The redshift factor $z$ is an invariant quantity of fundamental interest in post-Newtonian and self-force descriptions of compact binaries. It connects different approximation schemes, and plays a central role in the first law of binary black hole mechanics, which links local quantities to asymptotic measures of energy and angular momentum in these systems. Through this law, the redshift factor is conjectured to have a close relation to the surface gravity of the event horizons of black holes in circular orbits. We propose and implement a novel method for extracting the redshift factor on apparent horizons in numerical simulations of quasicircular binary inspirals. Our results confirm the conjectured relationship between $z$ and the surface gravity of the holes and that the first law holds to a remarkable degree for binary inspirals. The redshift factor enables tests of analytic predictions for $z$ in spacetimes where the binary is only approximately circular, giving a new connection between analytic approximations and numerical simulations.
We present a calculation of the Detweiler redshift factor in binary black hole simulations based on its relation to the surface gravity. The redshift factor has far-reaching applications in analytic approximations, gravitational self-force calculations, and conservative two-body dynamics. By specializing to non-spinning, quasi-circular binaries with mass ratios ranging from $m_A/m_B = 1$ to $m_A/m_B = 9.5$ we are able to recover the leading small-mass-ratio (SMR) prediction with relative differences of order $10^{-5}$ from simulations alone. The next-to-leading order term that we extract agree…
First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are …
We derive the first law of binary point-particle mechanics for generic bound (i.e. eccentric) orbits at the fourth post-Newtonian (4PN) order, accounting for the non-locality in time of the dynamics due to the occurence of a gravitational-wave tail effect at that order. Using this first law, we show how the periastron advance of the binary system can be related to the averaged redshift of one of the two bodies for a slightly non-circular orbit, in the limit where the eccentricity vanishes. Combining this expression with existing analytical self-force results for the averaged redshift, we recov…
We present a research of the time evolution of the redshift of light received from the binary star that moves in the external gravitational field of Kerr black hole. We formulate a method for the solution of inverse problem: calculating of the parameters of relative motion of stars in binary system using the redshift data. The considered formalism has no restrictions on the character of the motion of the center of mass of a compact binary star and can be applied even in the case of binary motion close to the event horizon of a supermassive black hole. The efficiency of the method is illustrate…
Detection of gravitational waves from the inspiral phase of binary neutron star coalescence will allow us to measure the effects of the tidal coupling in such systems. These effects will be measurable using 3rd generation gravitational wave detectors, e.g. the Einstein Telescope, which will be capable of detecting inspiralling binary neutron star systems out to redshift z=4. Tidal effects provide additional contributions to the phase evolution of the gravitational wave signal that break a degeneracy between the system's mass parameters and redshift and thereby allow the simultaneous measuremen…
One of approaches to quantum gravity is different models of a discrete pregeometry. An example of a discrete pregeometry on a microscopic scale is introduced. This is the particular case of a causal set. The causal set is a locally finite partially ordered set. The dynamics of this model is a stochastic sequential growth dynamics. New elements of causal set are added one by one. The probability of this addition of a new element depends on the structure of existed causal set. The particular case of the dynamics is considered. This dynamics is based on binary alternatives. Each directed path is …
The population-level distributions of the masses, spins, and redshifts of binary black holes (BBHs) observed using gravitational waves can shed light on how these systems form and evolve. Because of the complex astrophysical processes shaping the inferred BBH population, models allowing for correlations among these parameters will be necessary to fully characterize these sources. We hierarchically analyze the BBH population detected by LIGO and Virgo with a model allowing for correlations between the effective aligned spin and the primary mass and redshift. We find that the width of the effect…
Building upon the Noether charge formalism of Iyer and Wald, we derive a variational formula for spacetimes admitting a Killing vector field, for a generic energy-momentum distribution with compact support. Applying this general result to the particular case of a binary system of spinning compact objects moving along an exactly circular orbit, modelled using the multipolar gravitational skeleton formalism, we derive a first law of compact binary mechanics at dipolar order. We prove the equivalence of this new result with the canonical Hamiltonian first law previously derived for binary systems…
One of the goals of gravitational-wave astronomy is to quantify the evolution of the compact binary merger rate with redshift. The redshift distribution of black hole mergers would offer considerable information about their evolutionary history, including their progenitor formation rate, the dependence of black hole formation on stellar metallicity, and the time delay distribution between formation and merger. Efforts to measure the binary redshift distribution are currently limited, however, by the detection range of existing instruments, which can individually resolve compact binary merger e…
Inspiralling compact binaries as standard sirens will soon become an invaluable tool for cosmology when advanced interferometric gravitational-wave detectors begin their observations in the coming years. However, a degeneracy in the information carried by gravitational waves between the total rest-frame mass $M$ and the redshift $z$ of the source implies that neither can be directly extracted from the signal, but only the combination $M(1+z)$, the redshifted mass. Recent work has shown that for binary neutron star systems, a tidal correction to the gravitational-wave phase in the late-inspiral…
Optical appearance of a compact binary star in the field of a supermassive black hole is modeled in a strong field regime. Expressions for the redshift, magnification coefficient and pulsar extinction time are derived. By using the vierbein formalism we have derived equations of motion of compact binary star in the external gravitational field. We have analysed both the evolution of redshift of the optical ray from the usual star or white dwarf, and the times of arrival of pulses of pulsar. The results are illustrated by a calculation for a model binary system for the case of external gravitat…
Short gamma-ray bursts are believed to originate from the merger of two compact objects. If this scenario is correct, these bursts will be accompanied by the emission of strong gravitational waves, detectable by current or planned GW detectors, such as LIGO and Virgo. No detection of a gravitational wave has been made up to date. In this paper I will use a set of observed redshift measurements of short gamma-ray bursts to fit a model in order to determine the rate of such merger events in the nearby universe. Various corrections will be included in that calculation, as the field-of-view of the…
The mass and distance of a binary black hole (BBH) are fundamental parameters to measure in gravitational-wave (GW) astronomy. It is well-known that the measurement is affected by cosmological redshift, and recent works also showed that Doppler and gravitational redshifts could further affect the result if the BBH coalesces close to a supermassive black hole (SMBH). Here we consider the additional lensing effect induced by the nearby SMBH on the measurement. We compute the null geodesics originating within $10$ gravitational radii of a Kerr SMBH to determine the redshift and magnification of t…
For around 100 years, hydrogen spectral modelling has been based on Voigt profile fitting. The semi-classical Voigt profile is based on a 2-level atom approximation. Whilst the Voigt profile is excellent for many circumstances, the accuracy is insufficient for very high column density damped Lyman-$α$ absorption systems. We have adapted the quantum-mechanical Kramers-Heisenberg model to include thermal broadening, producing a new profile, the KHT profile. Interactions involving multiple discrete atomic levels and continuum terms, not accounted for in the Voigt model, generate asymmetries in th…
The noise produced by the inspiral of millions of white dwarf binaries in the Milky Way may pose a threat to one of the main goals of the space-based LISA mission: the detection of massive black hole binary mergers. We present a novel study for reconstruction of merger waveforms in the presence of Galactic confusion noise using dictionary learning. We discuss the limitations of untangling signals from binaries with total mass from $10^2 M_{\odot}$ to $10^4 M_{\odot}$. Our method proves extremely successful for binaries with total mass greater than $\sim 3\times 10^3$ $ M_{\odot}$ up to redshif…
We investigate the ability of current and third-generation gravitational wave (GW) detectors to determine the delay time distribution (DTD) of binary neutron stars (BNS) through a direct measurement of the BNS merger rate as a function of redshift. We assume that the DTD follows a power law distribution with a slope $Γ$ and a minimum merger time $t_{\rm min}$, and also allow the overall BNS formation efficiency per unit stellar mass to vary. By convolving the DTD and mass efficiency with the cosmic star formation history, and then with the GW detector capabilities, we explore two relevant regi…
We use the canonical Hamiltonian formalism to generalize to spinning point particles the first law of mechanics established for binary systems of non-spinning point masses moving on circular orbits [Le Tiec, Blanchet, and Whiting, Phys. Rev. D 85, 064039 (2012)]. We find that the redshift observable of each particle is related in a very simple manner to the canonical Hamiltonian and, more generally, to a class of Fokker-type Hamiltonians. Our results are valid through linear order in the spin of each particle, but hold also for quadratic couplings between the spins of different particles. The …
Gravitational wave detectors are starting to reveal the redshift evolution of the binary black hole (BBH) merger rate, $R_{\mathrm{BBH}}(z)$. We make predictions for $R_{\mathrm{BBH}}(z)$ as a function of black hole mass for systems originating from isolated binaries. To this end, we investigate correlations between the delay time and black hole mass by means of the suite of binary population synthesis simulations, COMPAS. We distinguish two channels: the common envelope (CE), and the stable Roche-lobe overflow (RLOF) channel, characterised by whether the system has experienced a common envelo…
Gravitational waves (GWs) are subject to gravitational lensing in the same way as electromagnetic radiation. However, to date, no unequivocal observation of a lensed GW transient has been reported. Independently, GW observatories continue to search for the stochastic GW signal which is produced by many transient events at high redshift. We exploit a surprising connection between the lensing of individual transients and limits to the background radiation produced by the unresolved population of binary back hole mergers: we show that it constrains the fraction of individually resolvable lensed b…
We show how LIGO is expected to detect coalescing binary black holes at $z>1$, that are lensed by the intervening galaxy population. Gravitational magnification, $μ$, strengthens gravitational wave signals by $\sqrtμ$, without altering their frequencies, which if unrecognised leads to an underestimate of the event redshift and hence an overestimate of the binary mass. High magnifications can be reached for coalescing binaries because the region of intense gravitational wave emission during coalescence is so small ($\sim$100km), permitting very close projections between lensing caustics and gra…
Reversible-jump MCMC reveals binary black hole subpopulations with distinct redshift evolution
Princeton University（普林斯顿大学）；Universitá degli Studi di Milano-Bicocca（米兰比可卡大学）；INFN, Sezione di Milano-Bicocca（意大利国家核物理研究所米兰-比可卡分部）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））
Observations of the gravitational radiation from well-localized, inspiraling compact object binaries can measure absolute source distances with high accuracy. When coupled with an independent determination of redshift through an electromagnetic counterpart, these standard sirens can provide an excellent probe of the expansion history of the Universe and the dark energy. Short gamma-ray bursts, if produced by merging neutron star binaries, would be standard sirens with known redshifts detectable by ground-based GW networks such as LIGO-II, Virgo, and AIGO. Depending upon the collimation of thes…
Atom interferometers allow the measurement of the acceleration of freely falling atoms with respect to an experimental platform at rest on Earth's surface. Such experiments have been used to test the universality of free fall by comparing the acceleration of the atoms to that of a classical freely falling object. In a recent paper, Müller, Peters and Chu [Nature {\bf 463}, 926-929 (2010)] argued that atom interferometers also provide a very accurate test of the gravitational redshift when considering the atom as a clock operating at the Compton frequency associated with the rest mass. We analy…
The Einstein Telescope (ET) is the future third generation gravitational wave detector consisting of three independent interferometers arranged in a triangular configuration, with the sensitivity large enough to be able to detect stellar mass black holes even beyond the redshift of 10. In this paper, we analyze the capabilities of ET as a standalone instrument and not part of a network. We show that the analysis of detection of binary coalescences in the three individual interferometers of the ET allow us to weakly constrain the sky location of the source. We present the analysis that leads to…
Probing the redshift evolution and sub-populations of binary neutron stars with the Einstein Telescope
Universitá degli Studi di Milano-Bicocca（米兰比可卡大学）；INFN, Sezione di Milano-Bicocca（意大利国家核物理研究所米兰比可卡分部）；Sorbonne Université（索邦大学）；Institut d’Astrophysique de Paris（巴黎天体物理学研究所）；Institut Universitaire de France（法兰西学院）
Strong-lensing effects in high-redshift massive black-hole binary population inference
University of Chinese Academy of Sciences（中国科学院大学）；Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）
The gravitational waves generated in the coalescence of massive binary black holes will be measurable by LISA to enormous distances. Redshifts z~10 or larger (depending somewhat on the mass of the binary) can potentially be probed by such measurements, suggesting that binary coalescences can be made into cosmological tools. We discuss two particularly interesting types of probes. First, by combining gravitational-wave measurements with information about the universe's cosmography, we can study the evolution of black hole masses and merger rates as a function of redshift, providing information …
The two binary black-hole (BBH) coalescences detected by LIGO, GW150914 and GW151226, were relatively nearby sources, with a redshift of ~0.1. As the sensitivity of Advanced LIGO and Virgo increases in the next few years, they will eventually detect stellar-mass BBHs up to redshifts of ~1. However, these are still relatively small distances compared with the size of the Universe, or with those encountered in most areas of astrophysics. In order to study BBH during the epoch of reionization, or black holes born from population III stars, more sensitive instruments are needed. Third-generation g…
Modified gravitational wave (GW) propagation is a generic phenomenon in modified gravity. It affects the reconstruction of the redshift of coalescing binaries from the luminosity distance measured by GW detectors, and therefore the reconstruction of the actual masses of the component compact stars from the observed (`detector-frame') masses. We show that, thanks to the narrowness of the mass distribution of binary neutron stars, this effect can provide a clear signature of modified gravity, particularly for the redshifts explored by third generation GW detectors such as Einstein Telescope and …
Nanohertz Gravitational-Wave Constraints on Supermassive Binary Black Holes at Cosmic Dawn
Beijing Normal University（北京师范大学）；Curtin University（科廷大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Monash University（蒙纳士大学）；University of Melbourne（墨尔本大学）
We present a method to include lensing selection effects due to the finite horizon of a given detector when studying lensing of gravitational wave (GW) sources. When selection effects are included, the mean of the magnification distribution is shifted from one to higher values for sufficiently high-redshift sources. This introduces an irreducible (multiplicative) bias on the luminosity distance reconstruction, in addition to the typical source of uncertainty in the distance determination. We apply this method to study lensing of GWs emitted by massive black hole binary mergers at high redshift…
Presently only 30% of short gamma ray bursts (SGRBs) have accurate redshifts, and this sample is highly biased by the limited sensitivity of {\it Swift} to detect SGRBs. We account for the dominant biases to calculate a realistic SGRB rate density out to $z = 0.5$ using the {\it Swift} sample of peak fluxes, redshifts, and those SGRBs with a beaming angle constraint from X-ray/optical observations. Assuming a significant fraction of binary neutron star mergers produce SGRBs, we calculate lower and upper detection rate limits of (1-180) per Yr by an advanced LIGO and Virgo coincidence search. O…
Coherent State Description of Gravitational Waves from Binary Black Holes
Kyushu University（九州大学）；Quantum and Spacetime Research Institute, Kyushu University（九州大学量子与时空研究所）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Kobe University（神户大学）
We explore the ability of gravitational-wave detectors to extract the redshift distribution of binary black hole (BBH) mergers. The evolution of the merger rate across redshifts $0 < z \lesssim 1$ is directly tied to the formation and evolutionary processes, providing insight regarding the progenitor formation rate together with the distribution of time delays between formation and merger. Because the limiting distance to which BBHs are detected depends on the masses of the binary, the redshift distribution of detected binaries depends on their underlying mass distribution. We therefore consid…
With one exception, previous analyses of the measurement accuracy of gravitational wave experiments for comparable-mass binary systems have neglected either spin-precession effects or subdominant harmonics and amplitude modulations. Here we give the first explicit description of how these effects combine to improve parameter estimation. We consider supermassive black hole binaries as expected to be observed with the planned space-based interferometer LISA, and study the measurement accuracy for several astrophysically interesting parameters obtainable taking into account the full 2PN waveform …
The recent detection of gravitational waves indicates that stellar-mass black hole binaries are likely to be a key population of sources for forthcoming observations. With future upgrades, ground-based detectors could detect merging black hole binaries out to cosmological distances. Gravitational wave bursts from high redshifts ($z \gtrsim 1$) can be strongly magnified by gravitational lensing due to intervening galaxies along the line of sight. In the absence of electromagnetic counterparts, the mergers' intrinsic mass scale and redshift are degenerate with the unknown magnification factor $μ…
The improvements in the sensitivity of the gravitational wave (GW) network enable the detection of several large redshift GW sources by third-generation GW detectors. These advancements provide an independent method to probe the large-scale structure of the universe by using the clustering of the binary black holes. The black hole catalogs are complementary to the galaxy catalogs because of large redshifts of GW events, which may imply that binary black holes (BBHs) are a better choice than galaxies to probe the large-scale structure of the universe and cosmic evolution over a large redshift r…
Gravitational waves (GWs) from supermassive binary black hole (BBH) inspirals are potentially powerful standard sirens (the GW analog to standard candles) (Schutz 1986, 2002). Because these systems are well-modeled, the space-based GW observatory LISA will be able to measure the luminosity distance (but not the redshift) to some distant massive BBH systems with 1-10% accuracy. This accuracy is largely limited by pointing error: GW sources generally are poorly localized on the sky. Localizing the binary independently (e.g., through association with an electromagnetic counterpart) greatly reduce…
The gravitational-wave (GW) events, produced by the coalescence of binary neutron-stars (BNS), can be treated as the standard sirens to probe the expansion history of the Universe, if their redshifts could be determined from the electromagnetic observations. For the high-redshift ($z\gtrsim 0.1$) events, the short $γ$-ray bursts (sGRBs) and the afterglows are always considered as the primary electromagnetic counterparts. In this paper, by investigating various models of sGRBs and afterglows, we discuss the rates and distributions of BNS mergers' multi-messenger observations with GW detectors i…
Understanding the origin of merging binary black holes is currently one of the most pressing quests in astrophysics. We show that if isolated binary evolution dominates the formation mechanism of merging binary black holes, one should expect a correlation between the effective spin parameter, $χ_\mathrm{eff}$, and the redshift of the merger, $z$, of binary black holes. This correlation comes from tidal spin-up systems preferentially forming and merging at higher redshifts due to the combination of weaker orbital expansion from low metallicity stars given their reduced wind mass loss rate, dela…
