Mock Catalogs of Strongly Lensed Gravitational Waves via a Halo Model Approach with Space-borne Detectors

Author(s)

Sun, Mingqi, Liao, Kai, Li, Youkai, Liu, Tonghua, Wu, Hengyu, Hou, Shaoqi, Yang, Tao, Fan, Xilong, Biesiada, Marek

Abstract

Future space-borne gravitational-wave (GW) detectors, such as LISA and DECIGO, are expected to detect a large number of GW events, a fraction of which may be strongly lensed by intervening galaxies or galaxy clusters. In this work, we develop a comprehensive framework to simulate strongly lensed GWs in the context of space-borne detectors. Based on realistic astrophysical models for both the source population and the lens distribution, we construct mock catalogs of lensed GW events, referred to as \textbf{GW-LMC-Space}. Our results show that, for a four-year LISA observation, the expected number of lensed events ranges from $0$ to $131$, depending on the adopted formation model of massive black hole binaries (MBHBs). The corresponding lensing probability for MBHBs can reach up to $\sim 0.3\%$. For DECIGO, we find that the number of lensed events in a one-year observation is expected to lie in the range of $0$--$44$, with a lensing probability of $\sim 0.15\%$ for stellar-mass binary black holes (BBHs), binary neutron stars (BNSs), and neutron star--black hole binaries (NSBHs). We further show that the overlap of lensed signals is a common feature in space-borne detectors, which can significantly affect both the signal-to-noise ratio (SNR) estimation and event identification. These results highlight the importance of accounting for signal overlap in the analysis of strongly lensed GW events in future space-borne GW observations.

Figures

SNR distribution (encoded in color map) of the intrinsic population of MBHBs for different formation models in 4-year LISA observation. The "x" axis is the total mass of the system, while the "y" axis denotes the redshift of the source (merging MBHB). For each model, we randomly selected 1000 events, considered their event rate from the intrinsic population and ploted the SNR distribution.
Caption SNR distribution (encoded in color map) of the intrinsic population of MBHBs for different formation models in 4-year LISA observation. The "x" axis is the total mass of the system, while the "y" axis denotes the redshift of the source (merging MBHB). For each model, we randomly selected 1000 events, considered their event rate from the intrinsic population and ploted the SNR distribution.
Distribution of the gravitational-wave signal duration for MBHBs during a four-year LISA mission. The durations are calculated for all 10,000 simulated events in each model using Eqs.~(\ref{frequency_evolution})-(\ref{ISCO}).
Caption Distribution of the gravitational-wave signal duration for MBHBs during a four-year LISA mission. The durations are calculated for all 10,000 simulated events in each model using Eqs.~(\ref{frequency_evolution})-(\ref{ISCO}).
Time delay distribution of the lensed events for different formation models in 100,000 samples of LISA observation. The upper panel shows the time delay distribution without considering the overlapping of lensed signals, while the lower panel shows the distribution when we consider the overlapping.
Caption Time delay distribution of the lensed events for different formation models in 100,000 samples of LISA observation. The upper panel shows the time delay distribution without considering the overlapping of lensed signals, while the lower panel shows the distribution when we consider the overlapping.
We present the distribution of lens redshift $z_l$ (redshift of the host halo) and source redshift $z_s$ for $10^5$ samples, together with a scatter plot of the maximum SNR ratio $r_{\mathrm{max}}$ (defined as the ratio of the SNR of the overlapped signal to that of the unlensed signal) for different formation models in a four-year LISA observation. The color bar indicates the value of $r_{\mathrm{max}}$, while the blue and green dashed lines denote the median source redshift and lens redshift, respectively.
Caption We present the distribution of lens redshift $z_l$ (redshift of the host halo) and source redshift $z_s$ for $10^5$ samples, together with a scatter plot of the maximum SNR ratio $r_{\mathrm{max}}$ (defined as the ratio of the SNR of the overlapped signal to that of the unlensed signal) for different formation models in a four-year LISA observation. The color bar indicates the value of $r_{\mathrm{max}}$, while the blue and green dashed lines denote the median source redshift and lens redshift, respectively.
Time delay difference distribution combined with SNR ratio distribution in the 100,000 samples in LISA observation. Different colors represent different formation models.
Caption Time delay difference distribution combined with SNR ratio distribution in the 100,000 samples in LISA observation. Different colors represent different formation models.
The joint distribution of the time delay difference and the magnification ratio for the model \texttt{HSnodSN} in 100,000 samples of LISA observation. The triple and quintuple plots (blue and orange points) represent the distribution of the baseline 3-image and 5-image systems, which are already shown in the Table~\ref{tab:original_compare_overlap}. The value shows in the top of the figure represents the median of each distribution.
Caption The joint distribution of the time delay difference and the magnification ratio for the model \texttt{HSnodSN} in 100,000 samples of LISA observation. The triple and quintuple plots (blue and orange points) represent the distribution of the baseline 3-image and 5-image systems, which are already shown in the Table~\ref{tab:original_compare_overlap}. The value shows in the top of the figure represents the median of each distribution.
The antenna pattern function of DECIGO in 1yr observation, with the source position fixed at $\bar{\theta}_S = \bar{\phi}_S =\pi/4$ and the orientation of the orbital angular momentum fixed at $\hat{\theta}_L = \hat{\phi}_L = 0$.
Caption The antenna pattern function of DECIGO in 1yr observation, with the source position fixed at $\bar{\theta}_S = \bar{\phi}_S =\pi/4$ and the orientation of the orbital angular momentum fixed at $\hat{\theta}_L = \hat{\phi}_L = 0$.
SNR distributions of the intrinsic populations of BBHs, BNSs, and NSBHs for the DECIGO case over a one-year observation period. In each subplot, the SNR distribution is divided into three regimes to distinguish different SNR ranges.
Caption SNR distributions of the intrinsic populations of BBHs, BNSs, and NSBHs for the DECIGO case over a one-year observation period. In each subplot, the SNR distribution is divided into three regimes to distinguish different SNR ranges.
Distribution of the gravitational-wave signal duration for BBHs, BNSs, and NSBHs during a one-year DECIGO mission. The durations are calculated for all events in each source type.
Caption Distribution of the gravitational-wave signal duration for BBHs, BNSs, and NSBHs during a one-year DECIGO mission. The durations are calculated for all events in each source type.
Time delay distribution of the lensed events for BBHs, BNSs, and NSBHs in the DECIGO band. The upper panel shows the time delay distribution without considering the overlapping of lensed signals, while the lower panel shows the distribution when we consider the overlapping.
Caption Time delay distribution of the lensed events for BBHs, BNSs, and NSBHs in the DECIGO band. The upper panel shows the time delay distribution without considering the overlapping of lensed signals, while the lower panel shows the distribution when we consider the overlapping.
The distribution of lens redshift $z_l$ and the source redshift $z_s$ for all samples in 1-yr observation with the scatter plot of the max SNR ratio $r_{\text{max}}$ for BBHs, BNSs, and NSBHs in the DECIGO band. The color bar represents the value of $r_{\mathrm{max}}$ while the blue and green dashed lines represent the median of the source redshift and lens redshift, respectively.
Caption The distribution of lens redshift $z_l$ and the source redshift $z_s$ for all samples in 1-yr observation with the scatter plot of the max SNR ratio $r_{\text{max}}$ for BBHs, BNSs, and NSBHs in the DECIGO band. The color bar represents the value of $r_{\mathrm{max}}$ while the blue and green dashed lines represent the median of the source redshift and lens redshift, respectively.
Time delay difference distribution combined with SNR ratio distribution in 1-year observation of DECIGO. Different colors represent different model types.
Caption Time delay difference distribution combined with SNR ratio distribution in 1-year observation of DECIGO. Different colors represent different model types.
The joint distribution of the time delay difference and the magnification ratio for the source \texttt{BBH} in 1-year DECIGO observation. The triple and quintuple plots (blue and orange points) represent the distribution of the baseline 3-image and 5-image systems, which are already shown in the Table~\ref{tab:original_compare_overlap}. The value shows in the top of the figure represents the median of each distribution.
Caption The joint distribution of the time delay difference and the magnification ratio for the source \texttt{BBH} in 1-year DECIGO observation. The triple and quintuple plots (blue and orange points) represent the distribution of the baseline 3-image and 5-image systems, which are already shown in the Table~\ref{tab:original_compare_overlap}. The value shows in the top of the figure represents the median of each distribution.
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