Lee S. Finn

The optimum design, construction, and use of the LIGO, VIRGO, or LAGOS gravitational radiation detectors depends upon accurate calculations of their sensitivity to different sources of radiation. Here I examine how to determine the sensitivity of these instruments to sources of gravitational radiation by considering the process by which data are analyzed in a noisy detector. The problem of detection (is a signal present in the output of the detector?) is separated from that of measurement (what are the parameters that characterize the signal in the detector output?). By constructing the probability that the detector output is consistent with the presence of a signal, I show how to quantify the uncertainty that the output contains a signal and is not simply noise. Proceeding further, I construct the probability distribution that the parameterization that characterizes the signal has a certain value. From the distribution and its mode I determine volumes in parameter space such that with probability [owing to the random nature of the detector noise, the volumes are are always different, even for identical signals in the detector output], thus quantifying the uncertainty in the estimation of the signal parameterization.

These techniques are suitable for analyzing the output of a noisy detector. If we are designing a detector, or determining the suitability of an existing detector for observing a new source, then we don’t have detector output to analyze but are interested in the “most likely” response of the detector to a signal. I exploit the techniques just described to determine the “most likely” volumes for detector output that would result in a parameter probability distribution with given mode.

Finally, as an example, I apply these techniques to determine the anticipated sensitivity of the LIGO and LAGOS detectors to the gravitational radiation from a perturbed Kerr black hole.

PACS numbers: 4.80.+z,98.60.a,97.60.Lf,6.20.Dk

etection, Measurement and Gravitational Radiation {instit} Department of Physics and Astronomy, Northwestern University, Evanston Illinois  60208 \receipt


1 Introduction

Under the present schedule, both the United States Laser Interferometer Gravitational Wave Observatory (LIGO[1, 2]) and the French/Italian VIRGO [3] will begin operation in the late 1990s. Long before that time, theorists must lay a foundation for the study of gravitational radiation sources. Part of this foundation involves the construction of detailed, parameterized models of the waveforms from expected sources; another part involves the calculation of the anticipated sensitivity of the detector to each of these sources. Calculation of these kinds are not only needed for LIGO and VIRGO: design and technology studies for a Laser Gravitational-wave Observatory in Space (LAGOS) are currently being pursued[4] and calculations of the sensitivity of LAGOS to appropriate sources are needed to guide these studies.

In this paper I address the problem of calculating the anticipated sensitivity of a detector, like LIGO, VIRGO, or LAGOS, to an arbitrary source of gravitational radiation. The problem breaks up into two parts which I term detection and measurement. To “detect” is to decide whether the observed detector output contains a signal from a particular source or is just an example of noise; to “measure” is to assume the presence of a signal in the detector output and to characterize the signal in terms of the parameter(s) that describe the source (and its orientation with respect to the detector).

Echeverria[5] recently examined some of these issues in the particular context of determining the precision with which one could characterize the mass and angular momentum of a perturbed Kerr black hole from observations in a gravitational radiation detector. The foundation of his analysis was the construction of a quantity similar to the signal-to-noise ratio (SNR), and he asserted that the parameters that characterize a signal observed in the output of the detector are those that maximized this quantity. This analysis is limited in two respects (Echeverria & Finn[6]):

  1. The validity of the formalism is restricted to the limit of high SNR; and

  2. The formalism cannot determine the amplitude of the signal.

In addition, the conceptual basis of this calculation is not compelling: the determination of the parameters characterizing a signal in a noisy detector does not proceed by maximizing the SNR-like quantity defined by Echeverria[5].

In contrast, the techniques developed here are all based upon the construction of probabilities and probability densities. For the problem of detection, I construct the probability that the observed detector output is consistent with the presence (or absence) of a signal. In the case of the measurement problem, where the detector output is assumed to include a signal, the quantity constructed is the probability density that describes the likelihood of a given signal parameterization.

In §2 I examine the twin processes of detection and measurement from the point of view of probability theory. The parameters characterizing a signal identified in the output of a noisy detector are defined to be those most likely to have resulted in the observed detector output. Some of the results described in this section are known elsewhere in the context of data analysis: they are included here for completeness and so that they may be compared with the techniques employed in Echeverria[5]. In §3 I show how these same techniques can be exploited to evaluate the anticipated sensitivity of an instrument to a signal: i.e., how precisely can the parameterization of a signal observed in the detector be determined. I find both exact and, in the interesting limit of a strong signal, approximate techniques for evaluating the expected precision with which an observed signal can be described. As an example, in §4 I apply the approximate techniques developed in §3 to the determination of the parameterization of the gravitational radiation from a perturbed black hole, especially the black hole mass and dimensionless angular momentum parameter . In §5 I briefly compare the methods and results of Echeverria[5] with my own. My conclusions are presented in §6

2 Detection, measurement and probability

In this section I consider two related problems that arise in the analysis of the output of a noisy detector: detection and measurement. The problem of detection is to determine whether or not a signal of known form (i.e., deterministic, though parameterized by one or several unknown parameters) is present in the detector output. The problem of measurement is to determine the values of some or all of the unknown parameters that characterize the observed signal.

Note that the distinction between detection and measurement separates the determination of the presence or absence of the signal from the determination of the parameters that characterize it: detection does not address the value of the unknown parameters, and measurement presumes the signal’s presence.

Detector noise can always conspire to appear as an example of the sought-for signal; alternatively, noise can mask the presence of a signal. In either case, noise interferes with our ability to determine the presence of the signal or the parameters that characterize it. Consequently, any claim of detection must be associated with a probability signifying the degree of certainty that the detected signal is not, in fact, an instance of noise. Similarly, when an observed signal is characterized it is appropriate to specify both a range of parameters and a probability that the signal parameters are in the given range.

For example, I can examine the data stream from a gravitational radiation detector to determine (with some probability) whether the radiation from the mode of a perturbed, rotating black hole is present, irrespective of the black hole mass, angular momentum, or orientation with respect to the detector. If I conclude that a signal is present in the data stream, then I can attempt to determine bounds on some or all of these parameters, such that I expect the actual parameters characterizing the signal to fall within those bounds with a given probability.

In the next several subsections I examine detection and measurement in more detail. I assume that the statistical properties of the detector noise are known, and also that the form of the sought-for signal is known up to one or several parameters. My discussion focuses on determining the probability that a signal of known form is present in the output of a noisy detector, and on determining the probability that the unknown parameters have particular values.

While the discussion in §4 is framed in the context of the measurement of gravitational radiation from astrophysical sources, the questions addressed in this (and the following) section are purely statistical ones and contain nothing that is specific to gravitational radiation, general relativity, or any particular physical system or theory. For more details, the reader may consult Wainstein & Zubakov[7].

2.1 Detection

Consider a data stream which represents the output of a detector. The data stream has a noise component and in addition may have a signal component . The signal component is parameterized by several unknown parameters (denoted collectively as , and individually as ); hence


Assume that is continuous, not discrete. I will describe how to determine the probability that , for undetermined , is present in , i.e.,


Begin by using Baye’s law of conditional probabilities to re-express as


where {mathletters}


Also re-express in terms of the two possibilities absent and present, and further re-express the probability that is present in terms of the probability that it is characterized by the particular :

where {mathletters}


Combining equations 3 and LABEL:eqn:unitarity, we find




In equation 12 all of the dependence of on the data stream has been gathered into the likelihood ratio . Aside from , depends only on the ratio of the a priori probabilities and . In turn, the likelihood ratio depends on two components: the a priori probability density and the ratio .

In order to determine we must assess the a priori probabilities and calculate the likelihood ratio. It is often the case that we know, or can make an educated guess regarding, the a priori probabilities. For example, the sources may be Poisson distributed in time [determining and ], and they may be homogeneously distributed in space [determining , where is the distance to the source]. At other times our assessment may be more subjective or based on imperfect knowledge, and in this case we can use the observed distribution of to test the validity of our assessments using the techniques of hypothesis testing (Winkler[8] §7).

Now turn to the evaluation of . To determine this ratio, first note that the conditional probability of measuring if the particular signal is present is the same as the conditional probability of measuring , assuming that the signal is not present in :


Consequently, we can focus on the conditional probability of measuring a data stream under the assumption that no signal is present [].

In the absence of the signal, is simply an instance of . Assume that the is a normal process with zero mean, characterized by the correlation function [or, equivalently, by the one-sided power spectral density (PSD) ]. In order to compute the ratio , consider the continuum limit of the case of discretely sampled data , with the correspondence {mathletters}


The probability that an individual is a sampling of the random process is given by


and the probability that the ordered set is a sampling of is


where is defined by




(Mathews & Walker [9] §14-6, Wainstein & Zubakov[7] eqn. 31.11). Note that the normalization constant in the denominator of equation 20 is independent of the ; consequently, it does not affect the ratio . Since it is this ratio that we are interested in, without loss of generality drop the normalization constant from the following.

To evaluate equation 20 in the continuum limit, first note that


Consequently, \widetext{mathletters}


To proceed from equation 24 to 25, use the Wiener-Khintchine (cf. Kittel[10] §28) theorem to relate the PSD to the correlation function and define


Consequently, as we approach the continuum limit, we have


With and Parseval’s Theorem, we can evaluate the continuum limit of the argument of the exponential in equation 20: \widetext


Here and henceforth we will denote the Fourier transform of as .

Since the detector output is real, . Define the symmetric inner product


for real functions and . In terms of this inner product,


The likelihood ratio is found by substituting eqn. 30 into eqn. 13.

To summarize, the probability that a signal of the class is present in the output of the detector can be expressed in terms of three a priori probabilities [, , and ] and the ratio of two conditional probabilities []. The a priori probabilities must be assessed, while the ratio of the conditional probabilities can be calculated. Often we know, or can make an educated guess regarding, the a priori probabilities; at other times our assessment is subjective or otherwise based on imperfect knowledge. Finally we establish a threshold for [or, equivalently, for , , or some other surrogate of ], and say that if the (or its surrogate) exceeds the threshold then we have detected the signal.

I will not discuss detection further, except to say that the choice of threshold is influenced by our strategy to minimize errors. The two kinds of errors we can make are to claim the presence of a signal when one is in fact not present (a “false alarm”), or to dismiss an observed as noise when a signal is present (a “false dismissal”). In order to minimize the probability of a false alarm (conventionally denoted ) we want a large threshold, while to minimize the probability of a false dismissal (conventionally denoted ) we want a small threshold. One obvious strategy for choosing the threshold is to minimize the sum , i.e., to minimize the probability of making an error. Alternatively, some other combination of and may be minimized, taking into account the relative seriousness of the different kinds of errors. Regardless, it is inadvisable to blindly choose a threshold for without careful consideration of the false alarm and false dismissal probabilities that arise and their relative severity.

2.2 Measurement

Turn now to the question of measurement. From equations 3, LABEL:eqn:unitarity and 14 we have


This conditional probability density is directly proportional to and, since the denominator in equation 31 is independent of , it is maximized where is maximized. If we assume that the signal is present, then the probability density that it is characterized by is


The goal of the measurement process is to determine a volume in parameter space such that with probability . This volume is “centered” on the mode of the distribution in a way we define later on. The mode of either or is the that maximizes . Denote the mode by .111While we assume in what follows that the distribution has a single mode, the generalization to a multi-modal distribution is trivial. While I will occasionally refer to as the “measured” parameterization of the signal, bear in mind that is only the most likely parameterization of the observed signal.

If we assume that the global maximum of is also a local extremum, then satisfies


equivalently, maximizes


i.e., it satisfies


This final set of equations is in general non-linear and may be satisfied by several different . Some will represent local maxima, while others will correspond to local minima or inflection points; thus, equation 33 is a necessary but not sufficient condition for .

An important characterization of the strength of the signal in a detector is the signal-to-noise ratio (SNR). The “actual” SNR depends on the true parameterization of the signal . We do not have access to ; however, we do know that the most likely value of is , and we define the SNR in terms of :


The factor of two arises because the power spectral density is one-sided while is two-sided. Note that is expressed in terms of the signal power (i.e., it is proportional to the square of the signal amplitude). There is some ambiguity in the literature over whether “SNR” refers to or . We avoid the ambiguity by referring to either or wherever the context demands it.

Having found the distribution (or ), we define the boundary of the volumes to be its iso-surfaces. The probability corresponding to the iso-surface is


Note that since the distribution is not generally symmetric, is not necessarily the mean of . Also, if the distribution has more than one local maximum then need not be simply connected.

To summarize, suppose we have an observation which we assume (or conclude) includes a signal (for unknown ). We construct the probability density according to equation 32, and identify iso-surfaces of as the boundary of probability volumes according to equation 37. Finally, we assert that with probability .

3 Measurement sensitivity

In §2 we saw how to decide whether a signal is present or absent from the output of a noisy detector, and, if present, how to determine bounds on the parameterization of the signal. Now I show how to anticipate the precision with which a detector can place bounds on the parameterization that characterizes a signal. In particular, consider an observed which contains a signal for unknown . We are interested ultimately in the distribution of


where is determined by the techniques discussed in §2. There are an infinity of possible that can lead to the same [corresponding to different instances of the noise ], and for each there is a different probability distribution (cf. eqn. 31) and a different set of probability volumes . We will find the probability volumes corresponding to


I show first how to do this exactly, and then show a useful approximation for strong signals.

The mode of the distribution satisfies


(cf. eqns. 1 and 35). Since is a normal variable with zero mean, so are each of the on the righthand side of equation 40. Denote these random variables :


The joint distribution of the is a multivariate Gaussian and its properties determine, through the equation 40, the properties of the distribution of . Consequently we can focus on the joint distribution of the .

Since the are normal, their distribution is determined completely by the means , which vanish, and the quadratic moments


To evaluate the average on the righthand side, we will use the ergodic principle to turn the ensemble average over the random process into a time average over a particular instance of . Recalling that a time translation affects the Fourier transform of a function by a change in phase,




Consequently, \widetext{mathletters}


In going from eqn. 45 to eqn. 46, we used the definition of the PSD of the detector noise :


(cf. Kittel[10] §28). With the result in equation 47, we have {mathletters}


In terms of the (i.e., the inverse of the ), the joint distribution of the is given by


This is also the joint distribution of the quantities


that appear on the lefthand side of equation 40; consequently, we expect that for an observation characterized by a given the probability volumes are given implicitly by \FL



This result is exact as long as the maximum of is also a local extremum of .

As the SNR becomes large the distribution becomes sharply peaked about and the determination of the volume is greatly simplified. Suppose that is so large that for for all of interest, the difference can be linearized in . We then obtain in place of equation 40


The random variables are related to the by a linear transformation,


consequently, the are normal with means


and quadratic moments


The probability distribution is a multivariate Gaussian (cf. eqn. 51): \FL


Note that the matrix now has acquired a physical meaning: in particular, we see that the variances of the are


and the correlation coefficients are given by


In this sense we say that is the covariance matrix of the random variables .

In the strong signal approximation, the surfaces bounding the volume are ellipsoids defined by the equation


where the constant is related to by


It is often the case that not all of the parameters that characterize the signal are of physical interest. In that case, we may integrate the probability distribution (eqn. 51 or 59) over the uninteresting parameters, leaving a distribution describing just the parameters of physical interest.

Finally we come to the question of when the linearization in equation 55 is a reasonable approximation. Two considerations enter here:

  1. It is important that the probability contours of interest (e.g., 90%) do not involve so large that the linearization of is a poor approximation; and

  2. It is important that the condition number (cf. Golub & Van Loan[11]) of the matrix be sufficiently small that the inverse is insensitive to this approximation in the neighborhood of .222Recall that the relative error in is the condition number times the relative error in : for a large condition number, small errors in introduced by the linearization approximation can result in large errors in .

These two conditions will depend on the problem addressed. If the validity of the linearization procedure is doubtful owing to the violation of either or both of these conditions, then we must fall-back on equation 40 and the exact results in equations LABEL:eqn:exact-dist and 54.

4 Application: A perturbed black hole

In this section, I show how to use the approximate techniques developed in §3 to find the precision with which the mass and angular momentum of a perturbed black hole can be determined through measurement in an interferometric gravitational wave detector. This problem was first considered by Echeverria[5].

Consider a single interferometric gravitational wave detector and a perturbed black hole of mass and dimensionless angular momentum parameter . Focus attention on a single oscillation mode of the black hole, e.g., the mode. The strain measured by the detector has the time dependence of an exponentially damped sinusoid characterized by the four parameters , , , and : \FL


For convenience, assume that the perturbation begins abruptly at the starting time . The frequency depends inversely on the mass of the black hole, and has a weak dependence on its angular momentum: for the quasi-normal oscillation mode, {mathletters}


is an accurate semi-empirical expression for the real part of the quasi-normal mode frequency (Echeverria[5] eqn. 4.4 and tbl. II). The quality is the damping time measured in units of the frequency :


For the oscillation mode of the black hole, depends entirely on :


(Echeverria[5] eqn. 4.3 and tbl. II). Finally, the amplitude of the waveform depends on the distance to the source, the size of the perturbation, and the relative orientation of the detector and the source.

This peculiar parameterization of the amplitude reflects our expectation that perturbed black holes are distributed uniformly throughout space (i.e., ) and that all relative orientations of the detector and the black hole source are equally probable. Additionally, it reflects an assumption that perturbations of any allowed amplitude are equally probable; consequently, the a priori distribution is uniform. Let us also assume that , and are uniform and that there is no a priori correlation of , , , or .

An interferometric gravitational wave detector is naturally a broadband receiver, though it can be operated in a narrow band mode (cf. Vinet, Meers, Man, & Brillet [12], Meers[13] and Krolak, Lobo & Meers[14]). Assume that the detector response function is uniform in the frequency domain over the bandwidth of the gravitational wave; consequently, the signal component in the output of the detector [] is equal to the waveform (cf. eqn. 64). Assume also that the noise PSD () of the detector is independent of frequency () in the bandwidth () of the signal (I will discuss the validity of this approximation below).

4.1 The signal-to-noise ratio

As a first step toward finding the precision with which , , and can be determined, we calculate the SNR . With given by equation 64, evaluate using equation 36 to obtain


This expression is valid to better than a percent as long as the signal is observed for a period of time .

In arriving at equation 69 we assumed that the noise PSD is constant over the bandwidth of the signal so that . The signal bandwidth is approximately


For small the bandwidth is approximately , while for large the signal is monochromatic. For small the approximation that is constant over the bandwidth of the signal is only a fair approximation for LIGO (cf. Vogt[1], Abramovici et al.[2]) or LAGOS (cf. Faller et al.[4]); however, it becomes a good approximation for both detectors when (corresponding to ).

The amplitude depends on the detector orientation with respect to the black hole, the amplitude of the perturbation, and the distance between the black hole and the detector. Average over all possible orientations of the detector with respect to the black hole (cf. Thorne[15] §9.5.3) to obtain


where is the total energy radiated by the mode of the black hole perturbation and is the distance of the source.


When operated as a broadband detector, the LIGO advanced detectors will be most sensitive to perturbed black holes with where (cf. Krolak, Lobo & Meers[14], Dhurandhar, Krolak & Lobo[16], Vogt et al.[1], Abramovici et al.[2]). LAGOS will be most sensitive to perturbed black holes in the range , where (cf. Faller et al.[4]). Consequently




For frequencies outside of the range to , the LIGO PSD scales with frequency: for frequencies greater than approximately  Hz (corresponding to ), scales as (cf. Thorne[15], Krolak, Lobo & Meers[14]), and for frequencies less than 100 Hz () it scales as (cf. Dhurandhar, Krolak, & Lobo[16]). Similarly, for frequencies outside the range to the LAGOS PSD scales with frequency: for it scales as , and for it scales as (cf. Faller et al.[4]). Consequently


for LIGO and


for LAGOS.

Little is known about the rate of, or the energy radiated during, black hole formation (cf. Rees[17], Kochanek, Shapiro & Teukolsky[18]); however, owing to the extreme sensitivity of both the LAGOS and LIGO detectors, it seems a conservative estimate that the formation of a black hole of mass anywhere in the universe will be detectable by LAGOS, and the formation of black holes with will be observable in LIGO at least to the distance of the Virgo cluster (10 Mpc). Additionally, note that the energy radiated in the mode during the radial infall of a test body (mass ) onto a Schwarzschild black hole (mass ) is given by


(Davis, Ruffini, Press, & Price[19], Oohara & Nakamura[20]; similar results hold for Kerr black holes: Sasaki & Nakamura[21], Kojima & Nakamura[22]). Consequently, the capture of a solar mass compact object (e.g., a black hole or neutron star) onto a black hole of mass (corresponding to ) may also be observable to a distance of (cf. eqn. 72)

Figure 6 shows the factor (cf. eqn. 73) as a function of . This figure may also be regarded as a plot of for fixed , , , , and detector-source orientation. With this interpretation, note how decreases with increasing . The reason for this behavior is that at fixed , the frequency and damping timescale both increase with ; consequently, a signal of smaller amplitude (i.e., smaller ) will yield the same radiated energy.

4.2 Precision of measurement

While the parameters and are convenient for characterizing the detector response, it is the determination of and that is of direct physical interest. If the perturbed black hole is also observed ele