlensing detection and highlighting the necessity of Bayesian model selection to avoid false negatives or biased parameter estimates in future data analysis.