Comprehensive analysis of time-domain Overlapping Gravitational Wave Transients

Overlapping Transients

Lensing Effects

Strong Lensing (Type II):

It is an effect of scaling the signal’s amplitude, a temporal delay and an overall phase shift. Resulting in non-trivial properties in the time domain for certain parameters.

Microlensing:

Exhibits frequency-dependent amplification due to wave-optics effects. Resemblance to the beating patterns in overlapping signals.

Parameters

Systematically vary key parameters influencing waveform evolution: Chirp mass ratio: $\mathcal{M}_B/\mathcal{M}_A$, SNR ratio: $\rm{SNR_B/SNR_A}$, Coalescence time difference: $\Delta t_c$.

Parameter estimation:

$\mathcal{M}_B/\mathcal{M}_A \in \{0.5,\,1,\,2\}$, $\mathrm{SNR_B/SNR_A} \in \{0.5,\,1\}$, $\Delta t_c \in [-1,\,1]\,$s, (60 signals)

Fitting factor:

$\mathcal{M}_B/\mathcal{M}_A \in [0.1,\,10]$, $\mathrm{SNR_B/SNR_A} \in [0.1,\,10]$, $\Delta t_c \in [-1,\,1]\,$s, ($\sim\mathcal{O}$(5000) signals)

Methods

Parameter Estimation

Bayesian inference: $\mathcal{L}(d|\theta)\propto\exp\Big[-\tfrac{1}{2}\langle d-h(\theta)|d-h(\theta)\rangle\Big]$ $\underbrace{\log_{10}\mathcal{B}^L_U}_{\text{Bayes Factor}} = \log_{10}\mathcal{Z}_L - \log_{10}\mathcal{Z}_U$ $\mathcal{Z}_M = \int d\theta \mathcal{L}(d | \theta, \mathcal{H}_M) \pi_M(\theta | \mathcal{H}_M)$

Fitting Factor

Maximizing waveform overlap: $\mathcal{M}[h_1, h_2] = \max_{t_c, \Phi_c}\frac{\langle h_1 | h_2 \rangle}{\sqrt{\langle h_1 | h_1 \rangle \langle h_2 | h_2 \rangle}}.$ $\mathcal{F}=\max_{\lambda}\,\mathcal{M}[h_1, h_2(\lambda)]$ $\log_{10}\mathcal{B}^L_U = (\mathcal{F}_L^2 - \mathcal{F}_U^2)\frac{\mathrm{SNR}^2}{2}$

Unlensed Singles:

Parameter Estimation:

• Overlap induces significant parameter biases (e.g., in chirp mass).
• Higher-SNR signals dominate recovery.
• Bimodal distributions emerge in strongly overlapping cases.

Fitting Factor:

• Confirms the individual case dependencies and biases.
• Evident biases increase at stronger overlaps and comparable chirp mass, SNRs.

Type II Lensed:

Parameter Estimation:

• A: Fixed Morse phase shows distinct Bayes factor differences over the unlensed case.
• B,C: Allowing the Morse phase to vary improves lensing characterization.

Fitting Factor:

• A: Fixed Morse phase presents mild support for Type II lensing recovery, except at comparable chirp mass, SNRs.
• B, C: Allowing the Morse phase to vary shows greater support for strong lensing.

Microlensed:

Parameter Estimation:

• Microlensed templates yield stronger support for strongly overlapping signals.
• Recovered lens parameters ($M_\ell^z$, $y$) show dependencies with $\Delta t_c$.
• For cases with significant overlap, the microlensed recovery tends to favour larger lens masses and smaller impact parameters.
• Decrease in support for lensing at higher difference in coalescence times.

Fitting Factor:

• Significant support for microlensing at specific parameter regimes.
• Larger lens mass and smaller impact parameters recovered at significant chirp mass and SNR ratios.

Conclusions

• Overlapping signals lead to significant biases in single signal unlensed parameter recovery.
• Further, we find that Type II lensing and microlensing signals can mimic overlapping effects, especially in the strong overlap regime.
• Advanced parameter estimation methods are essential to disentangle these effects.