Inferring binary black hole parameters with numerical relativity

In a new paper, my RIT graduate student Jacob Lange provides detailed demonstrations and descriptions of a new method to infer the properties of binary black holes: by direct comparison to solutions of Einstein's equations. As part of the LIGO Scientific Collaboration, we used this method to infer the properties of the first detected binary black hole (GW150914).

Why does this matter? Observations of binary black holes can tell us about the lives and deaths of massive stars. But only if we can confidently infer the binary's properties. For example, LIGO measurements could begin to differentiate between strongly precessing binaries (probably made in dynamical environments) from stellar siblings (born in isolated pairs). But inferring the properties of precessing binaries is hard.

Some of our earlier inferences have been made using approximations to solutions to Einstein's equations. These approximations are well-known to be omitting a lot of physics present in real solutions, most particularly for precessing mergers. By contrast, numerical solutions of Einstein's equations always include all available physics. So this method lets us make more precise inferences, without bias.

For experts: This methods paper describes the underpinnings and validation studies for the method used in our LVC-all paper comparing numerical solutions of Einstein's equations to GW150914. We describe and validate our one-step extraction procedure, applied to most simulations by default. We show most sources of systematic error are under control. We demonstrate our method using a variety of sources.

For more information

• Lange et al A Parameter Estimation Method that Directly Compares Gravitational Wave Observations to Numerical Relativity, submitted to PRD
• Abbott et al, Directly comparing GW150914 with numerical solutions of Einstein's equations for binary black hole coalescence, available here (summary on this article here)

Measuring BH natal kicks with GW

One of the coalescing binary black holes discovered by LIGO (GW151226 or "Boxing Day") is consistent with a scenario where the more massive black hole is spinning, with its spin misaligned with the orbital angular momentum. If true, some event or process must have imparted this misalignment. In one way the binary might have formed, this misalignment would arise when the biggest star ends its life and forms a black hole, via a "natal kick" imparted to the newborn BH. These natal kicks are known to be imparted to newborn black holes; some observations suggest they may also be imparted to BHs.

In a new paper, my collaborators and I tried to constrain the strength of these natal kicks, based on GW151226. Working under the assumption they formed from isolated pairs of stars, we found strong natal BH kicks were required.

For experts: We show how current gravitational wave observations provide new insight into how black holes form in stellar collapse. This paper is the first study to attempt to extract information about concrete astrophysical properties of supernova explosions from gravitational wave observations. It provides an easily-understoond and -generalized framework to enable the interpretation of future events. With just one event, our results begin to challenge standard supernova theory and black-hole binary formation models.

For more information, see the links below:

• ROS, Gerosa, Wysocki, "Inferences about supernova physics from gravitational-wave measurements: GW151226 spin misalignment as an indicator of strong black-hole natal kicks" (PRL 119, 011101)
, summarized in Physics
• LIGO's GW151226 paper

Surrogate models for faster GW inference

The detection of gravitational waves from massive binary black holes poses a theoretical challenge. Numerical relativity, the only accurate method to solve Einstein's equations for binary black holes, is slow, limiting the number of followup simulations that can be carried out to mimic new events. These highly-accurate solutions are also required: they include physics not incorporated in phenomenological models. Fortunately, NR solutions can be bridged by surrogate methods, which reliably interpolate between these solutions.

In a new paper, we introduce a fast method to compare the data to these NR surrogates. We demonstrate that physics uniquely available to NR or NR-surrogates significantly influences our interpretation of the gravitational wave data.

For experts: We introduce several technical improvements of use for GW parameter estimation. Our method circumvents ILE's previous need to use a discrete grid, enabling embarassingly-parallel PE for generic sources. Too, our extensions now mean ILE can use very accurate models anywhere in parameter space, being limited neither by systematic limitations of models nor discrete placement of NR simulations.

Too, we extended the gwsurrogate interface to provide direct access to the basis functions; developed tools to ``re-ROM" existing surrogates into the necessary linear basis; and built a new basis for an NR surrogate.

Most interesting, however, are the prospects for the future: this approach is very fast, scalable, accurate, and generic. With tighter integration (e.g., pre-filtering the data against a ROM basis) and GPU integration, this approach could enable streaming PE, providing the best parameter estimates in real time!

For more information, see

• ROS et al, An architecture for efficient multimodal parameter estimation with linear surrogate models, available at CQG
• Pankow et al, A novel scheme for rapid parallel parameter estimation of gravitational waves from compact binary coalescences (PRD 92 3002; arxiv:1502.04370), for comparison
• gwsurrogate, a surrogate waveform package used and extended for this work.

Do similar-looking galaxies host similar compact binary populations?

Epochs of low-metallicity star formation massively overproduce binary black holes. In any galaxy, most coalescing binary black holes often come from epochs of low-metallicity star formation, that occurred early in its lifetime. Low-mass galaxies, which are naturaly low-metallicity, will produce many more coalescing binary black holes per unit mass

The assembly history of galaxies can differ: galaxies with the same present-day properties will generally have different histories. I had guessed (incorrectly, it turns out) that this strong sensitivity meant that galaxies with similar present-day properties could host very differnt BBH properties. Using full galaxy assembly calculations, we found the opposite result: the star formation rate and metallicity vary in opposition, so the present-day BBH population is surprisingly stable to changes in the evolutionary trajectory.

For more information, see the links below, and prior posts

• Our paper O'Shaughnessy, Bellovary, et al, arxiv:1609.06715, in press at MNRAS

The origin of GW150914 via isolated binary evolution

In a paper appearing in Nature, my collaborators and I present a state-of-the-art synthetic universe (StarTrack) calculation, carefully accounting for the intrinsically rare but critical low-metallicity star-forming environments which produce binary black holes. This calculation shows how GW150914 could have formed from isolated stellar evolution. It introduces a framework with which to predict and interpret subsequent binary black hole gravitational wave events.

For context, this approach explains and produces all of the gravitational wave events and candidates seen to date, including the recently-announced GW151226. This model is being widely and rapidly adopted to prototype the future of gravitational wave astronomy. [For nonexperts, a short National Geographic article does a good job summarizing the current discussion among astrophysicists.]

For more information, see the links below, and prior posts

• Belczynski, Holz, Bulik, and O'Shaughnessy, Nature 2016
• Our history of predictions Dominik et al 2015, recently updated in Belczynski et al 2015

GW151226: a binary black hole detected via gravitational waves

Advanced LIGO has detected gravitational waves from the coalescence of two black holes, again. The new event, denoted GW151226, occurred on Dec 25th (US time).

What did we see?This event reveals another population of coalescing black holes. This time, the coalescing black holes were more similar in mass to black holes detected by other means (i.e., telescopes). Also, This coalesence of two smaller black holes

With two highly signifiant events, we can only begin to sketch the relative probabilities at which different binaries form...and only in the region we're reporting on so far.

• Masses: GW151226 was smaller and probably had one member bigger than another. Based on how far we can see sources like GW151226 versus GW150914, we can figure out how frequently the two objects occur. The lower mass object (GW151226) is most likely more frequent than the more massive one (GW150914). So, keeping in mind considerable uncertainty, the event rate is roughly consistent a population of more frequent binary mergers at lower mass, and somewhat fewer at higher mass.
• Spins: The new coalescing binary had at least one spinning black hole in it. But like before, the net effect of spin is small. And like before, we can't tell if the binary is precessing or not.
• Fundamental physics: The coalescence is still consistent with GR, as demonstrated by comparison with models and several NR similations.

What does it mean? (for experts) Personally, I think the low net aligned spins of GW150914 and GW151226, combined with the low mass of GW151226, place serious constraints on isolated chemical eveolution. Globular cluster models too are harder to reconcile with the low (and uneven) mass of GW151226.

For more information, see the links below, and prior posts

• papers.ligo.org: a one-stop location for the latest results!
• The new event: a paper; the LIGO Open Science Center data; and a summary of the latest news.

Directly comparing GW150914 to numerical relativity

GW150914 was produced by the coalescence of two black holes, in good agreement with the predictions of general relativity, as estimated by full numerical simulations of Einstein's equations. To infer the properties of GW150914, however, the LIGO Scientific Collaboration previously made systematic comparisons between the data and semianalytic models, tuned to these full numerical simulations. In a new paper, GW150914 is compared directly against a large suite of numerical simulations of Einstein's equations. These comparisons enable a completely independent reconstruction of GW150914's properties, without recourse to these approximations. Though these simulations include new physics not previously incorporated in our analysis, we find reassuringly similar conclusions regarding the source properties.

For nonexperts: The process of solving Einstein's equations on a supercomputer is called numerical relativity. All the understanding we have about the coalescence of binary black holes comes, in the end, from numerical relativity simulations. However, these simulations are notoriously challenging, requiring weeks to months of time on the world's fastest supercomputers. Working with four teams of experts in numerical relativity, including my RIT colleagues, we assembled a large collection of more than one thousand of their simulations of binary black holes.

Fortunately, because Einstein's theory of gravity works the same way for all objects, no matter their mass, we can scale each simulation to larger or smaller black holes. In other words, for every simulation, we can pick any total binary mass we want. Still, we only have one thousand simulations to cover all the other parameters needed to describe a binary black hole: the ratio between the two black hole masses, and any other intrinsic properties (spin) the black holes have.

The problem of reconstructing the black hole's properties turns into a jigsaw puzzle: we have only some of the pieces, but from experience we know the picture was drawn with a broad brush. We can boil the comparison between the data and a particular simulation and mass down to one number. This number (the marginalized likelihood) tells us the color of each piece of the puzzle. For an object like GW150914, with very few gravitational wave cycles, the colors of neighboring pieces of the puzzle are necessarily similar. So even though we're missing most of the pieces, we can still figure out what the whole puzzle looked like. Or, at least, the most important part of the puzzle: the part that describes black holes most like GW150914.

In our original analysis, which you can read about here, we were excited to identify the masses and spins of the two black holes. These provide pieces of another jigsaw puzzle: how do we make binary black holes in the first place? Our new analysis teased out a little more information about the two black holes that merged to form GW150914, using information uniquely available from numerical relativity simulations.

Perhaps most important, however, we found similar conclusions using an independent analysis that incorporated simulations of binary black hole coalescence.

This work also more explicitly illustrated a result we learned in our initial interpretation of GW150914: how binary black holes with very different dynamics can, under suitable circumstances, produce gravitational waves like GW150914. Some of the simulations which fit the data were simple, with the inspiral occurring entirely within one plane. In others, the orbital plane slews strongly (precesses) through the merger. The movies below illustrate the range of possible sources of GW150914.

Finally, by measuring how similar each simulation was to this event, we can now figure out where new simulations would fill in the gaps, for this and future events.

For experts: When higher harmonics are included (l=3), we can place tighter constraints on the mass ratio.

For more information, see the links below, and prior posts

• LIGO Science Summaries (summary on this article here)
• Abbott et al, Directly comparing GW150914 with numerical solutions of Einstein's equations for binary black hole coalescence, available here.
• APS April meeting presentation

Gravitational waves detected

Advanced LIGO has detected gravitational waves from the coalescence of two ``heavy" black holes -- source frame masses 36+29 solar masses. This amounts to three discoveries in one:

• The direct detection of gravitational waves
• The observation of a binary black hole merger, agreeing precisely with the strong-field predictions of Einstein's theory of general relativity
• The direct observation of binary black holes -- indeed, the frequent coalescence of these "heavy" binary binary black holes.
  • These black holes are significantly heavier than many people (though not everyone!) expected. This observation challenges and sharpens our understanding of how binary black holes form throughout the universe.

This report only covers some of the first science run -- the LIGO Scientific Collaboration continues to analyze the remaining O1 data, and prepare for O2. Based on this event, we expect many more detections over the next year.

For more information, see the links below, and prior posts

• papers.ligo.org: a one-stop location for the latest results!
• LIGO Open Science Center
• Event rates for aLIGO : Rodriguez et al 2015; Dominik et al 2015, recently updated in Belczynski et al 2015 , and the fiducial LIGO event rate review article

As expected...

My collaborators and I have claimed for years that advanced LIGO would frequently detect coalescing "heavy" binary black holes, most recently at the start of O1. GW150914 is consistent with our expectations, both recent and soon to be updated. Our predictions have been completely vindicated.

What's next? In our model, the binary black holes form from prompt collapse of massive stars. In this scenario, I personally expect their spins to be strictly and positively aligned with their orbital angular momentum. If GW150914 formed this way, however, LIGO data requires both black holes have surprisingly low spins.

Will future binary black hole observations support aligned spins? low spin magnitudes at birth? precessing spins? Eccentricity? A mixture of all of the above? All these options favor or disfavor different formation scenarios. Soon, LIGO observations will discriminate between them, converting a theorists' playground into an observer's paradise. Let the data decide!

For more information, see the links below, and prior posts

• Our history of predictions Dominik et al 2015, recently updated in Belczynski et al 2015.

What next?

Now that gravitational waves have been detected, what's next? We transform gravitational waves into a tool for astronomical discovery!

• The unexpected: Einstein's theory says any time-varying mass (quadrupole) radiates gravitational waves...so the universe is full of sources, some of which we may not have imagined.
• Coalescing binary black holes: LIGO will detect many more, soon, and measure their properties equally if not more precisely. We can build a large census of "heavy" black holes in the local universe -- a census we can correlate against other experience. We know how stars evolve; we see similar black holes (and their progenitors) in our own galaxy; and new large-scale surveys will complement insights from LIGO into binary star evolution. (We'll make a census of binaries including neutron stars too.)

  Next-generation instruments, if they have your support, can enhance our reach dramatically, enabling exquisitely precise measurements of hundreds of thousands of coalescing binaries per year. We'll see so many binaries, from all over the univese, that we can directly measure the formation rate versus time!

• Simultaneously seeing and hearing explosions: Multimessenger astronomy: The inspiral and merger of neutron stars, if sufficiently close by, can be detected via their gravitational waves too. But this merger often tears the neutron star apart, throwing highly radioactive material to large distances while releasing an enormous amount of gravitational energy. So we should be able to see it with telescope...and probably have: short gamma ray bursts and kilonovae.

  In other words, nature's provided us with a cosmic collider, throwing neutron stars onto other compact objects. Gravitational waves can tell telescopes where to look and identify what happened, particularly the inputs to the coalescence; electromagnetic observations tell us what comes out. Using these two probes, we'll be able to disassemble and reassemble the physics underlying these two longstanding cosmic mysteries, in a controlled fashion. And we'll learn quite a bit about nuclear physics, too.

• Neutron stars as a source and detector of gravitational waves: With roughly one billion in our galaxy, the closest and most persistent relics of stellar death are neutron stars, which uniquely are useful both as sources and detectors of gravitational waves:
  • Neutron stars as souces: If neutron stars irregular enough -- and all calculations suggest they could and indeed should be -- then we'll be able to detect these steady-state sources too. The strength of their gravitational wave signal will provide another perspective into nuclear physics.
  • The Milky Way as a detector? Neutron stars are often periodic radio sources: pulsars. The regular arrival time of these pulses can be distorted by an intervening gravitational wave. So, by timing sources, we can tell if nearby galaxies contain hidden supermassive black hole binaries: if so, we'll see distinctive patterns in their radiation.

Understanding parameter estimation for BH-NS binaries

What will gravitational waves tell us about merging compact binaries and why? For a theorist, that question can be easily albeit approximately answered by the Fisher matrix. Until recently surpassed by even-more-accurate (but slow) full Bayesian parameter estimation strategies like lalinference and ILE, the Fisher matrix has been the workhorse of gravitational wave astronomy for decades, allowing theorists to quickly calculate whether compact objects masses and spins; the nuclear equation of state; and even modifications to general relativity itself will be observationally accessible.

For nonprecesing binaries, the Fisher matrix is easily calculated from derivatives of the predicted GW signal. For precessing binaries, the intermediate quantities needed to compute the Fisher matrix were believed to be analytically intractable. (And, in many practical cases, numerically unstable.) In recent work, my collaborators and I showed how to carry out this calculation. Our extremely simple expressions agree well with the measurement accuracies found by detailed Bayesian calculations. Our estimates can be use to quickly and efficiently assess what parameters can be measured and why in astrophysically plausible sources.

For experts By expanding the spin-weighted harmonics using a corotating frame, we could perform the stationary-phase approximation term by term. If sufficiently many precession cycles occur in band, each term has a unique time-frequency trajectory and is effectively orthogonal to all others. Hence, at leading order the Fisher matrix separates into contributions from each harmonic, weighted by the power in each mode. Critically, each mode's contribution is as if from a nonprecessing binary, in a familiar and tractable form. Though our result formally applies to black hole-neutron star binaries, the a priori rarity of comparable spins implies our conclusions will be broadly applicable.

For more information, see

• Paper: A semianalytic Fisher matrix for precessing binaries with a single significant spin
• 2015 April APS Meeting: A Presentation on the method (April APS 2015)

Advanced LIGO begins operations

Advanced LIGO begins operations this week (September 18th), after 7 long years of enhancement. In O1 ("observing run 1") instruments will finally begin to confront the most optimistic predictions for how often compact binaries coalesce.

For more information, see the links below, and prior posts

• Caltech press release and RIT press release
• LIGO web site
• Event rates for aLIGO : Rodriguez et al 2015; Dominik et al 2015, and the fiducial LIGO event rate review article

Distinguishing resonant binaries via GW

Precessing binary black holes produce a complicated waveform, reflecting their time-dependent spins and the line of sight. As discussed below in ``Reliable parameter estimation with SpinTaylorF2" , particularly at high mass ratio the precession of two spinning black holes can be reasonably approximated as if the smaller body had no spin. To the extent this approximation holds, then, GW observatories would have no access to the smaller spin, and hence to unique strong-field effects associated with spin-spin interactions.

In a study led by Daniele Trifiro (Pisa), my collaborators and I demonstrate that if suitable sources occur, advanced LIGO can directly and robustly identify effects from both black hole spins in BH-BH binaries. These effects are not subtle; can be identified over a broad range of astrophysically plausible signal amplitudes and source orientations; and include the signature of conserved constants that reflect properties at compact binary birth.

For experts Using sources known to lie on one or the other post-Newtonian resonance, we construct posterior parameter distributions, each point in which can be classified as belonging to one of three classes. These classes are reliably measured. More broadly, as implied by our prior work, observationally-accessible properties of the nonlinear dynamics at merger encode conserved properties reflecting the compact binary's asymptotic initial conditions.

• Trifiro et al, Distinguishing black-hole spin-orbit resonances by their gravitational wave signatures. II: Full parameter estimation, submitted to PRD
• 2015 April APS Meeting: A presentation on the method

Instability of aligned-spin black holes

Binary black holes with exactly aligned spins remain aligned for all time. My collaborators and I have just discovered an instability: if more massive black hole has spin up (parallel to L) and the less massive black hole has spin down (antiparallel to L), then certain binaries will inevitably become unstable to spin precession, rapidly evolving to large misalignment. This instability could occur in observationally accessible astrophysical compact binaries, including supermassive binary black holes, with significant impact on the light and gravitational waves such an object would emit.

For experts Both analytically and via direct integration of the orbit-averaged 2PN spin precession equations, we have demonstrated that the up/down spin configuration is unstable; that the instability's nonlinear evolution can be understood analytically, via our previously-described nonlinear spin precession solution; that it occurs for a wide range of masses, spins, and mass ratios; and that, if nature creates up/down binaries, evidence of the instability will be present in the sensitive band of existing and future gravitational wave detectors like LIGO, LISA, and pulsar timing arrays.

For more information, see

• Gerosa et al 2015 Precessional instability in binary black holes with aligned spins, a PRL editor's suggestion
• Animations of the three figures in our paper (one, two, three); see Davide Gerosa's page.

Reliable parameter estimation with SpinTaylorF2

Multimessenger astronomy with gravitational waves relies on rapid decisions, as astronomers search for some type of transient afterglow left behind after compact binary mergers. Given limited telescope time, astronomers want to know where, when, and critically whether to point their telescopes. Astronomers in particular want to follow up mergers involving neutron stars, as these could be candidate gamma ray burst sources. Unfortunately, LIGO does not directly measure the components masses (let alone composition) via detected gravitational waves. Instead, these source parameters must be inferred by a process of systematically comparing all possible sources with the data. These exhaustive comparisons can be slow -- days to produce a reliable answer.

In a study led by Brandon Miller and I at RIT, building upon investigations started by my longstanding collaborators from the Chicago metro area, we demonstrated that a fast but approximate method for parameter estimation produces reliable answers to astrophysical questions. This method's speed advantage comes from simplified physics (ignoring the smaller object's spin) and, critically, a fast and accurate waveform model that Andy Lundgren and I introduced in 2013. Using existing codes and algorithms, this approach produces rapid and reliable-enough parameter estimates, often within an hour. Our approach shows LIGO can rapidly estimate the posterior probability that some gravitational wave data is consistent with a tidal disruption event, beamed towards us, and therefore a good candidate for followup with large electromagnetic telescopes.

For experts Using a fixed set of sources produced with SpinTaylorT2, we compare the performance of parameter estimation using several different approximants and physics, including SpinTaylorF2 (a single spin model); double-spin SpinTaylorT2; and single-spin SpinTaylorT2 and T4. Given the still-large systematic uncertainties in precessing waveform models, these approximations produce consistent predictions for posterior parameter distributions and astrophysically-motivated questions.

• Miller et al.,Systematic uncertainties in parameter estimation with the SpinTaylorF2 approximation, Submitted to PRD

Understanding double-spin evolution

As binary black holes (BBHs) spiral inward towards merger, both spin angular momenta precess, according to equations first solved analytically by my collaborators and I in 2014. In a followup to our breakthrough solution, we use our solution to understand how binary black holes precess and inspiral. We can efficiently predict and explain how spin orientations at infinity evolve into other spin configurations just prior to merger, an ingredient essential to many astrophysical processes (e.g., the evolution of supermassive black holes).

For experts In this work, we introduce a new way to describe precessing binaries that better respects the natural hierarchy of timescales of the problem; explain how these hierarchical coordinates evolve with time; and develop an extraordinarily efficient precession-averaged mechanism for evolving an ensemble of precessing binaries forward in time, treating the specific precession phase as a random variable. This exceptionally efficient approach can evolve precessing binaries from astrophysical seperations to the strong-field regime.

For more information, see ``Solving double-spin evolution" below, and

• Gerosa et al 2015 A multi-timescale analysis of phase transitions in precessing black-hole binaries, submitted to PRD, and lead author Davide Gerosa's animations
• Kesden et al 2015: Effective potentials and morphological transitions for binary black-hole spin precession (PRL 114 1103;arxiv:1411.0674)

100 years of relativity and the CCRG

This year marks 100 years since Einstein formulated his theory of general relativity. As one of our contributions to celebrating this event, the CCRG collaborated with APS TV to put together this short film about our research. At the Center for Computational Relativity and Gravitation, we are acutely aware of the enduring impact of Einstein's theory to relativistic astrophysics; to the burgoning field of gravitaitonal wave and multimessenger astronomy; and to cosmology and physics as a whole.

This film will be shown at the APS meetings in March and April, and has been featured on NSF's Science 360

ILE: an alternative architecture for parameter estimation

As noted below about lalinference, gravitational wave astronomy begins with inference : figuring out what kind of astrophysical source was responsible for the implausible event in our data. One conventional, robust, and easily generalized approach is Markov Chain Monte Carlo, where detector data is serially compared with a sequence of proposed model waveforms. For any ``reasonable" way of choosing these sequences (i.e., for any reasonable "jump proposal"), due to detailed balance, random samples from Markov Chain Monte Carlo asymptotically converge in distribution to the posterior parameter distribution. Like any Markov Chain Monte Carlo analysis, the challenge for gravitational wave parameter estimation is efficiency, scalability, and confidence in one's results, particularly given systematic errors and multiple secondary posterior maxima. Rapid and highly accurate parameter estimation is particularly important for:

• EM followup: Anyone wanting to point a telescope at a LIGO event immediately wants to know where to point and what to expect (e.g., binary black hole).
• Waveform systematics: When estimating source parameters of detected gravitational waves, the most accurate and comprehensive multimodal gravitational wave signal models are expensive to simulate and include.

In a recent work with colleagues from UWM, we proposed an alternative: the ILE architecture, short for integrate_likelihood_extrinsic. For each set of intrinsic parameters -- the parameters which specify a physically distinct binary, like component masses and spins -- our algorithm fully explores and properly weights all extrinsic parameters -- the parameters like distance, source orientation, and sky locations, which only influence how the binary appears in our instruments. Critically, our approach employs brute-force Monte Carlo integration of a low-cost likelihood function, enabling rapid processing with controlled statistical errors, even for expensive-to-compute waveforms.

For experts : By carefully reorganizing and caching waveform calculations that are used to compute the likelihood, we can efficiently marginalize over all extrinsic parameters by brute-force Monte Carlo. To further accelerate convergence, we use an adaptive Monte Carlo scheme that exploits information from the search (for sky location) and explicitly marginalize in time. With extremely low cost per likelihood, our method provides a useful complement to reduced-order methods for nonprecessing systems. However, further investigations are needed to extend this approach to explore more intrinsic dimensions, notably spin precession.

For more information, see

• Pankow et al, A novel scheme for rapid parallel parameter estimation of gravitational waves from compact binary coalescences (PRD 92 3002; arxiv:1502.04370)

Solving double-spin evolution

As binary black holes (BBHs) spiral inward towards merger, both spin angular momenta precess. This problem has long been simulated and explored numerically, and even solved when only one object has significant spin. When both objects have significant spin, however, the coupled nonlinear ODEs had been analytically intractable.

Astronomers and gravitational wave physicsts are deeply and broadly interested in carefully modeling how BBH spins precess

• Gravitational wave astronomers want to understand how to interpret the rich signal produced by the precession of stellar-mass BBHs' spins during the late stages of inspiral
• Extragalactic astronomers want to model the evolution of supermassive black holes, which merge repeatedly with companions as their host galaxies collide. At merger, the supermassive binary black hole radiates gravitational waves generally asymmetrically, causing recoil. Though the recoil strength depends precisely on the spin orientations at merger, occasionally the recoil can be strong enough to eject the remnant from its host galaxy.

In a recent study, we found a new semianalytic solution to this problem, providing new insight and faster computation for long-term spin evolution.

For experts : For relativists, our paper uses a known constant of motion of the 2PN orbit-averaged spin precession equations to solve for conservative motion, then combines that solution with a geometric ansatz for how the angular momenta evolve on radiation reaction timescales.

For more information, see

• Kesden et al 2015: Effective potentials and morphological transitions for binary black-hole spin precession (PRL 114 1103;arxiv:1411.0674)
• My slides from the 2015 Aspen winter conference "Black holes in dense star clusters"
• Articles in the popular press: UT Dallas website, Science Daily, phys.org, Media INAF (in Italian), Astroblogs (in Dutch), Science World Report, Tech Times, SpaceRef, ECN, R&D, The Daily Galaxy

LALInference: identifying sources of gravitational waves

Gravitational wave astronomy begins with inference : figuring out what kind of astrophysical source was responsible for the implausible event in our data. By exhaustively comparing that data against all candidate signals, we can reconstruct how consistent with that data any any proposed source is. In other words, we can use gravitational waves to measure the properties of the sources responsible for each gravitational wave signal we detect! These measurements will tell us how often different types of compact objects mege throughout the universe, revolutionizing our understanding of how stars and stellar systems evolve to produce these exotic binaries. And might also let us probe the nature of nuclear matter; resolve longstanding astrophysical mysteries like short gamma ray bursts; and even challenge our understanding of gravity itself.

Through the persistent efforts of several key colleagues (notably John Veitch; Vivien Raymond; and Ben and Will Farr), a subset of the LIGO Scientific Collaboration has consolidated its expertise in gravitational wave parameter estimation into a single authoritative library, lalinference.

For experts: LALInference provides a general library for accessing data and waveform libraries; evaluating the likelihood of the data, assuming gaussian noise; exploring the likelihood surface, via Markov Chain Monte Carlo or nested sampling; transforming the results of that exploration into robust statements about the posterior distribution of source parameters, given a model; and ranking different models, via their evidence. Though systematically limited by the accuracy of approximations to strong field binary motion and by ideosyncratic correlations unique to gravitational physics, gravitational wave measurements will constrain compact object masses and spins to a previously unattainable accuracy.

For more information, see

• Our paper: Veitch et al (2014)
• Sample applications of lalinference-based parameter estimation (PE):
  • Singer et al (2014), The First Two Years of Electromagnetic Follow-Up with Advanced LIGO and Virgo : PE for multimessenger astronomy
  • Wade et al (2014): PE to measure the equation of state of nuclear matter
  • Aasi et al (2014): PE applied to numerical relativity

LIGO Open Science Center

Do you want to try to find a gravitational wave? LIGO has publicly released two years of data taken by the three LIGO gravitational wave detectors during ``S5", an observing epoch from 2005-2007, along with all the tools and information needed to analyze it.

For more information, see

• The LIGO Open Science Center, a service of LIGO Laboratory and the LIGO Scientific Collaboration. LIGO is funded by the United States National Science Foundation.

Gravitational waves from eccentric binaries

Once a pair of compact objects are mutually gravitationally bound, they begin to lose energy and angular momentum to gravitational waves, a process that relatively rapidly circularizes their orbit. Over time, the orbit period decreases, until eventually the binary orbits tens to hundreds of times a second, the frequency range to which ground based gravitational wave detectors are sensitive. Since compact object binaries usually form in relatively wide orbits, with closest approach (perihelion) within a few orders of magnitude of a solar radius and hence orbital periods of minutes to days, by the time these binaries reach this sensitive band, they're almost exactly circular. But not always.

Even a slight eccentricity in the sensitive band of ground-based detectors produces a measurable impact. This impact will only impair our ability to detect gravitational waves if the eccentricity was truly extreme (e>0.4 at orbital frequency 5Hz). But eccentricity can complicate our ability to measure astrophysical parameters.

In this paper, Eliu Huerta at West Virginia University and his collaborators (myself included) introduce a simple, fast model for gravitational waves from eccentric binaries. This model will help quantify how much eccentricity matters, as well as contribute to strategies to detect eccentric sources.

For more information, see our paper: Huerta et al (2014)

What do I expect LIGO to detect?

The next gravitational wave detectors should soon operate at design sensitivity, frequently detecting gravitational waves. What will we see? What might we learn? Particularly when advanced detectors can see massive binaries out to significant redshift? In a recent paper, my collaborators and I try to figure that out.

LIGO and similar gravitational wave detectors are more sensitive to bigger (more massive) binaries. In previous work (Domink et al 2013), we showed ancient, low-metallicity star formation preferentially produced high-mass binaries at a high rate. In this work, we map out what we might detect and why:

• LIGO ought to preferentially detect more BH-BH binaries than anything else
• The detected mass distribution should be broad, including relatively massive objects.
• Detected binary black holes were formed in the early unverse. Detected double neutron star (NS-NS) binaries are formed recently.
• The detected (chirp) mass distribution for NS-NS and double black hole (BH-BH) varies significantly and on physical grounds as we change our assumptions.

For experts: For relativists, our detection rate calculations include high-mass black hole binaries, with realistic spin-dependent merger waveforms; sources at cosmologically significant distances and time-dependent merger rates; and reasonable corrections for beaming and multi-instrument sensitivity. We predict a high BH-BH merger and hence detection rate because we include the significant contribution from low-metallicity star-forming gas, integrated over all cosmic time.

For more information:

• Our current paper Dominik et al (2014) (arxiv:1405.7016)
• The two previous papers in the series: the merger rate per volume over cosmic time (Domink et al 2013) and updated massive star population synthesis, including revised common envelope and wind mass loss (Dominik et al 2012)
• What gravitational wave instruments will provide: the LIGO 'rates paper', Abadie et al 2010; an LVC survey of prospects for EM followup of GW events

RIT : Fall 2014

I am excited to be joining the faculty at the Rochester Institute of Technology in the fall!

I will be a member of the Center for Computational Relativity and Gravitation (CCRG) in the School of Mathematical Sciences (SMS) and affiliated with the Astrophysical Sciences and Technology (AST) Program in the School of Physics and Astronomy. Both SMS and AST provide students opportunities for graduate and undergraduate research.

Fraternal twins are distinguishable

Previously, my collaborators and I argued (a) that the spin and orbit orientations at birth reflect the processes that formed the binary, notably supernova, which misalign the spin and orbital plane; and (b) that some information about these orientations was preserved as the binary spiralled inwards for millions to billions of years until merger. Specifically, we claimed the relative spin orientations at merger may encode the birth order.

In a recent study, varying a limited set of parameters, we argue that the two resonant families seem to produce distinguishable gravitational waves. In other words, gravitational wave measurements might be able to identify which member of the binary was formed first.

We haven't performed an exhaustive study ("parameter estimation"); followup work is underway. But our results are a promising beginning.

For experts : For relativists, our paper uses two arguments to suggest that members of the twin resonant families can be distinguished using their gravitational waves. First, we argue based on dynamics: one resonance has both spins on the same side of J as the orbital angular momentum; the other has both spins opposite one another. In the first case, the orbital plane precesses little. In the second, it precesses a lot. Second, using a pessimistic choice of viewing direction (along J) for both source and template, minimizing precession-induced modulations, we calculate a fitting factor that is significantly different from unity.

For more information, see

• Gerosa et al, Capulets and Montagues: distinguishing the rival families of black-hole spin-orbit resonances by their gravitational-wave signatures arxiv.org:1403.7147
• Our prior work, arguing the two populations remain distinct : Gerosa et al 2013, PRD 87 4028

Measuring parameters of precessing BH-NS binaries

Precessing black hole-neutron star binaries produce a complicated and particularly informative gravitational wave signal, encoding information in two complementary channels. On the one hand, the orbit shrinks via gravitational radiation; on the other, the orbital plane wobbles, because the orbital angular momentum (and spins) precess through spin-orbit coupling.

The gravitational wave signal reflects these two complementary physical processes in two largely decoupled scales: a high-frequency, slowly-changing "carrier" signal, reflecting the slowly-shrinking orbit; and a low-frequency modulation in amplitude, phase, and polarization.

Gravitational wave detectors can constrain both processes independently, letting us measure the orbital plane and three-dimensional BH spin. Moreover, using simple estimates of how many orbits the binary makes and how much the orbit precesses, we can understand how well the binary's spins and masses can be measured.

For experts:

• Our analysis suggests simple Fisher-matrix-like studies can estimate how well detailed parameter-estimation calculations perform, even for precessing binaries.
• We found an example demonstrating that even for precessing binaries, higher harmonics can break degeneracies and rule out a significant subset of parameters.
• Our collaborators developed the currently-used-in-LIGO MCMC parameter estimation strategies for precessing binaries. Our analysis and companion studies were part of the calibration process, to gain confidence that we understood our simulations' results and hence how well parameters could be estimated.

For more information,

• ROS et al., Parameter Estimation of Gravitational Waves from Precessing BH-NS Inspirals with higher harmonics, arxiv.org:1403.0554
• Recent work by Vitale et al (1403.0554) uses the same lalinference codebase
• Recent work by the Montana State group (1402.3581) uses a different code and waveform model to assess parameter estimation of low-mass binaries, including double neutron stars.

Better coordinates for parameter estimation of precessing binaries

To draw inferences about gravitational waves from data, one must systematically compare models against data. The usual methods for doing so (Markov Chain Monte Carlo and nested sampling) wander through the parameter space, often nearby, to identify the next model to try. These methods are therefore invariably sensitive to the choice of coordinates on the set of models.

In this paper, we introduced coordinates defined the detector is sensitive, using angles relative to a conserved constant (J). These coordinates both reflect what's best-constrained by measurements and enable faster parameter estimation, by roughly an order of magnitude.

For more information, see our paper: Farr et al, "A more effective coordinate system for parameter estimation of precessing compact binaries from gravitational waves", arxiv:1404.7070

Midwest Relativity Meeting

UWM recently hosted the 23rd Midwest Relativity Meeting, with a satellite Compact Objects Meeting With more than 70 participants over three days, talks covered everything from gravitational theory to observations and astrophysics of compact binaries, showcasing the breadth of modern gravitational physics.

For more information, see

• 23rd Midwest Relativity Meeting: Participants and talks

Measuring parameters of BH-NS binaries with gravitational waves

Gravitational waves encode the properties of the source that emit them. But how, and how much? What information will gravitational waves provide to constrain the central engines of astrophysical processes (e.g., short gamma ray bursts)?

For nonexperts, our paper provides a concrete, end-to-end, and pedagogically-accessible illustration of how well existing gravitational wave detectors would do at measuring one specific source: a BH-NS binary. (Future detectors will do better, having better low-frequency sensitivity.) More important, our procedure shows that very simple approximations do a pretty good job at explaining the information gravitational waves provide.

For experts: Our paper shows that our effective Fisher matrix approximation adequately characterizes the performance of parameter estimation. Contrary to older work and in agreement with our earlier analytic study, we show that higher harmoncs provide relatively little astrophysically interesting information for typical nonprecessing BH-NS binaries. Using the output of our Markov-Chain Monte Carlo, we provide a method to quantify the number of degrees of freedom that gravitational wave measurements constrain.

For more information,

• ROS et al., Parameter Estimation of Gravitational Waves from Nonprecessing BH-NS Inspirals with higher harmonics, arxiv.org:1308.4704
• Cho et al., Gravitational waves from black hole-neutron star binaries: Effective Fisher matrices and parameter estimation using higher harmonics PRD 87 02400 (2013) : Our prior analytic work

Rare, ancient star formation and LIGO?

Previous calculations suggest star-forming environments very unlike our own ("low metallicity") formed very massive binary black holes exceptionally frequently. Because short GRBs and gravitational waves are detectable over vast distances, a careful prediction for mergers over cosmic time must include this trend.

We created a few plausible synthetic universes and populated them with merging binaries, accounting for when and where each was born; how long each binary lasts; and time-dependent star forming conditions. Because the rate at which stars form decreases and the composition of the universe gradually changes as stars produce more heavy elements, the intrinsic merger rate per unit volume increases rapidly with redshift out to the era of peak star formation (Domink et al 2013). In other words, rare ancient star formation matters.

For experts: The eight fiducial synthetic universes constructed here are used later to predict compact binary detection rates. Our calculations should also be useful when interpreting compact binary merger populations over cosmic time, including reconciling short GRB event rates and redshift distributions with galactic and (soon) LIGO constraints.

For more information, see

• Our current paper Domink et al 2013 ApJ 779 72.
• Our prior work: Dominik et al 2012 ApJ 759 52

Lectured at CGWAS summer school

At the Caltech Gravitational Wave Astrophysics summer school , my lectures provided students with a brief introduction to compact binary source populations -- that is, to what the first gravitational wave detections ought to be, and why.

Astrophysics with post-Newtonian resonances

Binary black holes spiral in very slowly through the emission of gravitational radiation; most mergers should occur millions to billions of years after the binary's birth. During this slow inspiral, coupling between angular momenta has been thought to scramble their relative orientations. That's unfortunate: the spin and orbit orientations at birth reflect the processes that formed the binary, notably supernova, which misalign the spin and orbital plane.

In Gerosa et al (2013), my collaborators and I suggest that the relative spin orientations are significantly less scrambled than previously assumed. In particular, we suggest the relative spin orientations may encode the birth order, potentially allowing gravitational wave measurements to identify which member of the binary was formed first.

For experts: For relativists, our paper demonstrates post-Newtonian resonances can be populated over a wide range of angles. For astrophysicists, our cartoon model of binary evolution proposes an alternative approach to spin alignment in binary evolution codes: tidal alignment during the Herzprung gap, prior to the common envelope phase.

For more information, see

• Our paper: Gerosa et al 2013, PRD 87 4028

A single-spin gravitational wave signal in closed form

In coming years, gravitational wave detectors should find black hole-neutron star binaries, potentially coincident with astronomical phenomena like short GRBs. These binaries are expected to precess. Precessing black hole binaries produce a complicated signal, different in each direction, that reflects how the orbital plane precesses and the relative orientation of the line of sight and that precessing plane. Gravitational wave science requires a tractable model for precessing binaries, to disentangle precession physics from other phenomena like modified strong field gravity, tidal deformability, or Hubble flow; and to measure compact object masses, spins, and alignments. Moreover, current searches for gravitational waves from compact binaries use templates where the binary does not precess and are ill-suited for detection of generic precessing sources.

In our paper , Andy Lundgren and I refactor existing expressions to provide a compact, accurate, intuitive expression for this precession, both as a function of time and a function of gravitational wave frequency. These analytic expressions enable previously-intractable analytic calculations like the Fisher matrix: a way of characterizing how similar neighboring signals are to one another and hence how well measurements can distinguish between them.

Our analysis provides an accessible foundation for the detection and interpretation of gravitational waves from generic compact binaries.

For experts By expressing the gravitational wave signal in radiation-frame coordinates, restricted to one line of sight, several authors following our work (e.g., Klein et al) have artificially introduced unnecessary coordinate-dependent complexity to the precessing gravitational wave signal. Our expressions are compact, correct, build on prior work, and enable new insight and calculations.

For more information, see

• Lundgren and O'Shaughnessy, arxiv:1304.3332 submitted to PRL
• Brown, Lundgren, and O'Shaughnessy PRD 86 4020 (2012) : A previous work easily explained using our simple result.

Precession during merger

The merger of two precessing black holes produces a complicated gravitational wave signal which in principle fully encodes its highly nonlinear dynamics. Intuition gained from analytic studies prior to merger provide a simple picture: the two black holes spiral inward, with their spin and orbital angular momenta precessing around one another. Analytic studies after merger show the isolated black hole loses its ``hair", radiating away energy and angular momentum in ``quasinormal modes," determined by eigenfunctions of the final black hole. How are these two epochs bridged for generic sources? How can observations probe those dynamics? In 2011, I identifed a robust ``corotating frame" , a way of choosing time-dependent angular coordinates at large distances using only the radiated signal. In this frame the binary's orbital plane is not precessing prior to merger. Prior to and during merger, the corotating-frame waveform signal is easy to understand and similar to previously-simulated nonprecessing results. We suspect differences can be modeled with perturbation theory. In our recent paper, we re-expressed the gravitational wave signal from simulated binary black hole mergers in this frame.

• our preferred frame seems to be physical, and connected to the line of sight polarization. Intuitively, the binary radiates cicularly polarized gravitational waves perpendicular to the instantaneous orbital plane. As the orbital plane precesses, radiation along the line of sight receives distinctive amplitude and phase modulations.
• the corotating-frame signal does not (and cannot) be identical to the signal from nonprecessing binaries. For example, corotating-frame modes generically have modulations on a precession timescale.
• the merger of precessing binaries coherently excites quasinormal modes, which "precess" around the total angular momentum direction. We speculate the details of this process provide a signature of strong-field gravity.

For more information, see

• Our paper: ROS et al 2013 PRD 87 4038
• My talk at the 2012 KITP meeting, "Rattle and Shine: Gravitational Wave and Electromagnetic Studies of Compact Binary Mergers"

How often do compact binaries merge?

I am often asked to explain how often compact binaries merge in the local universe; how often gravitational wave detectors will find the ripples in the fabric of the universe produced by these mergers; and what science these measurements enable. Recent presentations of this kind include

• January 2013, Princeton ; May, 2013, Caltech TAPIR group
• CGWAS 2013 summer school

The conservative answer involves only extrapolation from experience. We know pairs of neutron stars which will merge, in our galaxy; having lots of experience with finding neutron stars, we know how many exist in our galaxy, from what we've seen; and we know how galaxies similar to our own form stars in the nearby universe over the last few billion years. Combining these together, we find LIGO will almost certainly see the gravitational wave signal from merging NS-NS binaries.

The more aggressive answer involves substantial extrapolations away from known experience, using our best understanding of how compact binaries form. Compact binaries are known to form through binary evolution; "population synthesis" predictions involve using our best understanding of stellar evolution to generate synthetic binary populations. Compact binaries have also been observed in interacting stellar environments -- globular clusters. Both by extrapolating from experience and using our best knowledge, these interacting environments could contribute many mergers...but probably don't, based on our current expectations from the physics (e.g., Tsang et al 2013 review).

For more information, see

• Abadie et al 2010, CQG 27 173001 Predictions for the rates of compact binary coalescences : A slightly dated review of the compact binary literature provided by my collaborators and I in LIGO. My work features prominently.
• J. Aasi et al, arxiv:1304.0760 Prospects for localization of gravitational wave transients
• Dominik et al 2012 ApJ 759 52 and arxiv:1308.1456 : The most recent results

The big dog: Can gravitational wave detectors identify a signal?

The LIGO and Virgo Collaborations quizzed ourselves: a group hid a realistic signal in our data. We wanted to confirm we could find a gravitational wave and identify its properties.

This was the first time the collaborations had tested the parameter estimation codes in realistic conditions. I was the parameter estimation review co-chair: my job was to make sure any claims we made about parameters we could measure were correct. I had a particularly active role, devising specific statements which were astrophysically interesting that our data could support (e.g., on spin; whether a NS was present); organizing efforts to track down possible sources of error; coordinating between people who understood the detector and its idiosyncracies (e.g., Andy Lundgren) and our review; et cetera.

This process established new directions for the parameter estimation group, culminating in a unified code library (lalinference); a unified signal model interface and library (lalsimulation); and other technical improvements and consistency checks.

The often-torturous process of code validation has continued for years (to 2013, as of this post), long after my active role in the review. With a well-understood codebase, signal model, and detector, gravitational wave scientists are now in a position to ask pointed questions about what information future gravitational wave measurements will provide.

For more information

• GW20100916 : gravitational wave scientists quiz themselve. This is what first gravitational wave detection might look like. Descriptions are available for the public and other scientists
• LIGO publications: Science summaries