Is the Binary Black Hole Population Inference from Gravitational-Wave Data Robust?

Author(s)

Das, Upasana, Mukherjee, Suvodip

Abstract

Gravitational-wave observations are playing an instrumental role in understanding the population of binary compact objects in the Universe. These observations have begun to hint at the mass distribution of binary black holes (BBHs), with tentative evidence for features in the mass distribution beyond a simple power-law. Such features, hence, can be connected with different formation scenarios of BBHs and lead to important astrophysical conclusions. However, it is crucial to understand whether these features are truly astrophysical or connected with any unknown systematics. We show in this work that waveform modelling uncertainties can significantly distort inferred features in the BBH mass distribution, which can be more pronounced than the statistical uncertainty, even with the current generation detectors, which can peak close to the lower edge of the pair instability supernovae (PISN) mass gap, and also can impact the slope of the power-law distribution. So, in order to have a confirmed detection of astrophysical features in the BBH mass distribution and connecting them with BBH formation channels, it is important to consider waveform systematics in the astrophysical population analysis. We show the typical scaling of the systematic error and discuss a few avenues to mitigate this effect for robust measurements in the future.

Figures

\justifying{\textbf{Number of GW cycles per coalescence phase as a function of total mass}, for equal-mass, non-spinning BBH systems ($q = 1$, $\chi = 0$) observed above $f_{\min} = 20\,\mathrm{Hz}$. The inspiral (blue) is computed using post-Newtonian expressions up to the innermost stable circular orbit (ISCO), the merger (red) spans from ISCO to the peak GW amplitude, and the ringdown (green) is estimated from the dominant quasi-normal mode damping time. The total cycle count is shown as the grey dashed curve. For $M \gtrsim 60\,M_\odot$, the inspiral contributes fewer cycles than the ringdown, and the signal is dominated by the merger and ringdown phases, where the waveform models are least accurate.}
Caption \justifying{\textbf{Number of GW cycles per coalescence phase as a function of total mass}, for equal-mass, non-spinning BBH systems ($q = 1$, $\chi = 0$) observed above $f_{\min} = 20\,\mathrm{Hz}$. The inspiral (blue) is computed using post-Newtonian expressions up to the innermost stable circular orbit (ISCO), the merger (red) spans from ISCO to the peak GW amplitude, and the ringdown (green) is estimated from the dominant quasi-normal mode damping time. The total cycle count is shown as the grey dashed curve. For $M \gtrsim 60\,M_\odot$, the inspiral contributes fewer cycles than the ringdown, and the signal is dominated by the merger and ringdown phases, where the waveform models are least accurate.}
\justifying\textbf{Mass-dependent manifestation of waveform systematics.} 1D marginalized posteriors for the detector-frame mass primary($m_{1,\text{det}}$) recovered using \texttt{SEOBNRv5PHM} (blue), \texttt{IMRPhenomTPHM} (red) and \texttt{NRSur7dq4} (black) against an \texttt{NRSur7dq4} injection (black dashed line). \textbf{Top:} For a low-mass system ($M_{\text{tot}} \sim 20 M_\odot$), the inspiral-dominated signal allows for accurate, unbiased recovery. \textbf{Bottom:} For a high-mass system ($M_{\text{tot}} \sim 200 M_\odot$), the merger-ringdown dominance exposes fundamental modelling inaccuracies, resulting in a severe systematic shift away from the true source parameters and a broader posterior. Note that the noise seed is kept fixed for the runs demonstrated in the top and bottom panels.
Caption \justifying\textbf{Mass-dependent manifestation of waveform systematics.} 1D marginalized posteriors for the detector-frame mass primary($m_{1,\text{det}}$) recovered using \texttt{SEOBNRv5PHM} (blue), \texttt{IMRPhenomTPHM} (red) and \texttt{NRSur7dq4} (black) against an \texttt{NRSur7dq4} injection (black dashed line). \textbf{Top:} For a low-mass system ($M_{\text{tot}} \sim 20 M_\odot$), the inspiral-dominated signal allows for accurate, unbiased recovery. \textbf{Bottom:} For a high-mass system ($M_{\text{tot}} \sim 200 M_\odot$), the merger-ringdown dominance exposes fundamental modelling inaccuracies, resulting in a severe systematic shift away from the true source parameters and a broader posterior. Note that the noise seed is kept fixed for the runs demonstrated in the top and bottom panels.
\justifying\textbf{Mass-dependent manifestation of waveform systematics.} 1D marginalized posteriors for the detector-frame mass primary($m_{1,\text{det}}$) recovered using \texttt{SEOBNRv5PHM} (blue), \texttt{IMRPhenomTPHM} (red) and \texttt{NRSur7dq4} (black) against an \texttt{NRSur7dq4} injection (black dashed line). \textbf{Top:} For a low-mass system ($M_{\text{tot}} \sim 20 M_\odot$), the inspiral-dominated signal allows for accurate, unbiased recovery. \textbf{Bottom:} For a high-mass system ($M_{\text{tot}} \sim 200 M_\odot$), the merger-ringdown dominance exposes fundamental modelling inaccuracies, resulting in a severe systematic shift away from the true source parameters and a broader posterior. Note that the noise seed is kept fixed for the runs demonstrated in the top and bottom panels.
Caption \justifying\textbf{Mass-dependent manifestation of waveform systematics.} 1D marginalized posteriors for the detector-frame mass primary($m_{1,\text{det}}$) recovered using \texttt{SEOBNRv5PHM} (blue), \texttt{IMRPhenomTPHM} (red) and \texttt{NRSur7dq4} (black) against an \texttt{NRSur7dq4} injection (black dashed line). \textbf{Top:} For a low-mass system ($M_{\text{tot}} \sim 20 M_\odot$), the inspiral-dominated signal allows for accurate, unbiased recovery. \textbf{Bottom:} For a high-mass system ($M_{\text{tot}} \sim 200 M_\odot$), the merger-ringdown dominance exposes fundamental modelling inaccuracies, resulting in a severe systematic shift away from the true source parameters and a broader posterior. Note that the noise seed is kept fixed for the runs demonstrated in the top and bottom panels.
\justifying Recovered detector-frame $m_1$ plotted against true injected values for signals generated with \texttt{NRSur7dq4} and recovered independently with \texttt{SEOBNRv5PHM} (blue) and \texttt{IMRPhenomTPHM} (red) for our initial fixed bins test case. Violin plots show the full marginalised posterior for each event, while the filled markers (circles for \texttt{SEOBNRv5PHM}, squares for \texttt{IMRPhenomTPHM}) indicate the binned posterior medians. The black dashed line indicates perfect recovery. Error bars span the 68\% credible intervals.
Caption \justifying Recovered detector-frame $m_1$ plotted against true injected values for signals generated with \texttt{NRSur7dq4} and recovered independently with \texttt{SEOBNRv5PHM} (blue) and \texttt{IMRPhenomTPHM} (red) for our initial fixed bins test case. Violin plots show the full marginalised posterior for each event, while the filled markers (circles for \texttt{SEOBNRv5PHM}, squares for \texttt{IMRPhenomTPHM}) indicate the binned posterior medians. The black dashed line indicates perfect recovery. Error bars span the 68\% credible intervals.
\justifying \textbf{Systematic bias scaling with mass for fixed-bin injections.} The absolute systematic bias $|\Delta|$ between the injected \texttt{NRSur7dq4} signal and the recovered posterior median using using \texttt{IMRPhenomTPHM} and \texttt{SEOBNRv5PHM} is shown as a function of the true mass. The left panel shows this scaling with $m_1$, and the right panel shows the same with $m_2$. The injections were performed in fixed mass bins of width $5M_\odot$ with fixed extrinsic parameters, and the matched-filter SNR is kept constant. The solid lines represent power-law fits ($|\Delta| \propto M^\gamma$) in log-log space, indicating that waveform systematics grow monotonically with the mass of the binary components.
Caption \justifying \textbf{Systematic bias scaling with mass for fixed-bin injections.} The absolute systematic bias $|\Delta|$ between the injected \texttt{NRSur7dq4} signal and the recovered posterior median using using \texttt{IMRPhenomTPHM} and \texttt{SEOBNRv5PHM} is shown as a function of the true mass. The left panel shows this scaling with $m_1$, and the right panel shows the same with $m_2$. The injections were performed in fixed mass bins of width $5M_\odot$ with fixed extrinsic parameters, and the matched-filter SNR is kept constant. The solid lines represent power-law fits ($|\Delta| \propto M^\gamma$) in log-log space, indicating that waveform systematics grow monotonically with the mass of the binary components.
\justifying \textbf{Systematic bias scaling with mass for fixed-bin injections.} The absolute systematic bias $|\Delta|$ between the injected \texttt{NRSur7dq4} signal and the recovered posterior median using using \texttt{IMRPhenomTPHM} and \texttt{SEOBNRv5PHM} is shown as a function of the true mass. The left panel shows this scaling with $m_1$, and the right panel shows the same with $m_2$. The injections were performed in fixed mass bins of width $5M_\odot$ with fixed extrinsic parameters, and the matched-filter SNR is kept constant. The solid lines represent power-law fits ($|\Delta| \propto M^\gamma$) in log-log space, indicating that waveform systematics grow monotonically with the mass of the binary components.
Caption \justifying \textbf{Systematic bias scaling with mass for fixed-bin injections.} The absolute systematic bias $|\Delta|$ between the injected \texttt{NRSur7dq4} signal and the recovered posterior median using using \texttt{IMRPhenomTPHM} and \texttt{SEOBNRv5PHM} is shown as a function of the true mass. The left panel shows this scaling with $m_1$, and the right panel shows the same with $m_2$. The injections were performed in fixed mass bins of width $5M_\odot$ with fixed extrinsic parameters, and the matched-filter SNR is kept constant. The solid lines represent power-law fits ($|\Delta| \propto M^\gamma$) in log-log space, indicating that waveform systematics grow monotonically with the mass of the binary components.
Same as Fig. \ref{fig:bias_vs_mass_fixedbin}, but for injection catalog drawn from realistic PL+G (row 1). In this case, all the 15 parameters are allowed to vary according to the methods described in Sec. \ref{sec:3b}
Caption Same as Fig. \ref{fig:bias_vs_mass_fixedbin}, but for injection catalog drawn from realistic PL+G (row 1). In this case, all the 15 parameters are allowed to vary according to the methods described in Sec. \ref{sec:3b}
Same as Fig. \ref{fig:bias_vs_mass_fixedbin}, but for injection catalog drawn from realistic PL+G (row 1). In this case, all the 15 parameters are allowed to vary according to the methods described in Sec. \ref{sec:3b}
Caption Same as Fig. \ref{fig:bias_vs_mass_fixedbin}, but for injection catalog drawn from realistic PL+G (row 1). In this case, all the 15 parameters are allowed to vary according to the methods described in Sec. \ref{sec:3b}
\justifying \textbf{P-P Plot for recovered source parameters} Each curve shows the fraction of events with credible intervals (C.I.) containing the true value, shown for $\mathcal{M}$ (blue), $q$ (orange), $m_1$ and $m_2$ for injections drawn from the PLG population and recovered using \texttt{SEOBNRv5PHM} (solid) and \texttt{IMRPhenomTPHM} (dashed).
Caption \justifying \textbf{P-P Plot for recovered source parameters} Each curve shows the fraction of events with credible intervals (C.I.) containing the true value, shown for $\mathcal{M}$ (blue), $q$ (orange), $m_1$ and $m_2$ for injections drawn from the PLG population and recovered using \texttt{SEOBNRv5PHM} (solid) and \texttt{IMRPhenomTPHM} (dashed).
\justifying \textbf{Hyperparameter posteriors from Hierarchical Bayesian Inference for the PL+G mass model.} One and two-dimensional marginalised posteriors for the population hyperparameters $\{\alpha,\,\sigma_g,\,\mu_g,\,\beta,\,m_{\min},\,m_{\max},\,\lambda_g\}$ are shown for recovery using \texttt{SEOBNRv5PHM} (blue), \texttt{IMRPhenomTPHM} (orange) and idealised Gaussian posteriors (grey). Dashed vertical lines in the 1D panels indicate the 68\% credible interval, and contours in the 2D panels show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ regions. The truth lines are given in black.
Caption \justifying \textbf{Hyperparameter posteriors from Hierarchical Bayesian Inference for the PL+G mass model.} One and two-dimensional marginalised posteriors for the population hyperparameters $\{\alpha,\,\sigma_g,\,\mu_g,\,\beta,\,m_{\min},\,m_{\max},\,\lambda_g\}$ are shown for recovery using \texttt{SEOBNRv5PHM} (blue), \texttt{IMRPhenomTPHM} (orange) and idealised Gaussian posteriors (grey). Dashed vertical lines in the 1D panels indicate the 68\% credible interval, and contours in the 2D panels show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ regions. The truth lines are given in black.
\justifying\textbf{Summary plot of hierarchical Bayesian inference results} The panel shows results for the Power Law+Gaussian population model where the blue box plots correspond to the idealised mock Gaussian posterior validation runs, while the orange and purple box plots show the same for the \texttt{SEOBNRv5PHM} and \texttt{IMRPhenomTPHM} recovery, respectively. The x-axis represents all the inferred hyperparameters, and all posterior samples are divided by their respective median values to allow comparison across parameters on a common $y$-axis. The darker-shaded boxes span $1\sigma$ of the data around the median, while the lighter-shaded boxes extend to $2\sigma$. Whiskers indicate the full range of the posterior samples. Black triangles mark the injected values divided by the corresponding median. The grey boxes around each hyperparameter refer to the prior bounds for each in units normalised by the individual posterior median. The text represents the median value of the parameter and the 68\% confidence interval around it (refer to Table \ref{tab:hyperparams_combined}).
Caption \justifying\textbf{Summary plot of hierarchical Bayesian inference results} The panel shows results for the Power Law+Gaussian population model where the blue box plots correspond to the idealised mock Gaussian posterior validation runs, while the orange and purple box plots show the same for the \texttt{SEOBNRv5PHM} and \texttt{IMRPhenomTPHM} recovery, respectively. The x-axis represents all the inferred hyperparameters, and all posterior samples are divided by their respective median values to allow comparison across parameters on a common $y$-axis. The darker-shaded boxes span $1\sigma$ of the data around the median, while the lighter-shaded boxes extend to $2\sigma$. Whiskers indicate the full range of the posterior samples. Black triangles mark the injected values divided by the corresponding median. The grey boxes around each hyperparameter refer to the prior bounds for each in units normalised by the individual posterior median. The text represents the median value of the parameter and the 68\% confidence interval around it (refer to Table \ref{tab:hyperparams_combined}).
\justifying Events from the injected PL+G catalog are plotted with their injected source-frame primary mass versus their recovered primary mass using \texttt{SEOBNRv5PHM} (blue circles) and \texttt{IMRPhenomTPHM} (orange squares). The dashed $y=x$ line shows perfect recovery.
Caption \justifying Events from the injected PL+G catalog are plotted with their injected source-frame primary mass versus their recovered primary mass using \texttt{SEOBNRv5PHM} (blue circles) and \texttt{IMRPhenomTPHM} (orange squares). The dashed $y=x$ line shows perfect recovery.
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