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Jeff J. Andrews

Publications and source records attributed to Jeff J. Andrews.

At least 19 recordsLinked to original sources

Metallicity dependence of Wolf-Rayet binaries using detailed binary models: An absence of long-period systems at low metallicity

Observations of Wolf-Rayet (WR) stars in binaries in the SMC and LMC suggest a preference for short period ($P\lesssim30$ days) orbits, with an apparent absence of WRs in long-period systems. The Galactic population does extend to longer periods but shows a deficit in long-period WR binaries compared to their progenitors, the O-star population. Across these populations a nearly constant binary fraction has been observed. We aim to characterize the population of WR binaries across a range of metallicity and determine the metallicity dependence of the formation of these systems, specifically focusing on the period distribution. We use detailed binary evolution models from POSYDON to predict the population of WR binaries at 0.01, 0.1, 0.2, 0.45, and $1Z_\odot$, analyzing the resulting period distribution and formation channels, while comparing them against the observed population in the SMC, LMC, and Milky Way. We find that the absence of wider WR binaries in the SMC, and potentially in the LMC, can be explained by stable, Case-B mass transfer only partially stripping the donor star, combined with WR winds at low metallicity being insufficiently strong to strip the remaining envelope. As a result, the long-period peak from Case-B mass transfer, present at Zsun, disappears at low metallicity. We additionally find that stable mass transfer (SMT) produces a short-period peak through Case-A mass transfer (P~5-10 days) that closely matches observations across metallicity. Furthermore, SMT and non-interacting systems are the dominant formation channels of WR binaries at all metallicities, with their relative contribution showing no metallicity dependence. This result implies that the SMT channel has the same metallicity-dependence as isolated WR formation. At the same time, we find that common envelope evolution primarily produces short-period ($P<1$ day) WR binaries with black-hole companions.

astro-ph.SR

The Mass-Ratio Distribution of the Low-Mass Binary Black Hole Subpopulation: A Natural Outcome of Isolated Binary Evolution

The observed merging binary black hole population is increasingly consistent with being composed of a mixture of subpopulations, each likely the result of different formation mechanisms. In particular, the low-mass subpopulation, with primary black hole masses below $\approxeq 15\,M_{\odot}$, has been attributed to mergers formed through isolated binary evolution. We use the binary population synthesis code POSYDON to study the mass ratio $(q)$ distribution of binary black hole mergers from isolated binary evolution and compare to the observed low-mass subpopulation. We explore variations in supernova remnant prescriptions, common-envelope efficiency, and black hole accretion efficiency. We find that our models have a preference for asymmetric $q$, most with a broad peak near $q\approxeq0.5-0.7$, and a near-equal-mass component whose relative strength varies across models. This resultant $q$ distribution is consistent with the $q$ distribution of the low-mass subpopulation observed with gravitational waves. We find that for the majority of models these features arise from physically distinct formation subchannels: the asymmetric peak reflects contributions from common-envelope and stable mass-transfer systems, while the near-equal-mass component traces double-core common envelope and contact systems. We conclude that the features in the $q$ distribution of the observed low-mass subpopulation emerge naturally from isolated binary evolution across a range of model assumptions. As the gravitational-wave catalog continues to grow, the relative strength of these $q$ features will provide an increasingly powerful diagnostic of isolated binary evolution and its formation subchannels.

astro-ph.HE

High-mass binary black hole mergers from detailed binary evolution models

Gravitational-wave observations reveal a population of binary black hole (BBH) mergers with primary masses above ${\sim}40\,\mathrm{M}_\odot$, extending into and potentially beyond the pair-instability mass gap, with a possibly flat mass-ratio and broader χ_\mathrm{eff} distribution. We investigate whether super-Eddington accretion during stable mass transfer in isolated binary evolution can produce BBH mergers consistent with these properties across primary BH mass, mass-ratio, and χ_\mathrm{eff} distributions. Using POSYDON, we simulate BBH merger populations with primary BH masses above ${\sim}40\,\mathrm{M}_\odot$, under three BH accretion efficiencies: Eddington-limited, GRRMHD-informed, and fully conservative. We additionally vary the natal kick strength, including strong kicks at high BH masses. We find that super-Eddington accretion does not suppress BBH mergers in the high-mass regime. Fully-conservative accretion leads to an increase of BBH mergers in POSYDON with a strong kick-independent peak at $χ_\mathrm{eff}=0.6$ and a sharp mass-ratio peak at $q\sim0.5$, whereas observations favor $χ_\mathrm{eff}=0.0$ and a flatter mass-ratio distribution. The GRRMHD-informed and Eddington-limited accretion are compatible with the observed primary BH mass and mass ratio distribution, but require natal kicks to populate negative χ_\mathrm{eff}. A joint analysis of the primary BH mass, mass ratio, and χ_\mathrm{eff} distributions provides strong constraints on binary evolution physics, and disfavor fully-conservative BH accretion as the dominant formation mechanism for high-mass BBH mergers. The Eddington-limited and GRRMHD-informed prescriptions with modest kicks can explain part of the high-mass population, but an additional formation channel is still needed to account for the high fraction of negative χ_\mathrm{eff} systems and high secondary BH spins.

astro-ph.HE

Stellar Population Spectra Incorporating Detailed Binary Evolution using POSYDON

The accuracy of stellar population properties inferred through spectral energy distribution fitting hinges on the reliability of the underlying spectral models. Binary interactions are fundamental for massive star evolution, and ignoring their spectral contribution can lead to incorrect results. We use the POSYDON binary population synthesis code to generate spectral models of stellar populations that include binaries at solar metallicity. Our framework incorporates a collection of spectral libraries that is designed to address key outcomes of binary stellar evolution like Wolf-Rayet stars, stripped helium stars, and a treatment for stellar mergers. Our models confirm previous results showing that the inclusion of binary interactions has a significant effect on the UV and ionizing regime of the integrated spectrum. In particular we find that Wolf-Rayet stars and other massive stars dominate the production of ionizing radiation at earlier times, but after $\simeq$16 Myr stripped stars produced through mass transfer begin to dominate. Furthermore, we show that the production of ionizing He II photons is especially sensitive to the underlying population of stripped stars. While our results currently focus on high-mass stars ($\ge4~M_{\odot}$) at Solar metallicity, they provide the framework for binary spectral synthesis across a range of metallicities and masses and lay the foundation for calculations of the emergent emission-line spectra in the UV, optical, and IR regimes. We make the spectral models from this work publicly available for use in a format that can be integrated into fitting codes.

astro-ph.SR

Forming Double Neutron Stars using Detailed Binary Evolution Models with POSYDON: Comparison to the Galactic Systems

With over two dozen detections in the Milky Way, double neutron stars (DNSs) provide a unique window into massive binary evolution. We use the POSYDON binary population synthesis code to model DNS populations and compare them to the observed Galactic sample. By tracing their origins to underlying single and binary star physics, we place constraints on the detailed evolutionary stages leading to DNS formation. Our study reveals a bifurcation within the well-known common envelope (CE) formation channel for DNSs, which naturally explains an observed split in the orbital periods of the Galactic systems. The two subchannels are defined by whether the donor star has a helium core (Case B mass transfer) or a carbon-oxygen core (Case C) at the onset of the CE, with only the helium core systems eventually merging due to gravitational wave-modulated orbital decay. We find that across different treatments of the CE phase, the formation of DNSs through both subchannels requires either a generous core definition of $\simeq$ 30% H-fraction or a high CE ejection efficiency of $α_{\rm CE}\gtrsim1.2$. By testing different supernova kick velocity models, we find that galactic DNSs are best reproduced using a prescription that favors low velocity kicks ($\lesssim 50 \, \rm km/s$), in agreement with previous studies. Furthermore, our models indicate that merging DNSs are born from a stripped progenitor with a median pre-supernova envelope mass $\sim$ 0.2$M_{\odot}$. Our results highlight the value of detailed evolutionary models for improving our understanding of exotic binary star formation.

astro-ph.SR

Double Neutron Star Delay Times Across Cosmic Metallicities: The Role of Helium Star Progenitors

Metallicity can play a significant role in massive binary evolution through its impact on the opacity within stellar interiors and wind-driven mass loss. In this work, we investigate how the double neutron star (DNS) delay time distribution (DTD) is shaped by the metallicity-dependent evolution of the helium star$-$NS progenitor system. Drawing from insights rooted in single and binary star physics, we argue that at a given metallicity, the stellar radius during the helium main-sequence sets a lower limit on the size of the DNS orbit at birth. We then perform population synthesis with the detailed binary evolution code POSYDON to illustrate the resulting DTD across a range of metallicities. Our results indicate that, independent of the common envelope efficiency and reasonable natal kicks, the majority of DNS mergers across metallicities occur typically no earlier than $\simeq 40\,\rm{Myr}$ after star formation and peaks strongly between $80-250\,\rm{Myr}$. Roughly $15\%$ of DNSs merge within 80 Myr, which may explain $r$-process enrichment in environments with brief star formation histories, while $\gtrsim 20\%$ merge on delay times $>1$Gyr, providing an explanation for short gamma-ray bursts in old, metal-poor galaxies. The shape of the DTD can be complex, with a metallicity-dependent split in the dominant formation channel imprinting a characteristic double-peaked structure. Although ideally oriented natal kicks can produce very short merging DNS, we find that the required kick magnitudes are inconsistent with observations. Our work has implications for assessing the contribution of DNS mergers to $r$-process enrichment and gamma-ray bursts/kilonovae transients across cosmic time.

astro-ph.SR

A case for Case A: detailed look at binary black hole formation through stable mass transfer

In isolated binary evolution, binary black hole (BBH) mergers are generally formed through stable mass transfer (SMT) or common envelope evolution. In recent years, the SMT channel has received significant attention due to detailed binary models showing increased mass transfer stability compared to previous studies. In this work, we perform a full zero-age-main-sequence to compact object merger analysis using detailed binary models at eight metallicities between $10^{-4}Z_\odot$ and $2Z_\odot$ to self-consistently model the population properties of BBH mergers in the SMT channel, determined their progenitor initial conditional, and investigate the binary physics governing their formation and metallicity dependence. We use the population synthesis code POSYDON to determine the population of BBH mergers from SMT. Using its extended grids of MESA binary models, we determine the essential physics in the formation of BBH mergers. SMT produces BBH mergers predominantly from systems with $P_{ZAMS}\leq10$ days. In these systems, both the initial mass transfer between two stars and the subsequent interaction between the remaining star and the first-born BH take place while the respective donor star is on the main-sequence (Case A). We find a limited contribution from wider Case B/C systems. Without a natal kick, the SMT channel does not produce BBH mergers above $Z>0.2Z_\odot$ due to orbital widening from stellar wind mass loss. The primary BH mass distribution shows a strong dependence on metallicity, while the mass ratio prefers unity independent of metallicity due to mass ratio reversal. Additionally, the $χ_{eff}$ distributions contain peaks at $χ_{eff}=0$ and ~0.15 of which the former disappears at high metallicities. A mass-scaled natal kick leave this sub-population unchanged but introduce a low-mass, unequal mass ratio sub-population that merges due to their high eccentricity.

astro-ph.HE

Irregularly Sampled Time Series Interpolation for Binary Evolution Simulations Using Dynamic Time Warping

Binary stellar evolution simulations are computationally expensive. Stellar population synthesis relies on these detailed evolution models at a fundamental level. Producing thousands of such models requires hundreds of CPU hours, but stellar track interpolation provides one approach to significantly reduce this computational cost. Although single-star track interpolation is straightforward, stellar interactions in binary systems introduce significant complexity to binary evolution, making traditional single-track interpolation methods inapplicable. Binary tracks present fundamentally different challenges compared to single stars, which possess relatively straightforward evolutionary phases identifiable through distinct physical properties. Binary systems are complicated by mutual interactions that can dramatically alter evolutionary trajectories and introduce discontinuities difficult to capture through standard interpolation. In this work, we introduce a novel approach for track alignment and iterative track averaging based on Dynamic Time Warping to address misalignments between neighboring tracks. Our method computes a single shared warping path across all physical parameters simultaneously, placing them on a consistent temporal grid that preserves the causal relationships between parameters. We demonstrate that this joint-alignment strategy maintains key physical relationships such as the Stefan-Boltzmann law in the interpolated tracks. Our comprehensive evaluation across multiple binary configurations demonstrates that proper temporal alignment is crucial for track interpolation methods. The proposed method consistently outperforms existing approaches and enables the efficient generation of more accurate binary population samples for astrophysical studies.

astro-ph.SR

Numerically stable equations for the orbital evolution of compact object binaries

The orbital and eccentricity evolution for compact object binaries through gravitational wave emission first derived by Peters and Mathews are used extensively throughout the gravitational wave community for calculating the orbital evolution and merger time of compact binaries. While improved calculations of the binary merger time have been the focus of several investigations since, the orbital evolution has not received the same attention. As the equations lack a closed form solution, a numerical integrator is required, but standard methods typically break when the point of merger is overstepped. We present a rewrite of Peters' equations in $\ln$-space, which allows common numerical solvers to converge. This leads to a more numerically robust and computationally efficient method for evolving compact binaries due to gravitational wave emission, reducing the number of function evaluations by 60\% to 70\% in our tests.

astro-ph.HE

HST Deep Upper Limits Rule Out a Surviving Massive Binary Companion to the Type Ic Supernova 2012fh

Current explanations of the mass-loss mechanism for stripped-envelope supernovae remain divided between single and binary progenitor systems. Here we obtain deep ultraviolet (UV) imaging with the Hubble Space Telescope (HST) of the Type Ic SN 2012fh to search for the presence of a surviving companion star to the progenitor. We synthesize these observations with archival HST imaging, ground-based spectroscopy, and previous analyses from the literature to provide three independent constraints on the progenitor system. We fit the color-magnitude diagram of the surrounding population to constrain the most likely age of the system to be $<20$ Myr. Analysis of spectra of SN 2012fh provide an estimate of the He core mass of the progenitor star, $>5.6$ M$_{\odot}$. We analyze deep HST images at the precise location after the SN faded to constrain the luminosity of any remaining main-sequence binary companion to be $\log(L/L_{\odot}) \lesssim 3.35$. Combining observational constraints with current binary population synthesis models excludes the presence of a faint stellar companion to SN 2012fh at the $\lesssim10\%$ level. The progenitor was therefore either effectively isolated at the time of explosion or orbited by a black-hole companion. The latter scenario dominates if we only consider models that produce successful supernovae.

astro-ph.HE

POSYDON Version 2: Population Synthesis with Detailed Binary-Evolution Simulations across a Cosmological Range of Metallicities

Whether considering rare astrophysical events on cosmological scales or unresolved stellar populations, accurate models must account for the integrated contribution from the entire history of star formation upon which that population is built. Here, we describe the second version of POSYDON, an open-source binary population synthesis code based on extensive grids of detailed binary evolution models computed using the MESA code, which follows both stars' structures as a binary system evolves through its complete evolution from the zero-age main sequence, through multiple phases of mass transfer and supernovae, to their death as compact objects. To generate synthetic binary populations, POSYDON uses advanced methods to interpolate between our large, densely spaced grids of simulated binaries. In our updated version of POSYDON, we account for the evolution of stellar binaries across a cosmological range of metallicities, extending from $10^{-4}$ $Z_{\odot}$ to 2 $Z_{\odot}$, including grids specifically focused on the Small and Large Magellanic Clouds (0.2 $Z_{\odot}$ and 0.45 $Z_{\odot}$). In addition to describing our model grids and detailing our methodology, we outline several improvements to POSYDON. These include the incorporation of single stars in stellar populations, a treatment for stellar mergers, and a careful modeling of "reverse-mass transferring" binaries in which a once-accreting star later becomes a donor star. Our simulations are focused on binaries with at least one high-mass component, such as those that host neutron stars and black holes, and we provide post-processing methods to account for the cosmological evolution of metallicity and star formation as well as rate calculations for transient events.

astro-ph.GA

Mass-gap Black Holes in Coalescing Neutron Star Black Hole Binaries

The existence of a mass gap of $3-5\,M_{\odot}$ between the heaviest neutron stars (NSs) and the lightest black holes (BHs), inferred from the BH mass distribution in low mass X-ray binaries (LMXBs), has been suggested for decades. The recently reported gravitational-wave source GW230529 has been confidently identified as a NSBH merger, with the BH mass falling within this lower mass gap. This detection provides strong evidence against the existence of the latter and introduces new implications for the coalescing NSBH population, including a revised BH mass distribution and an updated local merger rate. In this study, we employ POSYDON, a binary population synthesis code that integrates detailed single- and binary-star models, to investigate coalescing NSBH binaries formed through isolated binary evolution. In particular, we focus on the BH mass distribution of the intrinsic NSBH merger population. We find that, with a high common-envelope efficiency of $α_{\rm{CE}} =2 $, the BH masses in NSBH mergers concentrate in the lower mass gap, aligning more closely with observations. However, after accounting for the constraints of the selection bias against mass-gap BHs in LMXBs, which suggests that the maximum NS birth mass is below $\simeq 2\,M_{\odot}$, we find that introducing a high $α_{\rm{CE}}$ is not required to match observations. Additionally, we explore the impact of core-collapse supernova kicks. Finally, we present the property distributions of observable NSBH mergers from our simulation and find that they match well with the observations. We find that the fraction of electromagnetic counterparts in observable populations is $\approx 4-30\%$, depending on different NS equations of state. Future detections of coalescing NSBH binaries would provide invaluable insights into SN mechanisms, common envelope evolution, and NS physics.

astro-ph.HE

Evidence for a Catastrophically Disrupted Open Cluster

Of the many discoveries uncovered by the Gaia astrometric mission, some of the most exciting are related to nearby dispersed stellar structures. We analyze one such structure in the Milky Way disk, OCSN-49, a coeval stellar stream with 257 identified members spanning approximately 30$^\circ$ across the sky. We obtained high-resolution spectroscopic data for four members that span the stream's extent, finding these four stars to have solar metallicities and remarkably homogeneous chemistry. Through a combination of isochrone fitting, lithium abundance analysis, and gyrochronology, we find a consistent stellar age of 400--600 Myr. Integrating stellar orbits backwards reveals that OCSN-49 converged to a single point at a much younger age. By integrating unbound model stars forward and comparing them to the current phase-space distribution of OCSN-49, we derive a dynamical age of 83$\pm$1 Myr, inconsistent with the age of the stellar population. The discrepancy between the kinematic and stellar age indicators is naturally explained by a disruptive event that unbound OCSN-49 roughly 500 Myr into its lifetime. Based on rate estimates, disruption due to a passing giant molecular cloud (GMC) is the most likely culprit. Assuming a single encounter, we find that a nearly head-on collision with a fairly massive GMC ($\sim$10$^5$ M$_\odot$) was necessary to unbind the cluster, although encounters with multiple GMCs may be responsible. To our knowledge, OCSN-49 serves as the first known remnant of a catastrophically disrupted open cluster and therefore serves as a benchmark for further investigating cluster disruption in the Milky Way.

astro-ph.GA

Emulators for stellar profiles in binary population modeling

Knowledge about the internal physical structure of stars is crucial to understanding their evolution. The novel binary population synthesis code POSYDON includes a module for interpolating the stellar and binary properties of any system at the end of binary MESA evolution based on a pre-computed set of models. In this work, we present a new emulation method for predicting stellar profiles, i.e., the internal stellar structure along the radial axis, using machine learning techniques. We use principal component analysis for dimensionality reduction and fully-connected feed-forward neural networks for making predictions. We find accuracy to be comparable to that of nearest neighbor approximation, with a strong advantage in terms of memory and storage efficiency. By providing a versatile framework for modeling stellar internal structure, the emulation method presented here will enable faster simulations of higher physical fidelity, offering a foundation for a wide range of large-scale population studies of stellar and binary evolution.

astro-ph.SR

Challenges in Forming Millisecond Pulsar-Black Holes from Isolated Binaries

Binaries harboring a millisecond pulsar (MSP) and a black hole (BH) are a key observing target for current and upcoming pulsar surveys. We model the formation and evolution of such binaries in isolation at solar metallicity using the next-generation binary population synthesis code POSYDON. We examine neutron star (NS)-BH binaries where the NS forms first (labeled NSBH), as the NS must be able to spin-up to MSP rotation periods before the BH forms in these systems. We find that NSBHs are very rare and have a birth rate < 1 Myr$^{-1}$ for a Milky Way-like galaxy in our typical models. The NSBH birth rate is 2-3 orders of magnitude smaller than that for NS-BHs where the BH forms first (labeled BHNS). These rates are also sensitive to model assumptions about the supernova (SN) remnant masses, natal kicks, and common-envelope efficiency. We find that 100% of NSBHs undergo a mass ratio reversal before the first SN and up to 64% of NSBHs undergo a double common envelope phase after the mass ratio reversal occurs. Most importantly, no NSBH binaries in our populations undergo a mass transfer phase, either stable or unstable, after the first SN. This implies that there is no possibility of pulsar spin-up via accretion, and thus MSP-BH binaries cannot form. Thus, dynamical environments and processes may provide the only formation channels for such MSP-BH binaries.

astro-ph.HE

Mass Transfer in Eccentric Orbits with Self-consistent Stellar Evolution

We investigate Roche lobe overflow mass transfer (MT) in eccentric binary systems between stars and compact objects (COs), modeling the coupled evolution of both the star and the orbit due to eccentric MT (eMT) in a self-consistent framework. We implement the analytic expressions for secular rates of change of the orbital semi-major axis and eccentricity, assuming a delta function MT at periapse, into the binary stellar evolution code MESA. Two scenarios are examined: (1) a simplified model isolating the effects of eMT on stellar and orbital evolution, and (2) realistic binary configurations that include angular momentum exchange (e.g., tides, mass loss, spin-orbit coupling, and gravitational wave radiation). Unlike the ad hoc approach of instant circularization that is often employed, explicit modeling of eMT reveals a large fraction of binaries can remain eccentric post-MT. Even binaries which naturally circularize during eMT have different properties (donor mass and orbital size) compared to predictions from instant circularization, with some showing fundamentally different evolutionary outcomes (e.g., stable versus unstable MT). We demonstrate that a binary's initial mass ratio and eccentricity are predictive of whether it will remain eccentric or circularize after eMT. These findings underscore the importance of eMT in understanding CO-hosting binary populations, including X-ray binaries, gravitational wave sources, and other high-energy transients.

astro-ph.SR

Irregularly Sampled Time Series Interpolation for Detailed Binary Evolution Simulations

Modeling of large populations of binary stellar systems is an intergral part of a many areas of astrophysics, from radio pulsars and supernovae to X-ray binaries, gamma-ray bursts, and gravitational-wave mergers. Binary population synthesis codes that employ self-consistently the most advanced physics treatment available for stellar interiors and their evolution and are at the same time computationally tractable have started to emerge only recently. One element that is still missing from these codes is the ability to generate the complete time evolution of binaries with arbitrary initial conditions using pre-computed three-dimensional grids of binary sequences. Here we present a highly interpretable method, from binary evolution track interpolation. Our method implements simulation generation from irregularly sampled time series. Our results indicate that this method is appropriate for applications within binary population synthesis and computational astrophysics with time-dependent simulations in general. Furthermore we point out and offer solutions to the difficulty surrounding evaluating performance of signals exhibiting extreme morphologies akin to discontinuities.

astro-ph.SR

An Empirical Framework Characterizing the Metallicity and Star-Formation History Dependence of X-ray Binary Population Formation and Emission in Galaxies

We present a new empirical framework modeling the metallicity and star-formation history (SFH) dependence of X-ray luminous ($L > 10^{36}$ ergs s$^{-1}$) point-source population luminosity functions (XLFs) in normal galaxies. We expect the X-ray point-source populations are dominated by X-ray binaries (XRBs), with contributions from supernova remnants near the low luminosity end of our observations. Our framework is calibrated using the collective statistical power of 3,731 X-ray detected point-sources within 88 Chandra-observed galaxies at $D <$ 40 Mpc that span broad ranges of metallicity ($Z \approx$ 0.03-2 $Z_\odot$), SFH, and morphology (dwarf irregulars, late-types, and early-types). Our best-fitting models indicate that the XLF normalization per unit stellar mass declines by $\approx$2-3 dex from 10 Myr to 10 Gyr, with a slower age decline for low-metallicity populations. The shape of the XLF for luminous X-ray sources ($L < 10^{38}$ ergs s$^{-1}$) significantly steepens with increasing age and metallicity, while the lower-luminosity XLF appears to flatten with increasing age. Integration of our models provide predictions for X-ray scaling relations that agree very well with past results presented in the literature, including, e.g., the $L_{\rm X}$-SFR-$Z$ relation for high-mass XRBs (HMXBs) in young stellar populations as well as the $L_{\rm X}/M_\star$ ratio observed in early-type galaxies that harbor old populations of low-mass XRBs (LMXBs). The model framework and data sets presented in this paper further provide unique benchmarks that can be used for calibrating binary population synthesis models.

astro-ph.GA