Institute of Astronomy


Astrophysical systems

Extreme-mass-ratio events

At the centre of many galaxies there exists a massive black hole (MBH) of 105–109 M. In quiescent galaxies, the MBH is surrounded by a cusp or core of 105–109 stellar mass compact objects (COs): main sequence stars, white dwarfs, neutron stars and stellar mass black holes. Mass segregation, resulting from gravitational interactions between COs, leads to the more massive objects migrating towards the centre. The capture of a CO by its MBH may be a frequent occurance in the Universe.

As a CO orbits the MBH it emits gravitational waves (GWs). As a result of scattering from other COs, the initial orbits is likely the be highly eccentric. The majority of the gravitational radiation is emitted as the object passes through periapsis. These signals are known as extreme-mass-ratio bursts (EMRBs; Rubbo, Holley-Bockelmann & Finn 2006), as a consequence of the extreme difference in the mases of the CO and MBH. As energy and angular momentum are carried away by GWs, the orbit shrinks and circularises. Eventually, significant radiation is emitted continuously, as the signal evolves to become an extreme-mass-ratio inspiral(EMRI).

EMRBs and EMRIs from MBH systems are of the right frequency range to be observed with a space-based detector, such as the Laser Interferometer Space Antenna (LISA) design concepts (Amaro-Seoane et al. 2007). LISA would be most sensitive to signals from systems with 105–106 M MBHs. EMRIs can be observed over many cycles, allowing high signal-to-noise ratios (SNRs) to accumulate. During the last year before plunge, an EMRI radiating at 3 × 10-3 Hz would complete about 105 cycles orbiting inside the LISA band. This makes EMRIs an exceptionally sensitive probe of the structure of the MBH's spacetime, this makes them ideal test of modified theories of gravity. It has been estimated that LISA could detect around 1–103 events per year, up to z ~ 1. EMRBs are short and, since they cannot be observed over multiple cycles, have lower SNRs; they can only be detected from nearby MBHs.

EMRI structure
(a) Side view of a typical EMRI orbit. (b) Emitted GW. The colour coding shows the dependence of the frequency upon distance from the MBH; there is also an overall modulation due to precession of the orbital plane. Credit: Gair (2008).

Modelling gravitational waveforms is challenging because of the non-linearity of general relativity: the background spacetime is a dynamical quantity that is distorted both by the presence of matter and the energy carried by the GWs themselves. For extreme-mass-ratio systems, the situation is simplified as we may approximate the background spacetime as that of the MBH. The CO, being much less massive, only produces a small perturbation. This allows us to use the Kerr geometry, and we can calculate many quantities analytically.

The effects of the CO's mass can be included through the introduction of the gravitational self-force (Pound 2004). This alters the motion of the CO away from a geodesic of the background spacetime. The self-force has both dissipative and conservative pieces. The former governs the evolution of an inspiral, while the latter is important to ensure the phase of the waveform is correct.

The theoretical formalism to compute the self-force has been fully developed, but the mathematical tools needed to implement that formalism have not yet been fully developed. Self-force computation is still an open problem. The simplest description of the CO is a structureless point-mass, but this introduces numerical singularities. In practice, we must introduce a spatial scale associated with the CO. This is much smaller than that associated with the MBH, making resolution difficult. Moreover, the extreme-mass-ratio ensures that there exist two timescales of significance, the orbital period and the inspiral time, that differ by many orders of magnitude. The large differences in scale make efficient computation complicated.

We work on developing new methods to provide efficient and accurate computation of the self-force. These are performed primarily in the time-domain (Canizares, Sopuerta & Jaramillo 2010Canizares & Sopuerta 2011). The goal is to make the intensive computations required for modelling astrophysical systems more tractable, such that it will be possible to build up the necessary template bank for data analysis.

Extreme-mass-ratio waveforms can be generated efficiently using the semi-relativistic approximation. We assume that the particle moves along a geodesic in the exact background geometry, but radiates as if it were in flat spacetime. This quick-and-dirty technique is known as a numerical kludge (NK). Comparison with more accurate techniques have shown NK waveforms to be fairly accurate in most cases (Babak, Fang, Gair, Glampedakis & Hughes 2007). By using the proper geodesic trajectory we ensure we have the correct frequency components, although these will not have exactly the right amplitudes.


Gravitational waves are a hitherto untapped resource for learning about the history of the Universe. It is possible to use them to investigate various questions of cosmological interest (Babak, Gair, Petiteau & Sesana 2011). They can be used to infer cosmological parameters such as the Hubble constant, or the dark-energy equation-of-state by observing standard sirens: binary systems, for example a neutron star-neutron star binary, with a characteristic frequency. These binaries are referred to as standard sirens because they can be used as milestones to measure cosmological distances. The same idea is true for Type Ia supernovae (often referred to as standard candles). However, the distances measured from GWs are directly imprinted in the waves themselves, making them a fundamental distance indicator.

Binary merger
The inspiral of a neutron star-neutron star binary system, caused by the emission of gravitational waves. This merger may have an electromagnetic counterpart. Credit: NASA.

By exploiting the inherent width of the binary neutron star mass distribution, it is possible to statistically constrain the redshift of the source. With distance and redshift, we can obtain cosmological parameter constraints using gravitational waves alone (Taylor, Gair & Mandel 2012Taylor & Gair 2012). GWs open up the possibility of investigating the nature of dark energy, as well as the star-formation rate at various epochs, without using light. The cosmological parameter constraints may be no tighter than those obtained using conventional techniques, such as the cosmic microwave background, but they are completely independent. They therefore provide a check for the cosmic distance ladder. Additionally, these systems may probe the distribution of delays between the binary formation and the final merger, as well as aspects of the neutron star mass distribution; these may be difficult to probe by conventional means.

These neutron star-neutron star systems are also regarded as the best candidate for the progenitor of a short gamma-ray burst (sGRB). These events typically emit a focused beam of high-energy electromagnetic radiation; we must be in the path of the beam to have any hope of observing it. In contrast to the electromagnetic observations, we should be able to measure the emitted GWs from most systems within the distance reach of our detectors. Observing gravitational waves from a merging neutron star-neutron star binary at the same time, or in the same sky position, as a sGRB would be convincing evidence of the link between these phenomena. We can also use the redshift information extracted from the afterglow of the gamma-ray burst with the GW-measured distances to probe cosmological parameters via the distance-redshift relation.

We not only aim to measure GWs from single sources, but also from a random background permeating the entire Universe. This stochastic gravitational wave background is similar in nature to the cosmic microwave background. It may have a variety of sources, including the superposition of waves from many inspiralling supermassive black hole binaries (Sesana et al. 2008), the GW decay of cosmic strings (Damour & Vilenkin 2005), and even fundamental early-Universe processes like inflation (Grishchuk 2005).

It is expected that this stochastic gravitational wave background may be observable using pulsar timing arrays (Foster & Backer 1990), which are sensitive in the 10-7–10-8 Hz window. The regularity of pulses from millisecond pulsars make these systems exceptionally stable clocks, such that we may measure small deviations from the expected pulse arrival-times due to perturbations caused by GWs. We use pulsars distributed across the sky because (if we have an isotropic background) the induced arrival-time deviations should be correlated between pulsars in different sky positions. This correlation is distinctive, and known as the Hellings and Downs curve (Hellings & Downs 1983). The International Pulsar Timing Array is currently organising data challenges to test if it is possible to extract information about the background.

Hellings and Downs curve
The theoretical correlation between the arrival-time deviations of pulses induced by an isotropic stochastic gravitational wave background. Image produced by Stephen Taylor.

We shall also be able to gain an insight into the history of massive black holes over cosmic times (Sesana, Gair, Berti & Volonteri 2011). Gravitational waves can be observed from both mergers of massive black holes binaries and from EMRIs about massive black holes. The signals encode properties such as the mass of the objects, and so will give insight into the mass function of the black holes (Gair, Tang & Volonteri 2010). This will inform our understanding of their growth mechanism. Since the properties of massive black holes are well correlated with those of their host galaxies, this may improve our knowledge of galaxy formation. We may be able to deduce the properties of the seed from which the black holes formed (Gair, Mandel, Sesana & Vecchio 2009). The formation mechanism is currently not well understood, but observations of quasars show that they can grow to billions of solar masses within a billion years.

Data analysis

The detection of EMRI gravitational signals has to face several issues, principally related to the fact that they will be concealed among the different kind of noises affecting the signal output, like instrumental noise and GW signals from the foreground (mainly from galactic binaries inside the EMRI frequency band). In this regard, matched filtering techniques can be employed to separate their signals from the total noise. With these methods, the signal power can be built up with each cycle, allowing for LISA detections. This in practice requires to have beforehand a bank of very accurate theoretical waveform templates to cross-correlate them with the detector data stream.

Matched filtering techniques will be used both to extract the EMRI signal from the noise, and to measure the parameters of the system associated with the signal. In order to do so, the detected signal (plus noise) is filtered out, i.e.~the detector output is cross-correlated with a filter, a modelled waveform that may match the desired signal. With this technique, when the filter matches the signal, there is a coherent contribution to the cross-correlation, since the noise contributes incoherently and is reduced in relation with the actual GW signal.

EMRI waveforms are very rich and complex signals, which are quite sensitive to the physical parameters of the system. During the last year before the plunge of the SCO onto the MBH, an EMRI system will perform around 105 orbits. Then, we would need a huge number of waveform templates to cover the EMRI signal space and perform a fully matched filter search of the parameters of the system. In turn, this will translate into a huge demand of computational resources, making necessary the obtention and computation of EMRI waveforms in an efficient and quick way. Consequently, it is very difficult to find an analytical solution to the EMRI equations of motion and to obtain accurate waveform templates. For this reason, the use and development of new numerical techniques to compute the EMRI dynamics without employing many simplifications becomes necessary.

Testing modified gravity

General relativity (GR) is an extremely successful theory. It has so far passed every experimental test we have devised. However, so far these tests have been confined to weak fields. The most exciting tests will be the ones that probe strong fields where spacetime is highly dynamic and the objects are extremely relativistic. These are the regions where GR is most likely to break down, and are exactly the region gravitational waves probe.

Tests of general relativity
Tests of GR plotting as functions of gravitational potential ε and spacetime curvature ξ. Current tests probe a wide range, but are yet to reach the most extreme regimes close to the surfaces of compact objects. The dashed line represents the event horizon of Schwarzschild black holes. Credit: Psaltis (2008).

We have considered a variety of ways in which we can test the nature of gravity. This can be done in two ways: we can consider a specific alternative theory of gravity and see what differences arise, or we can look for more general deviations from GR and see how large these need to be before we could measure them. Following the first approach, we have so far considered two alternative theories of gravity: f(R)-gravity, and dynamical Chern-Simons gravity.

In f(R)-gravity (Berry & Gair 2011) we start by modifying the action: GR can be derived starting from the Einstein-Hilbert action, this contains the Ricci scalar R, we change this to be a function of the Ricci scalar f(R). This allows us much greater freedom, and can be used to explain phenomena such as the accelerating expansion of the Universe. Since we are still using a function of the Ricci scalar, we inherit many of the useful properties and symmetries of GR. There are a number of interesting changes in f(R)-gravity. Gravitational radiation is modified to include an additional scalar mode in addition to the standard two transverse modes. If we could observe this, it would be conclusive proof that GR was incorrect. The scalar mode would only be excited above a critical cut-off frequency set by the function f(R). Some things remain the same in f(R)-gravity: light deflection, to first order, remains unchanged. The gravitational field of a massive object is modified in f(R)-gravity. This actually allows us to place the tightest constrains on this theory. We have exceedingly precise measurements in the laboratory for deviations from the Newtonian potential predicted from GR. Using these it seems that f(R) cannot deviate much from R, and it is unlikely that we could see any astrophysical effects. This assumes that the bounds derived on Earth are universal, which not necessarily be the case: f(R)-gravity could be an effective theory, which is different in the high curvature regions close to black holes than in the low curvature regions about the Earth, or the chameleon mechanism could come into play, masking the deviations in the high densities in the Solar System. Gravitational waves could thus gives us further insight.

Dynamical Chern-Simons modified gravity (Canizares, Gair & Sopuerta 2012) introduces an extra term, the Pontryagin invariant, and an extra scalar field to the gravitational action. These modifications are inspired by results from string theory and loop quantum gravity. The Pontryagin term leads to gravitational parity violation, and the black hole solution is no longer the Kerr form of GR. The energy-momentum tensor retains its usual GR form. It may be possible to observe the differences in the structure of spacetime from extreme-mass-ratio inspirals (in particular from the phase of the waveform), and so place constraints on the size of the Chern-Simons correction using gravitational wave observations. Fisher matrices can be used to quantify the amount of information encoded in a waveform. Results computed from approximate numerical kludge waveforms suggest that measurements from a LISA-like space-borne detector would be almost four orders of magnitude better than from Solar System measurements using Gravity Probe B.

While we can learn much about a proposed theory by studying it in depth and looking for observable consequences, it is difficult to do this for every possible theory. The true theory of gravity might be something we have never thought of. It is therefore useful to consider general deviations from GR. If these are not observed, we can place strong constraints on the form of any viable theory (and be more confident in our belief in GR), whilst if we do measure a deviation we can rule out many possible theories (including GR). In GR, black holes obey the no-hair theorem: astrophysical black holes can be described by just two parameters, their mass and spin. Higher multipole moments can be expressed in terms of these quantities. One potential correction to consider is that one of these higher order multipoles deviates from these relation (Gair & Yunes 2011). The resulting spacetime is known as a bumpy black hole. The change in the spacetime can leave an imprint on EMRIs. It is also possible that it may be possible to observe differences in the X-rays emitted from accretion discs about black holes.

Spinning test particles

The problem of motion in General Relativity (GR) has been extensively researched; indeed the theory was developed in part to explain observations of the motion of planets and photons in the Solar System. However, due to the complexity of the nonlinear field equations in GR, various approximation schemes must be used to calculate the evolution of a given system. In the test particle limit, we assume that the bodies under consideration do not contribute to the gravitational field and so simply move within a background spacetime. This approximation is likely to be more accurate for bodies with a mass and radius much smaller than the corresponding characteristic scales of the background, such as is found in EMRIs.

While the motion of structureless test particles is well-understood (the trajectory is simply a geodesic of the background metric), the motion of structured bodies is more uncertain. When the body is small, we may describe its structure by a series of multipole moments taken about some representative worldline (for a recent derivation see Steinhoff & Puetzfeld 2010). Restricting attention to just the pole and dipole terms leads to the Mathisson-Papapetrou (MP) equations of motion concerning the 4-momentum and the spin of the test particle (Papapetrou 1951). The MP equations must be supplied with three additional constraints in order to be deterministic. These are referred to collectively as the Spin Supplementary Condition (SSC) and are related to the choice of a representative worldline for the particle (Kyrian & Semerak 2007). Popular SSCs involve specifying the 4-velocity of some external observer, who measures the multipole moments about the centre of mass of the body.

We are interested in studying the motion of spinning test particles using different SSCs to characterise features of the trajectories in each case. We also hope to investigate the effect of the spin of the smaller body in EMRIs, calculating how this changes the emitted GWs from such a system. With a bank of GW templates, it might be possible to accurately measure the spin of the compact object or even provide evidence for one SSC over any others.

Page last updated: 17 February 2013 at 12:46