6 Prospects of Gravitational Radiation

In the coming decades, several gravitational wave observatories will begin detecting signals from relativistic binaries. It is likely that several of these binaries will have their origins within globular clusters. In this section, we review the detectors and their expected sources from globular clusters.

Ground-based interferometers such as the Laser Interferometer Gravitational-wave Observatory (LIGO) [299] in the United States, Virgo [476] in Europe, or the Large-scale Cryogenic Gravitational wave Telescope (LCGT) [287] in Japan, will be sensitive to gravitational radiation in the frequency range from a few Hz to a few kHz. In this frequency range, these detectors will be sensitive to the inspiral and merger of binaries containing neutron stars or black holes. These systems are somewhat rare in the Galactic globular cluster system, and the probability of a coalescence occurring within our Galaxy while the detectors are operating is vanishingly small. The enhanced design of LIGO was operational until the end of 2010, but did not detect any known gravitational wave signal. The enhanced design of Virgo continued operating into early 2012. Both Virgo and LIGO will be upgrading to advanced designs which will provide significant increases in sensitivity. Like advanced LIGO (aLIGO) and advanced Virgo (AdV), LCGT is a next-generation detector. It is unique in that it will be in an underground facility with and it will operate cryogenically. All of these detectors should be sensitive to double neutron star inspirals at distances of a few hundred Mpc, and double black hole mergers at distances of about a Gpc. At these distances, the number of extragalactic globular clusters is large enough that coalescences of neutron star or black hole binaries may be commonplace. Unfortunately, the angular resolution of the network of next-generation detectors will be much too coarse to identify any detected signal with a globular cluster. However, the gravitational waveform of the coalescence of these objects depends upon the orientation of the spins of the components relative to the orbital angular momentum. Although it is not entirely clear if field binaries will tend to have their spins aligned, there is no reason for this to be the case with binaries that have been formed dynamically through exchange of compact objects. Estimates of the event rates of compact object binary coalescence during the advanced detector era tend to weight the binary neutron star coalescences toward field binaries and the binary black hole coalescences toward cluster binaries. Successful parameter estimation for the properties of coalescing binaries in advanced detectors will require accurate modeling of the waveform, including spin effects for both black holes and neutron stars, and tidal effects for neutron stars.

Binary systems containing white dwarfs will be brought into contact at orbital periods of a few hundred seconds. At contact, the system will either coalesce on a dynamical timescale if the mass transfer is unstable, or it will begin to spiral out if the mass transfer is stable. In any case, these systems will never produce gravitational radiation in the frequency band of ground-based interferometers. Space-based interferometers are capable of achieving sufficient sensitivity in the millihertz band. Thus, the sources for space-based interferometers will include white dwarf, neutron star, and black hole binaries. At these orbital periods, the systems are not very relativistic and the gravitational waveforms can safely be described by the quadrapole formula of Peters & Mathews [371Jump To The Next Citation Point, 370]. An angle averaged estimate of the typical strain amplitude is [55Jump To The Next Citation Point]

( )2 ∕3 ( ) ( )5∕3 −21 --fgw--- 1 kpc- ℳ--- h0 = 1.5 × 10 10− 3 Hz r M ⊙ . (45 )
At a typical globular cluster distance of r ∼ 10 kpc and typical chirp mass of ℳ ∼ 0.5M ⊙, a relativistic WD–WD or WD–NS binary with Porb = 400 s will have a gravitational wave amplitude of 10− 22. This value is within the range of proposed space-based gravitational wave observatories of the LISA class [55Jump To The Next Citation Point].

Many globular clusters lie off the plane of the galaxy and are relatively isolated systems with known positions. The angular resolution of LISA improves with signal strength. By focusing the search for gravitational radiation using known positions of suspected sources, it is possible to increase the signal-to-noise ratio for the detected signal. Thus, the angular resolution of LISA for globular cluster sources can be on the order of the angular size of the globular cluster itself at fgw > 1 mHz. Consequently, the orbital period distribution of a globular cluster’s population of relativistic binaries can be determined through observations in gravitational radiation. We will discuss the prospects for observing each class of relativistic binaries covered in this review.

WD–WD binaries that are formed from a common envelope phase will be briefly visible while the recently revealed hot core of the secondary cools. These objects are most likely the “non-flickerers” of Cool et al. [82] and Edmonds et al. [116]. WD–WD binaries formed through exchange interactions may very well harbor white dwarfs which are too cool to be observed. In either case, hardening through dynamical interactions will become less likely as the orbit shrinks and the effective cross section of the binary becomes too small. These objects will then be effectively invisible in electromagnetic radiation until they are brought into contact and RLOF can begin. During this invisible phase, the orbital period is ground down through the emission of gravitational radiation until the orbital period is a few hundred seconds [51]. With a frequency of 1 to 10 mHz, gravitational radiation from such a binary will be in the band of LISA [55].

WD–NS binaries that are expected to be progenitors of the millisecond pulsars must pass through a phase of gravitational radiation after the degenerate core of the donor star emerges from the common envelope phase and before the spin-up phase begins with the onset of mass transfer from the white dwarf to the neutron star. The orbital period at the onset of RLOF will be on the order of 1 to 2 minutes and the gravitational wave signal will be received at LISA with a signal-to-noise of 50 – 100 at a frequency of around 20 mHz for a globular cluster binary.

Binaries with significant eccentricity will have a spectrum of harmonics of the orbital frequency, with the relative strength of the nth harmonic for eccentricity e given by [371]

{[ ]2 n4- 2- g(n,e) = 32 Jn−2(ne) − Jn−1(ne ) + n Jn(ne ) + Jn+1 (ne ) − Jn+2 (ne) 4 } +(1 − e2)[Jn−2(ne )− 2Jn(ne)+Jn+2 (ne )]2+ ----[Jn(ne)]2 , (46 ) 3n2
where Jn is the Bessel function. The higher harmonics of sufficiently eccentric binaries (e > 0.7) can be detected by LISA even though the fundamental orbital frequency is well below its sensitivity band of 1 – 100 mHz [53Jump To The Next Citation Point]. Dynamical interactions may produce an eccentric population of 30 – 140 white dwarf binaries that would be present in the LISA data after a 5 year observation [487].

The Galactic globular cluster population of NS–NS binaries is expected to be quite small (≲ 10), they may have high eccentricities. As such, their higher harmonics may have sufficient power to be detectable in the LISA band even if their orbital period is too low. The binary pulsar B2127+11C is an example of a NS–NS binary in a globular cluster. In terms of the unknown angle of inclination i, the companion mass to the pulsar is M2 sini ∼ 1 M ⊙ and its eccentricity is e = 0.68 [296]. These binaries may also be detectable by LISA. If the globular cluster systems of other galaxies follow similar evolution as the Milky Way population, these binaries may be potential sources for LIGO as gravitational radiation grinds them down to coalescence. With an expected reach of over 300 Mpc, aLIGO can be expected to observe 40 field DNS systems per year [7Jump To The Next Citation Point]. It is uncertain how many DNS coalescences would be added to this rate by globular clusters and it would be difficult to distinguish between the two populations.

With their high eccentricities and large chirp mass, black hole binaries will also be good potential LISA sources for gravitational radiation from the galactic globular cluster system [52, 53]. Although the orbits are expected to have circularized by the time the enter the advanced LIGO frequency band, aLIGO is expected to observe DBH systems out to a typical distance of 1 Gpc, with optimally oriented systems detectable out to over 2 Gpc. Thus, it will be able to probe an enormous population of globular cluster systems. Estimates of the expected rate of DBH coalescences of field binaries range from a pessimistic 0.4 per year to an optimistic 1000 per year, with a realistic estimate of 20 per year [7]. Given the wide range of predicted enhancements in the DBH population in globular clusters, it is possible that this rate may be significantly enhanced by globular cluster binaries. As noted previously, if there is a preference for spin alignments in field binaries, the spin distribution of observed binaries may be used to distinguish globular cluster binaries from field binaries. Finally, we note that IMBHs may be potential sources of gravitational radiation for both space-based and ground-based detectors. The low mass end of the IMBH class (∼ 100 M ⊙) will be observable from ground-based detectors out to very large distances, while the extreme mass ratio inspiral of stellar mass compact objects around the high mass end of the IMBH class will be observable from space-based detectors.

The relatively close proximity of the galactic globular cluster system and the separations between individual globular clusters allows for the identification of gravitational radiation sources with their individual host clusters. Although the expected angular resolution of LISA is not small enough to allow for the identification of individual sources, knowledge of the positions of the clusters will allow for focused searches of the relativistic binary populations of the majority of the galactic globular clusters. Armed with a knowledge of the orbital periods of any detected binaries, concentrated searches in electromagnetic radiation can be successful in identifying relativistic binaries that may have otherwise been missed.

There may be a few double black hole systems within the Galactic globular cluster system that are either within the LISA band, or are sufficiently eccentric that their peak gravitational wave power is in a high harmonic that is within the LISA band. In either case, such systems would be easily detectable by a space-based gravitational wave detector due to their large chirp masses. However, a greater number of double black hole systems are expected to have been ejected from their birth globular clusters and these would now be inhabitants of the Galactic halo. If these systems have been formed dynamically, they are likely to be composed of two relatively massive black holes, and they may have chirp masses that are high enough for them to be detected at extragalactic distances.

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