Relativistic Effects in Extreme Mass Ratio Gravitational Wave Bursts

TL;DR

This paper examines how relativistic effects influence gravitational wave signals from extreme mass ratio bursts, highlighting their significance for detection and black hole spin measurement.

Contribution

It introduces a geodesic-based relativistic model for EMRBs and assesses the impact of relativistic corrections on waveform accuracy and parameter estimation.

Findings

Relativistic EMRBs are common, constituting about half of the events.

Relativistic corrections significantly alter waveform amplitude and phase.

Dephasing could help measure black hole spin and test relativistic gravity.

Abstract

Extreme mass ratio bursts (EMRBs) have been proposed as a possible source for future space-borne gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA). These events are characterized by long-period, nearly-radial orbits of compact objects around a central massive black hole. The gravitational radiation emitted during such events consists of a short burst, corresponding to periapse passage, followed by a longer, silent interval. In this paper we investigate the impact of including relativistic corrections to the description of the compact object's trajectory via a geodesic treatment, as well as including higher-order multipole corrections in the waveform calculation. The degree to which the relativistic corrections are important depends on the EMRB's orbital parameters. We find that relativistic EMRBs (v_{max}}/c > 0.25) are not rare and actually account for approximately half of the events in our astrophysical model. The relativistic corrections tend to significantly change the waveform amplitude and phase relative to a Newtonian description, although some of this dephasing could be mimicked by parameter errors. The dephasing over several bursts could be of particular importance not only to gravitational wave detection, but also to parameter estimation, since it is highly correlated to the spin of the massive black hole. Consequently, we postulate that if a relativistic EMRB is detected, such dephasing might be used to probe the relativistic character of the massive black hole and obtain information about its spin.

Figures

Tables

Table 1. Table 1 SNR and overlap for different EMRB orbits.

$r_p$ [$\mu$pc]	$v_p/c$	$P$ [yrs]	$e$	$\#$ Bursts	$\frac{\left|\Delta \rho \right|}{\rho }$	$\frac{\left|\Delta {\rho }^{\left(1\right)}\right|}{{\rho }^{\left(1\right)}}$	$\left(h_1|h_2\right)$	${\left(h_1|h_2\right)}^{\left(1\right)}$
$7.8$	$0.196$	$0.0042$	$0.603$	$6$	$0.38$	$0.049$	$0.27$	$0.98$
$7.5$	$0.207$	$0.0055$	$0.684$	$5$	$0.13$	$0.0007$	$0.20$	$0.88$
$7.2$	$0.217$	$0.0074$	$0.749$	$4$	$0.22$	$0.057$	$0.31$	$0.90$
$7.1$	$0.220$	$0.0086$	$0.777$	$3$	$0.49$	$0.061$	$0.33$	$0.99$
$13$	$0.163$	$0.0450$	$0.859$	$1$	$\cdot$	$0.002$	$\cdot$	$0.90$
$8.0$	$0.217$	$0.1953$	$0.968$	$1$	$\cdot$	$0.033$	$\cdot$	$0.97$
$10$	$0.193$	$0.7683$	$0.984$	$1$	$\cdot$	$0.009$	$\cdot$	$0.72$
$12$	$0.173$	$3.0407$	$0.992$	$1$	$\cdot$	$0.015$	$\cdot$	$0.80$

Table 2. Table 2 Comparison between quadrupole and octopole waveforms.

Angle [degrees]	$x_{\text{obs}}$ [kpc]	$y_{\text{obs}}$ [kpc]	$z_{\text{obs}}$ [kpc]	Amp. diff.
$0$	$0$	$0$	$8$	$3.9\%$
$20$	$2.73$	$0$	$7.51$	$3.3\%$
$40$	$5.14$	$0$	$6.12$	$11.5\%$
$60$	$6.93$	$0$	$4$	$15.2\%$
$80$	$7.88$	$0$	$1.39$	$19.7\%$
$100$	$7.88$	$0$	$-1.39$	$21.6\%$
$120$	$6.93$	$0$	$-4$	$17.8\%$
$140$	$5.14$	$0$	$-6.13$	$43.3\%$
$160$	$2.74$	$0$	$-7.52$	$8.6\%$

Equations

$$R_{\bullet }$$

$$a^i=F^i[v^j;\text{mbox}{g}_{\mu \nu },{\partial }_j\text{mbox}{g}_{\mu \nu }],$$

$$\frac{T_{\text{orb}}}{T_{\text{rr}}}\sim 2\pi \mu {\left(\frac{R_{\bullet }}{2p}\right)}^{5/2},$$

$$r_p>(7.98\times 10^{-7}\text{pc})\frac{(1-e)}{0.1}{\left(\frac{M_{\bullet }}{M_{\text{mbox}MW}}\right)}^{1/3}{\left(\frac{T}{T_{\text{cut}}}\right)}^{2/3},$$

$$r_{\mathrm{cap}}=(1.91\times 10^{-7}\text{pc})\frac{M_{\bullet }}{M_{\text{mbox}MW}}\left[1+\sqrt{1-\frac{a_{\bullet }^2}{M_{\bullet }^2}}\right],$$

$$h_{+,\times }=\frac{2Gm}{r{c}^4}{ϵ}_{+,\times }^{ij}\left(a_i{z}_j+v_i{v}_j\right),$$

$$h_{+,\times }$$

$$z_0^i$$

$$v_0^i$$

$$r_p$$

$$|v_p|$$

$$z_0^i$$

$$v_0^i$$

$$(h_1|h_2)=\frac{{\int }_0^T{h}_1(t)h_2(t)dt}{\sqrt{{\int }_0^T{h}_1^2(t)dt}\sqrt{{\int }_0^T{h}_2^2(t)dt}}.$$

$${\rho }^2=4{\int }_0^{\infty }\frac{|\tilde{h}(f){|}^2}{S_n(f)}df,$$

Full text

1. Introduction

Low-frequency ($10^{-5}\lesssim f\lesssim 0.1$ Hz) gravitational wave interferometers, such as the proposed Laser Interferometer Space Antenna (LISA) (Bender et al., 1998; Danzmann & Rüdiger, 2003; Sumner & Shaul, 2004), will open a completely new window to the Universe. Through observations of low-frequency gravitational radiation we will be able to witness the inspiral and merger of massive black hole binaries; the inspiral of compact objects into massive black holes; and millions of quasi-stationary compact galactic binaries. Recently, a new source of low-frequency gravitational radiation has been suggested: extreme mass ratio bursts (EMRBs) (Rubbo et al., 2006, 2007).

EMRBs consist of a stellar-mass compact object (SCO) orbiting a massive black hole (MBH) of $10^{4-8}~{}\textrm{M}_{\odot}$ with orbital periods greater than $T_{{\textrm{cut}}}=3\times 10^{4}$ s. The defining orbital period cutoff is derived from LISA’s lower frequency limit of $f_{{\textrm{cut}}}=3\times 10^{-5}$ Hz. Systems with orbital periods less than $T_{{\textrm{cut}}}$ will radiate continuously inside the LISA band. Such continuous systems are more appropriately categorized as extreme mass ratio inspirals (EMRIs) and have been studied extensively elsewhere: recent estimations of the event rate are given by Gair et al. (2004) and Hopman & Alexander (2006); a discussion on a possible EMRI background is given by Barack & Cutler (2004a); accounts on the theoretical description of EMRIs can be found in the reviews by Poisson (2004) and Glampedakis (2005), while the recent review by Amaro-Seoane et al. (2007) describes the astrophysical and detection applications.

Although EMRB events are distinct from EMRI events, their evolutionary track could be connected. In the burst scenario, the SCO orbits the MBH emitting a beamed burst of gravitational radiation during pericenter passage. The emitted radiation carries away energy and angular momentum from the system so that after multiple pericenter passages the orbital period decreases, and possibly the system becomes an EMRI. However, this evolutionary track is most likely disrupted by scattering interactions with other stars and/or if the SCO plunges directly into the central MBH on one of its passages.

The EMRB event rate has recently been investigated using simplified galactic models and data analysis techniques (Rubbo et al., 2006, 2007; Hopman et al., 2006). Using a density profile described by an $\eta$-model (Tremaine et al., 1994), Rubbo et al. (2006, 2007) suggested an event rate of $\sim\!15~{}{\textrm{yr}}^{-1}$ for events with signal-to-noise ratios (SNRs) greater than five out to the Virgo cluster. When mass segregation and different inner cusp models are considered, the predicted rate decreases by an order of magnitude (Hopman et al., 2006). These preliminary studies were aimed at understanding if EMRB event rates are interesting for low-frequency gravitational wave detectors such as LISA. More work is still needed to improve the predicted event rate in the context of realistic galaxies, where the role of non-equilibrium dynamics, anisotropy, complex star formation histories, substructure, and non-sphericity may act to change the rate from these fiducial estimates by orders of magnitude (Holley-Bockelmann & Sigurdsson, 2006; Rubbo et al., 2006, 2007).

In addition to the astrophysical uncertainties, there are no investigations of the impact of relativistic corrections to EMRB dynamics. All EMRB studies have been carried out in a quasi-Newtonian framework. In this framework, one uses the Newtonian equations of motion and extracts the gravitational waveforms by means of the quadrupole formula. This approximation ignores the black hole nature of the central potential, including the black hole’s rotation (spin), and is technically valid only for orbits with non-relativistic velocities. However, a considerable number of EMRBs are characterized by large pericenter velocities ($v_{p}\gtrsim 0.25\,c$) and these relativistic EMRBs should produce gravitational wave signals with larger SNRs, as we will show later.

In this paper, we shall not consider the issue of EMRB event rates, but instead we shall study the effects of relativistic corrections to such events. For extreme-mass-ratio systems a simple way of introducing relativistic corrections is by using the so-called semi-relativistic approximation introduced by Ruffini & Sasaki (1981), and used recently in the context of EMRIs by Gair et al. (2005, 2006). In this approximation, the MBH and surrounding area are modeled using the Kerr solution to Einstein’s field equations, which describes a stationary spinning black hole (the Schwarzschild solution corresponds to the non-spinning case). The SCO is considered to be a point-like object (neglecting its own self-gravity) whose trajectory is described by a geodesic of the Kerr spacetime. In other words, relative to previous work, we have replaced the Newtonian equations of motion by relativistic geodesic equations of motion.

The relativistic description introduces effects such as orbital precession and frame dragging, but it does not account for effects due to the gravitational field induced by the SCO. These effects, for example, lead to changes in the (geodesic) constants of motion due to radiation reaction. Even though these effects introduce errors that scale with the system’s mass ratio (e.g. see Glampedakis, 2005), they cannot be neglected for EMRIs. This is because in the late stages of the EMRI the SCO spends a substantial fraction of cycles in the strong-field region of the MBH. On the other hand, in the case of EMRBs, the SCO sling-shots around the MBH and its interaction time during pericenter passage is relatively small ($<10^{5}$ s). Radiation reaction effects can then be neglected in EMRBs since the radiation reaction timescale is always much larger than the period of pericenter passage.

In this paper, we also improve on the semi-relativistic approximation by using a more precise gravitational wave extraction procedure. The procedure employed is the multipole-moment wave generation formalism for slow-motion objects with arbitrarily strong internal gravity (Thorne, 1980). We consider terms up to the mass-octopole and current-quadrupole multipoles, thus improving on the mass-quadrupole analysis of Rubbo et al. (2006, 2007) and Hopman et al. (2006). Higher multipoles will become important if the system becomes even more relativistic, but pericenter velocities for EMRBs are typically small to moderate relative to the speed of light (typically $0.1\lesssim v_{p}/c\lesssim 0.5$). Such higher multipolar corrections were taken into account for EMRIs by Babak et al. (2007), but for those sources the phase evolution must be tracked very accurately, requiring techniques from black hole perturbation theory (Poisson, 2004; Glampedakis, 2005) that we shall not consider here.

The study of the relativistic corrections considered in this work leads to the following conclusions. First, we find that relativistic effects are significant for approximately $50\%$ of the orbits contained in the EMRB phase space considered by Rubbo et al. (2006, 2007). These relativistic EMRB orbits differ from their Newtonian counterparts in such a way that the associated waveforms present a noticeably different structure. In particular, we find that there is a dephasing relative to Newtonian waveforms that is due to precessional effects and depend strongly on the MBH spin. These findings show that EMRB events are relativistic enough that they should be treated accordingly, as was previously found for EMRIs (Glampedakis, 2005).

Second, we find that the corrections to the trajectories affect the waveforms much more than the corrections in the waveform generation over several bursts. For example, for a given relativistic trajectory, we find that the difference between the SNR of a waveform obtain from the quadrupole formula to that obtained from the quadrupole-octopole formula is of the order of $10\%$ (depending on the location of the observer.) On the other hand, using the same waveform generation formula (quadrupole or quadrupole-octopole), the difference between the SNR of a Newtonian waveform to that of a Kerr waveform is of the order of $100\%$. These findings show that modeling EMRB waveforms with a quasi-Newtonian treatment might not be sufficient for certain data analysis applications.

Third, we find that the relativistic corrections accumulate with multiple bursts and, thus, they may have an important impact in improving the SNR. It is also conceivable that such corrections might be important for parameter estimation studies and, perhaps, may be used to determine or bound the spin of the MBH if a high SNR event is detected. Along this same lines, if parameters can be determined accurately enough, it might also be possible to use EMRB measurements to test deviations from General Relativity. We must note, however, that changing the orbital parameters in Newtonian waveforms could somewhat mimic some of the relativistic corrections, but a detailed Fisher analysis of such effects is beyond the scope of this paper.

The remainder of this paper is divided as follows: Section 2 deals with the dynamics of EMRBs in the semi-relativistic approximation and justifies the use of this approximation for these systems; Section 3 reviews the inclusion of higher-order multipolar corrections to the waveform generation formalism; Section 4 describes the numerical implementation of the equations of motion and the initial data used; Section 5 compares the orbital trajectories and waveforms; Section 6 concludes and points to future research.

In this paper, we denote the MBH mass by $M_{\bullet}$ and its gravitational radius by $R_{\bullet}=2GM_{\bullet}/c^{2}$, where $c$ is the speed of light and $G$ the Newtonian gravitational constant. To simplify some expressions we normalize masses with respect to $M_{\mbox{\tiny MW}}=4\times 10^{6}\,\textrm{M}_{\odot}$, which is of the same magnitude as the mass of the MBH at the center of the Milky Way (Ghez et al., 2005). The gravitational radius can then be written as:

[TABLE]

2. EMRB Dynamics

In this section, we discuss the description of the orbital motion. Newtonian dynamics usually provides an adequate description of many astrophysical sources of gravitational waves, at least from a qualitative point of view. However, for certain gravitational wave sources, such a description is insufficient and relativistic effects have to be considered. For EMRB sources with pericenter distances $r_{p}>4R_{\bullet}$ and velocities $v_{p}/c<0.5$, the semi-relativistic approximation to the equations of motion, in combination with a multipolar description of the gravitational radiation, can adequately model the dynamics and gravitational radiation, as we argue below.

The semi-relativistic approximation treats the motion of the SCO in the point-particle limit as a geodesic of the MBH geometry, which is justified based on the small mass-ratios associated with these systems. In this work, we adopt Cartesian Kerr-Schild coordinates, $\{t,x^{i}\}$ ($i=1,2,3$), in which the MBH geometry is time-independent, reflecting its stationary character, and tends to a flat-space geometry in Cartesian coordinates far from the MBH. We denote the geodesic trajectory by $z^{i}(t)$, its spatial velocity by $v^{i}(t)=dz^{i}/dt$, and its spatial acceleration by $a^{i}(t)=dv^{i}/dt$. The latter, in such a coordinate system, and by virtue of the geodesic equations of motion, has the following form (e.g. see Marck, 1996):

[TABLE]

where $\mbox{g}_{\mu\nu}$ ($\mu,\nu=0,1,2,3$) are the spacetime components of the MBH metric. These equations describe the influence of the spacetime curvature produced by the MBH and approach the Newtonian equations of motion in the regime where $v/c=|v^{i}|/c\ll 1$ and $GM_{\bullet}/(c^{2}\,r)\ll 1$ ($r=|x^{i}|$).

The effects from the self-gravity of the SCO can be neglected. To see this, consider the (Keplerian) orbital timescale, $T_{{\textrm{orb}}}$, in comparison to a characteristic radiation-reaction timescale, $T_{{\textrm{rr}}}$. For the latter, we can use the timescale associated with the rate of change of the semi-latus rectum, $p$, related to the pericenter distance by $r_{p}=p/(1+e)$, namely $T_{{\textrm{rr}}}\sim p/|dp/dt|$. The radiation-reaction timescales of the other orbital elements are comparable or larger (see, e.g., Glampedakis, 2005). The ratio of these timescales is

[TABLE]

where $\mu=m/M_{\bullet}$ is the mass ratio of the system and $m$ the SCO mass. It is evident that the radiation-reaction timescale is much greater than the orbital timescale due to the extreme mass ratio, $\mu\ll 1$, and because EMRBs have $p\gg\mu^{5/2}R_{\bullet}$. In the unlikely case that more accuracy is required, one could improve the analysis through the use of “Kludge” waveforms (Babak et al., 2007), which have been shown to reproduce numerical results in the adiabatic approximation accurately for EMRIs.

Formally, the orbital timescale used is not really the exact timescale of orbital motion. This is because the mass distribution of a MBH-embedded galaxy possesses a non-Keplerian potential that leads to non-Keplerian orbits. However, most EMRBs (by rate) have apocenters that do not extend far into the stellar population, implying that the contribution from the galaxy potential is minimal. The orbits we study in later sections have a contamination from the galactic potential that is less than $2\%$ of the MBH mass. Moreover, Hopman et al. (2006) rightly argue that the inner region is statistically empty of stars, which is due to finite effects realized at the small scales observed near the MBH.

Certain constraints may be derived on the size of $p$ and $r_{p}$ for EMRB events. The most important constraint is derived from the definition of EMRBs: orbits with sufficiently large orbital period $T_{{\textrm{orb}}}>T_{{\textrm{cut}}}$. Assuming a Keplerian orbit (which is a rough assumption), this constraint translates to pericenter distances as follows

[TABLE]

where we have rescaled quantities assuming a typical eccentricity of $e=0.9$ (Rubbo et al., 2006, 2007) and a typical MBH mass of $M_{\bullet}=M_{\mbox{\tiny MW}}$. In terms of geometrized units, such a constraint translates roughly to $r_{p}>2R_{\bullet}$.

This constraint can be compared with the requirement that the SCO does not get captured. Any object that enters the black hole event horizon is captured, where the horizon is located (in Boyer-Lindquist coordinates) at

[TABLE]

where $a_{\bullet}$ is the (Kerr) MBH spin parameter, related to its intrinsic angular momentum by $S_{\bullet}=GM_{\bullet}\,a_{\bullet}/c$, and bounded by $a_{\bullet}/M_{\bullet}\leq 1$. Thus, for a maximally spinning Kerr MBH ($a_{\bullet}=M_{\bullet}$), $r_{\rm cap}=0.5R_{\bullet}$, while for a Schwarzschild (non-spinning) MBH it is just $R_{\bullet}$. This condition tell us simply that $r_{p}>r_{\rm cap}$, which is a condition superseded by the constraint on the orbital period given in equation (4). One could explore other possible constraints (Rubbo et al., 2006, 2007) but they are in general superseded by equation (4).

These constraints clearly exclude the ergosphere of the MBH ($r_{\rm cap}<r\lesssim R_{\bullet}$) where frame dragging effects are most pronounced. However, for EMRIs it has been argued (Glampedakis, 2005) that orbits with $r_{p}<10R_{\bullet}$ cannot be considered Keplerian anymore, mainly due to precessional effects. This statement can be made more quantitative by looking at the ratio of first-order post-Newtonian (1 PN) predictions (Blanchet, 2006) to Newtonian ones (0 PN). For example, for the energy of a circular orbit, this ratio scales as $7R_{\bullet}/(8r_{p})$, while for the perihelion precession angle, the ratio scales as $3R_{\bullet}/(2r_{p})$, for extreme-mass ratios. Therefore, for orbits with pericenter passage $r_{p}\sim 5R_{\bullet}$, the 1 PN correction to the energy and the perihelion precession angle is approximately $20\%$ and $30\%$ respectively, relative to the Newtonian value. This indicates that, even for orbits outside the ergosphere, relativistic effects are not necessarily negligible.

The relativistic geodesic equations of motion introduce corrections to the Newtonian motion that can be interpreted in terms of a black hole effective potential. By comparing the Newtonian and relativistic potentials one can see that the relativistic corrections dominate over the centrifugal barrier at small distances from the black hole center. In this work we show that these relativistic corrections can be sampled by EMRBs and hence, one should model these systems accordingly. Nevertheless, as we argued above, the relativistic treatment of EMRBs does not need to be as sophisticated as in the case of EMRIs, since radiation-reaction can be neglected.

3. EMRB Waveforms

In this section we describe how we extract gravitational waveforms once we have integrated the geodesic equations of motion. We use a multipole-moment wave generation formalism for slowly-moving objects with arbitrarily strong internal gravity (Thorne, 1980; Flanagan & Hughes, 2005; Glampedakis, 2005). In quasi-Newtonian and semi-relativistic treatments, the radiation is modeled by the lowest non-vanishing multipole moment: the mass-quadrupole. To that order, and for the case of a point-like object orbiting a MBH at a fixed coordinate location, the plus and cross polarizations are given by (Misner et al., 1973; Thorne, 1980)

[TABLE]

where $r$ is the (flat-space) distance to the observer and $\epsilon^{ij}_{+,\times}$ are polarization tensors. This expression assumes, based on the slow motion approximation, that the change in the acceleration with respect to time, the jerk, $j^{i}=da^{i}/dt$, is a small quantity. More precisely, we are neglecting terms of order $(v/c)^{3}$, or in other words, since the (quadrupole) leading order terms are of order $(v/c)^{2}$, this implies a relative error of order $v/c$.

One can improve on this description for the gravitational radiation by accounting for higher-order multipole moments. In this paper, we consider the mass-octopole and current-quadrupole multipoles, which require the knowledge of one more time derivative of the trajectory, the jerk. Adding these contributions to equation (6), the gravitational waveforms are given by (Thorne, 1980)

[TABLE]

where $n^{i}=x^{i}/r$ is a unit vector that points to the observer and the vector product is the flat-space scalar product. In this case, we are neglecting terms of order $(v/c)^{4}$ and hence we are making a relative error of order $(v/c)^{2}$ with respect to the leading order quadrupole term.

The waveforms of equation (7) are a truncated multipole expansion, where we are neglecting the current-octopole, mass-hexadecapole, and higher multipole moments. This expansion is based on a slow-motion approximation which is valid for orbits whose pericenter velocity is small relative to the speed of light. For closed circular orbits, we can use the Virial theorem to argue that this is equivalent to requiring $r_{p}>M_{\bullet}$. For a relativistic EMRB event with $v_{p}/c=0.4$, the maximum relative contribution of the octopole to the quadrupole is of the order of $40\%$, since the octopolar term is of order $v/c$ smaller than the quadrupolar one. In this paper, we shall study EMRBs from the sample of Milky Way sources studied in Rubbo et al. (2006). These sources have initial pericenter distances of $r_{p}>8M_{\bullet}$, thus justifying the use of a low-multipolar expansion in the wave-generation formalism and the neglect of radiation reaction effects in the orbital motion.

4. Numerical Simulations

In this section we describe the EMRB simulations that were carried out, including the choice of initial conditions. The simulations involve integrating the equations for geodesic motion around a Kerr black hole, equation (2), forward in time. (For a detailed exposition of Kerr geodesics see Chandrasekhar, 1992). Since Cartesian Kerr-Schild coordinates are used, the initial conditions can be denoted by $(z^{i}_{0},v^{i}_{0})$. The numerical implementation does not use the Kerr geodesic constants of motion (energy, angular momentum, and Carter constant) in order to reduce the number of variables of the resulting system of ordinary differential equations. Instead, we have used the constants of motion to monitor the accuracy of the time integration. The integration accuracy is set so that we obtain fractional errors for the constants of motion smaller than one part in $10^{10}$. The code uses a Bulirsh-Stoer extrapolation method as the evolution algorithm (see, e.g. Press et al., 1992; Stoer & Bulirsch, 1993). We have also introduced in the code the possibility of switching between Kerr geodesics and Newtonian equations of motion. The gravitational waveforms are then obtained directly by applying expressions (6) and (7) to the numerically obtained trajectory $z^{i}(t)$.

Comparisons are carried out by choosing a representative relativistic orbit within the allowed phase space for EMRB’s. We made the following choices for the test case:

• •

The central MBH mass is $M_{\bullet}=M_{\mbox{\tiny MW}}$ and the SCO mass is $m=1~{}\textrm{M}_{\odot}$, such that the mass ratio is $\mu=m/M_{\bullet}=2.5\times 10^{-7}\ll 1$.

• •

The MBH spin parameter is either $a_{\bullet}=0$ (Schwarzschild) or $a_{\bullet}=0.998~{}M_{\bullet}$ (Kerr). The angular momentum is aligned along the $z$-axis and equal to either $S^{z}=0$ or $S^{z}=0.998GM_{\bullet}^{2}/c$.

• •

The observer is located at $r_{obs}=8$ kpc along the z-axis, which corresponds to the approximate distance from Earth to the center of the Milky Way (Eisenhauer et al., 2005).

Furthermore, we make the following choices for the orbital initial conditions:

[TABLE]

The initial conditions are such that $r_{0}=|z^{i}_{0}|=50R_{\bullet}=1.91\times 10^{-5}$ pc, and $|v^{i}_{0}|=0.11\,c=3.39\times 10^{4}~{}{\textrm{km s}}^{-1}$. The orbital plane is inclined by $10^{\circ}$ with respect to the $x-y$ plane to demonstrate the effects of orbital plane precession, which only occurs for spinning MBHs. Since this paper is concerned with the effect of relativistic corrections to EMRB events, we choose to give all orbits the same initial conditions. The possibility of relaxing this choice and its effect on the conclusions derived in this paper shall be discussed in a later section.

Although the test orbit is in the phase space of EMRB events studied in Rubbo et al. (2006), one might worry that it is too relativistic to actually have a significant probability to occur in nature. In particular, one can think that the SCO may be tidally disrupted. To address this question, let us consider a Newtonian description of the central potential, which leads to the following values of the pericenter distance and velocity:

[TABLE]

One might worry that stars might be tidally disrupted with such small pericenters. However, as shown by Hopman et al. (2006), most SCOs in EMRB scenarios consist of stellar-mass black holes, which cannot be tidally disrupted.

Such relativistic orbits are actually naturally occurring in the phase space of possible EMRBs studied in Rubbo et al. (2006). Of all orbits in the allowed EMRB phase space considered in Rubbo et al. (2006, 2007), $6\%$ is contained within a small $6$-dimensional phase space volume centered on the test orbit 111In other words, $6\%$ of the test orbits considered to be EMRBs in  Rubbo et al. (2006) are close in phase space to our test orbit. This does not imply that the probability of such a test orbit actually occurring in nature is $6\%$, since any element of phase space may have small overall probabilities, even down to $10^{-7}\%$.. Furthermore, the test case shown possesses a short orbital timescale, bursting approximately $150$ times per year. Such events with small orbital timescale were shown to dominate the EMRB event rate in Rubbo et al. (2006). It is in this sense that the test orbit studied here is typical or representative of EMRBs.

The relative location of the test orbit in the pericenter distance-eccentricity plane of the phase space of allowed EMRBs (Rubbo et al., 2006) is presented in Figure 1 (triangle.) The eccentricity was here calculated assuming a Newtonian orbit and the pericenter separation is given in gravitational radii, $R_{\bullet}$. Although the test orbit has a large eccentricity, its apocenter is small enough ($r_{a}\lesssim 150R_{\bullet}\approx 6\times 10^{-5}$ pc) that the contribution from the surrounding stellar population to the potential can be neglected. In general, the left side of the figure corresponds to highly relativistic orbits with large pericenter velocities and small pericenter distances. Orbits with pericenter velocities $|v_{p}|>0.25\,c=3\times 10^{4}~{}{\textrm{km s}}^{-1}$ account for approximately 50% of the possible orbits within the phase space studied in Rubbo et al. (2006).

Although the test orbit is a good source to demonstrate the differences between the Newtonian and relativistic treatments, we could have chosen an even more relativistic one. Such an event would still be classified as an EMRB in a Newtonian treatment, but it would border with the definition of a continuous source. An example of such an extreme orbit is shown with a circle in Figure 1, to the left of the test orbit (triangle). This extreme orbit possesses the following initial conditions:

[TABLE]

where $r_{p}=4\,R_{\bullet}=1.53\times 10^{-6}$ pc and $|v_{p}|=0.49c=1.46\times 10^{5}~{}{\textrm{km s}}^{-1}$ for a Newtonian potential. We will study such an extreme orbit at the end of the next section as an example of a limiting relativistic case.

5. Comparison of Trajectories and Waveforms

In this section we compare the results obtained for both the orbital motion and the gravitational radiation emitted by an EMRB event using both the Newtonian and relativistic description. Since the plus and cross polarization waveforms present similar features, we only plot the plus polarization waveforms. In the remainder of this section we use the following nomenclature: a quadrupolar (octopolar) Newtonian waveform is one that was calculated using the quadrupole (octopole) formula and Newtonian equations of motion; a quadrupole (octopole) Schwarzschild waveform is one that was calculated using the quadrupole (octopole) formula and the geodesic equations of motion with no spin ($a_{\bullet}=0$); a quadrupole (octopole) Kerr waveform is one that was calculated using the quadrupole (octopole) formula and the geodesic equations of motion with spin $a_{\bullet}=0.998M_{\bullet}$.

5.1. Orbital Trajectories

Let us begin by comparing the trajectories obtained in our simulations. In Figure 2, we plot the test orbit corresponding to a Newtonian treatment (solid line) and the one corresponding to a relativistic treatment without spin (dashed line) and with spin (dotted line). The dot and arrow indicate the initial location and velocity projected onto the $x$-$y$ plane. The MBH is located at the origin of the coordinates, and the vectors denoted by $L$ and $S$ describe the direction of the initial orbital angular momentum and the MBH spin respectively. In the relativistic description there are precessional effects in the SCO orbit that can be clearly observed in Figure 2. These precessional effects are: pericenter precession about the orbital angular momentum axis, which acts in the initial orbital plane; and frame-dragging precession about the total angular momentum axis, which acts out of the initial orbital plane. While the former always occurs in a relativistic treatment, the latter is present only in the spinning case.

Different relativistic precessional effects are generally of different magnitude. These effects are usually inversely proportional to the pericenter distance, or equivalently proportional to the pericenter velocity of the SCO. Precession out of the initial orbital plane, however, is smaller than precession in the orbital plane by a relative factor of order $v_{p}/c$ and it is directly proportional to the spin of the MBH. In terms of post-Newtonian theory (see, e.g. Blanchet, 2006), the pericenter advance is described by 1st-order post-Newtonian corrections to the equations of motion (order $(v/c)^{2}$ relative to the Newtonian acceleration), while precession off the orbital plane is due to spin-orbit and spin-spin couplings that correspond to $1.5$ and $2$-order corrections (order $(v/c)^{3}$ and $(v/c)^{4}$ relative to the Newtonian acceleration.) Therefore, since EMRBs are characterized as events with small to moderate pericenter velocities, precession out of the initial orbital plane is small to moderate relative to pericenter advance, even for maximally spinning MBHs.

We can estimate the precession rate by comparing the Newtonian and relativistic trajectories. For the test orbit considered, we find that the rate in the orbital plane is roughly $\pi/3$ radians per orbit for the non-spinning case and $2\pi/3$ radians per orbit for the spinning one. These precessional effects have been studied extensively in the context of EMRIs (see, e.g. Schmidt, 2002) and also specifically for S-stars in the central region of our Galaxy in Kraniotis (2007). Nonetheless, these effects have not been previously analyzed in the context of EMRBs, since previous studies employed a quasi-Newtonian treatment.

5.2. Waveforms

Let us now analyze how the differences in the SCO trajectories translate into different signatures in the waveforms. In Figure 3 we plot the quadrupole Newtonian and Schwarzschild waveforms (solid and dashed lines respectively), while in Figure 4 we plot the quadrupole Schwarzschild and Kerr waveforms (dashed and dot-dot-dashed lines respectively.) There are three main differences between the Newtonian and the relativistic waveforms: a modulated phase change, an amplitude change, and a time of arrival change. The changes in amplitude and time of arrival are due to the test particle experiencing a larger “force” of attraction as it approaches the black hole. Quantitatively, this increase in force is due to the presence of ($r_{p}^{-n}$)-contributions to the relativistic corrections to the central potential (with $n$ a real positive number.)

Gravitational wave interferometers are most sensitive to the phase, which is clearly different for Newtonian and relativistic waveforms. The dephasing present in Figures 3 and 4 parallels the orbital dephasing discussed earlier (the gravitational-wave and orbital frequencies are intimately related) and leads to an amplitude modulation. In terms of the gravitational wave phase, both Figures 3 and 4 show a dephasing of $\pi/6$ radians per cycle. This can be seen after the third burst where the waveforms are back in phase. In fact, there is a significant dephasing even between the relativistic waveforms due to the effect of the MBH spin. If a cursory examination by eye can detect the difference in the waveforms due to differences in the nature of the central potential, it stands to reason that strong-field EMRB waveforms might allow us to probe of the spacetime near a MBH. We should note, however, that no work has yet been done to find best-fit parameters for Newtonian waveforms that maximize the correlation with relativistic ones. In other words, it might be possible to mimic some of the relativistic corrections by choosing different initial data for the Newtonian waveforms, but such a study is beyond the scope of this paper.

The difference in dephasing can be better studied by calculating the signal overlap,

[TABLE]

The overlap indicates how well a signal $h_{1}$ can be extracted via matched filtering with a template $h_{2}$ 222The overlap is given here in the time domain, but an analogous representation in the frequency domain could also be used. Such an expression in the frequency domain can be derived through Parseval’s theorem.. In Figures 5 and 6 we plot the normalized integrand as a function of time, with $h_{1}$ given by the quadrupole Kerr waveform and $h_{2}$ given by either the quadrupole Newtonian or Schwarzschild waveforms. Observe that neither the Newtonian nor the Schwarzschild waveforms match well with the Kerr waveform. Moreover, note that the correlation with the Newtonian waveform deteriorates greatly after only the first cycle. The integral of equation (11) gives the correlation between different waveforms over nine days (four bursts): for the Newtonian and Kerr plus-polarized waveforms it is $9.6\%$; for the Schwarzschild and Kerr plus-polarized waveforms it is $-6.3\%$. As a point of comparison, a substantial signal overlap should be $\gtrsim 95\%$. As already mentioned, the same initial conditions were chosen for both the Newtonian and relativistic orbits, such that their waveforms would be both initially in-phase and any dephasing due to relativistic effects could be clearly studied. However, such a choice forces the SCO to pass through periapsis at slightly different times, because in the relativistic case this object experiences a “deeper” potential. Such a difference in timing degrades the overlap somewhat and could, in principle, be ameliorated by choosing different initial conditions for the Newtonian evolution, *i. e. * by maximizing the overlap over all orbital parameters, but such a study is beyond the scope of this paper.

Figures 5 and 6 provide some evidence that the use of a relativistic waveform might be required for the data analysis problem of extracting EMRB signals. Such expectations are somewhat confirmed in Table 1, where we present the correlation between Newtonian and Kerr plus-polarized waveforms integrated over both a single day (one burst) and nine days of data (several bursts) for a sample of different EMRB orbits 333Here we use the frequency representation of the correlation calculation, employing the Fourier transform of the waveforms.. Both a single and several bursts should be studied because, in principle, parameter adjustments might mitigate the between-burst dephasing, but probably not the in-burst dephasing. The orbits chosen were taken directly from the allowed EMRB phase space of Rubbo et al. (2006) and possess different initial positions and velocities, leading to different eccentricities, initial inclination and orbital periods. The orbital period can be used to classify the orbits into highly-bursting (burst more than once per week) and slowly-bursting (burst less than once per week.) For highly-bursting EMRBs, the average correlation is $\sim 27\%$ when all nine days of data (several bursts) are used, while it is $\sim 93\%$ when only one day of data (single burst) is used. For the slowly-bursting EMRBs we considered here, there is actually only one burst per week and its correlation is $\sim 85\%$. These results indicate that accumulating precession effects lead to a significant loss of correlation between Newtonian and Kerr waveforms if bursts are to be connected. Moreover, we see that even for a single burst the shape of the relativistic waveforms is sufficiently different from its Newtonian counterpart to lead to a significant mismatch (in-phase dephasing). If one maximizes the correlation over intrinsic orbital parameters it might be possible to increase the correlation somewhat, but again that is to be studied elsewhere.

Let us now focus on the differences in the waveforms when they are calculated with the quadrupolar approximation versus the quadrupolar-octopolar one. In Figure 7 we plot the absolute value of the difference between the octopole and quadrupole waveforms as a function of time for a Schwarzschild (upper panel) and a Kerr (lower panel) central potential. The difference is normalized to the maximum of the first peak of the quadrupole waveform, since other peaks have approximately the same maximum.

Observe that the inclusion of higher-order multipoles does not affect the phasing of the waveforms, but only the amplitude, which is in general different by roughly 4% relative to the quadrupole waveform for the Kerr test case. At first sight, this result is in disagreement with the expectation that the octopolar correction is at most $\lesssim 40\%$ of the quadrupolar one. Note, however, that the $40\%$ estimate is an order-of-magnitude upper limit, since the octopole correction is dependent on the location of the observer relative to the trajectory, velocity, acceleration and jerk vectors. For the test case, where the observer is located on the $\hat{z}$-axis and the orbit is initially inclined by $10^{\circ}$, the octopolar change is reduced by approximately an order of magnitude, since initially (\mbox{\boldmathn}\cdot\mbox{\boldmathz})\approx(\mbox{\boldmathn}\cdot\mbox{\boldmathv})\approx(\mbox{\boldmathn}\cdot\mbox{\boldmatha})\approx(\mbox{\boldmathn}\cdot\mbox{\boldmathj})\approx 0.1. In Table 2, we present the approximate maximum difference between octopolar and quadrupolar waveforms as a function of observer location, focusing only on the first burst of radiation. The location of the observer is rotated about the $\hat{y}$-axis on the $\hat{x}-{\hat{z}}$ plane, always at a fixed radial distance of 8 kpc. Note that for some observers the difference is larger and, in fact, of the order of 40%, since the dot products are closer to unity. These results are thus consistent with the expectation that the $n$-th multipolar contribution cannot in general be larger than order $(v/c)^{n}$ relative to the quadrupolar leading term.

5.3. Data Analysis

In order to quantify some of our statements about the change in phase and amplitude, we calculated the SNR for the relativistic waveforms via the standard formula

[TABLE]

where the tilde denotes the Fourier transform and $S_{n}(f)$ is the one-sided power spectral noise density. Here we employ the Online Sensitivity Curve Generator (Larson et al., 2000) with the standard LISA settings and the inclusion of the white-dwarf background contribution. When calculating SNRs, we set the observation time to roughly nine days, so as to include multiple bursts in our single SNR value.

The inclusion of relativistic corrections in the trajectories has a dramatic impact in the SNR. We find that the Schwarzschild waveform increases the SNR by a factor of approximately $59\%$, while the Kerr waveform increases it by $162\%$, relative to the Newtonian SNR. The difference in SNR is because the relativistic orbits experience a deeper effective potential and, thus, the interaction timescale is smaller. Therefore, the inverse of the interaction time, $f_{\star}=v_{p}/r_{p}$, is larger for the Schwarzschild and Kerr waveforms relative to the Newtonian one. As a result, the Fourier power is shifted to higher frequencies, where LISA is more sensitive.

The SNR behaves similarly for other EMRB orbits with different orbital periods, eccentricities and pericenter parameters. This can be observed in Table 1 where we present the SNR difference between Newtonian and Kerr waveforms for different EMRB orbits for a single and several bursts. Highly-bursting EMRBs lead to a large change in the SNR over a week of data, since they experience the depths of the effective potential several times. Per burst, the change in SNR can range from one, to ten or even ninety percent, depending on the inclination angle of the orbit, the pericenter distance and other orbital parameters. Also note that the orbits presented in the table are not as relativistic as the test case, which is why the SNR difference is smaller. This study seems to indicate that the SNR is in general somewhat larger for relativistic waveforms, specially when several burst are taken into account. 444Naive intuition might suggest that the change in the SNR should scale like the square root of the number of bursts, but this is not necessarily correct. First, different orbits possess different beaming patterns on the sky due to precession. Second, the starting frequencies of these bursts is very close to the limit of LISA’s sensitivity band [$10^{-5}$ Hz]. As the orbits burst, precession somewhat increases the frequency of the waveform, forcing different bursts to contribute different amounts to the SNR. Consequently, the event rate calculated in Rubbo et al. (2006, 2007), which in particular summed over all bursts in one year of data, is an underestimate for their galaxy model, because some of the systems with a Newtonian SNR $\lesssim 5$ should have been added to the detectable event rate. However, the uncertainty in the event rate is still probably dominated by astrophysical uncertainties and not by the dynamics modeling.

Conversely, the inclusion of higher multipole moments to the wave generation formalism has little to no effect in the SNR. In the previous section we showed that there was $\approx 4\%$ difference between the octopole and quadrupole waveforms relative to the maximum of the first peak of the quadrupole waveform. We further showed that this difference depends on the location of the observer (see table 2), but for the test case it does not exceed a maximum of 40%, which agrees with the multipole-ordering argument previously described. However, we also pointed out that the amplitude difference is confined to sharp peaks in the time domain. Such a confined change in the waveform amplitude leads to a Fourier power being dispersed over a large frequency region, including outside the LISA band. As a result, there is a correspondingly small change in the SNR: of the order of $\approx 1\%$ relative to the quadrupolar formalism. Such a result is in agreement with the analysis of Babak et al. (2007), which was carried out for EMRIs. Therefore, we see that the change in SNR is primarily dominated by the modifications introduced in the geodesic description of the equations of motion, and not in the octopolar correction to the waveform generation.

The analysis presented in this section, in particular Figure 5, makes it clear that relativistic corrections to the waveforms accumulate with multiple bursts. In other words, over a single burst (pericenter passage), a quadrupolar waveform calculation using Newtonian dynamics might be sufficient. However, if one wants to estimate parameters associated with the central MBH, then multiple bursts might be necessary to associate the events to a single SCO trajectory. In terms of data analysis, for a detection search it is simpler to look for a single burst using techniques such as excess power and wavelet decompositions (e.q. see Anderson et al., 2001; Klimenko et al., 2004; Stuver & Finn, 2006; Camarda & Ortolan, 2006). For estimating MBH parameters, the results of this paper suggest that multiple bursts might have to be connected. For this to occur, a single template may be used, but as our results indicate, the template will need to incorporate the effects of general relativity. 555A note of caution should be added here, since a detailed study of the maximization of the overlap with respect to orbital parameters in the Newtonian waveform has not yet been carried out. Indeed, it might be possible to mimic some of the relativistic effects with Newtonian waveforms with varying parameters, but such mimicking is most probably not possible for highly relativistic EMRBs.

At this junction, we should comment on some of the caveats in the conclusions derived from the analysis presented here. First, in this paper we have primarily concentrated on the question of characterizing the gravitational waves through the study of the SNR and overlap. In this study, however, we have not maximized these data analysis measures with respect to (intrinsic or extrinsic) orbital parameters. Although it might be possible to improve the SNR and overlap via parameter maximization, our study suggests that the introduction of relativistic effects, such as precession, lead to a clear imprint on the waveform that might be difficult to mimic with a purely Newtonian waveform irrespective of its parameters. Second, in this paper we have only touched the iceberg of the signal characterization and parameter estimation problem. A possible route to study this problem is through a numerical Fisher analysis, with the complications derived from the fact that the waveforms are known only numerically. Furthermore, increasing the complexity of the waveform will also increase the computational cost of these studies and, thus, it might be interesting to investigate whether it is possible or advantageous to search for individual bursts with similar frequency and identify them as belonging to the same physical system. These issues are beyond the scope of this paper, but they should be addressed in future investigations.

Setting these caveats momentarily aside, let us conclude with some discussion of the extreme relativistic case introduced earlier. As we have seen, relativistic corrections can introduce strong modifications to EMRB waveforms, which depend on how relativistic the EMRB event is and, in particular, on the pericenter velocity. The corrections are particularly strong for the class of EMRBs that inhabit the boundary between EMRBs and EMRIs, defined by the $T_{{\textrm{cut}}}=3\times 10^{4}$ s value, corresponding to the period between apocenter passages. An example of such an event is the extreme orbit discussed in Section 4, whose waveform is shown in Figure 8. Observe that a simplistic Newtonian description misses the rich structure, in which the SCO whirls twice about the black hole before zooming out to apocenter again. This behavior is missed entirely when we evolve the orbit with the Newtonian equations of motion, even though the same initial conditions were used. Even though in the previous cases a Newtonian waveform might be able to extract relativistic events by adjusting intrinsic parameters, such is definitely not the case for the highly-relativistic event of Fig. 8, since no choice of parameters in the Newtonian waveform can reproduce its multiple-peak structure.

Due to their whirling behavior, the extreme orbit waveform resembles the zoom-whirl events often mentioned in the EMRI literature (Hughes, 2001a, b; Glampedakis & Kennefick, 2002). However, the event is still an EMRB and not an EMRI because the period between apocenter passages is too long. For our galactic model, we find the probability of a small region of phase space around this orbit to be rather high, 10%. If this EMRB is detected with sufficiently high SNR it seems plausible that a parameter estimation analysis would allow for a determination of the background parameters, such as the black hole spin. Barack & Cutler (2004b) have already investigated LISA’s ability to measure MBH properties using approximate EMRI signals. They found that, depending on the actual orbital parameters, it will be possible to measure the MBH spin with fractional errors of $10^{-3}$ to $10^{-5}$. This high precision measurement is the result of observing up to $\sim\!10^{6}$ complete orbits. Conversely, it is very unlikely that EMRB measurements will be able to match the measurement capabilities of EMRI signals, since only a few bursts will probably be available. Whether accurate parameter extraction is possible can only be determined with a more detailed data analysis investigation of EMRBs.

6. Conclusions

We have studied the effects of relativistic corrections on the gravitational waves produced by EMRBs. These events originate from long period orbits of a SCO around a MBH, leading to large-amplitude, quasi-periodic gravitational wave bursts. Using a more accurate relativistic treatment of the phenomenon, we have improved on the waveforms and trajectories relative to previous work. The orbital trajectories were corrected by accounting for the spacetime curvature of the system for Schwarzschild and Kerr MBHs. The waveform generation was corrected by accounting for the next order term in the multipolar expansion of gravitational radiation.

We found that relativistic corrections change the waveform shape relative to its Newtonian counterpart. One of the most significant changes was found to be an amplitude-modulated dephasing, produced by the relativistic corrections to the orbital trajectory and, in particular, by relativistic precessional effects. Other effects included a change in the amplitude of the waveform, partially produced by the inclusion of higher-order terms in the gravitational wave generation scheme.

The magnitude of the relativistic corrections was found to be directly proportional to the pericenter velocity of the orbit, as expected. Surprisingly, we estimated that at least $50\%$ of the orbits analyzed in Rubbo et al. (2006, 2007) acquire relativistic velocities and, thus, non-negligible relativistic corrections. We investigated these corrections in detail by choosing a test orbit, that is representative of the kind that dominated the event rate calculation of Rubbo et al. (2006). We also studied a limiting case of a highly relativistic EMRB and found that it whirls more than once around the MBH before zooming back to apocenter and becoming silent again.

We have also discussed the possible consequences that relativistic effects might have on the detection and parameter estimation of gravitational waves from EMRBs by LISA, namely a change in the SNR and loss of overlap. These changes are mainly due to the relativistic treatment of the equations of motion, while a quadrupolar wave generation formalism seems to suffice. This finding is relevant particularly to match filtering searches, where a Newtonian treatment of the orbit might lead to a deterioration of confidence limits. Furthermore, our study suggests that, given an EMRB gravitational wave detection, it might be plausible to extract or bound the spin of the central potential with a template that takes into account the Kerr character of the MBH. Other astrophysical consequences include a possible increase in the event rate, which implies that the rates of Rubbo et al. (2006, 2007) and Hopman et al. (2006) might be lower limits, although these estimates are still dominated by uncertainties in the astrophysical modeling for the host galaxy.

In addition to the astrophysical modeling, future research should tackle the details of the EMRB data analysis and signal extraction issues put forward above. Based on the results of this paper, one may explore the possibility of testing alternative theories of gravity with EMRBs by performing matched filtering with templates from alternative theories (Will, 1998; Scharre & Will, 2002; Will & Yunes, 2004; Berti et al., 2005b, a). Another possible avenue for future research is the study of confidence limits with which the spin of the central MBH can be measured.

This research could then be used to examine whether EMRB events can distinguish between a pure Kerr MBH and some other spacetime. Such studies have already began with the analysis of Collins & Hughes (2004), Glampedakis & Babak (2006) and Barausse et al. (2007), where comparisons between a Kerr and other non-Kerr spacetimes were performed. Such studies could be extended to the perturbed Kerr solution found by Yunes & Gonzalez (2006), where the perturbation is parameterized by the Weyl tensor of the external universe and could represent some external accretion disk, planetary system or some other compact object. Ultimately, these investigations will decide whether EMRB events are worth studying in further detail by future gravitational wave observatories.

The authors would like to thank Ben Owen for reading the manuscript and providing useful comments, as well as the anonymous referee for providing insightful suggestions. We would also like to acknowledge the support of the Center for Gravitational Wave Physics funded by the National Science Foundation under Cooperative Agreement PHY-01-14375, and support from NSF grants PHY 05-55628, PHY 05-55436, PHY 02-18750, PHY 02-44788, PHY 02-45649 and PHY 00-99559. K. H. B. and L. R. also acknowledge the support of NASA NNG04GU99G, NASA NN G05GF71G. C. F. S. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.