Black Hole-neutron Star Systems Research Articles

A model-independent precision test of general relativity using bright standard sirens from ongoing and upcoming detectors

Abstract Gravitational waves (GWs) provide a new avenue to test Einstein’s General Relativity (GR) using the ongoing and upcoming GW detectors by measuring the redshift evolution of the effective Planck mass proposed by several modified theories of gravity. We propose a model-independent, data-driven approach to measure any deviation from GR in the GW propagation effect by combining multi-messenger observations of GW sources accompanied by EM counterparts, commonly known as bright sirens (Binary Neutron Star (BNS) and Neutron Star Black Hole systems (NSBH)). We show that by combining the GW luminosity distance measurements from bright sirens with the Baryon Acoustic Oscillation (BAO) measurements derived from galaxy clustering, and the sound horizon measurements from the Cosmic Microwave Background (CMB), we can make a data-driven reconstruction of deviation of the variation of the effective Planck mass (jointly with the Hubble constant) as a function of cosmic redshift. Using this technique, we achieve a precise measurement of GR with redshift (z) with a precision of approximately 7.9 % for BNSs at redshift z=0.075 and 10 % for NSBHs at redshift z=0.225 with 5 years of observation from LIGO-Virgo-KAGRA network of detectors. Employing Cosmic Explorer and Einstein Telescope for just 1 year yields the best precision of about 1.62 % for BNSs and 2 % for NSBHs at redshift z=0.5 on the evolution of the frictional term, and a similar precision up to z=1. This measurement can discover potential deviation from any kind of model that impacts GW propagation with ongoing and upcoming observations.

From ZAMS to merger: Detailed binary evolution models of coalescing neutron star – black hole systems at solar metallicity

Neutron star – black hole (NSBH) merger events bring us new opportunities to constrain theories of stellar and binary evolution and understand the nature of compact objects. In this work, we investigated the formation of merging NSBH binaries at solar metallicity by performing a binary population synthesis study of merging NSBH binaries with the newly developed code POSYDON. The latter incorporates extensive grids of detailed single and binary evolution models, covering the entire evolution of a double compact object progenitor. We explored the evolution of NSBHs originating from different formation channels, which in some cases differ from earlier studies performed with rapid binary population synthesis codes. In this paper, we present the population properties of merging NSBH systems and their progenitors such as component masses, orbital features, and BH spins, and we detail our investigation of the model uncertainties in our treatment of common envelope (CE) evolution and the core-collapse process. We find that at solar metallicity, under the default model assumptions, most of the merging NSBHs have BH masses in the range of 3 − 11 M⊙ and chirp masses within 1.5 − 4 M⊙. Independently of our model variations, the BH always forms first with dimensionless spin parameter ≲0.2, which is correlated to the initial binary orbital period. Some BHs can subsequently spin up moderately (χBH ≲ 0.4) due to mass transfer, which we assume to be Eddington limited. Binaries that experience CE evolution rarely demonstrate large tilt angles. Conversely, approximately 40% of the binaries that undergo only stable mass transfer without CE evolution contain an anti-aligned BH. Finally, accounting for uncertainties in both the population modeling and the NS equation of state, we find that 0 − 18.6% of NSBH mergers may be accompanied by an electromagnetic counterpart.

Constraining the charge of a black hole with electromagnetic radiation from a black hole-neutron star system

Constraining the charge of a black hole with electromagnetic radiation from a black hole-neutron star system

Electromagnetic Precursors to Black Hole–Neutron Star Gravitational Wave Events: Flares and Reconnection-powered Fast Radio Transients from the Late Inspiral

The presence of magnetic fields in the late inspiral of black hole–neutron star binaries could lead to potentially detectable electromagnetic precursor transients. Using general-relativistic force-free electrodynamics simulations, we investigate premerger interactions of the common magnetosphere of black hole–neutron star systems. We demonstrate that these systems can feature copious electromagnetic flaring activity, which we find depends on the magnetic field orientation but not on black hole spin. Due to interactions with the surrounding magnetosphere, these flares could lead to fast-radio-burst-like transients and X-ray emission, with as an upper bound on the luminosity, where B * is the magnetic field strength on the surface of the neutron star.

Measuring the Hubble Constant with Dark Neutron Star–Black Hole Mergers

Detection of gravitational waves (GWs) from neutron star-black hole (NSBH) standard sirens provides local measurements of the Hubble constant (H 0), regardless of the detection of an electromagnetic (EM) counterpart, given that matter effects can be exploited to break the redshift degeneracy of the GW waveforms. The distinctive merger morphology and the high-redshift detectability of tidally disrupted NSBH make them promising candidates for this method. Also, the detection prospects of an EM counterpart for these systems will be limited to z < 0.8 in the optical, in the era of future GW detectors. Using recent constraints on the equation of state of NSs from multi-messenger observations of NICER and LIGO/Virgo/KAGRA, we show the prospects of measuring H 0 solely from GW observation of NSBH systems, achievable by the Einstein telescope (ET) and Cosmic Explorer (CE) detectors. We first analyze individual events to quantify the effect of high-frequency (≥500 Hz) tidal distortions on the inference of NS tidal deformability parameter (Λ) and hence on H 0. We find that disruptive mergers can constrain Λ up to more precisely than nondisruptive ones. However, this precision is not sufficient to place stringent constraints on the H 0 from individual events. By performing Bayesian analysis on simulated NSBH data (up to N = 100 events, corresponding to a day of observation) in the ET+CE detectors, we find that NSBH systems enable unbiased 4%–13% precision on the estimate of H 0 (68% credible interval). This is a similar measurement precision found in studies analyzing NSBH mergers with EM counterparts in the LVKC O5 era.

GRB 211211A-like Events and How Gravitational Waves May Tell Their Origins

GRB 211211A is a rare burst with a genuinely long duration, yet its prominent kilonova association provides compelling evidence that this peculiar burst was the result of a compact binary merger. However, the exact nature of the merging objects, whether they were neutron star pairs, neutron star–black hole systems, or neutron star–white dwarf systems, remains unsettled. This Letter delves into the rarity of this event and the possibility of using current and next-generation gravitational wave detectors to distinguish between the various types of binary systems. Our research reveals an event rate density of for GRB 211211A-like gamma-ray bursts (GRBs), which, assuming GRB 211211A is the only example of such a burst, is significantly smaller than that of typical long- and short-GRB populations. We further calculated that if the origin of GRB 211211A is a result of a neutron star–black hole merger, it would be detectable with a significant signal-to-noise ratio (S/N), given the LIGO-Virgo-KAGRA designed sensitivity. On the other hand, a neutron star–white dwarf binary would also produce a considerable S/N during the inspiral phase at decihertz and is detectable by next-generation spaceborne detectors DECIGO and the Big Bang Observer. However, to detect this type of system with millihertz spaceborne detectors like LISA, Taiji, and TianQin, the event must be very close, approximately 3 Mpc in distance or smaller.

Modelling stellar evolution in mass-transferring binaries and gravitational-wave progenitors with metisse

ABSTRACT Massive binaries are vital sources of various transient processes, including gravitational-wave mergers. However, large uncertainties in the evolution of massive stars, both physical and numerical, present a major challenge to the understanding of their binary evolution. In this paper, we upgrade our interpolation-based stellar evolution code metisse to include the effects of mass changes, such as binary mass transfer or wind-driven mass loss, not already included within the input stellar tracks. metisse’s implementation of mass loss (applied to tracks without mass loss) shows excellent agreement with the sse fitting formulae and with detailed mesa tracks, except in cases where the mass transfer is too rapid for the star to maintain equilibrium. We use this updated version of metisse within the binary population synthesis code bse to demonstrate the impact of varying stellar evolution parameters, particularly core overshooting, on the evolution of a massive (25 and 15 M⊙) binary system with an orbital period of 1800 d. Depending on the input tracks, we find that the binary system can form a binary black hole or a black hole–neutron star system, with primary (secondary) remnant masses ranging between 4.47 (1.36) and 12.30 (10.89) M⊙, and orbital periods ranging from 6 d to the binary becoming unbound. Extending this analysis to a population of isolated binaries uniformly distributed in mass and orbital period, we show that the input stellar models play an important role in determining which regions of the binary parameter space can produce compact binary mergers, paving the way for predictions for current and future gravitational-wave observatories.

Neutron star-black hole mergers in next generation gravitational-wave observatories

Observations by the current generation of gravitational-wave detectors have been pivotal in expanding our understanding of the universe. Although tens of exciting compact binary mergers have been observed, neutron star-black hole (NSBH) mergers remained elusive until they were first confidently detected in 2020. The number of NSBH detections is expected to increase with sensitivity improvements of the current detectors and the proposed construction of new observatories over the next decade. In this work, we explore the NSBH detection and measurement capabilities of these upgraded detectors and new observatories using the following metrics: network detection efficiency and detection rate as a function of redshift, distributions of the signal-to-noise ratios, the measurement accuracy of intrinsic and extrinsic parameters, the accuracy of sky position measurement, and the number of early-warning alerts that can be sent to facilitate the electromagnetic follow-up. Additionally, we evaluate the prospects of performing multimessenger observations of NSBH systems by reporting the number of expected kilonova detections with the Vera C. Rubin Observatory and the Nancy Grace Roman Space Telescope. We find that as many as $\mathcal{O}(10)$ kilonovae can be detected by these two telescopes every year, depending on the population of the NSBH systems and the equation of state of neutron stars.

The influence of metallicity on helium and CO core masses in massive stars

We present the results of 58 detailed evolutionary models of massive single stars and close binary systems with the Solar and Small Magellanic Cloud (SMC) metallicity computed with the MESA (Modules for Experiments in Stellar Astrophysics) numerical code. Helium core masses of single stars (30 M? - 75 M?) with metallicities of 0.02 and 0.0021 are in the range of 9.26 M? - 29.56 M? and 11.62 M? - 33.96 M?, respectively. Their carbon-oxygen (CO) core masses are between 5.44 M? and 25.04 M? vs. 8.23 M? and 28.38 M? for the Solar vs. SMC metallicity, accounting for an average difference of 25%. To investigate the influence of metallicity on helium and carbon-oxygen core masses in massive close Case A binary systems, detailed evolutionary models of binary systems in the mass range of 30 M? to 40 M? are calculated. The initial orbital period is set to 3 days and the accretion efficiency to 10%. The helium core mass range for primary stars with lower metallicity is 10.61 - 16.21 M? vs. 7.94 - 11.69 M? for z = 0.02. The resulting CO core masses of primary stars for the SMC metallicity are on average about 50% larger than for the Solar metallicity, so the effect is more prominent than in the case of single stars. The black hole formation limit for primary stars with the SMC metallicity is under 30 M?. While the least massive primary stars with Solar metallicity end up as neutron stars, all primary stars with the SMC metallicity and all secondary stars complete their evolution as black holes. The double compact objects resulting from the presented models are of two types: mixed neutron star-black hole systems (4 models) and double black holes (18 models). We also derive the relation between the final helium core mass and the carbon-oxygen core mass and show that it does not depend on metallicity. We confirm the CO/helium core mass ratio to be larger in binary systems than for single stars.

Sensitivity of spin-aligned searches for neutron star-black hole systems using future detectors

Current searches for gravitational waves from compact-binary objects are primarily designed to detect the dominant gravitational-wave mode and assume that the binary components have spins which are aligned with the orbital angular momentum. These choices lead to observational biases in the observed distribution of sources. Sources with significant spin-orbit precession or unequal-mass-ratios, which have non-negligible contributions from sub-dominant gravitational-wave modes, may be missed; in particular, this may significantly suppress or bias the observed neutron star -- black hole (NSBH) population. We simulate a fiducial population of NSBH mergers and determine the impact of using searches that only account for the dominant-mode and aligned spin. We compare the impact for the Advanced LIGO design, A+, LIGO Voyager, and Cosmic Explorer observatories. We find that for a fiducial population where the spin distribution is isotropic in orientation and uniform in magnitude, we will miss $\sim 25\%$ of sources with mass-ratio $q > 6$ and up to $\sim 60 \%$ of highly precessing sources $(\chi_p > 0.5)$, after accounting for the approximate increase in background. In practice, the true observational bias can be even larger due to strict signal-consistency tests applied in searches. The observation of low spin, unequal-mass-ratio sources by Advanced LIGO design and Advanced Virgo may in part be due to these selection effects. The development of a search sensitive to high mass-ratio, precessing sources may allow the detection of new binaries whose spin properties would provide key insights into the formation and astrophysics of compact objects.

Investigating GW190425 with numerical-relativity simulations

The third observing run of the LIGO-Virgo Collaboration has resulted in many gravitational wave detections, including the binary neutron star merger GW190425. However, none of these events have been accompanied with an electromagnetic transient found during extensive follow-up searches. In this article, we perform new numerical-relativity simulations of binary neutron star and black hole-neutron star systems that have a chirp mass consistent with GW190425. Assuming that the GW190425's sky location was covered with sufficient accuracy during the electromagnetic follow-up searches, we investigate whether the nondetection of the kilonova is compatible with the source parameters estimated through the gravitational-wave analysis and how one can use this information to place constraints on the properties of the system. Our simulations suggest that GW190425 is incompatible with an unequal mass binary neutron star merger with a mass ratio $q&lt;0.8$ when considering stiff or moderately stiff equations of state if the binary was face on and covered by the observation. Our analysis shows that a detailed observational result for kilonovae will be useful to constrain the mass ratio of binary neutron stars in future events.

Inferring the Neutron Star Maximum Mass and Lower Mass Gap in Neutron Star–Black Hole Systems with Spin

Gravitational-wave (GW) detections of merging neutron star–black hole (NSBH) systems probe astrophysical neutron star (NS) and black hole (BH) mass distributions, especially at the transition between NS and BH masses. Of particular interest are the maximum NS mass, minimum BH mass, and potential mass gap between them. While previous GW population analyses assumed all NSs obey the same maximum mass, if rapidly spinning NSs exist, they can extend to larger maximum masses than nonspinning NSs. In fact, several authors have proposed that the ∼2.6 M ⊙ object in the event GW190814—either the most massive NS or least massive BH observed to date—is a rapidly spinning NS. We therefore infer the NSBH mass distribution jointly with the NS spin distribution, modeling the NS maximum mass as a function of spin. Using four LIGO–Virgo NSBH events including GW190814, if we assume that the NS spin distribution is uniformly distributed up to the maximum (breakup) spin, we infer the maximum nonspinning NS mass is (90% credibility), while assuming only nonspinning NSs, the NS maximum mass must be >2.53 M ⊙ (90% credibility). The data support the mass gap’s existence, with a minimum BH mass at . With future observations, under simplified assumptions, 150 NSBH events may constrain the maximum nonspinning NS mass to ±0.02 M ⊙, and we may even measure the relation between the NS spin and maximum mass entirely from GW data. If rapidly rotating NSs exist, their spins and masses must be modeled simultaneously to avoid biasing the NS maximum mass.

Modelling the host galaxies of binary compact object mergers with observational scaling relations

ABSTRACT The merger rate density evolution of binary compact objects and the properties of their host galaxies carry crucial information to understand the sources of gravitational waves. Here, we present galaxy$\mathcal {R}$ate, a new code that estimates the merger rate density of binary compact objects and the properties of their host galaxies, based on observational scaling relations. We generate our synthetic galaxies according to the galaxy stellar mass function. We estimate the metallicity according to both the mass–metallicity relation (MZR) and the fundamental metallicity relation (FMR). Also, we take into account galaxy–galaxy mergers and the evolution of the galaxy properties from the formation to the merger of the binary compact object. We find that the merger rate density changes dramatically depending on the choice of the star-forming galaxy main sequence, especially in the case of binary black holes (BBHs) and black hole neutron star systems (BHNSs). The slope of the merger rate density of BBHs and BHNSs is steeper if we assume the MZR with respect to the FMR, because the latter predicts a shallower decrease of metallicity with redshift. In contrast, binary neutron stars (BNSs) are only mildly affected by both the galaxy main sequence and metallicity relation. Overall, BBHs and BHNSs tend to form in low-mass metal-poor galaxies and merge in high-mass metal-rich galaxies, while BNSs form and merge in massive galaxies. We predict that passive galaxies host at least ∼5–10 per cent, ∼15–25 per cent, and ∼15–35 per cent of all BNS, BHNS, and BBH mergers in the local Universe.

New pseudospectral code for the construction of initial data

Numerical studies of the dynamics of gravitational systems, e.g., black hole-neutron star systems, require physical and constraint-satisfying initial data. In this article, we present the newly developed pseudo-spectral code Elliptica, an infrastructure for construction of initial data for various binary and single gravitational systems of all kinds. The elliptic equations under consideration are solved on a single spatial hypersurface of the spacetime manifold. Using coordinate maps, the hypersurface is covered by patches whose boundaries can adapt to the surface of the compact objects. To solve elliptic equations with arbitrary boundary condition, Elliptica deploys a Schur complement domain decomposition method with a direct solver. In this version, we use cubed sphere coordinate maps and the fields are expanded using Chebyshev polynomials of the first kind. Here, we explain the building blocks of Elliptica and the initial data construction algorithm for a black hole-neutron star binary system. We perform convergence tests and evolve the data to validate our results. Within our framework, the neutron star can reach spin values close to breakup with arbitrary direction, while the black hole can have arbitrary spin with dimensionless spin magnitude $\sim 0.8$.

Modelling the formation of the first two neutron star–black hole mergers, GW200105 and GW200115: metallicity, chirp masses, and merger remnant spins

ABSTRACT The two neutron star–black hole mergers (GW200105 and GW200115) observed in gravitational waves by advanced LIGO and Virgo, mark the first ever discovery of such binaries in nature. We study these two neutron star–black hole systems through isolated binary evolution, using a grid of population synthesis models. Using both mass and spin observations (chirp mass, effective spin, and remnant spin) of the binaries, we probe their different possible formation channels in different metallicity environments. Our models only support LIGO data when assuming the black hole is non-spinning. Our results show a strong preference that GW200105 and GW200115 formed from stars with sub-solar metallicities Z ≲ 0.005. Only two metal-rich (Z = 0.02) models are in agreement with GW200115. We also find that chirp mass and remnant spins jointly aid in constraining the models, while the effective spin parameter does not add any further information. We also present the observable (i.e. post-selection effects) median values of spin and mass distribution from all our models, which may be used as a reference for future mergers. Further, we show that the remnant spin parameter distribution exhibits distinguishable features in different neutron star–black hole sub-populations. We find that non-spinning, first born black holes dominate significantly the merging neutron star–black hole population, ensuring electromagnetic counterparts to such mergers a rare affair.

Population Properties of Gravitational-wave Neutron Star–Black Hole Mergers

Abstract Over the course of the third observing run of the LIGO–Virgo–KAGRA Collaboration, several gravitational-wave (GW) neutron star–black hole (NSBH) candidates have been announced. By assuming that these candidates are real signals with astrophysical origins, we analyze the population properties of the mass and spin distributions for GW NSBH mergers. We find that the primary BH mass distribution of NSBH systems, whose shape is consistent with that inferred from the GW binary BH (BBH) primaries, can be well described as a power law with an index of α = 4.8 − 2.8 + 4.5 plus a high-mass Gaussian component peaking at ∼ 33 − 9 + 14 M ⊙ . The NS mass spectrum could be shaped as a nearly flat distribution between ∼1.0 and 2.1 M ⊙. The constrained NS maximum mass agrees with that inferred from NSs in our Galaxy. If GW190814 and GW200210 are NSBH mergers, the posterior results of the NS maximum mass would be always larger than ∼2.5 M ⊙ and significantly deviate from that inferred in Galactic NSs. The effective inspiral spin and effective precession spin of GW NSBH mergers are measured to potentially have near-zero distributions. The negligible spins for GW NSBH mergers imply that most events in the universe should be plunging events, which support the standard isolated formation channel of NSBH binaries. More NSBH mergers to be discovered in the fourth observing run would help to more precisely model the population properties of cosmological NSBH mergers.

Multimessenger Constraints on Magnetic Fields in Merging Black Hole–Neutron Star Binaries

Abstract The LIGO–Virgo–KAGRA Collaboration recently detected gravitational waves (GWs) from the merger of black hole–neutron star (BHNS) binary systems GW200105 and GW200115. No coincident electromagnetic (EM) counterparts were detected. While the mass ratio and BH spin in both systems were not sufficient to tidally disrupt the NS outside the BH event horizon, other, magnetospheric mechanisms for EM emission exist in this regime and depend sensitively on the NS magnetic field strength. Combining GW measurements with EM flux upper limits, we place upper limits on the NS surface magnetic field strength above which magnetospheric emission models would have generated an observable EM counterpart. We consider fireball models powered by the black hole battery mechanism, where energy is output in gamma rays over ≲1 s. Consistency with no detection by Fermi-GBM or INTEGRAL SPI-ACS constrains the NS surface magnetic field to ≲1015 G. Hence, joint GW detection and EM upper limits rule out the theoretical possibility that the NSs in GW200105 and GW200115, and the putative NS in GW190814, retain dipolar magnetic fields ≳1015 G until merger. They also rule out formation scenarios where strongly magnetized magnetars quickly merge with BHs. We alternatively rule out operation of the BH-battery-powered fireball mechanism in these systems. This is the first multimessenger constraint on NS magnetic fields in BHNS systems and a novel approach to probe fields at this point in NS evolution. This demonstrates the constraining power that multimessenger analyses of BHNS mergers have on BHNS formation scenarios, NS magnetic field evolution, and the physics of BHNS magnetospheric interactions.

Neutron Star–Neutron Star and Neutron Star–Black Hole Mergers: Multiband Observations and Early Warnings

Abstract The detections of gravitational waves (GWs) from binary neutron star systems and neutron star–black hole systems provide new insights into dense matter properties in extreme conditions and associated high-energy astrophysical processes. However, currently, information about the neutron star equation of state (EoS) is extracted with very limited precision. Meanwhile, the fruitful results from the serendipitous discovery of the γ-ray burst alongside GW170817 show the necessity of early warning alerts. Accurate measurements of the matter effects and sky location could be achieved by joint GW detection from space and ground. In our work, based on two example cases, GW170817 and GW200105, we use the Fisher information matrix analysis to investigate the multiband synergy between the space-borne decihertz GW detectors and the ground-based Einstein Telescope (ET). We especially focus on the parameters pertaining to the spin-induced quadrupole moment, tidal deformability, and sky localization. We demonstrate that (i) only with the help of multiband observations we can constrain the quadrupole parameter; and (ii) with the inclusion of decihertz GW detectors, the errors of tidal deformability would be a few times smaller, indicating that many more EoSs could be excluded; (iii) with the inclusion of ET, the sky localization improves by about 1 order of magnitude. Furthermore, we have systematically compared the different limits from four planned decihertz detectors and adopting two widely used waveform models.

Constraints on compact binary merger evolution from spin-orbit misalignment in gravitational-wave observations

ABSTRACT The identification of the first confirmed neutron star–black hole (NS-BH) binary mergers by the LIGO, Virgo, and KAGRA collaboration provides the opportunity to investigate the properties of the early sample of confirmed and candidate events. Here, we focus primarily on the tilt angle of the BH’s spin relative to the orbital angular momentum vector of the binary, and the implications for the physical processes that determine this tilt. The posterior tilt distributions of GW200115 and the candidate events GW190426_152155 and GW190917_114630 peak at significantly anti-aligned orientations (though display wide distributions). Producing these tilts through isolated binary evolution would require stronger natal kicks than are typically considered (and preferentially polar kicks would be ruled out), and/or an additional source of tilt such as stable mass transfer. The early sample of NS-BH events are less massive than expected for classical formation channels, and may provide evidence for efficient mass transfer that results in the merger of more massive NS-BH binaries before their evolution to the compact phase is complete. We predict that future gravitational-wave detections of NS-BH events will continue to display total binary masses of ≈7 M⊙ and mass ratios of q ∼ 3 if this interpretation is correct. Conversely, the high mass of the candidate GW191219_163120 suggests a dynamical capture origin. Large tilts in a significant fraction of merging NS-BH systems would weaken the prospects for electromagnetic detection. However, EM observations, including non-detections, can significantly tighten the constraints on spin and mass ratio.

Bayesian parameter estimation using conditional variational autoencoders for gravitational-wave astronomy

Gravitational wave (GW) detection is now commonplace and as the sensitivity of the global network of GW detectors improves, we will observe $\mathcal{O}(100)$s of transient GW events per year. The current methods used to estimate their source parameters employ optimally sensitive but computationally costly Bayesian inference approaches where typical analyses have taken between 6 hours and 5 days. For binary neutron star and neutron star black hole systems prompt counterpart electromagnetic (EM) signatures are expected on timescales of 1 second -- 1 minute and the current fastest method for alerting EM follow-up observers, can provide estimates in $\mathcal{O}(1)$ minute, on a limited range of key source parameters. Here we show that a conditional variational autoencoder pre-trained on binary black hole signals can return Bayesian posterior probability estimates. The training procedure need only be performed once for a given prior parameter space and the resulting trained machine can then generate samples describing the posterior distribution $\sim 6$ orders of magnitude faster than existing techniques.