How many coalescing binaries

are there waiting to be detected?

Vassiliki Kalogera, Harvard-Smithsonian Center for Astrophysics
vkalogera@cfa.harvard.edu

The inspiral and coalescence of close binaries with two compact objects, neutron stars (NS) or black holes (BH), are considered to be some of the most important sources of gravitational waves. Assessment of their detectability is crucial and depends on two factors: (i) The strength of the inspiral gravitational radiation signal in the frequency range of interest, which determines the maximum distance ( $D_{\rm max}$) out to which coalescing binaries could be detected given a certain detection system. For LIGO II (and I), the most recent estimates of $D_{\rm max}$ are reported in the latest version of the LSC White Paper on Detector Research and Development [1]: 450Mpc (20Mpc) for NS-NS binaries, 1000Mpc (40Mpc) for NS-BH binaries, and 2000Mpc (100Mpc) for BH-BH binaries (assuming 10M$_\odot$ BH). (ii) The rate of coalescence events out to these maximum distances. This rate depends on our expectation of the Galactic coalescence rates and their extragalactic extrapolation. Using the above $D_{\rm max}$ estimates and the method of extrapolation to galaxies other than the Milky Way developed by Phinney (1991) [2] (based on the blue-light luminosities associated with galaxy star formation history), it can be estimated that the Galactic coalescence rates required for a LIGO II detection rate of 2-3 events per year are $\sim 10^{-6}$yr^-1 (NS-NS coalescence) and $\sim 10^{-8}$yr^-1 (BH-BH coalescence).

On the issue of detectability then the main question concerns estimates of the Galactic coalescence rates derived based on our current astrophysical understanding of coalescing binaries. This question has occupied the astrophysics community for about ten years now. A number of studies have appeared in the literature with a wide range of results that often create a confusing picture for the outside reader. In this article I will try to present an up-to-date review focusing on our best current bet for a coalescence rate estimate and its most important uncertainties.

Purely theoretical coalescence rates can be predicted using population synthesis models of the formation of coalescing binaries, given an evolutionary formation path. The basic idea is that an ensemble of primordial binaries, formed at a rate in accordance with the Galactic star formation rate, is followed as it evolves through a long sequence of evolutionary stages, including multiple phases of mass and angular-momentum losses, stable or unstable mass transfer, supernovae or stellar collapse events. The details of these physical processes are not very well understood at present, so a number of assumptions are necessary to obtain coalescence rate estimates and exhaustive parameter studies are essential in assessing the robustness of the results. Recent studies [3],[4],[5],[6] have mainly focused on the effect of kicks imparted to compact objects at birth, as well other uncertain factors at various levels of detail. The results obtained by varying the kick magnitudes solely lie in the ranges $< 10^{-7}-5\times 10^{-4}$yr^-1,   < 10^-7-10^-4yr^-1, and   < 10^-7-10^-5yr^-1, for NS-NS, NS-BH, and BH-BH coalescence events, respectively. Other uncertain factors can further change the estimates by factors of 10-100. Given such wide ranges of predicted rates, it becomes evident that population synthesis calculations have a rather limited predictive power and provide fairly loose constraints on coalescence rates.

The observed sample of NS-NS binaries with coalescence times shorter than 10¹⁰yr consists of only two systems, PSR B1913+16 and PSR B1534+12, but provides us with an alternative way of estimating the NS-NS coalescence rate. Phinney (1991) [2] and Narayan et al. (1991) [7] obtained the first empirical estimates based on models for radio-pulsar selection effects and estimates of the lifetimes of the observed systems. Both studies obtained an estimate of 10^-6yr^-1 assuming a NS-NS Galactic scale height of 1kpc. Since then, the increase of the Galactic volume covered by radio pulsar surveys and an upward revision of the distance estimate to PSR B1534+12 have lead to a reduction of the NS-NS coalescence rate. On the other hand, upward corrections have been applied, which account for beaming effects and the faint end of the pulsar luminosity function. Recent estimates [8],[9],[10],[11] lie in the range $6\times 10^{-7}$yr^-1 to $8\times 10^{-6}$yr^-1. I am currently involved in a study [12] in which the issues of NS-NS scale height, pulsar lifetimes, beaming, and small-number sample and faint-pulsar corrections are examined in detail. Our best estimate for the Galactic coalescence rate is $1-2\times 10^{-5}$yr^-1. Uncertainties dominated by the faint-pulsar luminosity correction (which is typically large and uncertain because of the small-number sample of close NS-NS) could decrease this estimate to $\sim 10^{-6}$yr^-1 or raise it up to $\sim 10^{-4}$yr^-1. Although a significant uncertainty in the estimate persists, it is clear that the empirical estimates of the NS-NS coalescence rate are more robust than those calculated purely theoretically.

Recently, a new candidate NS-NS system (PSR J1141-6545) was discovered by the ongoing Parkes Multibeam pulsar survey [13]. Although the nature of the pulsar companion needs confirmation (it could be a white dwarf) and the associated selection effects have not been modeled yet, a lower limit to its contribution to the empirical NS-NS coalescence rate can be estimated based solely on the pulsar lifetime [12]. Unlike the other two systems, PSR J1141-6545 is young with a characteristic age of only 1.45Myr and its total lifetime is estimated to 30.5Myr. Even if it is the only such pulsar in the Galaxy, this newly discovered system can contribute to the coalescence rate by at least $\simeq 3\times 10^{-8}$yr^-1. Taking into account all the corrections, a 10-fold upward revision of the rate would require that 50 to 200 such pulsars exist in our Galaxy.

Information about the detectability of coalescing NS-NS systems can also be obtained if robust limits to the rate can be derived. So far a safe upper limit of $\sim 10^{-4}$yr^-1 has been derived based on two different arguments: (i) the absence (until recently) of any young pulsars in close NS-NS binaries [14],[10] (this upper limit will be increased by a multiplication factor equal to the estimated number of pulsars similar to PSR J1141-6545 in the Galaxy), and (ii) the maximum ratio of the formation frequencies of coalescing NS-NS and isolated pulsars similar to those found in NS-NS systems (freed at the second supernova) and an empirical estimate of the birth rate of such isolated pulsars [15].

If we compare the estimated coalescence rates to the requirement for a LIGO II detection rate of 2-3 events per year, then we can expect a detection rate in the range of 1-10 (based on the more robust empirical estimates) or even up to $\sim 100$ per year, based on the derived upper limits. For NS-BH and BH-BH coalescence, we can only rely on purely theoretical estimates. Despite the large uncertainties (typically 3-4 orders of magnitude), the ranges for their most part lie above the requirements for a couple of events detected per year by LIGO II and imply detection rates of a few up to even 100-1000 per year. For LIGO I, a simple volume scaling shows that detection of NS-NS inspiral is rather unlikely, while BH binaries could be detected provided that the upper ends of the ranges are closer to reality.

So far we have dealt with coalescing binaries formed in galactic fields. Formation of coalescing binaries in globular clusters involves a whole range of very different processes mostly dominated by stellar interactions and also differs because of the absence of ongoing star formation over timescales comparable to the lifetimes of these binaries. The contribution of clusters to NS-NS coalescence has been found to be negligible [2]. However, a recent study [16] examined the formation of BH-BH binaries with coalescence times shorter than 10¹⁰yr and concluded that their formation rates are quite high possibly leading to LIGO II detection rates of $\sim 100$ per year (one event per two years for LIGO I). Although these predicted rates may be lower because of necessary cosmological corrections and loss of systems with very short coalescence timescales, they are still more than encouraging!

Overall, it seems fair to say that, despite the uncertainties in the rate estimates, the prospects for gravitational wave detection from the inspiral of compact binaries appear to be quite promising, especially for the upgraded LIGO interferometers.

References:

[1]  Gustafson, E., Shoemaker, D., Strain, K., and Weiss, R. 1999, LSC White Paper on Detector Research and Development (LIGO-Project document, September 11).

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[11]  Evans, T., et al. 2000, to appear in the proceedings of the XXXIVth Rencontres de Moriond on ``Gravitational Waves and Experimental Gravity", Les Arcs, France.

[12]  Kalogera, V., Narayan, R., Spergel, D., & Taylor, J. 2000, to be submitted to ApJ.

[13]  Manchester, R.N., et al. 2000, to appear in Pulsar Astronomy - 2000 and Beyond, eds. N. Wex, M. Kramer, & R. Wielebinski.

[14]  Bailes M. 1996, in Compact Stars in Binaries, IAU Symp. No. 165, eds. J. van Paradijs, E.P.J. van den Heuvel, and E. Kuulkers (Dordrecht: Kluwer Academic Publishers), 213

[15]  Kalogera, V., and Lorimer, D.R. 2000, ApJ, 530, in press, astro-ph/9907426.

[16]  Portegies-Zwart, S.F., and McMillan, S.L.W. ApJ Letters, 528, L17 (2000).