Gravitational lensing rarely produces high-mass outliers to the compact binary population

TL;DR

This work shows that gravitational lensing rarely produces high-mass outliers in the gravitational-wave CBC population, because lensing-induced outliers require intrinsically high masses and large magnifications that are themselves rare. The authors develop a probabilistic framework combining a redshift-dependent magnification distribution $p(\mu|z)$, CBC population models for BBH and BNS, and a population-outlier criterion with a 99th percentile threshold to constrain the strong-lensing optical depth $\tau^{\geq5}$. Applying the method to GWTC-3 yields only weak constraints, with $\tau^{\geq5}(z=1) \le 3.5\times10^{-2}$, underscoring that absence of outliers provides limited information given measurement noise and the intrinsic mass distribution. Future detectors and longer observing runs will further improve constraints only modestly unless multiple outliers are observed; the authors advocate a hierarchical approach that jointly infers the CBC population, the magnification distribution, and event-level magnifications to optimally extract lensing information from GW catalogs.

Abstract

All gravitational-wave signals are inevitably gravitationally lensed by intervening matter as they propagate through the Universe. When a gravitational-wave signal is magnified, it \emph{appears} to have originated from a closer, more massive system. Thus, high-mass outliers to the gravitational-wave source population are often proposed as natural candidates for strongly lensed events. However, when using a data-driven method for identifying population outliers, we find that high-mass outliers are not necessarily strongly lensed, nor will the majority of strongly-lensed signals appear as high-mass outliers. This is both because statistical fluctuations produce a larger effect on observed binary parameters than does lensing magnification, and because lensing-induced outliers must originate from intrinsically high-mass sources, which are rare. Thus, the appearance of a single lensing-induced outlier implies the existence of many other lensed events within the catalog. We additionally show that it is possible to constrain the strong lensing optical depth, which is a fundamental quantity of our Universe, with the detection or absence of high-mass outliers. However, constraints using the latest gravitational-wave catalog are weak$\unicode{x2014}$we obtain an upper limit on the optical depth of sources at redshift $1$ magnified by a factor of $5$ or more of $τ(μ\geq5,z=1)\leq 0.035 \unicode{x2014}$and future observing runs will not make an outlier-based method competitive with other probes of the optical depth. However, the full inferred population of compact binaries may be more informative of the distribution of lenses in the Universe, opening a unique opportunity to access the high-redshift Universe and constrain cosmic structures.

Figures (10)

• Figure 1: Simulated observed BBH population with an O3-like detector with (left panel) and without (right panel) the effects of gravitational lensing. To clearly demonstrate these effects, the magnification distribution used for this plot has unrealistically high support for large magnifications. The same systems are plotted in both panels. In both panels, we show the detector horizon (black line), the maximum true source frame mass (orange line), and the region in which we define events to be outliers, as described in Section \ref{['sec:methods-ffh']} (green shading). The separation between the maximum mass of the true population and the outlier region is due to the large measurement uncertainties typical of GW observations. In the left panel, binaries are colored by their lens magnification. Magnification causes the binaries' apparent redshifts ($y$-axis of left panel) to be lower than their true redshifts ($y$-axis of right panel). This allows for the detection of events whose true distances lie beyond the detector horizon and results in larger apparent source frame masses. Thus, most high-mass outliers (squares) have large magnifications. However, events with large true redshifts and source frame masses are more likely to become outliers than low-mass events, as indicated by the cluster of squares in the upper-right corner of the right plot. "False positives" are also possible, as evidenced by the two events whose true parameters lie in the outlier region.
• Figure 2: Magnification distribution at $z=1$ for various choices of the tail width parameter, $t^0_c$. These directly correspond to different optical depths, which are denoted by the colorbar. The black line indicates our fiducial model, which is the one presented in dai_effect_2017 and calibrated to cosmological simulations. It corresponds to $\tau^{\geq 5}=1.47\times 10^{-5}$ (black dashed line on colorbar).
• Figure 3: Similar to the left panel of Fig. \ref{['fig:schematic']}, but with two different values of the high-magnification optical depth, $\tau^{\geq 5}$, and an expanded colorbar range. Larger values of $\tau^{\geq 5}$ allow for a higher fraction of events to receive large magnifications, causing more observed outliers. This allows the number of outliers to constrain the width of the high-magnification tail. Additionally, the observation of even one high-mass outlier implies the existence of many lower-mass events with comparable or higher magnifications.
• Figure 4: Expected number of outliers in GWTC-3 (dashed line, right $y$-axis) and the resulting probability of observing $k=0$ outliers in GWTC-3 (solid line, left $y$-axis) as functions of the high-magnification optical depth, $\tau^{\geq 5}$. Though the expected number of outliers increases monotonically with the optical depth, it is still less than unity even for extreme values of $\tau^{\geq 5}$. For all allowed $\tau^{\geq 5}$, the probability of not having observed an outlier is $\gtrsim 85\%$, meaning that meaningful constraints on the optical depth are not possible with current GW data. Our fiducial value for $\tau^{\geq 5}$ is indicated by the black tickmark.
• Figure 5: Probability of detecting zero (left panel) and one (right panel) high-mass outliers in current and future observing runs, as a function of $\tau^{\geq 5}$. On the left panel, 95% upper limits on $\tau^{\geq 5}$ for each observing run are shown by vertical dashed lines. While increased observing time with no outlier detections will improve the upper limits on $\tau^{\geq 5}$, these limits are orders of magnitude away from those expected for our Universe (black tickmark on both $x$-axes). If one high-mass outlier is detected, weak constraints (rather than upper limits) on $\tau$ are possible. Note the different $y$-axis ranges between the two panels.