Self-force on extreme mass ratio inspirals via curved spacetime effective field theory
Chad R. Galley, B. L. Hu
备注：
22 pages, 5 figures; references added, revised Appendices B & C, corrected typos, revisions throughout for clarification particularly in Section IV.B; submitted to PRD
In this series we construct an effective field theory (EFT) in curved spacetime to study gravitational radiation and backreaction effects. We begin in this paper with a derivation of the self-force on a compact object moving in the background spacetime of a supermassive black hole. The EFT approach utilizes the disparity between two length scales, which in this problem are the size of the compact object and the radius of curvature of the background spacetime, to treat the orbital dynamics of the compact object, described as an effective point particle, separately from its tidal deformations. Ultraviolet divergences are regularized using Hadamard's {\it partie finie} to isolate the non-local finite part from the quasi-local divergent part. The latter is constructed from a momentum space representation for the graviton retarded propagator and is evaluated using dimensional regularization in which only logarithmic divergences are relevant for renormalizing the parameters of the theory. As a first important application of this framework we explicitly derive the first order self-force given by Mino, Sasaki, Tanaka, Quinn and Wald. Going beyond the point particle approximation, to account for the finite size of the object, we demonstrate that for extreme mass ratio inspirals the motion of a compact object is affected by tidally induced moments at $O(ε^4)$, in the form of an Effacement Principle. The relatively large radius-to-mass ratio of a white dwarf star allows for these effects to be enhanced until the white dwarf becomes tidally disrupted, a potentially $O(ε^2)$ process, or plunges into the supermassive black hole. This work provides a new foundation for further exploration of higher order self force corrections, gravitational radiation and spinning compact objects.
The inspirals of stellar-mass compact objects into supermassive black holes are some of the most important sources for LISA. Detection techniques based on fully coherent matched filtering have been shown to be computationally intractable. We describe an efficient and robust detection method that utilizes the time-frequency evolution of such systems. We show that a typical extreme mass ratio inspiral (EMRI) source could possibly be detected at distances of up to ~2 Gpc, which would mean ~10s of EMRI sources can be detected per year using this technique. We discuss the feasibility of using this …
Compact remnants on orbits with peri-apses close to the Schwarzschild radius of a massive black hole (MBH) lose orbital energy by emitting gravitational waves (GWs) and spiral in. Scattering with other stars allows successful inspiral of such extreme mass ratio inspiral sources (EMRIs) only within small distances, a < few \times 0.01 pc from the MBH. The event rate of EMRIs is therefore dominated by the stellar dynamics and content in the inner few \times 0.01 pc. I discuss the relevant dynamical aspects and resulting estimated event rates of EMRIs. Subjects considered include the loss-cone tr…
In this review article I attempt to summarise past and present-ongoing-work on the problem of the inspiral of a small body in the gravitational field of a much more massive Kerr black hole. Such extreme mass ratio systems, expected to occur in galactic nuclei, will constitute prime sources of gravitational radiation for the future LISA gravitational radiation detector. The article's main goal is to provide a survey of basic celestial mechanics in Kerr spacetime and calculations of gravitational waveforms and backreaction on the small body's orbital motion, based on the traditional `flux-balanc…
In its final year of inspiral, a stellar mass ($1 - 10 M_\odot$) body orbits a massive ($10^5 - 10^7 M_\odot$) compact object about $10^5$ times, spiralling from several Schwarzschild radii to the last stable orbit. These orbits are deep in the massive object's strong field, so the gravitational waves that they produce probe the strong field nature of the object's spacetime. Measuring these waves can, in principle, be used to ``map'' this spacetime, allowing observers to test whether the object is a black hole or something more exotic. Such measurements will require a good theoretical understa…
An extreme mass ratio inspiral consists of two parts: adiabatic inspiral and plunge. The plunge trajectory from the innermost stable circular orbit (ISCO) is special (somewhat independent of initial conditions). We write an expression for its solution in closed-form and for the emitted waveform. In particular we extract an expression for the associated black-hole ringdown amplitudes, and evaluate them numerically.
We consider extreme mass ratio inspirals during which an electrically charged compact object with mass $m_p$ and the charge to mass ratio $q$ inspirals around a Schwarzschild black hole of mass $M$. Using the Teukolsky and generalized Sasaki-Nakamura formalisms for the gravitational and electromagnetic perturbations around a Schwarzschild black hole, we numerically calculate the energy flux of both gravitational and electromagnetic waves induced by a charged particle moving in circular orbits. With one year observation of these extreme mass ratio inspirals, we show that space-based gravitation…
An extreme mass ratio inspiral takes place when a compact stellar object is inspiraling into a supermassive black hole due to gravitational radiation reaction. Gravitational waves (GWs) from this system can be calculated using the Teukolsky equation (TE). In our case, we compute the asymptotic GW fluxes of a spinning body orbiting a Kerr black hole by solving numerically the TE both in time and frequency domain. Our ultimate goal is to produce GW templates for space-based detectors such as LISA.
We describe the hyperboloidal compactification for Teukolsky equations in Kerr spacetime. We include null infinity on the numerical grid by attaching a hyperboloidal layer to a compact domain surrounding the rotating black hole and the orbit of an inspiralling point particle. This technique allows us to study, for the first time, gravitational waveforms from large- and extreme-mass-ratio inspirals in Kerr spacetime extracted at null infinity. Tests and comparisons of our results with previous calculations establish the accuracy and efficiency of the hyperboloidal layer method.
The inspirals of stellar-mass compact objects into supermassive black holes constitute some of the most important sources for LISA. Detection of these sources using fully coherent matched filtering is computationally intractable, so alternative approaches are required. In a previous paper (Wen and Gair 2005, gr-qc/0502100), we outlined a detection method based on looking for excess power in a time-frequency spectrogram of the LISA data. The performance of the algorithm was assessed using a single `typical' trial waveform and approximations to the noise statistics. In this paper we present resu…
We study Extreme Mass Ratio Inspirals (EMRIs), during which a small body spirals into a supermassive black hole, in gravity theories with additional scalar fields. We first argue that no-hair theorems and the properties of known theories that manage to circumvent them introduce a drastic simplification to the problem: the effects of the scalar on supermassive black holes, if any, are mostly negligible for EMRIs in vast classes of theories. We then exploit this simplification to model the inspiral perturbatively and we demonstrate that the scalar charge of the small body leaves a significant im…
To investigate the imprint on the gravitational-wave emission from extreme mass-ratio inspirals in non-pure Kerr spacetimes, we have studied the ``kludge'' waveforms generated in highly-accurate, numerically-generated spacetimes containing a black hole and a self-gravitating, homogeneous torus with comparable mass and spin. In order to maximize their impact on the produced waveforms, we have considered tori that are compact, massive and close to the central black hole, investigating under what conditions the LISA experiment could detect their presence. Our results show that for a large portion…
The Laser Interferometer Space Antenna, LISA, will detect gravitational wave signals from Extreme Mass Ratio Inspirals, where a stellar mass compact object orbits a supermassive black hole and eventually plunges into it. Here we report on LISA's capability to detect whether the smaller compact object in an Extreme Mass Ratio Inspiral is endowed with a scalar field, and to measure its scalar charge -- a dimensionless quantity that acts as a measure of how much scalar field the object carries. By direct comparison of signals, we show that LISA will be able to detect and measure the scalar charge…
Inspirals of stellar mass compact objects into massive black holes are an important source for future gravitational wave detectors such as Advanced LIGO and LISA. Detection of these sources and extracting information from the signal relies on accurate theoretical models of the binary dynamics. We cast the equations describing binary inspiral in the extreme mass ratio limit in terms of action angle variables, and derive properties of general solutions using a two-timescale expansion. This provides a rigorous derivation of the prescription for computing the leading order orbital motion. As shown…
The post-Newtonian approximation is useful to discuss gravitational waveforms for binary inspirals. In this note, for retrograde circular orbits in the equatorial plane, we discuss the region of validity of the post-Newtonian approximation by using the gravitational energy flux derived numerically and analytically in the black hole perturbation approach. It is found that the edge of the allowable region behaves quite intricately, including prograde orbits even if we treat higher post-Newtonian orders.
We study binary systems in which a stellar mass compact object spirals into a massive black hole, known as extreme mass ratio inspirals, in scenarios with a new fundamental scalar field. Earlier work has shown that, in most interesting such scenarios and to leading order in the mass ratio, the massive black holes can be adequately approximated by the Kerr metric and the imprint of the scalar field on the waveform is fully controlled by the scalar charge of the stellar mass object. Here we use this drastic simplification in the inspiral modelling and consider eccentric equatorial orbits. We stu…
The extreme-mass-ratio inspirals (EMRIs) of stellar mass compact objects into massive black holes in the centres of galaxies are an important source of low-frequency gravitational waves for space-based detectors. We discuss the prospects for detecting these sources with the evolved Laser Interferometer Space Antenna (eLISA), recently proposed as an ESA mission candidate under the name NGO. We show that NGO could observe a few tens of EMRIs over its two year mission lifetime at redshifts z < 0.5 and describe how the event rate changes under possible alternative specifications of the eLISA desig…
The bumblebee gravity model, with a vector field nonminimally coupled to gravity, is a natural extension of the Einstein-Maxwell theory. In this theory, a black hole can carry a vector hair, making the metric deviate from the Schwarzschild metric. To investigate the detectability of the vector hair, we consider an Extreme Mass Ratio Inspiral (EMRI) system, where a stellar-mass black hole inspiraling into a supermassive black hole. We find that, with a one-year observation of an EMRI by a space-based gravitational-wave detector, we can probe the vector charge as small as $Q\sim 10^{-3}$ in the …
In this article we consider prospects for detecting extreme mass ratio inspirals (EMRIs) using gravitational wave (GW) observations by a future space borne interferometric observatory eLISA. We start with a description of EMRI formation channels. Different formation scenarios lead to variations in the expected event rate and predict different distributions of the orbital parameters when the GW signal enters the eLISA sensitivity band. Then we will briefly overview the available theoretical models describing the GW signal from EMRIs and describe proposed methods for their detection.
I review the status of research, conducted by a variety of independent groups, aimed at the eventual observation of Extreme Mass Ratio Inspirals (EMRIs) with gravitational wave detectors. EMRIs are binary systems in which one of the objects is much more massive than the other, and which are in a state of dynamical evolution that is dominated by the effects of gravitational radiation. Although these systems are highly relativistic, with the smaller object moving relative to the larger at nearly light-speed, they are well described by perturbative calculations which exploit the mass ratio as a n…
In the Galaxy, extremely large mass-ratio inspirals(X-MRIs) composed of brown dwarfs and the massive black hole at the Galactic Center are expected to be promising gravitational wave sources for space-borne detectors. In this work, we simulate the gravitational wave signals from twenty X-MRI systems by an axisymmetric Konoplya-Rezzolla-Zhidenko metric with varied parameters. We find that the mass, spin, and deviation parameters of the Kerr black hole could be determined accurately ( $\sim 10^{-5} - 10^{-6}$ ) with only one X-MRI event with a high signal-to-noise ratio. The measurement of the a…
The inspirals of stellar-mass mass compact objects into massive black holes in the centres of galaxies are one of the most important sources of gravitational radiation for space-based detectors like LISA or eLISA. These extreme-mass-ratio inspirals (EMRIs) will enable an ambitious research program with implications for astrophysics, cosmology, and fundamental physics. This article is a summary of the talks delivered at the plenary session on EMRIs at the 10th International LISA Symposium. It contains research updates on the following topics: astrophysics of EMRIs; EMRI science potential; and E…
The observation of the inspiral and merger of compact binaries by the LIGO/Virgo collaboration ushered in a new era in the study of strong-field gravity. We review current and future tests of strong gravity and of the Kerr paradigm with gravitational-wave interferometers, both within a theory-agnostic framework (the parametrized post-Einsteinian formalism) and in the context of specific modified theories of gravity (scalar-tensor, Einstein-dilaton-Gauss-Bonnet, dynamical Chern-Simons, Lorentz-violating, and extra dimensional theories). In this contribution we focus on (i) the information carri…
It is not currently clear how important it will be to include conservative self-force (SF) corrections in the models for extreme-mass-ratio inspiral (EMRI) waveforms that will be used to detect such signals in LISA (Laser Interferometer Space Antenna) data. These proceedings will address this issue for circular-equatorial inspirals using an approximate EMRI model that includes conservative corrections at leading post-Newtonian order. We will present estimates of the magnitude of the parameter estimation errors that would result from omitting conservative corrections, and compare these to the e…
The "flux-balance formulae" that determine the averaged evolution of energy, azimuthal angular momentum, and Carter constant in terms of the averaged asymptotic gravitational-wave fluxes for inspirals of small bodies into Kerr black holes were first derived about 15 years ago. However, this derivation is restricted to the case that the background Kerr geodesics are non-resonant (i.e., the radial and angular motions are always incommensurate), and excludes the resonant case that can be important for the radiative dynamics of extreme mass-ratio inspirals. We give here a new derivation of the flu…
How to calculate the gravitational waves (GWs) of Extreme-mass-ratio-inspirals (EMRIs) in a highly accurate and efficient way still keeps a challenge. In this paper, we present a so-called fully recalibrated waveforms for EMRIs with high accuracy. Based on the numerical data by solving the Teukolsky equations, we recalibrate all mass-ratio independent coefficients of the factorized waveforms which are used in the effective-one-body (EOB) models. Due to these new coefficients, the precision of waveforms is improved enormously, and is much higher than the original forms and at the same time high…
Intermediate/Extreme mass ratio inspiral (I/EMRI) system provides a good tool to test the nature of gravity in strong field. Based on the method of osculating orbits, we compute the orbital evolutions of I/EMRIs on quasi-elliptic orbits in both Einstein's general relativity and Brans-Dicke theory. The extra monopolar and dipolar channels in Brans-Dicke theory accelerate the orbital decay, so it is important to consider the effects of monopolar and dipolar emissions on the waveform. With the help of accurate orbital motion, we generate waveform templates which include both monopolar and dipolar…
Recurrence analysis is a well settled method allowing to discern chaos from order, and determinism from noise. We apply this tool to study time series representing geodesic and inspiraling motion of a test particle in a deformed Kerr spacetime, when deterministic chaos and different levels of stochastic noise are present. In particular, we suggest a recurrence-based criterion to reveal whether the time series comes from a deterministic source and find a noise-level threshold of its applicability.
The Kerr geometry admits the Carter symmetry, which ensures that the geodesic equations are integrable. It is shown that gravitational waveforms associated with extreme-mass-ratio inspirals involving a non-integrable compact object display `glitch' phenomena, where the frequencies of gravitational waves increase abruptly, when the orbit crosses certain spacetime regions known as Birkhoff islands. The presence or absence of these features in data from upcoming space-borne detectors will therefore allow not only for tests of general relativity, but also of fundamental spacetime symmetries.
One of the main targets of the Laser Interferometer Space Antenna (LISA) is the detection of extreme mass-ratio inspirals (EMRIs) and extremely large mass-ratio inspirals (X-MRIs). Their orbits are expected to be highly eccentric and relativistic when entering the LISA band. Under these circumstances, the inspiral time-scale given by Peters' formula loses precision and the shift of the last-stable orbit (LSO) caused by the massive black hole spin could influence the event rates estimate. We re-derive EMRIs and X-MRIs event rates by implementing two different versions of a Kerr loss-cone angle …
We study the effects of a thin gaseous accretion disk on the inspiral of a stellar--mass black hole into a supermassive black hole. We construct a phenomenological angular momentum transport equation that reproduces known disk effects. Disk torques modify the gravitational wave phase evolution to detectable levels with LISA for reasonable disk parameters. The Fourier transform of disk-modified waveforms acquires a correction with a different frequency trend than post-Newtonian vacuum terms. Such inspirals could be used to detect accretion disks with LISA and to probe their physical parameters.
Future space-based laser interferometry experiments such as LISA are expected to detect $\cal O$(100--1000) stellar-mass compact objects (e.g., black holes, neutron stars) falling into massive black holes in the centers of galaxies, the so-called extreme-mass-ratio inspirals (EMRIs). If dark matter forms a "spike" due to the growth of the massive black hole, it will induce a gravitational drag on the inspiraling object, changing its orbit and gravitational-wave signal. We show that detection of even a single dark matter spike from the EMRIs will severely constrain several popular dark matter c…
A compact object with a mass $\mathcal{O}(1 \sim 1000) M_{\odot}$, such as a black hole of stellar or primordial origin or a neutron star, and a much lighter exotic compact object with a subsolar mass could form a non-standard mini extreme mass ratio inspiral (EMRI) and emit gravitational waves within the frequency band of ground-based gravitational-wave detectors. These systems are extremely interesting because detecting them would definitively point to new physics. We study the capability of using LIGO/Virgo to search for mini-EMRIs and find that a large class of exotic compact objects can b…
We describe a new class of resonances for extreme mass-ratio inspirals (EMRIs): tidal resonances, induced by the tidal field of nearby stars or stellar-mass black holes. A tidal resonance can be viewed as a general relativistic extension of the Kozai-Lidov resonances in Newtonian systems, and is distinct from the transient resonance already known for EMRI systems. Tidal resonances will generically occur for EMRIs. By probing their influence on the phase of an EMRI waveform, we can learn about the tidal environmental of the EMRI system, albeit at the cost of a more complicated waveform model. O…
Extreme mass ratio inspirals (EMRIs) show a strong separation of timescales, with the time characterizing inspiral, $T_{\rm i}$, much longer than any time $T_{\rm o}$ characterizing orbital motions. The ratio of these timescales (which is essentially an EMRI's mass ratio) can be regarded as a parameter that controls a perturbative expansion. Here we describe the value and limitations of an "adiabatic" description of these binaries, which uses only the leading terms arising from such a two-timescale expansion. An adiabatic approach breaks down when orbits evolve through resonances, with importa…
Extreme-mass-ratio inspirals will be prized sources for the upcoming space-based gravitational-wave observatory LISA. The hunt for these is beset by many open theoretical and computational problems in both source modeling and data analysis. We draw attention here to one of the most poorly understood: the phenomenon of non-local correlations in the space of extreme-mass-ratio-inspiral signals. Such correlations are ubiquitous in the continuum of possible signals (degeneracy), and severely hinder the search for actual signals in LISA data. However, they are unlikely to manifest in a realistic se…
We investigate the possibility of sustained orbital resonances in extreme mass ratio inspirals. Using a near-identity averaging transformation, we reduce the equations of motion for a particle moving in Kerr spacetime with self-force corrections in the neighbourhood of a resonant geodesic to a one dimensional equation for a particle moving in an effective potential. From this effective equation we obtain the necessary and sufficient conditions that the self-force needs to satisfy to allow inspiralling orbits to be captured in sustained resonance. Along the way we also obtain the full non-linea…
Using black hole perturbation theory, we calculate the gravitational waves produced by test particles moving on bound geodesic orbits about rotating black holes. The orbits we consider are generic - simultaneously eccentric and inclined. The waves can be described as having radial, polar, and azimuthal "voices", each of which can be made to dominate by varying eccentricity and inclination. Although each voice is generally apparent in the waveform, the radial voice is prone to overpowering the others. We also compute the radiative fluxes of energy and axial angular momentum at infinity and thro…
The space based interferometer LISA will be capable of detecting the gravitational waves emitted by stellar mass black holes or neutron stars slowly inspiralling into the supermassive black holes found in the centre of most galaxies. The gravitational wave signal from such an extreme mass ratio inspiral (EMRI) event will provide a unique opportunity to test whether the spacetime metric around the central black hole is well described by the Kerr solution. In this paper a well studied model for EMRIs around Kerr black holes is extended to a family of parametrically deformed bumpy black holes whi…
In general relativity, there is no dispersion in gravitational waves, while some modified gravity theories predict dispersion phenomena in the propagation of gravitational waves. In this paper, we demonstrate that this dispersion will induce an observable deviation of waveforms if the orbits have large eccentricities. The mechanism is that the waveform modes with different frequencies will be emitted at the same time due to the existence of eccentricity. During the propagation, because of the dispersion, the arrival time of different modes will be different, then produce the deviation and deph…
Black hole binaries with extreme ($\gtrsim 10^4:1$) or intermediate ($\sim 10^2-10^4:1$) mass ratios are among the most interesting gravitational wave sources that are expected to be detected by the proposed Laser Interferometer Space Antenna. These sources have the potential to tell us much about astrophysics, but are also of unique importance for testing aspects of the general theory of relativity in the strong field regime. Here we discuss these sources from the perspectives of astrophysics, data analysis, and applications to testing general relativity, providing both a description of the c…
Bayesian inference for tidal heating with extreme mass ratio inspirals
Hunan Normal University（湖南师范大学）；Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院）；Yangzhou University（扬州大学）；Universidade de São Paulo（圣保罗大学）；Universidade Estadual Paulista（圣保罗州立大学）
