### 3.4 Radiation from a binary star system

#### 3.4.1 Radiation from a binary system and its backreaction

A binary star system can also be treated as a “centrifuge”. Two stars of the same mass in a circular orbit of radius have all their mass in nonspherical motion, so that

where is the orbital angular velocity. The gravitational wave amplitude can then be written

Since the internal radius of the orbit is not an observable, it is sometimes convenient to replace by the orbital angular frequency using the above orbit equation, giving

The gravitational wave luminosity of such a system is, by a calculation analogous to that for bumps on neutron stars (assuming that four components of to be significant),

in units given by the fundamental luminosity in Equation (16). This shows that self-gravitating systems always emit at a fraction of , since is always smaller than 1, but it can approach for highly-relativistic systems where .

The radiation of energy by the orbital motion causes the orbit to shrink. The shrinking will make any observed gravitational waves increase in frequency with time. This is called a chirp. The timescale for this in a binary system with equal masses is

As the binary evolves, the frequency and amplitude of the wave grow and this drives the binary to evolve even more rapidly. The signal’s frequency, however, will not increase indefinitely; the slow inspiral phase ends either when the stars begin to interact and merge or (if they are very compact) when the distance between the stars is roughly at the last stable orbit (LSO) , which corresponds to a gravitational wave frequency of
where we have normalized this to a binary with . This is the last stable orbit (LSO) frequency.

A compact-object binary that coalesces after passing through the last stable orbit is a powerful source of gravitational waves, with a luminosity that approaches the limiting luminosity . This is called a coalescing binary in gravitational wave searches. Since a typical search might last on the order of one year, a coalescing binary can be defined as a system that has a chirp time smaller than one year. In Figure 2 the coalescence line is shown as a straight line with slope 3/4 (set to a constant in Equation (28)). Binary systems below this line have a chirp time smaller than one year. It is evident from the figure that all binary systems observable by ground-based detectors will coalesce in less than a year.

As mentioned for gravitational wave pulsars, the raw amplitude of the radiation from a long-lived system is not by itself a good guide to its detectability, if the waveform can be predicted. Data analysis techniques like matched filtering are able to eliminate most of the detector noise and allow the recognition of weaker signals. The improvement in amplitude sensitivity is roughly proportional to the square root of the number of cycles of the waveform that one observes. For neutron stars that are observed from a frequency of 10 Hz until they coalesce, there could be on the order of 104 cycles, meaning that the sensitivity of a second-generation interferometric detector would effectively be 100 times better than its broadband (prefiltering) sensitivity. Such detectors could see typical coalescences at  200 Mpc. The event rate for second-generation detectors is estimated at around 40 events per year, with rather large error bars [103212244].

#### 3.4.2 Chirping binaries as standard sirens

When we consider real binaries we must do the calculation for systems that have unequal masses. Still assuming for the moment that the binary orbit is circular, the quadrupole amplitude turns out to be

where we define the chirp mass as
with the reduced mass, the total mass and the symmetric mass ratio. We have left out of Equation (30) a factor of order one that depends on the angle from which the binary system is viewed. The two polarization amplitudes can be found in Equation (132).

Remarkably, the other observable, namely the shrinking of the orbit as measured by the rate of change of the orbital frequency also depends on the masses just through  [291]:

In this case, the chirp time is
This is just the equal-mass chirp time of Equation (28) scaled inversely with the symmetric mass ratio . From this equation it is clear that systems with large mass ratios between the components can spend a long time in highly relativistic orbits, whereas equal-mass binaries can be expected to merge after only a few orbits in the highly relativistic regime.

If one observes and , one can infer from Equation (32). Then, from the observed amplitude in Equation (30), the only remaining unknown is the distance to the source. Gravitational wave observations of orbits that shrink because of gravitational energy losses can therefore directly determine the distance to the source [332]. By analogy with the “standard candles” of electromagnetic astronomy, these systems are now being called “standard sirens”. Although our calculation here assumed an equal-mass circular system, the conclusion is robust: any binary, even with ellipticity and extreme mass ratio, encodes its distance in its gravitational wave signal.

This is another way in which gravitational wave observations are complementary to electromagnetic ones, providing information that is hard to obtain electromagnetically. One consequence is the possibility that observations of coalescing compact object binaries could allow one to measure the Hubble constant [332] or other cosmological parameters. This will be particularly interesting for the LISA project, whose observations of black hole binaries could contribute an independent measurement of the acceleration of the universe [19713250].

Because chirping systems are so interesting we have also drawn, in Figure 2, a line where the chirp time can be measured in one year. This means that the change in frequency due to the chirp must be larger than the frequency resolution 1 yr–1. A little algebra shows that the condition for the chirp to be resolved in an observation time in a binary with period is

Since , this condition leads to a line of slope 7/11 in the logarithmic plot in Figure 2. The line drawn there corresponds to a resolution time of one year. All binaries below this line will chirp in a short enough time for their distances to be measured.

#### 3.4.3 Binary pulsar tests of gravitational radiation theory

The most famous example of the effects of gravitational radiation on an orbiting system is the Hulse–Taylor Binary Pulsar, PSR B1913+16. In this system, two neutron stars orbit in a close eccentric orbit. The pulsar provides a regular clock that allows one to deduce, from post-Newtonian effects, all the relevant orbital parameters and the masses of the stars. The key to the importance of this binary system is that all of the important parameters of the system can be measured before one takes account of the orbital shrinking due to gravitational radiation reaction. This is because a number of post-Newtonian effects on the arrival time of pulses at the Earth, such as the precession of the position of the periastron and the time-dependent gravitational redshift of the pulsar period as it approaches and recedes from its companion, can be measured accurately, and they fully determine the masses, the semi-major axis and the eccentricity of their orbit [394347].

Equation (28) for the chirp time predicts that this system would change its orbital period on the timescale (taking and )

From this one can infer that . But this has to be corrected for our oversimplification of the orbit as circular: an eccentric orbit evolves much faster because, during the phase of closest approach, the velocities are much higher, and the emitted luminosity is a very strong function of the velocity. Using equations first computed by Peters and Mathews [291], for the actual eccentricity of 0.62, one finds (see Equation (109) below) . Observations [394388] currently give . There is a significant discrepancy between these, but it can be removed by realizing that the binary system is accelerating toward the center of our galaxy, which produces a small period change. Taking this into account gives a corrected prediction of , and this agrees with the observation within the uncertainties [394358]. This is the most sensitive test that we have of the correctness of Einstein’s equations with respect to gravitational radiation, and it leaves little room for doubt in the validity of the quadrupole formula for other systems that may generate detectable radiation.

A number of other binary systems are now known in which such tests are possible [347]. The most important of the other systems is the “double pulsar” in which both neutron stars are seen as pulsars [248]. This system will soon overtake the Hulse–Taylor binary as the most accurate test of gravitational radiation.

#### 3.4.4 White-dwarf binaries

Binary systems at lower frequencies are much more abundant than coalescing binaries, and they have much longer lifetimes. LISA will look for close white-dwarf binaries in our galaxy, and will probably see thousands of them. White dwarfs are not as compact as black holes or neutron stars. Although their masses can be similar to that of a neutron star their sizes are much larger, typically 3,000 km in radius. As a result, white-dwarf binaries never reach the last stable orbit, which would occur at roughly 1.5 kHz for these masses. We will discuss the implications of multi-messenger astronomy for white-dwarf binaries in Section 7.4.

The maximum amplitude of the radiation from a white-dwarf binary will be several orders of magnitude smaller than that of a neutron star or black hole binary at the same distance but close to coalescence. However, a binary system with a short period is long lived, so the effective amplitude (after matched filtering) improves as the square root of the observing time. Besides that, these sources are nearer: there are many thousands of such systems in our galaxy radiating in the LISA frequency window above about 1 mHz, and LISA should be able to see all of them. Below 1 mHz there are even more sources, so many that LISA will not resolve them individually, but will see them blended together in a stochastic background of radiation, as shown in Figure 5.

#### 3.4.5 Supermassive black hole binaries

Observations indicate that the center of every galaxy probably hosts a black hole whose mass is in the range of  [307], with the black holes mass correlating well with the mass of the galactic bulge. A black hole whose mass is in the above range is called a supermassive black hole (SMBH). There is now abundant observational evidence that galaxies often collide and merge, and there are good reasons to believe that when that happens, friction between the SMBHs and the stars and gas of the irregular merged galaxy will lead the SMBHs to spiral into a common nucleus and (on a timescale of some 108 yr) even get close enough to be driven into complete orbital decay by gravitational radiation reaction. In many systems this should lead to a black hole merger within a Hubble time [222]. For a binary with two nonspinning black holes, the frequency of emitted gravitational waves at the last stable orbit is, from Equation (29), ; during and after the merger the frequency rises from 4 mHz to the quasi-normal-mode frequency of 24 mHz (if the spin of the final black hole is negligible). (Naturally, all these frequencies simply scale inversely with the mass for other mass ranges.) This is in the frequency range of LISA, and observing these mergers is one of the central purposes of the mission.

SMBH mergers are so spectacularly strong that they will be visible in LISA’s data stream even before applying any matched filter, although good models of the inspiral and particularly the merger radiation will be needed to extract source parameters. Because the masses of such black holes are so large, LISA can see essentially any merger that happens in its frequency band anywhere in the universe, even out to extremely high redshifts. It can thereby address astrophysical questions about the origin, growth and population of SMBHs. The recent discovery of an SMBH binary [222] and the association of X-shaped radio lobes with the merger of SMBH binaries [256] has further raised the optimism concerning SMBH merger rates, as has the suggestion that an SMBH has been observed to have been expelled from the center of its galaxy, an event that could only have happened as a result of a merger between two SMBHs [223]. The rate at which galaxies merge is about 1 yr–1 out to a red-shift of  [187], and LISA’s detection rate for SMBH mergers might be roughly the same.

Modelling of the merger of two black holes requires numerical relativity, and the accuracy and reliability of numerical simulations is now becoming good enough that they will soon become an integral part of gravitational wave searches.

#### 3.4.6 Extreme and intermediate mass-ratio inspiral sources

The SMBH environment of our own galaxy is known to contain a large number of compact objects and white dwarfs. Near the central SMBH there is a disproportionately large number of stellar-mass black holes, which have sunk there through random gravitational encounters with the normal stellar population (dynamical friction). Three body interaction will occasionally drive one of these compact objects into a capture orbit of the central SMBH. The compact object will sometimes be captured [307341340] into a highly eccentric trajectory () with the periastron close to the last stable orbit of the SMBH. Since the mass of the captured object will be about , while the SMBH will have a far greater mass, we essentially have a “test mass” falling in the geometry of a Kerr black hole. By Equation (33) we would expect that the small body would spend many orbits in the relativistic regime near the horizon of the large black hole: a black hole falling into a black hole would require on the order of 105 orbits. The emitted gravitational radiation [3201801796917159] would consist of a very long wave train that carries information about the nearly geodesic trajectory of the test body, thereby providing a very clean probe to survey the spacetime geometry of the central object (which could be a Kerr black hole or some other compact object) and test whether or not this geometry is as predicted by general relativity [32120017817770].

This kind of event happens occasionally to every SMBH that lives in the center of a galaxy. Indeed, since the SNR from matched filtering builds up in proportion to the square root of the observation time [cf. Equation (33)] and the inherent amplitude of the radiation is linear in [cf. Equation (30)], the SNR varies with the symmetric mass ratio as and typical SNR will be about ten to a thousand times smaller than an SMBH binary at the same distance. LISA will, therefore, be able to see such sources only to much smaller distances, say between 1 to 10 Gpc depending on the mass ratio. For events at such distances LISA’s SNR after matched filtering could be in the range 10 – 100, but matched filtering will be very difficult because of the complexity of the orbit, especially of its evolution due to radiation effects. However, this volume of space contains a large number of galaxies, and the event rate is expected to be several tens to hundreds per year [69]. There will be a background from more distant sources that might in the end set the limit on how much sensitivity LISA has to these events.

Due to relativistic frame dragging, for each passage of the apastron the test body could execute several nearly circular orbits at its periastron. Therefore, long periods of low-frequency, small-amplitude radiation will be followed by several cycles of high-frequency, large-amplitude radiation [3201801796917159]. The apastron slowly shrinks, while the periastron remains more or less at the same location, until the final plunge of the compact object before merger. Moreover, if the central black hole has a large spin then spin-orbit coupling leads to precession of the orbital plane thereby changing the polarization of the wave as seen by LISA.

Thus, there is a lot of structure in the waveforms owing to a number of different physical effects: contribution from higher-order multipoles due to an eccentric orbit, precession of the orbital plane, precession of the periastron, etc., and gravitational radiation backreaction plays a pivotal role in the dynamics of these systems. If one looks at the time-frequency map of such a signal one notices that the signal power is greatly smeared across the map [323], as compared to that of a sharp chirp from a nonspinning black-hole binary. For this reason, this spin modulated chirp is sometimes referred to as a smirch [325]. More commonly, such sources are called extreme mass ratio inspirals (EMRIs) and represent systems whose mass ratio is in the range of  10–3 – 10–6. Inspirals of systems with their mass ratio in the range  10–2 – 10–3 are termed intermediate mass ratio inspirals or IMRIs. These latter systems correspond to the inspiral of intermediate mass black holes of mass and might constitute a prominent source in LISA provided the central SMBH grew in mass as a result of a number of mergers of small black holes [333435].

While black hole perturbation theory with a careful treatment of radiation reaction is necessary for the description of EMRIs, IMRIs may be amenable to a description using a hybrid scheme of post-Newtonian approximations and perturbation theory. This is an area that requires more study.