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Register a new userInferring the properties of a population of compact binaries in presence of selection effects
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Shortly after a new class of objects is discovered, the attention shifts from the properties of the individual sources to the question of their origin: do all sources come from the same underlying population, or several populations are required? What are the properties of these populations? As the detection of gravitational waves is becoming routine and the size of the event catalog increases, finer and finer details of the astrophysical distribution of compact binaries are now within our grasp. This Chapter presents a pedagogical introduction to the main statistical tool required for these analyses: hierarchical Bayesian inference in the presence of selection effects. All key equations are obtained from first principles, followed by two examples of increasing complexity. Although many remarks made in this Chapter refer to gravitational-wave astronomy, the write-up is generic enough to be useful to researchers and graduate students from other fields.
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Tidal interactions can play an important role as compact white dwarf (WD) binaries are driven together by gravitational waves (GWs). This will modify the strain evolution measured by future space-based GW detectors and impact the potential outcome of the mergers. Surveys now and in the near future will generate an unprecedented population of detached WD binaries to constrain tidal interactions. Motivated by this, I summarize the deviations between a binary evolving under the influence of only GW emission and a binary that is also experiencing some degree of tidal locking. I present analytic relations for the first and second derivative of the orbital period and braking index. Measurements of these quantities will allow the inference of tidal interactions, even when the masses of the component WDs are not well constrained. Finally, I discuss tidal heating and how it can provide complimentary information.
Excess energy method is used in searches of gravitational waves (GWs) produced from sources with poorly modeled characteristics. It identifies GW events by searching for a coincidence appearance of excess energy in a GW detector network. While it is sensitive to a wide range of signal morphologies, the energy outliers can be populated by background noise events (background), thereby reducing the statistical confidence of a true signal. However, if the physics of the source is partially understood, weak model dependent constraints can be imposed to suppress the background. This letter presents a novel idea of using the reconstructed chirp mass along with two goodness of fit parameters for suppressing background when search is focused on GW produced from the compact binary coalescence.
We perform a hierarchical Bayesian inference to investigate the population properties of the coalesc- ing compact binaries involving at least one neutron star (NS). With the current observation data, we can not rule out either of the Double Gaussian, Single Gaussian and Uniform NS mass distribution models, although the mass distribution of the Galactic NSs is slightly preferred by the gravitational wave (GW) observations. The mass distribution of black holes (BHs) in the neutron star-black hole (NSBH) population is found to be similar to that for the Galactic X-ray binaries. Additionally, the ratio of the merger rate densities between NSBHs and BNSs is estimated to be about 3 : 7. The spin properties of the binaries, though constrained relatively poor, play nontrivial role in reconstructing the mass distribution of NSs and BHs. We find that a perfectly aligned spin distribution can be ruled out, while a purely isotropic distribution of spin orientation is still allowed.
We present ${tt nimbus}$ : a hierarchical Bayesian framework to infer the intrinsic luminosity parameters of kilonovae (KNe) associated with gravitational-wave (GW) events, based purely on non-detections. This framework makes use of GW 3-D distance information and electromagnetic upper limits from a given survey for multiple events, and self-consistently accounts for finite sky-coverage and probability of astrophysical origin. The framework is agnostic to the brightness evolution assumed and can account for multiple electromagnetic passbands simultaneously. Our analyses highlight the importance of accounting for model selection effects, especially in the context of non-detections. We show our methodology using a simple, two-parameter linear brightness model, taking the follow-up of GW190425 with the Zwicky Transient Facility (ZTF) as a single-event test case for two different prior choices of model parameters -- (i) uniform/uninformative priors and (ii) astrophysical priors based on surrogate models of Monte Carlo radiative transfer simulations of KNe. We present results under the assumption that the KN is within the searched region to demonstrate functionality and the importance of prior choice. Our results show consistency with ${tt simsurvey}$ -- an astronomical survey simulation tool used previously in the literature to constrain the population of KNe. While our results based on uniform priors strongly constrain the parameter space, those based on astrophysical priors are largely uninformative, highlighting the need for deeper constraints. Future studies with multiple events having electromagnetic follow-up from multiple surveys should make it possible to constrain the KN population further.
Focusing on the remnant black holes after merging binary black holes, we show that ringdown gravitational waves of Population III binary black holes mergers can be detected with the rate of $5.9-500~{rm events~yr^{-1}}~({rm SFR_p}/ (10^{-2.5}~M_odot~{rm yr^{-1}~Mpc^{-3}})) cdot ({rm [f_b/(1+f_b)]/0.33})$ for various parameters and functions. This rate is estimated for the events with SNR$>8$ for the second generation gravitational wave detectors such as KAGRA. Here, ${rm SFR_p}$ and ${rm f_b}$ are the peak value of the Population III star formation rate and the fraction of binaries, respectively. When we consider only the events with SNR$>35$, the event rate becomes $0.046-4.21~{rm events~yr^{-1}}~({rm SFR_p}/ (10^{-2.5}~M_odot~{rm yr^{-1}~Mpc^{-3}})) cdot ({rm [f_b/(1+f_b)]/0.33})$. This suggest that for remnant black holes spin $q_f>0.95$ we have the event rate with SNR$>35$ less than $0.037~{rm events~yr^{-1}}~({rm SFR_p}/ (10^{-2.5}~M_odot~{rm yr^{-1}~Mpc^{-3}})) cdot ({rm [f_b/(1+f_b)]/0.33})$, while it is $3-30~{rm events~yr^{-1}}~({rm SFR_p}/ (10^{-2.5}~M_odot~{rm yr^{-1}~Mpc^{-3}})) cdot ({rm [f_b/(1+f_b)]/0.33})$ for the third generation detectors such as Einstein Telescope. If we detect many Population III binary black holes merger, it may be possible to constrain the Population III binary evolution paths not only by the mass distribution but also by the spin distribution.
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