PHYSICAL REVIEW D, VOLUME 59, 102001

Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities

Bruce Allen* and Joseph D. Romano† Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201 ͑Received 28 October 1997; published 31 March 1999͒ We analyze the signal processing required for the optimal detection of a stochastic background of gravita- tional radiation using laser interferometric detectors. Starting with basic assumptions about the statistical properties of a stochastic gravity-wave background, we derive expressions for the optimal filter function and signal-to-noise ratio for the cross-correlation of the outputs of two gravity-wave detectors. Sensitivity levels required for detection are then calculated. Issues related to ͑i͒ calculating the signal-to-noise ratio for arbitrarily large stochastic backgrounds, ͑ii͒ performing the data analysis in the presence of nonstationary detector noise, ͑iii͒ combining data from multiple detector pairs to increase the sensitivity of a stochastic background search, ͑iv͒ correlating the outputs of 4 or more detectors, and ͑v͒ allowing for the possibility of correlated noise in the outputs of two detectors are discussed. We briefly describe a computer simulation that was used to ‘‘experi- mentally’’ verify the theoretical calculations derived in the paper, and which mimics the generation and detection of a simulated stochastic gravity-wave signal in the presence of simulated detector noise. Numerous graphs and tables of numerical data for the five major interferometers ͑LIGO-WA, LIGO-LA, VIRGO, GEO- 600, and TAMA-300͒ are also given. This information consists of graphs of the noise power spectra, overlap reduction functions, and optimal filter functions; also included are tables of the signal-to-noise ratios and sensitivity levels for cross-correlation measurements between different detector pairs. The treatment given in this paper should be accessible to both theorists involved in data analysis and experimentalists involved in detector design and data acquisition. ͓S0556-2821͑99͒02708-3͔

PACS number͑s͒: 04.80.Nn, 04.30.Db, 07.05.Kf, 95.55.Ym

I. INTRODUCTION gravity-wave detectors. One might also be able to detect a faint stochastic background of gravitational radiation, pro- The design and construction of a number of new and more duced very shortly after the big bang. These detections may sensitive detectors of gravitational radiation is currently un- happen soon after the detectors go ‘‘on-line’’ or they may derway. These include the two Laser Interferometric Gravi- require a decade of further work to increase the sensitivity of tational Wave Observatory ͑LIGO͒ detectors being built in the instruments. But it is fairly safe to say that eventually, Hanford, WA and Livingston, LA by a joint Caltech-MIT when their sensitivity passes some threshold value, the collaboration ͓1͔, the VIRGO detector being built near Pisa, gravity-wave detectors will find sources. Even more exciting Italy by an Italian-French collaboration ͓2͔, the GEO-600 is the prospect that the detectors will discover new sources of detector being built in Hanover, Germany by an Anglo- gravitational radiation—sources which are different from those mentioned above, and which we had not expected to German collaboration ͓3͔, and the TAMA-300 detector being find. It promises to be an exciting time. built near Tokyo, Japan ͓4͔. There are also several resonant The subject of this paper is a stochastic ͑i.e., random͒ bar detectors currently in operation, and several more refined background of gravitational radiation, first studied in detail bar and interferometric detectors presently in the planning by Michelson ͓5͔, Christensen ͓6͔, and Flanagan ͓7͔. and proposal stages. Roughly speaking, it is the type of gravitational radiation The operation of these detectors will have a major impact produced by an extremely large number of weak, indepen- on the field of gravitational physics. For the first time, there dent, and unresolved gravity-wave sources. The radiation is will be a significant amount of experimental data to be ana- stochastic in the sense that it can be characterized only sta- lyzed, and the ‘‘ivory tower’’ relativists will be forced to tistically. As mentioned above, a stochastic background of interact with a broad range of experimenters and data ana- gravitational radiation might be the result of processes that lysts to extract the interesting physics from the data stream. took place very shortly after the big bang. But since we ‘‘Known’’ sources such as coalescing neutron star ͑or black know very little about the state of the universe at that time, it hole͒ binaries, pulsars, supernovae, and other periodic and is impossible to say with any certainty. A stochastic back- transient ͑or burst͒ sources should all be observable with ground of gravitational radiation might also arise from more recent processes ͑e.g., radiation from many unresolved bi- nary star systems͒, and this more recent contribution might *Email address: [email protected] overwhelm the parts of the background that contain informa- †Permanent address: Department of Physical Sciences, University tion about the state of the early universe. In any case, the of Texas at Brownsville, Brownsville, Texas 78521 ͑Email address: properties of the radiation will be very dependent upon the [email protected]͒ source. For example, one would expect a stochastic back-

0556-2821/99/59͑10͒/102001͑41͒/$15.0059 102001-1 ©1999 The American Physical Society BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001 ground of cosmological origin to be highly isotropic, tiple detector pairs when searching for a stochastic back- whereas that produced by white dwarf binaries in our own ground of gravitational radiation. Section VI A also includes galaxy would be highly anisotropic. We will just have to a graph of the ‘‘enhanced’’ LIGO detector noise curves, wait and see what the detectors reveal before we can decide which track the projected performance of the LIGO detector between these two possibilities. design over the next decade. This paper will focus on issues related to the detection of In Sec. VII, we describe a computer simulation that mim- a stochastic background of gravitational radiation. ͑We will ics the generation and detection of a simulated stochastic not talk much about possible sources.͒ We give a complete gravity-wave signal in the presence of simulated detector and comprehensive treatment of the problem of detecting a noise. The simulation was used to verify some of the theo- stochastic background, which should be accessible to both retical calculations derived in the previous sections. theorists involved in the data analysis and experimentalists Section VIII concludes the paper with a brief summary involved in detector design and data acquisition. and lists some topics for future work. The outline of the paper is as follows: Note that throughout the paper, we use c to denote the In Sec. II, we begin by describing the properties of a speed of light and G to denote Newton’s gravitational 10 Ϫ8 stochastic background of gravitational radiation—its spec- constant (cϭ2.998ϫ10 cm/sec and Gϭ6.673ϫ10 3 2 trum, statistical assumptions, and current observational con- cm /g sec ). straints. In Sec. III, we describe how one can correlate the outputs II. STOCHASTIC BACKGROUND: PROPERTIES of two gravity-wave detectors to detect ͑or put an upper limit A. Spectrum on͒ a stochastic gravity-wave signal. Section III B includes a detailed derivation of the overlap reduction function that A stochastic background of gravitational radiation is a covers the case where the two arms of a detector are not random gravity-wave signal produced by a large number of perpendicular ͑e.g., GEO-600͒ and corrects a typographical weak, independent, and unresolved gravity-wave sources. In error that appears in the literature. Section III C includes a many ways it is analogous to the cosmic microwave back- rigorous derivation of the optimal signal processing strategy. ground radiation ͑CMBR͓͒9͔, which is a stochastic back- Most of the material in Secs. II and III has already appeared ground of electromagnetic radiation. As with the CMBR, it in the literature. Interested readers should see Ref. ͓8͔ for is useful to characterize the spectral properties of the gravi- more details, if desired. tational background by specifying how the energy is distrib- In Sec. IV, we ask the following questions: ͑i͒ How do we uted in frequency. Explicitly, one introduces the dimension- decide, from the experimental data, if we’ve detected a sto- less quantity chastic gravity-wave signal? ͑ii͒ Assuming that a stochastic gravity-wave signal is present, how do we estimate its 1 d␳gw ⍀gw͑ f ͒ª , ͑2.1͒ strength? ͑iii͒ Assuming that a stochastic gravity-wave signal ␳critical d ln f is present, what is the minimum value of ⍀0 required to detect it 95% of the time? This leads to a discussion of signal where d␳gw is the energy density of the gravitational radia- detection, parameter estimation, and sensitivity levels for tion contained in the frequency range f to f ϩdf, and ␳critical stochastic background searches, adopting a frequentist point is the critical energy density required ͑today͒ to close the 95%,5% universe: of view. The calculation of ⍀0 in Sec. IV D corrects an error that has appeared in the literature. 3c2H2 ergs In Sec. V, we return to the problem of detection by ad- 0 Ϫ8 2 ␳critical ϭ Ϸ1.6ϫ10 h100 . ͑2.2͒ dressing a series of subtle issues initially ignored in Sec. III. 8␲G cm3 These include ͑i͒ calculating the signal-to-noise ratio for ar- bitrarily large stochastic backgrounds, ͑ii͒ performing the H0 is the Hubble expansion rate ͑today͒, data analysis in the presence of nonstationary detector noise, km 1 ͑iii͒ combining data from multiple detector pairs to increase Ϫ18 H0ϭh100ϫ100 ϭ 3.2ϫ10 h100 the sensitivity of a stochastic background search, ͑iv͒ corre- sec Mpc sec lating the outputs of 4 or more detectors, and ͑v͒ allowing for 1 the possibility of correlated noise in the outputs of two de- Ϫ28 ϭ1.1ϫ10 ch100 , ͑2.3͒ tectors. The material presented in these sections extends the cm initial treatment of these issues given, for example, in Refs. and h is a dimensionless factor, included to account for ͓6,7͔. 100 the different values of H that are quoted in the literature.1 It Section VI consists of a series of graphs and tables of 0 is this dimensionless function of frequency, ⍀ ( f ), that we numerical data for the five major interferometers ͑LIGO- gw will use to describe the spectrum of a stochastic background WA, LIGO-LA, VIRGO, GEO-600, TAMA-300͒. The noise of gravitational radiation. It follows directly from the above power spectra, overlap reduction functions, optimal filter 2 functions, signal-to-noise ratios, and sensitivity levels for definitions that ⍀gw( f )h100 is independent of the actual cross-correlation measurements between different detector pairs ͑not just LIGO͒ are given. This information allows us 1 to determine the optimal way of combining data from mul- h100 almost certainly lies within the range 1/2Ͻh100Ͻ1.

102001-2 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

Hubble expansion rate. For this reason, we will often focus the universe is roughly 20 orders of magnitude larger than attention on this quantity, rather than on ⍀gw( f ) alone. the characteristic period of the waves that LIGO, VIRGO, Two remarks are in order: etc. can detect, and 9 orders of magnitude larger than the ͑i͒ There appears to be some confusion about ⍀gw( f )in longest realistic observation times. It seems very unlikely the literature. Some authors assume that ⍀gw( f )is that a stochastic background of gravitational radiation would constant—i.e., independent of frequency. Although this is have statistical properties that vary over either of these time true for some cosmological models, it is not true for all of scales. But unlike the stochastic gravity-wave background, them. The important point is that any spectrum of gravita- the noise intrinsic to the detectors will change over the tional radiation can be described by an appropriate ⍀gw( f ). course of the observation times. This poses a problem for the With the correct dependence on frequency, ⍀gw( f ) can de- data analysis, which we initially ignore in Sec. III. We return scribe a flat spectrum, a blackbody spectrum, or any other to this problem in Sec. V B where we discuss nonstationary distribution of energy with frequency. detector noise. ͑ii͒⍀gw( f ) is the ratio of the stochastic gravity-wave en- ͑iv͒ The final assumption is that the stochastic gravity- ergy density contained in a bandwidth ⌬ f ϭ f to the total wave background is a Gaussian random process. This means energy density required to close the universe. For the that the joint probability density function of the gravitational CMBR, one can define an analogous quantity: strains hi(ti),hj(tj), . . . in detectors i, j, . . . is a multivari- ate Gaussian ͑i.e., normal͒ distribution. In this case, the mean 1 d␳em values ͗hi(t)͘ and the second-order moments ͗hi(ti)h j(t j)͘ ⍀ ͑ f ͒ . ͑2.4͒ em ª ␳ d ln f completely specify the statistical properties of the signal. For critical many early-universe processes, or even for more recent Since the 2.73 K blackbody spectrum has a peak value of sources of a gravity-wave background, this is a reasonable assumption. It can be justified by the central limit theorem, ⍀ ( f )h2 Ϸ10Ϫ5 at f ϭ1012 Hz, the CMBR contains ͑in em 100 which says that the sum of a large number of statistically the vicinity of 1012 Hz) approximately 10Ϫ5 of the total independent random variables is a Gaussian random vari- energy density required to close the universe. A similar in- able, independent of the probability distributions of the origi- terpretation applies to ⍀ ( f ). gw nal variables. This will be the case for the stochastic back- ground if it is the sum of gravity-wave signals produced by a B. Statistical assumptions large number of independent gravity-wave sources. This as- sumption will not be true, however, if the stochastic back- The spectrum ⍀gw( f ) completely specifies the stochastic background of gravitational radiation provided we make ground is the sum of the radiation produced, e.g., by only a enough additional assumptions. We will assume that the sto- few unresolved binary star systems radiating in a given fre- chastic background is ͑i͒ isotropic, ͑ii͒ unpolarized, ͑iii͒ sta- quency interval at any instant of time. ͑See, e.g., Ref. ͓11͔.͒ tionary, and ͑iv͒ Gaussian. Since these properties might not The above four properties form the basis for the statistical hold in general, it is worthwhile to consider each one of them analysis that we will give in the following sections. We will in turn. assume that they hold throughout, unless we explicitly state ͑i͒ Since it is now well established that the CMBR is otherwise. highly isotropic ͓9͔, it is not unreasonable to assume that a stochastic background of gravitational radiation is also iso- C. Expectation value tropic. But this assumption might not be true. For example, Using the definition of the spectrum ⍀gw( f ) and the sta- as mentioned in Sec. I, if the dominant source of the stochas- tistical assumptions described in the previous subsection, we tic gravity-wave background is a large number of unresolved can derive a useful result for the expectation value of the white dwarf binary star systems within our own galaxy, then Fourier amplitudes of a stochastic background of gravita- the stochastic background will have a distinctly anisotropic tional radiation. This result will be needed in Sec. III when distribution, which forms a ‘‘band in the sky’’ distributed we discuss signal detection and optimal filtering. roughly in the same way as the Milky Way galaxy. It is also The starting point of the derivation is a plane wave ex- possible for a stochastic gravity-wave background of cosmo- pansion for the gravitational metric perturbations in a trans- logical origin to be anisotropic, although one would then verse, traceless gauge: have to explain why the CMBR is isotropic but the gravity- ϱ wave background is not. In either case, such anisotropies can ˆ ជ ជ ˆ ˆ i2␲f͑tϪ⍀•x/c͒ A ˆ be searched for in the data stream. ͑See Ref. ͓10͔ for details.͒ hab͑t,x͒ϭ͚ df d⍀hA͑f,⍀͒e eab͑⍀͒. A ͵Ϫϱ ͵S2 ͑ii͒ The second assumption is that the stochastic gravity- ͑2.5͒ wave background is unpolarized. This means that the gravi- tational radiation incident on a detector has statistically Here ⍀ˆ is a unit vector specifying a direction on the two- equivalent ‘‘plus’’ and ‘‘cross’’ polarization components. ជ ˆ A ˆ We see no strong reason why this should not be the case. sphere, with wave vector kª2␲ f ⍀/c. Also, eab(⍀) are the ͑iii͒ The assumption that the stochastic background is sta- spin-2 polarization tensors for the ‘‘plus’’ and ‘‘cross’’ po- tionary ͑i.e., that all statistical quantities depend only upon larization states Aϭϩ,ϫ. Explicitly, the difference between times, and not on the choice of time ϩ ˆ ˆ ˆ ˆ ˆ origin͒ is almost certainly justified. This is because the age of eab͑⍀͒ϭmambϪnanb , ͑2.6͒

102001-3 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

ϫ 2 e ͑⍀ˆ ͒ϭmˆ nˆ ϩnˆ mˆ , ͑2.7͒ 3H0 ab a b a b H͑ f ͒ϭ ͉f͉Ϫ3⍀ ͉͑f͉͒. ͑2.15͒ 32␲3 gw where Thus, ⍀ˆ ϭcos ␾ sin ␪xˆ ϩsin ␾ sin ␪yˆ ϩcos ␪zˆ, ͑2.8͒ ˆ ˆ ͗hA*͑ f ,⍀͒hAЈ͑ f Ј,⍀Ј͒͘ mˆ ϭsin ␾xˆ Ϫcos ␾yˆ , ͑2.9͒ 3H2 0 2 ˆ ˆ Ϫ3 ϭ 3 ␦ ͑⍀,⍀Ј͒␦AAЈ␦͑fϪfЈ͉͒f͉ ⍀gw͉͑f͉͒, ͑2.16͒ nˆ ϭcos ␾ cos ␪xˆ ϩsin ␾ cos ␪yˆ Ϫsin ␪zˆ, ͑2.10͒ 32␲ and (␪,␾) are the standard polar and azimuthal angles on the which is the desired result. ˆ two-sphere. The Fourier amplitudes hA( f ,⍀) are arbitrary ˆ ˆ D. Observational constraints complex functions that satisfy hA(Ϫ f ,⍀)ϭhA*( f ,⍀), where * denotes complex conjugation. This last relation follows as At present, there are three observational constraints on the a consequence of the reality of hab(t,xជ). stochastic gravity-wave spectrum ⍀gw( f ). These constraints The assumptions that the stochastic background is isotro- are quite weak in the frequency range of interest for ground- pic, unpolarized, and stationary imply that the expectation based interferometers (1 HzϽ f Ͻ103 Hz) and for proposed value ͑i.e., ensemble average͒ of the Fourier amplitudes space-based detectors (10Ϫ4 HzϽ f Ͻ10Ϫ1 Hz). There are h ( f ,⍀ˆ ) satisfies tighter constraints on the spectrum in two frequency ranges, A and one ‘‘wideband’’ but very weak constraint. In this paper, ˆ ˆ 2 ˆ ˆ we simply state the constraints. For a more complete discus- ͗hA*͑ f ,⍀͒hAЈ͑ f Ј,⍀Ј͒͘ϭ␦ ͑⍀,⍀Ј͒␦AAЈ␦͑fϪfЈ͒H͑f͒, ͑2.11͒ sion, see Ref. ͓8͔ and the references mentioned therein. ͑i͒ The strongest observational constraint on ⍀gw( f ) 2 ˆ ˆ comes from the high degree of isotropy observed in the where ␦ (⍀,⍀Ј)ª␦(␾Ϫ␾Ј)␦(cos ␪Ϫcos ␪Ј) is the covari- ant Dirac delta function on the two-sphere, and H( f )isa CMBR. In particular, the 1-yr ͓13,14͔, 2-yr ͓15͔, and 4-yr 16 data sets from the Cosmic Background Explorer real, non-negative function, satisfying H( f )ϭH(Ϫ f ).2 If we ͓ ͔ further assume that the stochastic background has zero mean, ͑COBE͒ satellite place very strong restrictions on ⍀gw( f )at then very low frequencies: 2 H0 ͗h ͑ f ,⍀ˆ ͒͘ϭ0. ͑2.12͒ ⍀ ͑ f ͒h2 Ͻ7ϫ10Ϫ11 for H Ͻ f Ͻ30H . A gw 100 ͩ f ͪ 0 0 Finally, since we are assuming that the stochastic back- ͑2.17͒ ground is Gaussian, the expectation values ͑2.11͒ and ͑2.12͒ Note that the above constraint does not apply to any gravi- completely specify its statistical properties. tational wave, but only to those of cosmological origin that H( f ) is related to the spectrum ⍀ ( f ) of the stochastic gw were already present at the time of last scattering of the gravity-wave background. This follows from the expression Ϫ18 CMBR. Also, since H0ϭ3.2ϫ10 h100 Hz, this limit ap- Ϫ18 c2 plies only over a narrow band of frequencies (10 Hz ˙ ˙ab Ϫ16 ␳gwϭ ͗hab͑t,xជ͒h ͑t,xជ͒͘ ͑2.13͒ Ͻ f Ͻ10 Hz), which is far below any frequency band 32␲G accessible to investigation by either Earth-based or space- based interferometers. Thus, although this constraint is se- for the energy density in gravitational waves see, e.g., p. ͑ vere, it is not directly relevant for any of the present-day 955 of Ref. 12 . By differentiating the plane wave expan- ͓ ͔͒ gravity-wave experiments. sion ͑2.5͒ with respect to t, forming the contraction in Eq. ͑ii͒ The second observational constraint comes from al- 2.13 , and calculating the expectation value using Eq. ͑ ͒ most a decade of monitoring the radio pulses arriving from a ͑2.12͒,wefind number of stable millisecond pulsars ͓17͔. These pulsars are remarkably stable clocks, and the regularity of their pulses 4 2c2 ϱ ϱ d ␲ 2 ␳gw ␳gw ϭ df f H͑f͒ ϭ: df . places tight constraints on ⍀gw( f ) at frequencies on the order G ͵0 ͩ ͵0 df ͪ of the inverse of the observation time of the pulsars ͑2.14͒ (ϳ10Ϫ8 Hz):

Using Eqs. ͑2.1͒ and ͑2.2͒ for ⍀gw( f ) then yields Ϫ8 2 Ϫ8 ⍀gw͑ f ϭ10 Hz͒h100Ͻ10 . ͑2.18͒

Like the constraint on the stochastic gravity-wave back- 2If the stochastic background is anisotropic, we should replace ground from the isotropy of the CMBR, the millisecond pul- H( f ) by a function that depends on ⍀ˆ in addition to f. If the sto- sar timing constraint is irrelevant for current gravity-wave chastic background is polarized, we should replace H( f ) by a func- experiments. The frequency f ϭ10Ϫ8 Hz is 10 orders of tion that depends on the polarization Aϭϩ,ϫ as well. magnitude smaller than the band of frequencies accessible to

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LIGO, VIRGO, etc., and 4 orders of magnitude smaller than detectors are coincident and coaligned ͑i.e., have identical that for proposed space-based detectors. locations and arm orientations͒, the gravitational strains are ͑iii͒ The third and final observational constraint on identical: ⍀gw( f ) comes from the standard model of big-bang nucleo- synthesis ͓9͔. This model provides remarkably accurate fits to the observed abundances of the light elements in the uni- h͑t͒ªh1͑t͒ϭh2͑t͒. ͑3.3͒ verse, tightly constraining a number of key cosmological pa- rameters. One of the parameters constrained in this way is the expansion rate of the universe at the time of nucleosyn- But the noises n1(t) and n2(t) are not equal to one another. thesis. This places a constraint on the energy density of the As mentioned above, we will assume that they are stationary, universe at that time, which in turn constrains the energy Gaussian, statistically independent of one another and of the density in a cosmological background of gravitational radia- gravitational strains, and much larger in magnitude than the tion: gravitational strains.4 Given the detector outputs s1(t) and s2(t), we can form a 2 Ϫ5 product ‘‘signal’’ S by multiplying them together and inte- d ln f ⍀gw͑ f ͒h100Ͻ10 . ͑2.19͒ ͵f Ͼ10Ϫ8 Hz grating over time: Although this bound constrains the spectrum of gravitational radiation ⍀gw( f ) over a broad range of frequencies, it is not T/2 very restrictive. Sª dt s1͑t͒s2͑t͒. ͑3.4͒ ͵ϪT/2

III. STOCHASTIC BACKGROUND: DETECTION In this section, we begin our detailed discussion of the This quantity is proportional to the ͑zero-lag͒ cross- detection of a stochastic background of gravitational radia- correlation of s1(t) and s2(t) for an observation time T. tion. We explain how one can correlate the outputs of two Since s1(t) and s2(t) are random variables, so too is S.Ithas gravity-wave detectors to detect ͑or put an upper limit on͒ a a mean value stochastic background signal. In Sec. III B, we give a de- tailed derivation of the overlap reduction function that covers the case where the two arms of a detector are not perpen- ␮ª͗S͘ ͑3.5͒ dicular ͑e.g., GEO-600͒. In Sec. III C, we give a rigorous derivation of the optimal signal processing strategy. The sta- tistical assumptions that we will use for the stochastic and variance gravity-wave background are those described in Sec. II B. In addition, we will assume that the noises intrinsic to the de- 2 2 2 tectors are ͑i͒ stationary, ͑ii͒ Gaussian, ͑iii͒ statistically inde- ␴ ª͗S ͘Ϫ͗S͘ , ͑3.6͒ pendent of one another and of the stochastic gravity-wave background, and ͑iv͒ much larger in magnitude than the sto-

chastic gravity-wave background. The modifications that are which are related to the variances of n1(t), n2(t), and necessary when one relaxes most of these assumptions will h(t).5 The goal is to calculate ␮ and ␴, and then to construct be discussed in Sec. V. the signal-to-noise ratio

A. Coincident and coaligned detectors ␮ To begin, let us consider the simplest possible case. Let us SNR . ͑3.7͒ suppose that we have two coincident and coaligned gravity- ª␴ wave detectors with outputs

s1͑t͒ªh1͑t͒ϩn1͑t͒, ͑3.1͒ As we shall see in Sec. IV, the value of the signal-to-noise ratio enters the decision rule for the detection of a stochastic s2͑t͒ªh2͑t͒ϩn2͑t͒. ͑3.2͒ gravity-wave signal.

Here h1(t) and h2(t) denote the gravitational strains in the two detectors due to the stochastic background, and n1(t) 4The assumption that the noises intrinsic to the detectors are sta- and n2(t) denote the noises intrinsic to the first and second detectors, respectively.3 Since we are assuming that the two tistically independent of one another is unrealistic for the case of coincident and coaligned detectors. But it is a reasonable assump- tion for widely separated detector sites. ͑See Sec. V E for more details.͒ 3 5 We will assume throughout that the detector outputs are not whit- The mean values of n1(t), n2(t), and h(t) are equal to zero, ened. either by assumption or by definition.

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Let us start with the mean value ␮. By definition, is the response of either detector to a zero frequency, unit amplitude, Aϭϩ,ϫ polarized gravitational wave. Using Eq. T/2 2.16 for the expectation value h*( f , ˆ )h ( f , ˆ ) , the ␮ ͗S͘ϭ dt ͗s ͑t͒s ͑t͒͘ ͑3.8͒ ͑ ͒ ͗ A ⍀ AЈ Ј ⍀Ј ͘ ª ͵ 1 2 2 ϪT/2 above expression for ␴h simplifies to

2 T/2 3H ϱ 2 2 0 Ϫ3 ˆ A ˆ A ˆ ϭ dt ͗h ͑t͒ϩh͑t͒n ͑t͒ ␴hϭ 3 df ͉f͉ ⍀gw͉͑f͉͒ d⍀ F ͑⍀͒F ͑⍀͒ 2 ͵ ͚ ͵ 2 ͵ϪT/2 32␲ Ϫϱ A S ͑3.15͒ ϩn1͑t͒h͑t͒ϩn1͑t͒n2͑t͒͘ ͑3.9͒ 3H2 ϱ 0 Ϫ3 T/2 ϭ 2 df ͉f͉ ⍀gw͉͑f͉͒, ͑3.16͒ ϭ dt ͗h2͑t͒͘ ͑3.10͒ 20␲ ͵Ϫϱ ͵ϪT/2 where we used 2 2 ϭT͗h ͑t͒͘ϭ:T␴h , ͑3.11͒ 8␲ d⍀ˆ FA͑⍀ˆ ͒FA͑⍀ˆ ͒ϭ ͑3.17͒ 2 6 ͚ ͵ 2 where ␴h denotes the ͑time-independent͒ variance of h(t). A S 5 Note that we used the statistical independence of n1(t), n2(t), and h(t) to obtain the third line, and the sta- to obtain the last line. Thus, for coincident and coaligned tionarity of h(t) to obtain the last. detectors, the mean value of the cross-correlation signal S is 2 2 To express the variance ␴h ͗h (t)͘ in terms of the fre- ª 2 ϱ 3H0 quency spectrum ⍀gw( f ), we will make use of the plane Ϫ3 ␮ϭ 2 T df ͉f͉ ⍀gw͉͑f͉͒. ͑3.18͒ wave expansion ͑2.5͒ and the expectation value ͑2.16͒. Since 20␲ ͵Ϫϱ

1 a b a b This is the first of our desired results. h͑t͒ h ͑t,xជ ͒ ͑Xˆ Xˆ ϪYˆ Yˆ ͒ ͑3.12͒ 2 ª ab 0 2 To evaluate the variance ␴ , we will make use of the assumption that the noises intrinsic to the detectors are much larger in magnitude than the gravitational strains. Then ͑where xជ 0 is the common position vector of the central sta- tion of the two coincident and coaligned detectors, and Xˆ a 2 2 2 2 ␴ ª͗S ͘Ϫ͗S͘ Ϸ͗S ͘ ͑3.19͒ and Yˆ a are unit vectors pointing in the directions of the de- tector arms͒,7 it follows that T/2 T/2 ϭ dt dtЈ͗s1͑t͒s2͑t͒s1͑tЈ͒s2͑tЈ͒͘ ͵ϪT/2 ͵ϪT/2 ϱ ϱ ␴2ϭ d⍀ˆ d⍀ˆ Ј df dfЈ ͑3.20͒ h ͚ ͚ ͵ 2 ͵ 2 ͵ ͵ A AЈ S S Ϫϱ Ϫϱ T/2 T/2 ˆ ជ ˆ ˆ Ϫi2␲f͑tϪ⍀•x0 /c͒ ϫ͗h*͑f,⍀͒hA ͑fЈ,⍀Ј͒͘e Ϸ dt dtЈ͗n1͑t͒n2͑t͒n1͑tЈ͒n2͑tЈ͒͘ A Ј ͵ϪT/2 ͵ϪT/2 Ϫ ˆ ជ ͑3.21͒ ϫei2␲fЈ͑t ⍀Ј•x0 /c͒FA͑⍀ˆ ͒FAЈ͑⍀ˆ Ј͒, ͑3.13͒

T/2 T/2 where ϭ dt dtЈ͗n1͑t͒n1͑tЈ͒͗͘n2͑t͒n2͑tЈ͒͘, ͵ϪT/2 ͵ϪT/2 1 ͑3.22͒ FA͑⍀ˆ ͒ eA ͑⍀ˆ ͒ ͑Xˆ aXˆ bϪYˆ aYˆ b͒ ͑3.14͒ ª ab 2 where we used the statistical independence of n1(t) and 8 n2(t) to obtain the last line. By definition,

6The dimensions of 2 and 2 are different: 2 has dimensions of 1 ϱ ␴h ␴ ␴h i2␲f͑tϪtЈ͒ strain2, while ␴2 has dimensions of strain4 sec2. ͓See, e.g., Eq. ͗ni͑t͒ni͑tЈ͒͘ ϭ: df e Pi͉͑f͉͒, ͑3.23͒ 2 ͵Ϫϱ ͑3.20͒.͔ 7xជ and Xˆ a,Yˆ a are actually functions of time due to the Earth’s 0 where Pi(͉f͉) is the ͑one-sided͒ noise power spectrum of the motion with respect to the cosmological rest frame. They can be ith detector (iϭ1,2). Pi(͉f͉) is a real, non-negative func- treated as constants, however, since ͑i͒ the velocity of the Earth with respect to the cosmological rest frame is small compared to the speed of light and ͑ii͒ the distance that the central stations and arms 8 move during the correlation time between the two detectors is small Equation ͑3.23͒ can also be written in the frequency domain: compared to the arm length. ͑The correlation time equals zero for 1 coincident and coaligned detectors; it equals the light travel time ˜ ˜ ͗ni*͑ f ͒ni͑ f Ј͒͘ϭ ␦͑ f Ϫ f Ј͒Pi͉͑f͉͒. between the two detectors when the detectors are spatially sepa- 2 rated.͒ See Ref. ͓10͔ for more details. See the discussion surrounding Eq. ͑3.64͒ for more details.

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is a finite-time approximation to the Dirac delta function ␦( f ). In the limit T ϱ, ␦ ( f ) reduces to ␦( f ), but for a → T finite observation time T, one has ␦T(0)ϭT. Since in prac- tice the observation time T will be large enough so that ␦T( f Ϫ f Ј) is sharply peaked over a region in f Ϫ f Ј whose size Ϸ1/T is very small compared to the scale on which the 10 functions P1(͉f͉) and P2(͉f͉) are varying, we can replace one of the finite-time delta functions ␦T( f Ϫ f Ј) by an ordi- nary Dirac delta function, and evaluate the other at f ϭ f Ј. Doing this yields

T ϱ 2 ␴ Ϸ df P1͉͑f͉͒P2͉͑f͉͒, ͑3.28͒ 4 ͵Ϫϱ

which is the second of our desired results. Using Eq. ͑3.18͒ and ͑3.28͒, we can now form the signal- 11 FIG. 1. A log-log plot of the predicted noise power spectra for to-noise ratio the initial and advanced LIGO detectors. The data for these noise ϱ power spectra were taken from the published design goals ͓1͔. df f Ϫ3⍀ f 2 ͵ ͉ ͉ gw͉͑ ͉͒ ␮ 3H0 Ϫϱ tion, defined with a factor of 1/2 to agree with the standard SNRª Ϸ ͱT . ␴ 10␲2 ϱ 1/2 ͑one-sided͒ definition used by instrument builders. It df P f P f 9 1͉͑ ͉͒ 2͉͑ ͉͒ satisfies ͫ ͵Ϫϱ ͬ ͑3.29͒ ϱ 2 2 ␴n ª͗ni ͑t͒͘ϭ df Pi͑f͒, ͑3.24͒ The multiplicative factor of ͱT means that we can always i ͵0 exceed any prescribed value of the signal-to-noise ratio by correlating the outputs of two gravity-wave detectors for a and so the total noise power is the integral of Pi( f ) over all 12 positive frequencies f from 0 to ϱ, not from Ϫϱ to ϱ. long enough period of time. We will have more to say ͑Hence the reason for the name one-sided.͒ Graphs of the about signal detection, parameter estimation, and sensitivity predicted noise power spectra for the initial and advanced levels for stochastic background searches in Sec. IV. LIGO detectors are shown in Fig. 1. Graphs of the predicted noise power spectra for the other major interferometers ͑i.e., B. Overlap reduction function VIRGO, GEO-600, and TAMA-300͒ and for the ‘‘en- To provide a rigorous treatment of the signal analysis for hanced’’ LIGO detectors are shown in Figs. 11–15 in Sec. a stochastic background of gravitational radiation, we must VI A. take into account the fact that the two gravity-wave detectors Inserting Eq. ͑3.23͒ into Eq. ͑3.22͒ yields will not necessarily be either coincident or coaligned. There will be a reduction in sensitivity due to ͑i͒ the separation 1 T/2 T/2 ϱ ϱ ␴2Ϸ dt dtЈ df dfЈei2␲f͑tϪtЈ͒ time delay between the two detectors and ͑ii͒ the non-parallel 4 ͵ϪT/2 ͵ϪT/2 ͵Ϫϱ ͵Ϫϱ alignment of the detector arms. These two effects imply that h (t) and h (t) are no longer equal; the overlap between the Ϫi2␲fЈ͑tϪtЈ͒ 1 2 ϫe P1͉͑f͉͒P2͉͑fЈ͉͒, ͑3.25͒ gravitational strains in the two detectors is only partial. Sta- tistically, these effects are most apparent in the frequency where we used the reality of n (t) and P ( f ) to produce 2 2 ͉ ͉ domain. the minus sign in the power of the second exponential. If we integrate this expression over t and tЈ, we find

1 ϱ ϱ 10 2 2 Typically, an observation time T will be on the order of months ␴ Ϸ df dfЈ␦T͑fϪfЈ͒P1͉͑f͉͒P2͉͑fЈ͉͒, ͑i.e., 107 sec͒, while the noise power spectra P ( f ) vary on a scale 4 ͵Ϫϱ ͵Ϫϱ i ͉ ͉ ͑3.26͒ of greater than a few Hz. 11Be warned that the signal processing strategy described above, where leading to Eq. ͑3.29͒,isnot optimal even for the case of coincident and coaligned detectors. The optimal signal processing strategy, T/2 sin fT which is described in Sec. III C, leads to the signal-to-noise ratio Ϫi2 ft ͑␲ ͒ ␦ ͑ f ͒ dt e ␲ ϭ ͑3.27͒ given by Eq. ͑3.75͒. Setting ␥( f )ϭ1 in Eq. ͑3.75͒ yields an expres- T ª ͵ ␲f ϪT/2 sion for the optimally filtered signal-to-noise ratio for the case of coincident and coaligned detectors. 12This assumes that there is no systematic source of correlated 9Unlike ␴2 and ␴2, ␴2 and ␴2 have the same dimensions h h ni detector noise. In Sec. V E, we discuss the limits that correlated (strain2). detector noise imposes.

102001-7 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

FIG. 3. The surface of the Earth (15°ϽlatitudeϽ75°,Ϫ130° ϽlongitudeϽ20°) including the LIGO detectors in Hanford, WA ͑L1͒ and Livingston, LA ͑L2͒, the VIRGO detector ͑V͒ in Pisa, FIG. 2. The overlap reduction function ␥( f ) for the Hanford, Italy, and the GEO-600 ͑G͒ detector in Hanover, Germany. The WA and Livingston, LA LIGO detector pair. ͑The horizontal axis of perpendicular arms of the LIGO detectors are also illustrated the left-hand graph is linear, while that of the right-hand graph is ͑though not to scale͒. A plane gravitational wave passing by the log10 .) The overlap reduction function has its first zero at 64 Hz, as Earth is indicated by successive minimum, zero, and maximum of explained in the text. It falls off rapidly at higher frequencies. the wave. As this wave passes by the pair of LIGO detectors, it excites the two in coincidence at the moment shown, since both The overlap reduction function ␥( f ), first calculated in detectors are driven negative by the wave. During the time when the closed form by Flanagan ͓7͔, quantifies these two effects. zero is between L1 and L2, the two detectors respond in anti- This is a dimensionless function of frequency f, which is coincidence. Provided that the wavelength of the incident gravita- determined by the relative positions and orientations of a pair tional wave is larger than twice the separation (dϭ3001 km) be- of detectors. Explicitly, tween the detectors, the two detectors are driven in coincidence more of the time than in anti-coincidence. 5 ˆ ជ ˆ i2␲ f ⍀•⌬x/c A ˆ A ˆ ␥͑ f ͒ d⍀e F1 ͑⍀͒F2 ͑⍀͒, two-sphere is an isotropic average over all directions ⍀ˆ of ª 8␲͚ ͵ 2 A S the incoming radiation. ͑iv͒ The exponential phase factor is ͑3.30͒ the phase shift arising from the time delay between the two where ⍀ˆ is a unit vector specifying a direction on the two- detectors for radiation arriving along the direction ⍀ˆ . In the ជ ជ ជ limit f 0, this phase shift also goes to zero, and the two sphere, ⌬xªx1Ϫx2 is the separation vector between the → central stations of the two detector sites, and detectors become effectively coincident. ͑v͒ The quantity A ˆ A ˆ ͚AF1 (⍀)F2 (⍀) is the sum of products of the responses of 1 the two detectors to the ϩ and ϫ polarization waves. For A ˆ A ˆ ab A ˆ ˆ a ˆ b ˆ a ˆ b Fi ͑⍀͒ªeab͑⍀͒di ªeab͑⍀͒ ͑Xi Xi ϪYi Yi ͒ A ˆ A ˆ 2 coaligned detectors, F1 (⍀)ϭF2 (⍀) and the integral of this ͑3.31͒ quantity over the two-sphere equals the inverse of the overall normalization factor. ͓See Eq. ͑3.17͒.͔ is the response of the ith detector (iϭ1,2) to the Aϭϩ,ϫ Figure 2 shows a graph of the overlap reduction function polarization. ͓See also Eq. ͑3.14͒.͔ The symmetric, trace-free ␥( f ) for the Hanford, WA and Livingston, LA LIGO detec- ab 13 tensor di specifies the orientation of the two arms of the ith tor pair. Note that the overlap reduction function for the detector. The overlap reduction function ␥( f ) equals unity LIGO detector pair is negative as f 0. This is because the for coincident and coaligned detectors. It decreases below arm orientations of the two LIGO detectors→ are not parallel to unity when the detectors are shifted apart ͑so there is a phase one another, but are rotated by 90°. If, for example, the shift between the signals in the two detectors͒ or rotated out Livingston, LA detector arms were rotated by 90° in the of coalignment ͑so the detectors are sensitive to different clockwise direction, only the overall sign of ␥( f ) would polarizations͒. In Sec. III C, we will see that ␥( f ) arises change. Note also that the magnitude of ␥(0) is not unity, naturally when evaluating the expectation value of the prod- because the planes of the Hanford, WA and Livingston, LA 14 uct of the gravitational strains at two different detectors detectors are not identical. Thus, the arms of the two de- when they are driven by an isotropic and unpolarized sto- tectors are not exactly parallel, and ͉␥(0)͉ϭ0.89, which is chastic background of gravitational radiation. less than 1. To get a better feeling for the meaning of ␥( f ), let us look at each term in Eq. ͑3.30͒ separately: ͑i͒ The overall normalization factor 5/8␲ is chosen so that for a pair of 13Figures 16–20 in Sec. VI B show graphs of the overlap reduc- coincident and coaligned detectors ␥( f )ϭ1 for all frequen- tion functions for different detector pairs. cies f. ͑ii͒ The sum over polarizations A is appropriate for an 14The two LIGO detectors are separated by an angle of 27.2° as unpolarized stochastic background. ͑iii͒ The integral over the seen from the center of the Earth.

102001-8 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

From Fig. 2, one also sees that the overlap reduction func- ⌫ ͑␣,sˆ͒ϭA͑␣͒␦ ␦ ϩB͑␣͒͑␦ ␦ ϩ␦ ␦ ͒ tion for the two LIGO detectors has its first zero at 64 Hz. abcd ab cd ac bd bc ad This can be explained by the fact that a gravitational plane ϩC͑␣͒͑␦abscsdϩ␦cdsasb͒ wave passing by the Earth excites a pair of detectors in co- incidence when the positive ͑or negative͒ amplitude part of ϩD͑␣͒͑␦acsbsdϩ␦adsbsc the wave is passing by both detectors at the same time; it ϩ␦bcsasdϩ␦bdsasc͒ϩE͑␣͒sasbscsd . excites the two detectors in anti-coincidence when the posi- tive ͑or negative͒ amplitude part of the wave is passing by ͑3.35͒ one detector, and the negative ͑or positive͒ amplitude part of We then contract Eq. ͑3.35͒ with ␦ab␦cd,(␦ac␦bd the wave is passing by the other detector. ͑See Fig. 3.͒ Pro- bc ad a b c d vided that the wavelength of the incident gravitational wave ϩ␦ ␦ ),...,s s s s to obtain a linear system of equa- is larger than twice the distance between the two detectors, tions for the functions A,B,...,E: the detectors will be driven in coincidence ͑on average͒. For 96641 A p the case of the LIGO detector pair, this means that the Han- ford, WA and Livingston, LA detectors will be driven in 6 24 4 16 2 B q coincidence ͑on the average͒ by an isotropic and unpolarized 64882 C ͑␣͒ϭ r ͑␣͒, ͑3.36͒ stochastic background of gravitational radiation having a fre- quency of less than f ϭc/(2d)ϭ50 Hz. The actual fre- 4 16 8 24 4 D s quency of the zero ( f ϭ64 Hz) is slightly larger than this, ͫ12241ͬͫEͬ ͫ t ͬ since ␥( f ) is a sum of three spherical Bessel functions, which does not vanish at exactly 50 Hz. where In Appendix B of Ref. ͓7͔, Flanagan outlines a derivation ˆ ab cd of a closed-form expression for the overlap reduction func- p͑␣͒ª⌫abcd͑␣,s͒␦ ␦ , tion ␥( f ). The resulting expression applies to any pair of ˆ ac bd bc ad gravity-wave detectors, including interferometers with non- q͑␣͒ª⌫abcd͑␣,s͒͑␦ ␦ ϩ␦ ␦ ͒, perpendicular arms and/or arbitrary orientations. This is a useful result, because, e.g., the arms of the GEO-600 detector ˆ ab c d cd a b r͑␣͒ ⌫abcd͑␣,s͒͑␦ s s ϩ␦ s s ͒, are separated by 94.33°. Below we give a more detailed ª version of the derivation that appears in Ref. ͓7͔, and correct s ␣͒ ⌫ ␣,sˆ͒ ␦acsbsdϩ␦adsbscϩ␦bcsasd a typographical error that appears in Eq. ͑B6͒ of that paper. ͑ ª abcd͑ ͑ We take, as our starting point for the derivation, the inte- ϩ␦bdsasc͒, gral expression ͑3.30͒ for ␥( f ). To simplify the notation in what follows, we also define ˆ a b c d t͑␣͒ª⌫abcd͑␣,s͒s s s s . ͑3.37͒

2␲ fd From Eq. ͑3.34͒, we see that the functions p,q,...,t are ⌬xជ dsˆ and ␣ , ͑3.32͒ ª ª c scalar integrals that involve contractions of the spin-two po- A ˆ larization tensors eab(⍀). To evaluate these integrals, we choose ͑without loss of ˆ where s is a unit vector that points in the direction connect- generality͒ a coordinate system where the unit vector sˆ co- ing the two detectors, and d is the distance between the two ˆ detectors. In terms of these quantities, we can write incides with unit vector z. Then ˆ ˆ ˆ ˆ ˆ ˆ ⍀•sϭcos ␪, m•sϭ0, n•sϭϪsin ␪, ͑3.38͒ ab cd ˆ ␥͑ f ͒ϭd1 d2 ⌫abcd͑␣,s͒, ͑3.33͒ and where p͑␣͒ϭ0,

5 1 i␣⍀ˆ sˆ A A i␣x ⌫ ͑␣,sˆ͒ d⍀ˆ e • e ͑⍀ˆ ͒e ͑⍀ˆ ͒. q͑␣͒ϭ10 dx e ϭ20j0͑␣͒, abcd ª ͚ 2 ab cd ͵ 8␲ A ͵S Ϫ1 ͑3.34͒ r͑␣͒ϭ0, ( ,sˆ) is a tensor which is symmetric under the inter- ⌫abcd ␣ 1 40 i␣x 2 changes a b,c d,ab cd. It is also trace-free with re- s͑␣͒ϭ10 dx e ͑1Ϫx ͒ϭ j1͑␣͒, spect to the↔ab and↔ cd index↔ pairs. ͵Ϫ1 ␣ To evaluate ⌫ (␣,sˆ), we begin by writing down the abcd 5 1 20 most general tensor constructed from ␦ab and sa that has the i␣x 2 2 t͑␣͒ϭ dx e ͑1Ϫx ͒ ϭ 2 j2͑␣͒, ͑3.39͒ above-mentioned symmetry properties: 4 ͵Ϫ1 ␣

102001-9 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001 where j0(␣), j1(␣), and j2(␣) are the standard spherical start by writing the cross-correlation signal S between the Bessel functions: outputs of the two detectors in the following form:

sin ␣ T/2 T/2 j0͑␣͒ϭ , ␣ Sª dt dtЈ s1͑t͒s2͑tЈ͒Q͑t,tЈ͒, ͑3.45͒ ͵ϪT/2 ͵ϪT/2 sin ␣ cos ␣ j ͑␣͒ϭ Ϫ , where as before 1 ␣2 ␣

sin ␣ cos ␣ sin ␣ s1͑t͒ªh1͑t͒ϩn1͑t͒, ͑3.46͒ j ͑␣͒ϭ3 Ϫ3 Ϫ . ͑3.40͒ 2 ␣3 ␣2 ␣ s2͑t͒ªh2͑t͒ϩn2͑t͒, ͑3.47͒ Note that p(␣)ϭ0 and r(␣)ϭ0 are immediate conse- ˆ quences of the trace-free property of ⌫abcd(␣,s). but now Q(t,tЈ) is a filter function, which is not necessarily The above linear system of equations ͑3.36͒ can be in- equal to ␦(tϪtЈ) as we assumed in Sec. III A. Because we verted for the functions A,B,...,E. The results are are assuming in this section that the statistical properties of

2 the stochastic gravity-wave background and noise intrinsic to A Ϫ5␣ 10␣ 5 the detectors are both stationary, the best choice of filter 2 B 5␣ Ϫ10␣ 5 j function Q(t,tЈ) can depend only upon the time difference 0 ⌬t tϪt . The goal is to find the optimal choice of filter 1 5␣2 Ϫ10␣ Ϫ25 ª Ј C ͑␣͒ϭ j1 ͑␣͒. function Q(tϪtЈ) Q(t,tЈ) in a rigorous way. 2␣2 2 ª D Ϫ5␣ 20␣ Ϫ25 ͫ j ͬ The optimal choice of filter function Q(tϪtЈ) will depend 2 2 upon the locations and orientations of the detectors, as well ͫ E ͬ 5␣ Ϫ50␣ 175 ͫ ͬ as on the spectrum of the stochastic gravity-wave back- ͑3.41͒ ground and the noise power spectra of the detectors. It falls off rapidly to zero for time delays ⌬tϭtϪtЈ whose magni- Finally, to obtain an expression for the overlap reduction tude is large compared to the light travel time d/c between function ␥( f ), we substitute Eq. ͑3.35͒ into Eq. ͑3.33͒. Since 15 ab the two sites. ͑See Fig. 5, which is located at the end of this di (iϭ1,2) is trace-free, it follows that section.͒ Since a typical observation time T will be ӷd/c, we are justified in changing the limits on one of the integra- ϭ ab ϩ ab c ␥͑ f ͒ 2B͑␣͒d1 d2ab 4D͑␣͒d1 d2a sbsc tions in Eq. ͑3.45͒ to obtain ab cd ϩE͑␣͒d1 d2 sasbscsd . ͑3.42͒ T/2 ϱ Sϭ dt dtЈ s1͑t͒s2͑tЈ͒Q͑tϪtЈ͒. ͑3.48͒ Substituting the expressions for the functions B,D,E given ͵ϪT/2 ͵Ϫϱ by Eq. ͑3.41͒ into Eq. ͑3.42͒ yields This change of limits simplifies the mathematical analysis ab ab c that follows. ␥͑ f ͒ϭ␳1͑␣͒d1 d2abϩ␳2͑␣͒d1 d2a sbsc We can also write Eq. ͑3.48͒ in the frequency domain. ab cd ϩ␳3͑␣͒d1 d2 sasbscsd , ͑3.43͒ Using the convention where ϱ ˜ Ϫi2␲ft g͑ f ͒ª ͵ dt e g͑t͒ ͑3.49͒ 2 Ϫϱ ␳1 10␣ Ϫ20␣ 10 j0 1 2 ␳2 ͑␣͒ϭ Ϫ20␣ 80␣ Ϫ100 j1 ͑␣͒. for the Fourier transform of g(t), it follows that 2␣2 ͫ ␳ ͬ ͫ 5␣2 Ϫ50␣ 175ͬͫj ͬ 3 2 ϱ ϱ ˜ ˜ ˜ ͑3.44͒ Sϭ df dfЈ␦T͑fϪfЈ͒s1*͑f͒s2͑fЈ͒Q͑fЈ͒, ͵Ϫϱ ͵Ϫϱ This is the desired result. ͑3.50͒ Note that in Eq. ͑B6͒ of Ref. ͓7͔, the factor multiplying ˜ ˜ ˜ j 1(␣)in␳1(␣)isϪ2/␣. As shown in Eq. ͑3.44͒, this factor where s1( f ), s2(f), and Q( f ) are the Fourier transforms of should equal Ϫ10/␣. s1(t), s2(t), and Q(tϪtЈ), and ␦T( f Ϫ f Ј) is the finite-time approximation to the Dirac delta function ␦( f Ϫ f Ј) defined C. Optimal filtering by Eq. ͑3.27͒. Note also that for a real Q(tϪtЈ), Q˜(Ϫf) ˜ Using the techniques developed in the previous two sub- ϭQ*(f). sections, we are now in a position to give a rigorous deriva- tion of the optimal signal processing required for the detec- tion of a stochastic background of gravitational radiation. We 15d/cϭ10Ϫ2 sec for the LIGO detector pair.

102001-10 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

The optimal choice of filter function also depends on the cross-correlation signal S defined by Eqs. ͑3.5͒ and ͑3.6͒. quantity that we want to maximize. As we shall see in Sec. The techniques that we will use to evaluate ␮ and ␴2 are IV, in the context of stochastic background searches, it is very similar to those that we used in Sec. III A for the case of natural to maximize the signal-to-noise ratio coincident and coaligned detectors. The calculation of the mean value ␮ is straightforward. ␮ Since we are assuming that the noises intrinsic to the two SNR , ͑3.51͒ ª␴ detectors are statistically independent of each other and of the gravitational strains, it follows immediately from Eq. where ␮ and ␴2 are the mean value and variance of the ͑3.50͒ that

ϱ ϱ ˜ ˜ ˜ ␮ª͗S͘ϭ df dfЈ␦T͑fϪfЈ͒͗h1*͑f͒h2͑fЈ͒͘Q͑fЈ͒. ͑3.52͒ ͵Ϫϱ ͵Ϫϱ

˜ ˜ To calculate the expectation value ͗h1*( f )h2( f Ј)͘, we again make use of the plane wave expansion ͑2.5͒ and the expectation value ͑2.16͒. Since

ˆ ជ ˜ ˆ ˆ Ϫi2␲ f ⍀•xi /c A ˆ hi͑ f ͒ϭ d⍀ hA͑ f ,⍀͒e Fi ͑⍀͒, ͑3.53͒ ͚A ͵S2 where iϭ1,2 labels the two detectors, we find

ˆ ជ ˆ ជ ˜ ˜ ˆ ˆ ˆ ˆ i2␲ f ⍀•x1 /c Ϫi2␲ f Ј⍀Ј•x2 /c A ˆ AЈ ˆ ͗h*͑ f ͒h2͑ f Ј͒͘ϭ d⍀ d⍀Ј͗h*͑ f ,⍀͒hA ͑ f Ј,⍀Ј͒͘e e F ͑⍀͒ F ͑⍀Ј͒ ͑3.54͒ 1 ͚ ͚ ͵ 2 ͵ 2 A Ј 1 2 A AЈ S S

2 3H0 ˆ ជ Ϫ3 ˆ i2␲ f ⍀•⌬x/c A ˆ A ˆ ϭ 3 ␦͑ f Ϫ f Ј͉͒f͉ ⍀gw͉͑f͉͒ d⍀e F1 ͑⍀͒F2 ͑⍀͒ ͑3.55͒ 32␲ ͚A ͵S2

2 3H0 ϭ ␦͑ f Ϫ f Ј͉͒f͉Ϫ3⍀ ͉͑f͉͒␥͉͑f͉͒, ͑3.56͒ 20␲2 gw

where we used Eq. ͑2.16͒ to obtain the second equality and 2 3H0 the definition ͑3.30͒ of the overlap reduction function to ob- H ͑ f ͒ϭ ͉f͉Ϫ3⍀ ͉͑f͉͒␥͉͑f͉͒. ͑3.59͒ 12 20␲2 gw tain the third. Substituting Eq. ͑3.56͒ into Eq. ͑3.52͒ yields

In other words, H12( f ) is just the Fourier transform of the cross-correlation of the gravitational strains h1(t) and h2(tЈ) 3H2 ϱ 0 Ϫ3 ˜ at the two detector sites. Moreover, since the noises intrinsic ␮ϭ 2 T df ͉f͉ ⍀gw͉͑f͉͒␥͉͑f͉͒Q͑ f ͒. ͑3.57͒ 20␲ ͵Ϫϱ to the two detectors are statistically independent of one an- other and of the gravitational strains, it follows that

The factor of T on the right hand side ͑RHS͒ arises from ͗h1͑t͒h2͑tЈ͒͘ϭ͗s1͑t͒s2͑tЈ͒͘. ͑3.60͒ evaluating ␦T(0). Before calculating the variance ␴2, it is worthwhile to Thus, H12( f ) is the Fourier transform of the cross-correlation make a slight digression and study in more detail the expec- of the outputs of the two detectors. But this correlation is something that we can measure ͑or at least estimate͒ given tation value ͑3.56͒ derived above. It turns out that this equa- 16 tion has an important physical implication. In terms of the enough data. This in turn implies that we can measure ͑or at least estimate͒⍀gw(͉f͉). Explicitly, given the measured time domain variables h1(t) and h2(tЈ), Eq. ͑3.56͒ can be rewritten as

16For example, suppose we measure the outputs of the two detec- ϱ tors for a total observation time of 1 yr. To estimate the cross- i2␲f͑tϪtЈ͒ ͗h1͑t͒h2͑tЈ͒͘ϭ df e H12͑ f ͒, ͑3.58͒ correlation s (t)s (tЈ) , we simply form the products of s (t) and ͵Ϫϱ ͗ 1 2 ͘ 1 s2(tЈ) for all t and tЈ having the same ⌬tϭtϪtЈ ͑e.g., 1 msec͒, and then average the results. We then repeat this procedure for ⌬t where ϭ2 msec, 3 msec, etc.

102001-11 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001 values of ͗s1(t)s2(tЈ)͘, we take their Fourier transform f Ͻ1/T, the Fourier transform lacks sufficient time domain ͑with respect tϪtЈ), multiply by ͉f͉3, and divide by data to provide useful information; for f Ͼc/d, the overlap 2 2 (3H0/20␲ )␥(͉f͉), to determine ⍀gw(͉f͉). This will yield a reduction function quickly approaches zero, and so the divi- good approximation to the real stochastic gravity-wave spec- sion by ␥( f ) makes ⍀gw(͉f͉) ill-behaved. trum provided that the noise intrinsic to the detectors is not Let us return now to the calculation of the signal-to-noise 2 too large or, equivalently, if we measure the detector outputs ratio SNRª␮/␴. To calculate the variance ␴ , we assume for a long enough period of time T. Also, the approximation ͑as in Sec. III A͒ that the noises intrinsic to the two detectors of ⍀gw(͉f͉) will be best for frequencies 1/TϽ f Ͻc/d ͑where are much larger in magnitude than the stochastic gravity- d/c is the light travel time between the two detectors͒: For wave background. Then

2 2 2 2 ␴ ª͗S ͘Ϫ͗S͘ Ϸ͗S ͘ ͑3.61͒

ϱ ϱ ϱ ϱ ˜ ˜ ˜ ˜ ˜ ˜ Ϸ df dfЈ dk dkЈ␦T͑fϪfЈ͒␦T͑kϪkЈ͒͗n1*͑f͒n2͑fЈ͒n1*͑k͒n2͑kЈ͒͘Q͑fЈ͒Q͑kЈ͒ ͑3.62͒ ͵Ϫϱ ͵Ϫϱ ͵Ϫϱ ͵Ϫϱ

ϱ ϱ ϱ ϱ ˜ ˜ ˜ ˜ ˜ ˜ Ϸ df dfЈ dk dkЈ␦T͑fϪfЈ͒␦T͑kϪkЈ͒͗n1*͑f͒n1͑Ϫk͒͗͘n2*͑ϪfЈ͒n2͑kЈ͒͘Q͑fЈ͒Q͑kЈ͒, ͑3.63͒ ͵Ϫϱ ͵Ϫϱ ͵Ϫϱ ͵Ϫϱ

where we used the statistical independence and reality of The problem now is to find the filter function Q˜ ( f ) that 2 n1(t) and n2(t) to obtain the last line. maximizes the signal-to-noise ratio ͑3.51͒, with ␮ and ␴ as In Sec. III A, we defined the noise power spectrum given above. This turns out to be remarkably simple if we Pi(͉f͉) in terms of the expectation value ͗ni(t)ni(tЈ)͘ of the first introduce an inner product (A,B) for any pair of com- time domain random variables ni(t). ͓See Eq. ͑3.23͒.͔ An plex functions A( f ) and B( f ). The inner product of A( f ) analogous expression holds in the frequency domain as well. and B( f ) is a complex number defined by ˜ Using definition ͑3.49͒ for the Fourier transform ni( f ) and definition ͑3.23͒ for the noise power spectrum Pi(͉f͉), it ϱ follows that ͑A,B͒ª df A*͑f͒B͑f͒P1͉͑f͉͒P2͉͑f͉͒. ͑3.69͒ ͵Ϫϱ 1 ˜ ˜ Since P (͉f͉)Ͼ0, it follows that (A,A)у0, and (A,A)ϭ0if ͗ni*͑ f ͒ni͑ f Ј͒͘ϭ ␦͑ f Ϫ f Ј͒Pi͉͑f͉͒. ͑3.64͒ i 2 and only if A( f )ϭ0. In addition, (A,B)ϭ(B,A)* and (A,Bϩ␭C)ϭ(A,B)ϩ␭(A,C) for any complex number ␭. Substituting this result into Eq. ͑3.63͒ then yields Thus, (A,B)isapositive-definite inner product. It satisfies

1 ϱ ϱ 2 2 ␴ Ϸ df dfЈ␦T͑fϪfЈ͒ 4 ͵Ϫϱ ͵Ϫϱ ˜ ˜ ϫP1͉͑f͉͒P2͉͑fЈ͉͒Q͑f͒Q*͑fЈ͒ ͑3.65͒

T ϱ ˜ 2 Ϸ df P1͉͑f͉͒P2͉͑f͉͉͒Q͑f͉͒ , ͑3.66͒ 4 ͵Ϫϱ where we replaced one of the finite-time delta functions ␦T( f Ϫ f Ј) by an ordinary Dirac delta function, and evaluated the other at f ϭ f Ј to obtain the last line. To summarize,

3H2 ϱ 0 Ϫ3 ˜ ␮ϭ 2 T df ͉f͉ ⍀gw͉͑f͉͒␥͉͑f͉͒Q͑ f ͒, ͑3.67͒ 20␲ ͵Ϫϱ FIG. 4. Optimal filter functions Q˜ ( f ) for the initial and ad- vanced LIGO detector pairs, for a stochastic background having a T ϱ 2 ˜ 2 constant frequency spectrum ⍀ ( f )ϭ⍀ . Both filters are normal- ␴ Ϸ df P1͉͑f͉͒P2͉͑f͉͉͒Q͑f͉͒ . ͑3.68͒ gw 0 4 ͵Ϫϱ ized to have maximum magnitude equal to unity.

102001-12 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001 all of the properties of an ordinary dot product of vectors in two detectors. We would then analyze the outputs of the two three-dimensional Euclidean space. detectors for each of these filters separately. Figure 4 shows In terms of this inner product, the mean value ␮ and the optimal filter functions ͑displayed in the frequency do- variance ␴2 can be written as main͒ for both the initial and advanced LIGO detector pairs,

2 for a stochastic background having a constant frequency 3H ␥͉͑f͉͒⍀ ͉͑f͉͒ 19 0 ˜ gw spectrum ⍀gw( f )ϭ⍀0 (i.e., ␣ϭ0). Figure 5 shows these ␮ϭ 2 T Q, , ͑3.70͒ 20␲ ͩ ͉f͉3P ͉͑f͉͒P ͉͑f͉͒ͪ same optimal filter functions displayed in the time domain— 1 2 i.e., as a function of the lag tϪtЈ. T Having found the optimal choice of filter function Q˜ ( f ), ␴2Ϸ ͑Q˜ ,Q˜ ͒. ͑3.71͒ 4 it is now straightforward to calculate the signal-to-noise ratio for a given pair of detectors. Substituting Eq. ͑3.73͒ into Eq. The problem is to choose Q˜ ( f ) so that it maximizes the ͑3.72͒ and taking the square root gives signal-to-noise ratio ͑3.51͒ or equivalently, the squared 1/2 signal-to-noise ratio 2 ϱ 2 2 3H0 ␥ ͉͑f͉͒⍀gw͉͑f͉͒ 2 SNRϷ 2ͱT df . ͑3.75͒ ␥ f ͒⍀ f ͒ 10␲ ͫ ͵Ϫϱ f 6P ͉͑f͉͒P ͉͑f͉͒ͬ ͉͑ ͉ gw͉͑ ͉ 1 2 Q˜ , ␮2 3H2 2 ͩ ͉f͉3P ͉͑f͉͒P ͉͑f͉͒ͪ 2 0 1 2 We will use this result in later sections to calculate signal- SNR ϭ 2 Ϸ 2 T . ␴ ͩ 10␲ ͪ ͑Q˜ ,Q˜ ͒ to-noise ratios and sensitivity levels for different detector ͑3.72͒ pairs, assuming that the stochastic gravity-wave background has a constant frequency spectrum ⍀gw( f )ϭ⍀0 . Tables I–V But this is trivial. For suppose we are given a fixed three- in Sec. VI D contain the results of these calculations. dimensional vector Aជ , and are asked to find the three- IV. DETECTION, ESTIMATION, ជ ជ ជ 2 ជ dimensional vector Q that maximizes the ratio (Q•A) /Q AND SENSITIVITY LEVELS •Qជ . Since this ratio is proportional to the squared cosine of the angle between the two vectors, it is maximized by choos- Once the detectors have gone ‘‘on-line’’ and are generat- ing data that needs to be analyzed, we will be confronted ing Qជ to point in the same direction as Aជ . The problem of with the following questions: ͑i͒ How do we decide, from the maximizing Eq. ͑3.72͒ is identical. The solution is experimental data, if we have detected a stochastic gravity- wave signal? ͑ii͒ Assuming that a stochastic gravity-wave ␥͉͑f͉͒⍀gw͉͑f͉͒ signal is present, how do we estimate its strength? ͑iii͒ As- Q˜ ͑ f ͒ϭ␭ , ͑3.73͒ 3 suming that a stochastic gravity-wave signal is present, what ͉f͉ P1͉͑f͉͒P2͉͑f͉͒ is the minimum value of ⍀0 required to detect it 95% of the time? In this section, we answer these questions, using a where ␭ is a ͑real͒ overall normalization constant. frequentist approach to the theory of probability and One of the curious things about expression ͑3.73͒ for the statistics.20 optimal filter Q˜ ( f ) is that it depends upon the spectrum ⍀gw( f ) of the stochastic gravity-wave background. This is a A. Statistical considerations function that we do not know a priori.17 In practice this When performing a search for a stochastic background of means that we cannot use a single optimal filter when per- gravitational radiation, it is convenient to break the data set forming the data analysis; we will need to use a set of such filters. For example, within the bandwidth of interest for the ground-based interferometers, it is reasonable to assume that 19 the spectrum is given by a power law ⍀ ( f )ϭ⍀ f ␣ ͑where Figures 21–30 in Sec. VI C show the analogous optimal filter gw ␣ functions for different detector pairs. ⍀ ϭconst).18 We could then construct a set of optimal fil- ␣ 20There are actually two approaches that one can take when ana- ˜ ters Q␣( f ) ͑say, for ␣ϭϪ4,Ϫ7/2,...,7/2,4) with the over- lyzing data: the frequentist ͑or classical frequency probability͒ ap- all normalization constants ␭␣ chosen so that proach, which is adopted in this paper, and the Bayesian ͑or sub- jective probability͒ approach, which is adopted in Refs. ͓18,19͔. Although we will not describe the similarities and differences of ␮ϭ⍀␣T. ͑3.74͒ these two approaches in any detail in this paper, it is important to emphasize that the frequentist and Bayesian approaches are in- With this choice of normalization, the optimal filter functions equivalent methods of analyzing data. Frequentists and Bayesians ˜ Q␣( f ) are completely specified by the exponent ␣, the over- ask different questions about data and hypotheses, and consequently lap reduction function, and the noise power spectra of the obtain different answers and draw different conclusions. In fact, there are certain questions that one can ask and answer in the Baye- sian approach that are ill-defined for a frequentist. Interested readers should see Refs. ͓18,19͔ for a detailed discussion of the frequentist 17See, however, the discussion surrounding Eqs. ͑3.58͒ and ͑3.59͒. and Bayesian approaches applied to gravitational-wave data analy- 18 ␣ The ␣ in ⍀␣ and f is just a number; it is not an index label. sis with multiple detectors.

102001-13 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

experiment. From these samples, we can construct the sample mean

1 n ˆ ␮ ª Si ͑4.1͒ ni͚ϭ1

and the sample variance

1 n ˆ 2 ˆ 2 ␴ ª ͑SiϪ␮͒ . ͑4.2͒ nϪ1 i͚ϭ1

Given the values of these estimators ␮ˆ and ␴ˆ 2, we would like to decide, in some reliable way, whether or not we have detected a stochastic gravity-wave signal. To make such a decision, we will apply a standard theo- rem from the theory of probability and statistics ͑see, e.g., p. FIG. 5. Optimal filter functions Q(tϪtЈ) for the initial and ad- 178 of Ref. ͓20͔͒. The theorem states that if ␮ˆ is the sample vanced LIGO detector pairs, for a stochastic background having a mean of a set of n independent samples drawn from a normal 2 constant frequency spectrum ⍀gw( f )ϭ⍀0 . Both filters are normal- distribution having mean ␮ and variance ␴ , then ized to have maximum magnitude equal to unity. ␮ˆ Ϫ␮ ͑which might be hours, days, or weeks in length͒ into shorter t ϭ ͑4.3͒ stretches of N points, which we can then fast Fourier trans- ␴ˆ /ͱn form ͑FFT͒ and correlate with data from other detectors. De- pending on the choice of N and on the sampling rate of the is the value of a random variable having Student’s detector, these shorter segments of data will typically last on t-distribution with parameter ␯ϭnϪ1. This is the classic the order of seconds.21 From Eq. ͑3.50͒, we see that in a Student’s t-test. It is used to compare the means of two nor- mal distributions that have the same variance, when the vari- measurement over a single observation period TϷ4 sec, the 2 signal S is a sum ͑over f and f Ј) of approximately 400 sta- ance ␴ is unknown. tistically independent random variables ͑products of the Fou- Since tables of Student’s t-distribution and its associated ˜ cumulative probability distribution function can be found in rier amplitudes of the signals͒. This is because s1( f ) and most handbooks on statistics ͑see, e.g., ͓21͔͒, we could do all ˜ s2( f Ј) are correlated only when ͉f Ϫ f Ј͉Ͻ1/TϷ.25 Hz, and of the remaining calculations in this section in terms of the the bandwidth over which the integral in Eq. ͑3.50͒ gets its t-distribution. The drawback to this approach, however, is major contribution is ϳ100 Hz wide. Thus, by virtue of the that the t-distribution depends on the parameter ␯ϭnϪ1. central limit theorem, S is well approximated by a Gaussian This means that all of our results would depend on the num- random variable, provided we are not too far away from the ber of observations n that constitute a single experiment. For mode of the distribution. Equivalently, the values of S in a stochastic background searches, this undesirable feature can set of measurements over statistically independent time in- be avoided by choosing n large enough so that the tervals ͑each of length T) are normally distributed. The mean t-distribution and standard normal distribution are virtually value of this distribution is ␮ ª ͗S͘, and the variance is indistinguishable. Since a typical total observation time will 2 2 2 ␴ ª ͗S ͘Ϫ͗S͘ . be on the order of months or years (ϳ107 sec), while T ͑the Let s ª (S1 ,S2 ,...,Sn) be a set of such measurements duration of a single observation period͒ is typically on the over statistically independent time intervals, each of length order of seconds, it is no problem to choose nϳ103 or 22 T. We can think of these measurements as n independent more.23 For such large n, Eq. ͑4.3͒ can be rewritten as samples drawn from a normal distribution having mean ␮ and variance ␴2. The set s represents the outcome of a single ␮ˆ Ϫ␮ z Ϸ , ͑4.4͒ ␴ˆ /ͱn 21 16 For example, for Nϭ65536ϭ2 and a sampling rate of 20 kHz, where z is the value of a random variable having the standard Tϭ3.2768 sec. For the same N and a sampling rate of 14 normal distribution—i.e., z is a Gaussian random variable 16.384 kHzϭ2 Hz, Tϭ4.0 sec. Since the relevant bandwith having zero mean and unit variance. Note that the approxi- for stochastic background detection is f Ͻ300 Hz, the data stream ˆ could, in principle, be decimated to sampling rates which are sub- mation becomes a strict equality if ␴ is replaced by its ex- stantially smaller—i.e., 1024 Hz. pected value ␴. This is because a linear combination of n 22In order that the measurements be statistically independent, the time intervals should be non-overlapping, and T should be ӷ the light travel time d/c between the two detectors. For the LIGO de- 23In fact, even for nϭ30, the 2␴ values for the t-distribution and tector pair, this corresponds to Tӷ10Ϫ2 sec, which is satisfied for standard normal distribution differ by less than 5%. As n ϱ, this → TϷ4 sec. difference goes to zero.

102001-14 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

Gaussian random variables ͑e.g., ␮ˆ ) is also a Gaussian ran- signal is present, characterized by the fixed mean value ␮ dom variable, independent of n. Ͼ0. In the calculations that follow, we will want to assign From Eq. ͑4.4͒, it follows that probabilities to different events. From a frequentist point of n 2 view, this means that we should perform ͑or imagine per- Si p͑s͉0͒ϭ͑2␲␴ˆ 2͒Ϫn/2 exp Ϫ͚ , ͑4.6͒ forming͒ some fixed experiment many, many times. The ͫ iϭ1 2␴ˆ 2ͬ probability of an event is then defined as the frequency of occurrence of that event, in the limit of an infinite number of n 2 ͑SiϪ␮͒ repeated, independent experiments. p͑s͉␮͒ϭ͑2␲␴ˆ 2͒Ϫn/2 exp Ϫ͚ . ͫ iϭ1 2␴ˆ 2 ͬ ͑4.7͒ B. Signal detection The fact that we can use the sample variance ␴ˆ 2 ͑instead of In order to decide whether or not we have detected a 2 stochastic gravity-wave signal, we need a rule that, given a the true variance ␴ ) on the right-hand sides of Eqs. ͑4.6͒ set of measured data, will select for us one of two alternative and ͑4.7͒ follows from the large n approximation that we hypotheses: used to obtain Eq. ͑4.4͒. A decision rule that, given the outcome of an experiment, H0 : A stochastic gravity-wave signal is absent. selects for us either H0 or H1 is equivalent to a division of H1 : A stochastic gravity-wave signal is present, charac- terized by some fixed, but unknown, mean value ␮Ͼ0. the space of all possible experimental outcomes into two Moreover, we would like this rule to be ‘‘optimal’’ with disjoint regions R0 and R1 :Ifs෈R0, then H0 is chosen; if respect to some chosen set of criteria. This method of deci- s෈R1 , then H1 is chosen. The success and failure of such a sion making, or ‘‘hypothesis testing’’ as it is more formally rule are characterized by two types of errors: A type I ͑or called, is a well-studied branch of frequentist statistics. As false alarm͒ error occurs when the decision rule chooses H1 such, we will not go into any of the details here. After mak- when H0 is really true. A type II ͑or false dismissal͒ error ing the appropriate definitions, we will simply state the rule occurs when the decision rule chooses H0 when H1 is really that we adopt for our stochastic background searches and true. In terms of p(s͉0), p(s͉␮), R0 , and R1 , we have explain in what sense it is optimal. Interested readers should see Ref. ͓22͔ for a much more thorough discussion of the ␣ϵfalse alarm rate ª ͵ ds p͑s͉0͒, ͑4.8͒ statistical theory of signal detection. R1 To begin, let us note that the two hypotheses H0 and H1 are exhaustive and mutually exclusive—i.e., a stochastic ␤͑␮͒ϵfalse dismissal rate ª ͵ ds p͑s͉␮͒. ͑4.9͒ gravity-wave signal is either absent or present. And, if R0 present, it will be characterized by some fixed mean value ␮, which is proportional to ⍀0 for a stochastic background of Note that, for the complex hypothesis H1 , the false dismissal gravitational radiation having a constant frequency spectrum rate is actually a function of the mean value ␮Ͼ0. Note also ⍀gw( f )ϭ⍀0 . The only requirement is that ␮Ͼ0. H0 is a that 1Ϫ␣ is the fraction of experimental outcomes that the simple hypothesis, since it does not depend on any unknown decision rule correctly identifies the absence ͑not presence͒ parameters. H1 is a complex ͑or composite͒ hypothesis, since of a stochastic gravity-wave signal. If we want to talk about it depends on a range of the unknown parameter ␮. Explic- detection, we should evaluate itly, ␥͑␮͒ϵdetection rate ª 1Ϫ␤͑␮͒, ͑4.10͒ ϱ H1ϭ ഫ Hnd␮,d␮ , ͑4.5͒ which is the fraction of experimental outcomes that the de- nϭ0 cision rule correctly identifies the presence of a stochastic gravity-wave signal, characterized by the fixed mean value ␮Ͼ0.24 where H␮,d␮ is the hypothesis that a stochastic gravity-wave signal is present, characterized by a fixed mean value lying In order for the decision rule to choose the regions R0 and in the range (␮,␮ϩd␮͔. R1 in an ‘‘optimal’’ way, we must first select some set of criteria with respect to which ‘‘optimal’’ can be defined. For As before, let s ª (S1 ,S2 ,...,Sn) be a set of n statisti- cally independent measurements of the cross-correlation sig- stochastic background searches, where one does not know a nal S. Because of the noise intrinsic to the detectors and priori the ‘‘costs’’ that one should associate with false alarm errors inherent to the measurement process, the outcome of and false dismissal errors, it is reasonable to choose a deci- an experiment s is a random variable. It is described statisti- sion rule that minimizes the false dismissal rate ␤(␮) for a cally by the probability density functions: fixed value of the false alarm rate ␣. Equivalently, one p(s͉0): Probability density function for the outcome of an chooses a decision rule that maximizes the probability of experiment to be s, given that a stochastic gravity-wave sig- nal is absent. p(s͉␮): Probability density function for the outcome of 24The detection rate ␥(␮) should not be confused with the overlap an experiment to be s, given that a stochastic gravity-wave reduction function ␥( f ), which was defined in Sec. III B.

102001-15 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001 detecting a stochastic gravity-wave signal, while keeping the Given the above decision rule for signal detection, we can false alarm rate fixed. This decision criterion is known in the now calculate the false alarm and false dismissal rates de- literature as the Neyman-Pearson criterion. In general, for a fined by Eqs. ͑4.8͒ and ͑4.9͒. First, for the false alarm rate, complex hypothesis, the Neyman-Pearson criterion yields re- one finds gions R0 and R1 that depend on the unknown parameter ␮. ˆ ˆ But for this case, where H1 involves all parameter values ␣ϭProb͑␮уz␣␴/ͱn͉␮ϭ0͒ ͑4.16͒ 25 ␮Ͼ0, R0 and R1 are actually independent of ␮. Without going into details here, let us simply state the result: Namely, ϭProb͑zуz␣͒ ͑4.17͒ in the context of stochastic background searches as described above, the Neyman-Pearson criterion is satisfied if, given the 1 z␣ outcome of an experiment s, we form the estimators ˆ and ϭ erfc , ͑4.18͒ ␮ 2 ͩ ͱ2 ͪ ␴ˆ 2 according to Eqs. ͑4.1͒ and ͑4.2͒, and then where we used Eq. ͑4.4͒ with ␮ϭ0 to obtain the second equality. Thus, our two uses of the symbol ␣ ͑for the false choose H if ␮ˆ Ͻz ␴ˆ /ͱn, 0 ␣ alarm rate and for the area under the standard normal distri- bution to the right of z␣) are consistent. ˆ ˆ Second, for the false dismissal rate, choose H1 if ␮уz␣␴/ͱn. ͑4.11͒ ˆ ˆ ␤͑␮͒ϭProb͑␮Ͻz␣␴/ͱn͉␮Ͼ0 fixed͒ ͑4.19͒ Here z␣ is that value of the random variable z for which the area under the standard normal distribution to its right is ϭProb͑zϽz Ϫͱn␮/␴ˆ ͒ ͑4.20͒ equal to ␣. ͑See Fig. 6.͒ In terms of the complementary error ␣ function ˆ 1 z␣Ϫͱn␮/␴ ϭ1Ϫ erfc , ͑4.21͒ 2 ͩ ͱ2 ͪ 2 ϱ Ϫx2 erfc͑z͒ ª dx e ͑4.12͒ ͱ␲ ͵z where we again used Eq. ͑4.4͒͑but this time with ␮Ͼ0) to obtain the second equality. A graph of the detection rate ␥(␮) ª 1Ϫ␤(␮) is shown in Fig. 8. From this graph, we ͑see Fig. 7͒, see that the detection rate ␥(␮) approaches ␣ as ␮ 0; it approaches 1 as ␮ ϱ. Also, the detection rate equals→ 0.50 Ϫ1 ˆ → ˆ z␣ϭͱ2 erfc ͑2␣͒, ͑4.13͒ for ␮ϭz␣␴/ͱn ͑or, equivalently, for ͱn␮/␴ϭz␣). Thus, if we replace ␴ˆ with its expected value ␴, we see that the detection rate is only 50% for a signal having a theoretical where erfcϪ1 denotes the inverse of erfc. Since ␮ˆ and ␴ˆ are signal-to-noise ratio after n observation periods equal to z . functions of the outcome of the experiment s, the inequalities ␣ It is also interesting to note that the Neyman-Pearson de- ˆ ˆ ˆ ˆ ␮Ͻz␣␴/ͱn and ␮уz␣␴/ͱn define R0 and R1 . Note also tection criterion, when applied to stochastic background that the decision rule can be restated as searches, is equivalent to the maximum-likelihood detection criterion ͓22͔. In other words, we will obtain the same deci- ˆ sion rule given above if we first construct the likelihood ratio choose H0 if ͱnSNRϽz␣ , p͑s͉␮͒ ⌳͑s͉␮͒ ; ͑4.22͒ ˆ ª p s 0 choose H1 if ͱnSNRуz␣ , ͑4.14͒ ͑ ͉ ͒ maximize ⌳(s͉␮) with respect to variations of the parameter where ␮Ͼ0,

⌳ ͑s͒ max ⌳͑s͉␮͒; ͑4.23͒ ␮ˆ max ª ˆ ␮Ͼ0 ͱnSNR ª ͱn ͑4.15͒ ␴ˆ and then divide the space of all possible experimental out- comes s into two regions R0 and R1 according to the rule is the measured signal-to-noise ratio after n observation pe- riods. The fact that the signal-to-noise ratio enters the above choose H0 if ⌳max͑s͒Ͻ⌳0 , inequalities is one of the main reasons why we paid so much attention to evaluating it in Sec. III. choose H1 if ⌳max͑s͒у⌳0 , ͑4.24͒

where ⌳0 is chosen so that the false alarm rate equals ␣. 25See pp. 152–153 of Ref. ͓22͔ for more details. Explicitly, from Eqs. ͑4.6͒ and ͑4.7͒, it follows that

102001-16 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

FIG. 7. The complementary error function. FIG. 6. The standard normal probability distribution for a Gaussian random variable having zero mean and unit variance. z␣ is is the value of a Gaussian random variable having zero mean that value of z for which the area under the standard normal distri- and unit variance. Thus, if z is that value of the random bution to its right is equal to ␣. Typical values for 1Ϫ␣ are 0.90, ␣/2 variable z for which the area under the standard normal dis- 0.95, and 0.99. The corresponding values for z are z ␣ 0.10 tribution to its right is equal to ␣/2 ͑see Fig. 9͒, then (1 ϭ1.28, z0.05ϭ1.65, and z0.01ϭ2.33. These are the threshold values appropriate for the one-sided test described in the text. Ϫ␣)ϫ100% of the time

n Ϫz␣/2ϽzϽz␣/2 ͑4.29͒ n␮ 1 1 n␮ 1 ⌳͑s ␮͒ϭexp S Ϫ ␮ ϭexp ␮ˆ Ϫ ␮ . ͉ 2 ͚ i 2 or, equivalently, ͫ ␴ˆ ͩ niϭ1 2 ͪͬ ͫ ␴ˆ ͩ 2 ͪͬ ͑4.25͒ ˆ ˆ ˆ ˆ ␮Ϫz␣/2␴/ͱnϽ␮Ͻ␮ϩz␣/2␴/ͱn. ͑4.30͒ ⌳(s͉␮) is maximized when ␮ equals the sample mean ␮ˆ :26 Said another way, in an ensemble of observations of the same stochastic background, a fraction 1Ϫ␣ of the intervals 1 n␮ˆ 2 ˆ ⌳max͑s͒ϭ⌳͑s͉␮͒ϭexp . ͑4.26͒ ͫ 2 ␴ˆ 2 ͬ I ͓␮ˆ Ϫz ␴ˆ /ͱn,␮ˆ ϩz ␴ˆ /ͱn͔, ͑4.31͒ ␣ ª ␣/2 ␣/2 ˆ constructed from the measured data, will contain the value of The decision surface ⌳0 is obtained by setting ␮ equal to the ˆ the true mean ␮. Equivalently, ␣ is the fraction of intervals threshold value z␣␴/ͱn: I␣ that fail to contain the value of the true mean ␮.Of course, given the outcome of a single experiment s, the in- ˆ 1 2 ⌳0ϭ⌳max͑z␣␴/ͱn͒ϭexp͓ 2 z␣͔. ͑4.27͒ terval I␣ either contains or does not contain the value of ␮. And the value of ␮, if it is contained in I␣ , need not be any C. Parameter estimation closer to the center of the interval than to either of its edges. Thus, the confidence that one associates with the above esti- Assuming that a stochastic gravity-wave signal is present, mation procedure is not equivalent to our degree of belief characterized by some fixed, but unknown, mean value ␮ that the true mean ␮ lies within a given interval. ͑This is a Ͼ0, parameter estimation attempts to answer the question, Bayesian interpretation of probability.͒ Rather, it is the frac- what is the value of ? It does this by first constructing the ␮ tion of experimental outcomes that our estimation procedure ˆ ˆ 2 sample mean ␮ and sample variance ␴ of a set s will produce an interval that contains the true mean ␮, in the ª (S1 ,S2 ,...,Sn)ofnstatistically independent measure- limit of an infinite number of repeated, independent experi- ments of the cross-correlation signal S, as described in Sec. ments. IV A. From Eq. ͑4.4͒, we then know that D. Sensitivity levels ␮ˆ Ϫ␮ zϷ ͑4.28͒ Let us assume once again that a stochastic gravity-wave ˆ / n ␴ ͱ signal is present, characterized by some fixed, but unknown, mean value ␮Ͼ0. Then it is reasonable to ask what is the minimum value of ␮ required so that our decision rule cor- 26The sample mean ␮ˆ is said to be the maximum-likelihood esti- rectly identifies the presence of a signal at least ␥ϫ100% of mator for this problem. the time?

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FIG. 8. The detection rate ␥(␮) ª 1Ϫ␤(␮) plotted as a func- FIG. 9. The standard normal probability distribution for a tion of the mean value ␮. Gaussian random variable having zero mean and unit variance. z␣/2 is that value of z for which the area under the standard normal The answer to this question can be obtained by applying distribution to its right is equal to ␣/2. The area under the standard the results of Sec. IV B. Namely, we require that the detec- normal distribution between Ϫz␣/2 and z␣/2 is thus 1Ϫ␣. Typical tion rate ␥(␮) ª 1Ϫ␤(␮) be greater than or equal to the values for 1Ϫ␣ are 0.90, 0.95, and 0.99. The corresponding values desired rate ␥, and then solve the resulting inequality for ␮. for z␣/2 are z0.05ϭ1.65, z0.025ϭ1.96, and z0.005ϭ2.58. Explicitly, from Eq. ͑4.21͒, it follows that Ϫ ˆ ence of a signal was 1 ␣. But as mentioned in Sec. IV B, 1 z␣Ϫͱn␮/␴ 1Ϫ␤͑␮͒ϭ erfc у␥ ͑4.32͒ 1Ϫ␣ is the probability that the decision rule correctly iden- 2 ͩ ͱ2 ͪ tifies the absence of a signal. To talk about detection ͑and the associated minimum value of ␮), one needs to include the or, equivalently, additional parameter ␥. Ϫ1 For stochastic background searches, we can rewrite Eq. z␣Ϫͱn␮/␴рͱ2 erfc ͑2␥͒, ͑4.33͒ ͑4.35͒ in terms of ⍀0 , ␥( f ), and the noise power spectra where we replaced the estimator ␴ˆ by its expected value ␴ to P1( f ) and P2( f ) of the two detectors: obtain the LHS of the above equation. Thus, 2 2 Ϫ1/2 ␴ 1 10␲ ϱ ␥ ͉͑f͉͒ Ϫ1 ⍀ у df ␮у ͓z␣Ϫͱ2 erfc ͑2␥͔͒ ͑4.34͒ 0 2 ͵ 6 ͱT 3H0 ͫ Ϫϱ f P1͉͑f͉͒P2͉͑f͉͒ͬ ͱn tot ϫͱ2͓erfcϪ1͑2␣͒ϪerfcϪ1͑2␥͔͒, ͑4.37͒ ␴ ϭ ͱ2͓erfcϪ1͑2␣͒ϪerfcϪ1͑2␥͔͒. ͑4.35͒ ͱn where Ttot ª nT is the duration of a single experiment s. Equivalently, This result follows from Eq. ͑3.75͒ for a stochastic back- ground of gravitational radiation having a constant frequency ͱn SNRуͱ2͓erfcϪ1͑2␣͒ϪerfcϪ1͑2␥͔͒, ͑4.36͒ spectrum ⍀gw( f )ϭ⍀0 . Let us now evaluate the minimum value of ⍀0 for 4 months of observation ͑i.e., Ttot where ͱnSNR ͱn␮/␴ is the theoretical signal-to-noise ϭ107 sec), for a false alarm rate ␣ϭ0.05, and for a detec- ª 95%,5% ratio after n observation periods. Note that the right-hand tion rate ␥ϭ0.95. Denoting the solution by ⍀0 ,we sides of Eqs. ͑4.35͒ and ͑4.36͒ depend on both ␣ and ␥.27 find28 95%,5% 2 Ϫ6 This means that to calculate the minimum value of ␮ ͑or the ͑i͒⍀0 h100ϭ5.74ϫ10 for the initial LIGO detector minimum signal-to-noise ratio͒, we must specify the desired pair, 95%,5% 2 Ϫ11 detection rate ␥ in addition to the false alarm rate ␣. In the ͑ii͒⍀0 h100ϭ5.68ϫ10 for the advanced LIGO past ͑see, e.g., Refs. ͓7,8͔͒, physicists have only specified the detector pair. false alarm rate ␣. It seems that they have mistakenly as- sumed that the probability of correctly identifying the pres- 28There is nothing special about the choice ␣ϭ0.05 and ␥ ϭ0.95. For example, ␣ and ␥ need not sum to 1. We could equally 27 The dependence on ␣ is via the threshold value z␣ , which de- well have chosen ␣ϭ0.10 and ␥ϭ0.95, and calculated a different 95%,10% fines the decision rule ͑4.14͒. minimum value ⍀0 .

102001-18 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

Note, however, that these values disagree with those wave signal should always be made with these error rates in quoted in the literature ͑see, e.g., Refs ͓7,8͔͒. In Refs. ͓7,8͔, mind. the minimum value of ⍀0 is determined by the equation Moreover, one has to be very careful about trying to de- fine ‘‘termination’’ criteria. For example, it would be mis-

2 2 Ϫ1/2 leading to try to terminate an experiment by correlating the 1 10␲ ϱ ␥ ͉͑f͉͒ outputs of two gravity-wave detectors until the measured ⍀0ϭ 2 ͵ df 6 ͱT 3H0 ͫ Ϫϱ f P1͉͑f͉͒P2͉͑f͉͒ͬ signal-to-noise ratio for the total observation period exceeds tot some threshold value z␣ . One can show that the false alarm ϫͱ2 erfcϪ1͑␣͒. ͑4.38͒ rate associated with such a rule is 100%. In other words, the conclusion drawn from such an experiment would always be There are two mistakes: First, the argument of the inverse that a stochastic gravity-wave signal is present, even in the complementary error function is ␣ instead of 2␣. But this is absence of a signal. Noises intrinsic to the detectors and appropriate for a two-sided test, which would be correct if errors inherent to the measurement process are sufficient to the mean value ␮ could be either positive or negative. Sec- guarantee that the measured signal-to-noise ratio for the total ond, and more importantly, there is no term proportional to observation period will eventually exceed any z␣ . erfcϪ1(2␥). As mentioned above, it seems that the authors If, however, the value of the threshold level increases mistakenly assumed that the probability of correctly identi- with observation time in an appropriate manner, then one fying the presence of a signal was 1Ϫ␣. Even the ␣ can define termination criteria that have false alarm rates less ϭ0.10,␥ϭ0.90 value of Eq. ͑4.37͒ does not agree with the than 100%. A famous theorem of probability and statistics, 90% 90% called the law of the iterated logarithm 23 , states that if values of ⍀ quoted in Refs. ͓7,8͔. Thus, by ⍀ the ͓ ͔ 0 0 S ,S , . . . are statistically independent and identically dis- authors do not mean the minimum value of ⍀ for a false 1 2 0 tributed random variables with zero mean and finite variance alarm rate equal to 10% and a detection rate equal to 90%. ␴2, then29 Alternatively, the absence of the term erfcϪ1(2␥) from Eq. ͑4.38͒ is equivalent to calculating the minimum value of Ϫ1 S1ϩS2ϩ•••ϩSn ⍀0 for a detection rate ␥ϭ0.50. This is because erfc (1) Prob lim sup ϭ1 ϭ1. ͑4.39͒ 90% ϭ0. Thus, the values of ⍀ quoted in Refs. ͓7,8͔ are for a ͩ n ϱ ␴ͱ2n log͓log͑n͔͒ ͪ 0 → false alarm rate equal to 5% and a detection rate equal to 50%—not 90% as they claim. This is why the minimum This means that if ␭Ͻ1, there is unit probability that values of ⍀0 quoted in those papers are smaller than those S1ϩS2ϩ•••ϩSn found here. Ͼ␭͑4.40͒ Table I in Sec. VI D contains theoretical signal-to-noise ␴ͱ2n log͓log͑n͔͒ ratios after 4 months of observation, for a stochastic back- ground having a constant frequency spectrum ⍀gw( f )ϭ⍀0 for infinitely many n.If␭Ͼ1, there is unit probability that 6 Ϫ2 Eq. ͑4.40͒ holds for only finitely many n. In terms of the ϭ6ϫ10 h100 . 2 measured signal-to-noise ratio after n observation periods, Table II in Sec. VI D contains minimum values of ⍀0h100 for 4 months of observation, for a false alarm rate equal to Eq. ͑4.40͒ can be rewritten as 5%, and for a detection rate equal to 95%. ͱnSNRˆϾ␭ͱ2 log͓log͑n͔͒, ͑4.41͒

E. Summary where we have replaced ␴ˆ in the definition of SNRˆ by its We started this section by asking a series of questions. To expected value ␴. Thus, if we want to define a termination conclude, we summarize the answers obtained above, and criterion with a false alarm rate less than 100%, we should ˆ address a couple of other related issues. compare ͱnSNR with threshold levels tn satisfying ͑i͒ How do we decide, from the experimental data, if we have detected a stochastic gravity-wave signal? tn lim ϭϱ. ͑4.42͒ Answer: We compare the measured signal-to-noise ratio n ϱͱlog͓log͑n͔͒ → after n observation periods to the threshold value z␣ .If ˆ ͱnSNRϽz␣ , we conclude that a stochastic gravity-wave This is consistent with the claim made in the previous para- ˆ graph that constant threshold levels t ϭz will always be signal is absent. If ͱnSNRуz␣ , we conclude that a stochas- n ␣ tic gravity-wave signal is present, characterized by some exceeded for some n. Unfortunately, we have not been able fixed, but unknown, mean value ␮Ͼ0. to write down a simple analytic expression for the false Note, however, that we can never conclude, with 100% alarm rate, for arbitrary threshold levels tn satisfying Eq. confidence, that a stochastic gravity-wave signal is absent or present. Our decision rule leads us to infer one of these two possibilities, but the rule is not perfect. The false alarm rate 29 The lim supn ϱ of a sequence x1 ,x2 , . . . is defined as follows: → ␣ and false dismissal rates ␤(␮) ͓defined by Eqs. ͑4.8͒ and Let am equal the least upper bound of the subsequence ͑4.9͔͒ are the error rates associated with the rule. Thus, xm ,xmϩ1 ,... . ͑Note that a1уa2у . . . .) Then lim supn ϱ xn claims about the absence or presence of a stochastic gravity- lim a . → ª m ϱ m → 102001-19 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

ˆ 2 2 Ϫ1/2 ͑4.42͒. The probability that ͱnSNRуtn for some n involves 1 10␲ ϱ ␥ ͉͑f͉͒ integrals of products of the Gaussian probability density ⍀0ϭ 2 ͵ df 6 ͱT 3H0 ͫ Ϫϱ f P1͉͑f͉͒P2͉͑f͉͒ͬ function with the complementary error function. tot ͑ii͒ Assuming that a stochastic gravity-wave signal is ϫͱ2 erfcϪ1͑2␣͒. ͑4.45͒ present, how do we estimate its strength? Answer: Assuming that a stochastic gravity-wave signal This alternative upper limit is the maximum value of the test is present, characterized by some fixed, but unknown, mean statistic leading to the conclusion that a stochastic gravity- value ␮Ͼ0, we estimate ␮ by constructing the interval wave signal is absent. The decision rule—by its very construction—will never allow one to conclude that a sto- ˆ ˆ ˆ ˆ chastic gravity-wave signal is present with an ⍀ less than I␣ ª ͓␮Ϫz␣/2␴/ͱn,␮ϩz␣/2␴/ͱn͔. ͑4.43͒ 0 this upper limit. The above two definitions of upper limit agree when the In an ensemble of observations of the same stochastic back- detection rate ␥ϭ0.50. This is because erfcϪ1(1)ϭ0. This ground, 1Ϫ␣ is the fraction of intervals I␣ constructed in means that, if a stochastic gravity-wave signal is present, this way that contain the value of the true mean ␮. with an ⍀0 less than or equal to this maximum value, then We should emphasize that parameter estimation assumes there is more than a 50% chance of falsely dismissing it. the presence of a signal. First, as mentioned above, we can Note also that if we change the argument of the inverse never be 100% certain that a stochastic gravity-wave signal complementary error function from ␣ to 2␣ in Eq. ͑4.38͒, is present. Second, it does not make any sense to try to esti- then Eqs. ͑4.38͒ and ͑4.45͒ agree. Thus, the minimum values mate the parameters of something that we assume does not of ⍀0 quoted in Refs. ͓7,8͔ can be interpreted as either the exist. ␥ϭ0.50,␣ϭ0.05 upper limit defined by Eq. ͑4.44͒ or the ␣ ͑iii͒ Assuming that a stochastic gravity-wave signal is ϭ0.05 upper limit defined by Eq. ͑4.45͒. present, what is the minimum value of ⍀0 required to detect it ␥ϫ100% of time? Answer: Assuming that a stochastic gravity-wave signal V. COMPLICATIONS is present, characterized by some fixed, but unknown, mean In Sec. III, we discussed optimal filtering under the as- value ␮Ͼ0, the minimum value of ⍀ , for an observation 0 sumptions that the noises intrinsic to the detectors were ͑i͒ time T , false alarm rate ␣, and detection rate ␥, is given tot stationary, ͑ii͒ Gaussian, ͑iii͒ statistically independent of one by another and of the stochastic gravity-wave background, and ͑iv͒ much larger in magnitude than the stochastic gravity- 1 10␲2 ϱ ␥2͉͑f͉͒ Ϫ1/2 wave background. In this section, we describe the modifica- ⍀ ϭ df tions that are necessary when most of these assumptions are 0 3H2 ͵ f6P ͉͑f͉͒P ͉͑f͉͒ ͱTtot 0 ͫ Ϫϱ 1 2 ͬ removed.30 We will also describe how one can correlate the outputs of 4 or more detectors, and how one can combine ϫͱ2͓erfcϪ1͑2␣͒ϪerfcϪ1͑2␥͔͒. ͑4.44͒ data from multiple detector pairs to increase the sensitivity of a stochastic background search. Ϫ1/2 For fixed ␣ and ␥, the factor of Ttot implies that the mini- mum value of ⍀ decreases with increasing observation 0 A. Signal-to-noise ratios for arbitrarily large time. Thus, the sensitivity improves as the total observation stochastic backgrounds time increases. This means that as Ttot increases we can put a tighter ‘‘upper limit’’ on the value of ⍀0 for a stochastic In Sec. III C, we calculated the signal-to-noise ratio for a gravity-wave signal that we will falsely dismiss more than stochastic background of gravitational radiation, assuming (1Ϫ␥)ϫ100% of the time. that the noises intrinsic to the detectors were much larger in Alternatively, for a fixed ⍀0 and a fixed false alarm rate magnitude than the gravitational strains. Although this as- Ϫ1/2 sumption is most likely valid, there are at least two reasons ␣, the factor of Ttot implies that the false dismissmal rate 1Ϫ␥, for a stochastic gravity-wave signal having a strength why we want to consider stochastic background signals whose magnitudes are comparable to ͑or larger than͒ the equal to ⍀0 , decreases with increasing observation time. For example, suppose that a particular theory—such as cosmic noise intrinsic to the detectors: ͑i͒ Computer simulations for Ϫ7 stochastic background searches allow one to ‘‘dial-in’’ arbi- strings—predicts a value of ⍀0ϭ10 . Moreover, suppose that, in successive years of observation, we fail to detect the trarily large signal strengths. Thus, in order to compare the- presence of a signal at this level of sensitivity. Then we can oretical predictions with the results of computer simulations, still say that the probability of our falsely dismissing a sto- we need to be able to analyze the large signal case. ͑ii͒ Fu- ture generations of gravity-wave detectors might have intrin- chastic gravity-wave signal having a strength equal to ⍀0 ϭ10Ϫ7 has decreased over the course of the observation. sic detector noise levels comparable to the level of a real The above upper limit on stochastic background signal strengths ͓defined by Eq. ͑4.44͔͒ is different from that ob- tained by setting the measured signal-to-noise ratio for the 30We will always assume that the noise intrinsic to a detector is total observation period equal to the threshold value z␣ : Gaussian and statistically independent of the gravitational strains.

102001-20 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001 stochastic background signal. For this case, optimal filtering ϱ of large signal data will also be necessary. ͑A,B͒ ª ͵ df A*͑f͒B͑f͒P1͉͑f͉͒P2͉͑f͉͒, ͑5.5͒ To begin, let us recall the main results of Sec. III C. Un- Ϫϱ der the assumption that the noises intrinsic to the detectors and writing ␮ and ␴2 in terms of this inner product. The were much larger than the gravitational strains, the calcula- squared signal-to-noise ratio then took the form 2 2 2 tion of the variance ␴ ª ͗S ͘Ϫ͗S͘ simplified consider- ably. For the large noise case, ␴2 was dominated by the pure 2 ␥͉͑f͉͒⍀gw͉͑f͉͒ detector noise contribution: Q˜ , 3H2 2 ͩ ͉f͉3P ͉͑f͉͒P ͉͑f͉͒ͪ 2 0 1 2 T ϱ SNR ϭ 2 T , ͑5.6͒ 2 ˜ 2 ͩ 10␲ ͪ ͑Q˜ ,Q˜ ͒ ␴ Ϸ df P1͉͑f͉͒P2͉͑f͉͉͒Q͑f͉͒ . ͑5.1͒ 4 ͵Ϫϱ which was maximized by choosing This is Eq. ͑3.68͒ from Sec. III C. If, however, the magnitude of the stochastic background ␥͉͑f͉͒⍀gw͉͑f͉͒ Q˜ ͑ f ͒ϭ␭ . ͑5.7͒ 3 signal is comparable to the noises intrinsic to the detectors, ͉f͉ P1͉͑f͉͒P2͉͑f͉͒ the calculation of the signal-to-noise ratio SNR ª ␮/␴ is ˜ more involved. Although the mean value ␮ is independent of Although Q( f ) depended on ⍀gw( f ), we could construct a the relative size of the stochastic background signal and the ˜ 31 set of optimal filters Q␣( f ) for stochastic backgrounds hav- detector noise, ␣ ing power-law spectra ⍀gw( f )ϭ⍀␣ f ͑where ⍀␣ϭconst). The proportionality constants ⍀␣ could always be absorbed 2 ϱ 3H0 Ϫ3 ˜ into the normalization constants ␭␣ . The resulting set of ␮ϭ 2 T df ͉f͉ ⍀gw͉͑f͉͒␥͉͑f͉͒Q͑ f ͒, ͑5.2͒ 20␲ ͵Ϫϱ optimal filters was then completely specified by the exponent ␣, the overlap reduction function, and the noise power spec- the variance ␴2 is not. Explicitly, tra of the two detectors. For the case where the stochastic background is compa- T ϱ rable to the noise intrinsic to the detectors, we can try to do ␴2ϭ df͉Q˜͑f͉͒2R͑f͒, ͑5.3͒ something similar. We can define a new inner product 4 ͵Ϫϱ ϱ where ͑A,B͒ ª df A*͑f͒B͑f͒R͑f͒, ͑5.8͒ ͵Ϫϱ

2 3H0 ⍀gw͉͑f͉͒ where R( f ) is given by Eq. ͑5.4͒. Although this inner prod- R͑ f ͒ P ͑ f ͒P ͑ f ͒ϩ ª 1 ͉ ͉ 2 ͉ ͉ 2 3 uct is more complicated in form than the original inner prod- ͫ ͩ 10␲ ͪ ͉f͉ uct ͑5.5͒, it is still positive-definite, since R( f ) is real and 2 2 2 positive. In terms of Eq. ͑5.8͒, 3H0 ⍀gw͉͑f͉͒ ϫ P ͉͑f͉͒ϩP ͉͑f͉͒ ϩ „ 1 2 … ͩ 10␲2 ͪ 6 2 f 3H0 T ␮ϭ T͑Q˜ ,A͒ and ␴2ϭ ͑Q˜ ,Q˜ ͒, ͑5.9͒ 20␲2 4 ϫ 1ϩ␥2͉͑f͉͒ . ͑5.4͒ „ …ͬ where Ϫ3 Ϫ1 Note the additional terms that contribute to the variance. A͑ f ͒ ª ͉f͉ ⍀gw͉͑f͉͒␥͉͑f͉͒R ͑ f ͒. ͑5.10͒ Roughly speaking, they can be thought of as two ‘‘signal The squared signal-to-noise ratio is thus ϩnoise’’ cross-terms and a ‘‘pure signal’’ variance term.32 When ⍀gw(͉f͉) is negligible compared to the detector noise 3H2 2 Q˜ ,A 2 2 2 0 ͑ ͒ power spectra Pi(͉f͉)(iϭ1,2), Eqs. ͑5.3͒ and ͑5.4͒ for ␴ SNR ϭ 2 T , ͑5.11͒ reduce to the ‘‘pure noise’’ variance term ͑5.1͒ as they ͩ 10␲ ͪ ͑Q˜ ,Q˜ ͒ should. which has the same form as Eq. ͑5.6͒, although with a much At this stage of the analysis, the filter function Q˜ ( f )in more complicated expression for A( f ). But the same argu- Eqs. ͑5.2͒ and ͑5.3͒ is arbitrary. For the case of large detector ment for maximizing the squared signal-to-noise ratio still ˜ noise, we were able to make an optimal choice for Q( f ), goes through. The optimal choice of filter function is which maximized the signal-to-noise ratio. This was facili- tated by introducing an inner product Q˜ ͑ f ͒ϭ␭A͑ f ͒, ͑5.12͒

where ␭ is a ͑real͒ overall normalization constant. 31See the discussion surrounding Eqs. ͑3.52͒–͑3.57͒ in Sec. III C. But this is where the similarity with the large detector 32 2 noise case ends, and where the complications start to arise. These are the terms proportional to ⍀gw(͉f͉) and ⍀gw(͉f͉), re- spectively. The main problem is that the optimal filter function Q˜ ( f ) has

102001-21 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001 a complicated functional dependence on the stochastic ͑3.73͔͒ so that the theoretical mean ␮ is equal to some fixed gravity-wave spectrum ⍀gw( f ). For the large detector noise value ͓e.g., ␮ϭ⍀␣T for a stochastic background having a ␣ case, this problem did not exist. As mentioned earlier, we power-law spectrum ⍀gw( f )ϭ⍀␣ f ], then the quiet periods could always consider stochastic backgrounds having power- will have a correspondingly larger signal-to-noise ratio. ␣ law spectra ⍀gw( f )ϭ⍀␣ f , and then construct a set of op- Thus, a natural question that arises in this context is, can one ˜ combine the different sets of measurements, corresponding timal filters Q␣( f ) labeled by the different values of ␣. But for the optimal filter function Q˜ ( f ) given by Eqs. ͑5.12͒, to the quiet and noisy periods of detector operation, so as to ͑5.10͒, and ͑5.4͒, this idea of constructing a set of filters maximize the overall signal-to-noise ratio? The answer to labeled by only the power-law exponents ␣ fails. The pro- this question is yes, and the proof is sketched below. In order to handle the most general case, let us consider m portionality constants ⍀ cannot be absorbed into the nor- ␣ different sets of measurements malization constants ␭␣ . For example, if we consider a sto- chastic background having a constant frequency spectrum ͑i͒S ,͑i͒S ,...,͑i͒S ͑5.14͒ 1 2 ni ⍀gw( f )ϭ⍀0 ͑i.e., ␣ϭ0), then corresponding to m different levels of detector noise or m ˜ Ϫ3 Q͑ f ͒ϭ␭⍀0͉f͉ ␥͉͑f͉͒ P1͉͑f͉͒P2͉͑f͉͒ different periods of detector operation. ͑Here i ͫ ϭ1,2,...,m.) Each of these measurements is taken over an identical time interval of length T. These measurements can 2 (i) 3H0 ⍀0 be thought of as realizations of m random variables S, ϩ P ͉͑f͉͒ϩP ͉͑f͉͒ ͩ10␲2 ͪ ͉f͉3 „ 1 2 … each having the same theoretical mean

͑i͒ ͑i͒ 2 2 2 Ϫ1 ␮ ª ͗ S͘ϭ:␮, ͑5.15͒ 3H0 ⍀0 ϩ 1ϩ␥2͉͑f͉͒ . ͑5.13͒ ͩ 10␲2 ͪ f 6 „ …ͬ but different theoretical variances

͑i͒ 2 ͑i͒ 2 ͑i͒ 2 Although the factor of ⍀0 in the numerator can be absorbed ␴ ͗ S ͘Ϫ͗ S͘ . ͑5.16͒ 2 ª into the normalization constant ␭, the factors of ⍀0 and ⍀0 in the denominator cannot. In other words, if we want to The equality of the mean values follows because we construct a set of optimal filters for arbitrarily large stochas- assume identical normalization conventions on the optimal ␣ tic backgrounds having power-law spectra ⍀gw( f )ϭ⍀␣ f , filters—e.g., ␮ϭ⍀␣T for a power-law spectrum ⍀gw( f ) ␣ we need to specify the proportionality constants ⍀␣ in addi- ϭ⍀␣ f . We also assume that all of the measurements(1)S ,(1)S ,...,(m)S are statistically inde- tion to the exponents ␣. The ‘‘space of optimal filters’’ thus 1 2 nm 34 becomes a much larger set, parametrized by (␣,⍀␣). Al- pendent of one another. though in principle this poses no problem, in practice it re- For each set of measurements, we can construct the quires a more sophisticated search algorithm; the detector sample mean ͑or estimator͒ outputs will have to be analyzed for each of the filters n Q˜ ( f ) separately. 1 i (␣,⍀␣) ͑i͒ ˆ ͑i͒ ␮ ª ͚ S j . ͑5.17͒ ni jϭ1 B. Nonstationary detector noise Viewed as a random variable in its own right, (i)␮ˆ has mean It is not at all uncommon for the power spectra of the value noises intrinsic to the detectors to change over the course of time. There will be periods of time when the detectors ␮ ͗͑i͒␮ˆ ͘ϭ␮ ͑5.18͒ are relatively ‘‘quiet,’’ and other periods of time when i ª the detectors are relatively ‘‘noisy.’’33 These changes in the and variance power spectra will, in turn, lead to measurements whose sta- tistical properties also change with time. For example, during ͑i͒␴2 2 ͑i͒ ˆ 2 ͑i͒ ˆ 2 the quiet periods of detector operation, the ␴i ª ͗ ␮ ͘Ϫ͗ ␮͘ ϭ . ͑5.19͒ ni measurements (1)S ,(1)S ,...,(1)S will have an associated 1 2 n1 variance (1)␴2 that will be smaller than the variance (2)␴2 What we want to do now is combine all the meas- (2) (2) (2) (1) (1) (m) associated with the measurements S , S ,..., S urements S1 , S2 ,..., Sn ͑or, equivalently, combine 1 2 n2 m taken during the noisy periods. Moreover, if the optimal filter the sample means (1)␮ˆ ,(2)␮ˆ ,...,(m)␮ˆ) so as to maximize function is normalized ͓by an appropriate choice of ␭ in Eq. the overall signal-to-noise ratio. We thus use a weighted av- erage to define the estimator

33Although the frequency of these more ‘‘noisy’’ periods should decrease as the detectors are gradually improved over the course of 34As mentioned in Sec. IV A, this means that the measurements months or years, variations in the detector noise power spectra will are taken over distinct, nonoverlapping periods of operation, with still inevitably occur. Tӷ the light travel time d/c between the two detectors.

102001-22 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

m ˆ ˆ ␮optimal ␮͉␭ ϭ␴Ϫ2 ͑5.27͒ ͑i͒ ˆ ª i i ␭i ␮ i͚ϭ1 ˆ has a very simple form ␮ ª m , ͑5.20͒ m m ␭ j j͚ϭ1 Ϫ2 Ϫ2 ͑i͒ Ϫ2 ␴optimal ϭ ␴i ϭ ni ␴ . ͑5.28͒ i͚ϭ1 i͚ϭ1 and then choose ␭iϾ0 to maximize the signal-to-noise ratio ˆ ˆ This result says that the variances for the optimal estimator of ␮. From Eq. ͑5.20͒, it follows that ␮ has mean value add like electrical resistors in parallel. The squared signal-to- noise ratio of the optimal estimator also has a very simple ␮ ˆ ͗␮ˆ ͘ϭ␮ ͑5.21͒ ␮ ª form

͑which is independent of the choice of ␭i) and variance m m 2 2 ͑i͒ 2 m SNRoptimal ϭ SNRi ϭ ni SNR , ͑5.29͒ i͚ϭ1 i͚ϭ1 2 2 ͚ ␭i ␴i iϭ1 where 2 ˆ 2 ˆ 2 ␴␮ˆ ª ͗␮ ͘Ϫ͗␮͘ ϭ m 2 . ͑5.22͒ ␮i ␮ SNR ϭ ͑5.30͒ ͚ ␭ j i ª ␴ ␴ ͩ jϭ1 ͪ i i The squared signal-to-noise ratio is thus and m 2 ͑i͒␮ ␮ ͑i͒ SNR ª ϭ . ͑5.31͒ 2 ͚ ␭ j ͑i͒␴ ͑i͒␴ ␮ ˆ ͩ jϭ1 ͪ ␮ 2 2 SNR␮ˆ ª ϭ␮ m . ͑5.23͒ Thus, the squared signal-to-noise ratio of the optimal estima- ␴2 ␮ˆ 2 2 tor is simply a sum of the squared signal-to-noise ratios for ͚ ␭i ␴i (i) iϭ1 each S after ni observation periods, each of length T. It is instructive to compare the results for the optimal 2 To find the ␭ which maximize SNR ˆ ͑and hence which ˆ i ␮ estimator ␮optimal with those for the ‘‘naive’’ estimator tell us how to optimally combine data from periods of quiet and noisy detector operation͒ is quite easy. This is because 1 m ni ␮ˆ ͑i͒S , ͑5.32͒ Eq. ͑5.23͒ can be written as a ratio of inner products, just as naive ª n ͚ ͚ j we were able to write the squared signal-to-noise ratio ͑3.72͒ tot iϭ1 jϭ1 in Sec. III C as a ratio of inner products. Explicitly, which simply averages all of the signal estimates, paying no Ϫ2 2 attention to the different variances (i)␴2 associated with the 2 ͑␭,␴ ͒ SNR ϭ␮2 ͑5.24͒ (i) ˆ ␮ˆ S. The naive estimator ␮naive corresponds to Eq. ͑5.20͒ ͑␭,␭͒ m with ␭iϭni and ntot ª ͚iϭ1ni . One can show that where the inner product (␣,␤) is defined by 1 m 1 m m 2 2 2 ͑i͒ 2 2 ␴naiveϭ ni ␴i ϭ ni ␴ ͑5.33͒ 2 ͚ϭ 2 ͚ϭ ͑␣,␤͒ ª ␣i*␤i␴i , ͑5.25͒ ntot i 1 ntot i 1 i͚ϭ1 and for any pair of ͑complex͒ sequences ␣i and ␤i (i ϭ1,2,...,m). The inner product (␣,␤) is positive-definite, m m 1 1 since ␴2 is real and positive, and it satisfies all of the prop- Ϫ2 2 Ϫ2 ͑i͒ Ϫ2 i SNRnaiveϭ ni SNRi ϭ ni SNR . 2 ͚ϭ 2 ͚ϭ erties of the ordinary dot product of vectors in three- ntot i 1 ntot i 1 dimensional Euclidean space. Thus, choosing ͑5.34͒

Ϫ2 2 ␭iϰ␴i ͑5.26͒ Note that in addition to the factors of 1/ntot , the signs of the exponents for the variances and signal-to-noise ratios in Eqs. maximizes Eq. ͑5.24͒ in the same way that choosing Aជ pro- ͑5.33͒ and ͑5.34͒ are opposite to those in Eqs. ͑5.28͒ and 2 portional to Bជ maximizes the ratio (Aជ •Bជ ) /(Aជ•Aជ). By aver- ͑5.29͒. aging each set of measurements with the inverse of its asso- To give a numerical example, suppose we have just two ciated theoretical variance, we give more weight to signal different periods of detector operation, with the second pe- values that are measured when the detectors are quiet than to riod twice as noisy as the first—i.e., (2)␴ϭ2x(1)␴. Since 2 (i) 2 signal values that are measured when the detectors are noisy. ␴i ϭ ␴ /ni , it follows that if there are 4 times as many This weighting maximizes the overall signal-to-noise ratio. measurements during the noisy period ͑i.e., n2ϭ4n1), then Ϫ2 2 2 For the optimal choice of weights ␭iϭ␴i , the inverse ␴1ϭ␴2 . Intuition suggests that the optimal way of combin- of the variance of the optimal estimator ing the measurements for this case is to weight the estimators

102001-23 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

(1) (2) Ϫ2 ␣ ␮ˆ and ␮ˆ equally ͑i.e., ␭ ϭ␭ ϭ␴ ). This agrees having a power-law spectrum ⍀gw( f )ϭ⍀␣ f ], the math- 1 2 1 36 with the above mathematical analysis, and yields a squared ematical analysis of Sec. V B goes through unchanged. The signal-to-noise ratio for the optimal estimator equal to optimal way of combining the estimators 1 nij 2 ͑ij͒ ˆ ͑ij͒ ␮ ␮ª Sk ͑5.41͒ 2 ͑1͒ 2 n ͚ϭ SNRoptimalϭ2n1 SNR ϭ2n1 . ͑5.35͒ ij k 1 ͑1͒␴2 for each detector pair is For the naive estimator, l l ͑ij͒ ˆ ͚ ͚ ␭ij ␮ 25 25 ␮2 iϭ1 jϽi 2 ͑1͒ 2 SNRnaiveϭ n1 SNR ϭ n1 . ͑5.36͒ ␮ˆ , ͑5.42͒ 17 17 ͑1͒␴2 ª l l ͚ ͚ ␭ij Thus, for this particular example, the signal-to-noise ratio for iϭ1 jϽi the optimal estimator is ͱ34/25ϭ1.17 times larger than that where37 for the naive estimator. Ϫ2 ͑ij͒ Ϫ2 ␭ijϭ␴ij ϭnij ␴ . ͑5.43͒ C. Multiple detector pairs The inverse of the variance for the optimal estimator Combining measurements from multiple detector pairs in ˆ ˆ ␮optimal ␮͉␭ ϭ␴Ϫ2 ͑5.44͒ order to increase the sensitivity of a stochastic background ª ij ij search is identical to combining different sets of measure- is given by ments corresponding to quiet and noisy periods of detector l l operation in order to maximize the overall signal-to-noise Ϫ2 ͑ij͒ Ϫ2 ␴optimalϭ͚ ͚ nij ␴ . ͑5.45͒ ratio. The only difference between the two is one of inter- iϭ1 jϽi pretation and notation. In Sec. V B, The squared signal-to-noise ratio for the optimal estimator is ͑i͒ ͑i͒ ͑i͒ given by S1 , S2,... Sn ͑5.37͒ i l l 2 ͑ij͒ 2 denoted ni different measurements, each of length T,ofthe SNRoptimal ϭ nij SNR . ͑5.46͒ i͚ϭ1 ͚jϽi optimally filtered cross-correlation signal (i)S when the level of detector noise was characterized by the variance These equations should be compared with Eqs. ͑5.28͒ and ͑5.29͒ in Sec. V B. ͑i͒ 2 ͑i͒ 2 ͑i͒ 2 ␴i ª ͗ S ͘Ϫ͗ S͘ . ͑5.38͒ Using Eq. ͑5.45͒ for the inverse variance of the optimal estimator, we can derive an expression for the minimum In this section, for multiple detector pairs, value of ⍀0 after 4 months of observation ͑i.e., Ttot ϭ107 sec), for a false alarm rate equal to 5% and a detec- ͑ij͒S ,͑ij͒S ,...,͑ij͒S ͑5.39͒ 1 2 nij denote nij different measurements, again each of length T,of 36 (ij) (kl) (ij) Although cross-correlation signals S and S taken during the optimally filtered cross-correlation signal S between the same time interval T are correlated, i.e., the ith and jth detectors.35 As usual, cov͕͑ij͒S,͑kl͒S͖ ͗͑ij͒S ͑kl͒S͘Ϫ͗͑ij͒S͗͑͘kl͒S͘Þ0, ͑ij͒ 2 ͑ij͒ 2 ͑ij͒ 2 ª ␴i ª͗ S ͘Ϫ͗ S͘ ͑5.40͒ the variance-covariance matrix C (ij) (kl) is (ij) (ij)(kl) ª cov͕ S, S͖ denotes the variance of the cross-correlation signal S. (ij) (ij) dominated by the diagonal terms C(ij)(ij)ϭcov͕ S, S͖ Provided that each cross-correlation signal measurement ϭ:(ij)␴2 in the large noise approximation. Thus, in practice, one occurs over the same time interval T ͑which should be ӷ the can treat all of the measurements (12)S ,(12)S ,...,(lϪ1,l)S ,as light travel time between any pair of detectors͒, and that the 1 2 nlϪ1,l effectively uncorrelated. This is because the detector noises ͑which optimal filter functions for each detector pair have identical are statistically independent of one another͒ are the only contribu- (ij) normalizations ͓e.g., ␭ in Eq. ͑3.73͒ is chosen so that ␮ tors to (ij)␴2 in this approximation. (ij) ϭ͗ S͘ϭ⍀␣T for a stochastic gravity-wave background 37For completeness, we note that the optimal way of combining ˆ correlated random variables x1 ,...,xm is given by ␮ m m m Ϫ1 Ϫ1 ª ͚iϭ1␭ixi /͚ jϭ1␭ j , where ␭iϭ͚ jϭ1(C )ij and (C )ij is the inverse of the variance-covariance matrix Cij ª cov͕xi ,x j͖. When the variance-covariance matrix is dominated by the diagonal terms 35 2 In Sec. V B, iϭ1,2,...,mlabeled m different levels of detector Ciiϭcov͕xi ,xi͖ϭ:␴i , the optimal combination of data reduces Ϫ2 noise or m different periods of detector operation. In this section, ͑approximately͒ to the uncorrelated result ␭iϭ␴i . As argued in i, jϭ1,2,...,l label l different detectors, and m ª l(lϪ1)/2 is the the previous footnote, this is what happens for the cross-correlation number of different detector pairs. signals x (ij)S. i↔ 102001-24 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001 tion rate equal to 95%, where we optimally combine data the outputs of 4 detectors ͑or, in general, 2N detectors͒ in a from different detector pairs. In order to simplify the analy- manner analogous to the single 2-detector correlation de- 38 sis, we will assume that TtotϭnT, where n ª nij is the same scribed in Sec. III C. In fact, it turns out that one can write for all pairs of detectors. down expressions for the optimally filtered squared signal- 95%,5% For a single pair of detectors, the minimum value of ⍀0 is to-noise ratio and the inverse of the square of ⍀0 for given by the 4-detector correlation in terms of a simple sum of prod- ucts of the corresponding quantities for the individual detec- 2ϫ1.65 ͑ij͒ 95%,5% ͑ij͒ tor pairs. As noted above, the analysis given here can easily ⍀0 ϭ ͱn ␴, ͑5.47͒ 107 sec be generalized to the case of 2N detectors. At the end of the where section, we summarize our results by writing down the key equations in the general 2N-detector form. 10␲2 2 ϱ ␥2͉͑f͉͒ Ϫ1 ͑ij͒ 2 To begin, we define the 4-detector correlation signal S to ␴ ϷT 2 df 6 . ͩ 3H ͪ ͵Ϫϱ f Pi͉͑f͉͒Pj͉͑f͉͒ 0 ͫ ͬ be the integrated product ͑5.48͒ The above expression for (ij)␴2 follows from Eqs. ͑3.68͒ and T/2 T/2 S ª ͵ dt1•••͵ dt4 s1͑t1͒•••s4͑t4͒Q͑t1 ,...,t4͒, ͑3.73͒, together with the normalization condition ␮ϭ⍀0T ϪT/2 ϪT/2 for a stochastic background having a constant frequency ͑5.51͒ spectrum ⍀ ( f )ϭ⍀ . The factor of 2ϫ1.65 in Eq. ͑5.47͒ gw 0 where comes from the choice of the false alarm and detection rates. ͓See, e.g., Eq. ͑4.37͒.͔ Table II in Sec. VI D contains values si͑t͒ ª hi͑t͒ϩni͑t͒ ͑5.52͒ (ij) 95%,5% 2 of ⍀0 h100 for different detector pairs. are the outputs of the detectors (iϭ1,...,4), and For the optimal combination of data from multiple detec- Q(t1,...,t4) is an arbitrary filter function, which we will tor pairs, we have determine shortly. To save some writing in what follows, we Ϫ2 2ϫ1.65 l l will use the shorthand notation 95%,5% Ϫ2 Ϫ1 ͑ij͒ Ϫ2 ͑⍀0 ͉optimal͒ ϭ n ͚ ͚ ␴ . T/2 T/2 T/2 ͩ 107 secͪ iϭ1 jϽi 4 d tϵ dt1 ... dt4 . ͑5.53͒ ͑5.49͒ ͵ϪT/2 ͵ϪT/2 ͵ϪT/2 This can also be written as Throughout, we will assume that the gravitational strains l l hi(t) satisfy the statistical properties listed in Sec. II B. We 95%,5% Ϫ2 ͑ij͒ 95%,5% Ϫ2 will also assume that the noises n (t) intrinsic to the detec- ͑⍀0 ͉optimal͒ ϭ͚ ͚ ͑ ⍀0 ͒ . ͑5.50͒ i iϭ1 jϽi tors are ͑i͒ stationary, ͑ii͒ Gaussian, ͑iii͒ statistically indepen- Tables III and IV in Sec. VI D contain values of dent of one another and of the gravitational strains, and ͑iv͒ 95%,5% much larger in magnitude than the gravitational strains. The ⍀0 ͉optimal for the optimal combination of data from mul- tiple detector pairs, for all possible triples and quadruples of goal is to determine the filter function Q(t1 ,...,t4) that the five major interferometers. maximizes the signal-to-noise ratio SNR ª ␮/␴ of S. Let us start by calculating the variance ␴2. Since we are assuming that the noises intrinsic to the detectors are statis- D. Four-detector correlation tically independent of one another and of the gravitational Rather than combine data from multiple detector pairs as strains, and that they are much larger in magnitude than the described in the previous section, one can directly correlate gravitational strains, it follows that

␴2:ϭ͗S2͘Ϫ͗S͘2Ϸ͗S2͘ ͑5.54͒

T/2 T/2 4 4 ϭ d t d tЈ͗s1͑t1͒•••s4͑t4͒s1͑t1Ј͒•••s4͑t4Ј͒͘Q͑t1 ,...,t4͒Q͑t1Ј,...,t4Ј͒ ͑5.55͒ ͵ϪT/2 ͵ϪT/2

T/2 T/2 4 4 Ϸ d t d tЈ͗n1͑t1͒•••n4͑t4͒n1͑t1Ј͒•••n4͑t4Ј͒͘Q͑t1 ,...,t4͒Q͑t1Ј,...,t4Ј͒ ͑5.56͒ ͵ϪT/2 ͵ϪT/2

T/2 T/2 4 4 ϭ d t d tЈ͗n1͑t1͒n1͑t1Ј͒͘•••͗n4͑t4͒n4͑t4Ј͒͘Q͑t1 ,...,t4͒Q͑t1Ј,...,t4Ј͒. ͑5.57͒ ͵ϪT/2 ͵ϪT/2

͓See Eqs. ͑3.19͒–͑3.22͒.͔ Using the definition ͑3.23͒ of the noise power spectra Pi(͉f͉), Eq. ͑5.57͒ can be rewritten as

38The correlation of 3 detectors ͑or, in general, 2Nϩ1 detectors͒ yields a signal that has zero mean.

102001-25 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

1 4 ϱ T/2 2 4 4 i2␲f1t1 i2␲f4t4 ␴ Ϸ d fP1͉͑f1͉͒•••P4͉͑f4͉͒ d te •••e Q͑t1 ,...,t4͒ ͩ 2ͪ ͵Ϫϱ ͵ϪT/2

T/2 4 Ϫi2␲ f 1t1Ј Ϫi2␲f4t4Ј ϫ d tЈ e •••e Q͑t1Ј,...,t4Ј͒, ͑5.58͒ ͵ϪT/2 where ϱ ϱ ϱ 4 d f ϵ df1••• df4. ͑5.59͒ ͵Ϫϱ ͵Ϫϱ ͵Ϫϱ If we further define the ͑finite-time͒ Fourier transform T/2 ˜ 4 Ϫi2␲f1t1 Ϫi2␲f4t4 Q͑ f 1 ,...,f4͒ª d te •••e Q͑t1 ,...,t4͒, ͑5.60͒ ͵ϪT/2 which has as its inverse ͑for ϪT/2Ͻt1 ,...,t4ϽT/2) ϱ 4 i2␲f1t1 i2␲f4t4˜ Q͑t1 ,...,t4͒ϭ d fe •••e Q͑f1 ,...,f4͒, ͑5.61͒ ͵Ϫϱ we obtain 1 4 ϱ 2 4 ˜ 2 ␴ Ϸ d fP1͉͑f1͉͒•••P4͉͑f4͉͉͒Q͑f1 ,...,f4͉͒ . ͑5.62͒ ͩ 2ͪ ͵Ϫϱ This can be written in an even more convenient form if we define an inner product

ϱ 4 ͑A,B͒ ª d fP1͉͑f1͉͒•••P4͉͑f4͉͒A*͑f1 ,...,f4͒B͑f1,...,f4͒, ͑5.63͒ ͵Ϫϱ

where A( f 1 ,...,f4) and B( f 1 ,...,f4) are any two ͗x1x2x3x4͘ϭ͗x1x2͗͘x3x4͘ϩ͗x1x3͗͘x2x4͘ϩ͗x1x4͗͘x2x3͘ complex-valued functions of four variables. ͓See Eq. ͑3.69͒.͔ ͑5.69͒ Note that the inner product is positive-definite since Pi(͉f͉) Ͼ0. Using Eq. ͑5.63͒ it follows that for Gaussian random variables x1 ,x2 ,x3 ,x4 each having zero mean.39 1 4 To express ␮ in terms of the inner product ͑5.63͒, we first ␴2Ϸ ͑Q˜ ,Q˜ ͒. ͑5.64͒ use Eq. ͑5.61͒ to rewrite Q(t ,...,t ) in terms of its Fou- 2 1 4 ͩ ͪ ˜ rier transform Q( f 1 ,...,f4):

To calculate the mean value ␮, we proceed in a similar T/2 4 manner. Since the noises intrinsic to the detectors are statis- ␮ϭ d t ͑͗h1͑t1͒h2͑t2͒͗͘h3͑t3͒h4͑t4͒͘ tically independent of one another and of the gravitational ͵ϪT/2 strains, it follows that ϩ͗13͗͘24͘ϩ͗14͗͘23͒͘ ␮ ª ͗S͘ ͑5.65͒ ϱ 4 i2␲f1t1 i2␲f4t4˜ ϫ d fe •••e Q͑f1 ,...,f4͒. ͵Ϫϱ T/2 4 ϭ d t ͗s1͑t1͒•••s4͑t4͒͘Q͑t1 ,...,t4͒ ͑5.66͒ ͑5.70͒ ͵ϪT/2 Then by interchanging the order of integrations and rearrang- T/2 4 ing terms in the integrand, we see that ϭ ͵ d t ͗h1͑t1͒•••h4͑t4͒͘Q͑t1 ,...,t4͒. ϪT/2 ϱ T/2 4 4 i2␲f1t1 i2␲f4t4 ͑5.67͒ ␮ϭ d f d te •••e „͗h1͑t1͒h2͑t2͒͘ ͵Ϫϱ ͵ϪT/2 This can be expanded to ϫ͗h3͑t3͒h4͑t4͒͘ϩ͗13͗͘24͘ϩ͗14͗͘23͘… T/2 4 ϫQ˜ ͑ f ,...,f ͒ ͑5.71͒ ␮ϭ d t „͗h1͑t1͒h2͑t2͒͗͘h3͑t3͒h4͑t4͒͘ 1 4 ͵ϪT/2

ϩ͗13͗͘24͘ϩ͗14͗͘23͘…Q͑t1 ,...,t4͒ ͑5.68͒ 39In Eq. ͑5.68͒, ͗13͗͘24͘ is used as a shorthand notation for by using the ‘‘factorization’’ property ͗h1(t1)h3(t3)͗͘h2(t2)h4(t4)͘, etc.

102001-26 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001 or, equivalently, Also, since the noise power spectra Pi(͉f i͉) and filter func- ˜ tion Q( f 1 ,...,f4) are not expected to vary much over the ␮ϭ A,Q˜ , ͑5.72͒ ͑ ͒ support, 1/T,of␦T(f1Ϫf) and ␦T( f 3Ϫ f Ј), we are justified in approximating these two finite-time delta functions by or- where dinary Dirac delta functions ␦( f 1Ϫ f ) and ␦( f 3Ϫ f Ј). This approximation allows us to eliminate the integrations over f 1 T/2 4 and f Ј, yielding A͑ f 1 ,...,f4͒ª d t P1͉͑f1͉͒•••P4͉͑f4͉͒͵ϪT/2 H12͑ f 1͒ A͑ f ,...,f ͒Ϸ ␦ ͑ f ϩ f ͒ Ϫi2␲ f 1t1 Ϫi2␲f4t4 1 4 T 1 2 ϫe •••e „͗h1͑t1͒h2͑t2͒͘ P1͉͑f 1͉͒P2͉͑f 2͉͒

ϫ͗h3͑t3͒h4͑t4͒͘ϩ͗13͗͘24͘ϩ͗14͗͘23͘…. H34͑ f 3͒ ϫ ␦T͑ f 3ϩ f 4͒ ͑5.73͒ P3͉͑f 3͉͒P4͉͑f 4͉͒ ϩ13,24ϩ14,23. ͑5.79͒ To simplify this expression for A( f 1 ,...,f4), we expand 2 the expectation values ͗hi(ti)h j(t j)͘ as Given the above expressions for ␴ , ␮, and A( f 1 ,...,f4), it is now a simple matter to evaluate the ϱ squared signal-to-noise ratio of the 4-detector correlation S, ͗h ͑t ͒h ͑t ͒͘ϭ df ei2␲f͑tiϪtj͒H ͑f͒, ͑5.74͒ i i j j ͵ ij and to determine the filter function Q˜ ( f ,...,f ) that maxi- Ϫϱ 1 4 mizes this ratio. In terms of the inner product ͑5.63͒, we have where ␮2 ͑A,Q˜ ͒2 2 SNR2 Ϸ24 . ͑5.80͒ 3H0 ª ␴2 ˜ ˜ H ͑f͒ϭ ͉f͉Ϫ3⍀ ͉͑f͉͒␥ ͉͑f͉͒, ͑5.75͒ ͑Q,Q͒ ij 20␲2 gw ij As we saw already in Secs. III C and V A, such a ratio of and ␥ij(f) denotes the overlap reduction function between inner products is maximized by choosing the ij detector pair. ͓See Eqs. ͑3.58͒ and ͑3.59͒.͔ Substituting Q˜ ͑ f ,...,f ͒ϭ␭A͑f ,...,f ͒, ͑5.81͒ Eq. ͑5.74͒ into ͑5.73͒ yields 1 4 1 4 where ␭ is an arbitrary ͑real͒ overall normalization constant. ˜ 1 For this choice of Q( f 1 ,...,f4), the value of the squared A͑ f 1 ,...,f4͒ª signal-to-noise ratio is given by P1͉͑f1͉͒•••P4͉͑f4͉͒ SNR2 Ϸ24͑A,A͒. ͑5.82͒ ϱ ϱ optimal ϫ 2 ͵ df͵ dfЈ H12͑ f ͒H34͑ f Ј͒ To find an explicit expression for SNRoptimal , we substi- Ϫϱ Ϫϱ tute Eq. ͑5.79͒ into the RHS of Eq. ͑5.82͒ and expand the T/2 product of A( f 1 ,...,f4) with itself. This leads to nine dif- 4 Ϫi2␲ft Ϫi2␲f t ϫ ͵ d te 11•••e 4 4 ferent terms: three diagonal ͑i.e., squared͒ terms and six off- ϪT/2 diagonal terms. The diagonal terms are given by i2␲f͑t Ϫt ͒ i2␲fЈ͑t Ϫt ͒ ϫe 1 2 e 3 4 ϩ13,24ϩ14,23, ϱ 2 H12͑ f 1͒ diagonal termsϭ24 d4 f ͑5.76͒ ͵ Ϫϱ P1͉͑f 1͉͒P2͉͑f 2͉͒ where 13,24 and 14,23 denote the analogous terms with the H2 f 2 34͑ 3͒ appropriate interchange of detector indices 1,...,4.Since ϫ␦T͑ f 1ϩ f 2͒ P3͉͑f 3͉͒P4͉͑f 4͉͒ T/2 2 i2␲ft ϫ␦T͑ f 3ϩ f 4͒ϩ13,24ϩ14,23. ͑5.83͒ ␦T͑ f ͒ϭ dt e ͑5.77͒ ͵ϪT/2 A typical off-diagonal term is given by see Eq. 3.27 , we can explicitly integrate over the time vari- ͓ ͔ ϱ 1 4 4 ables t1 ,...,t4: off-diagonal termϭ2 d f ͵Ϫϱ P1͉͑f 1͉͒•••P4͉͑f4͉͒ 1 ϫ A͑f1,...,f4͒ª H12͑ f 1͒H34͑ f 3͒H13͑ f 1͒H24͑ f 2͒ P1͉͑f1͉͒•••P4͉͑f4͉͒ ϫ␦ ͑ f ϩ f ͒␦ ͑ f ϩ f ͒ ϱ ϱ T 1 2 T 3 4 ϫ df dfЈH12͑ f ͒H34͑ f Ј͒ ϫ f ϩ f f ϩ f . 5.84 ͵Ϫϱ ͵Ϫϱ ␦T͑ 1 3͒␦T͑ 2 4͒ ͑ ͒

ϫ␦ ͑ f Ϫ f ͒␦ ͑ f ϩ f ͒␦ ͑ f Ϫ f Ј͒ Let us evaluate each of these terms separately. First, for T 1 T 2 T 3 the diagonal terms, by approximating one of the two finite- 2 ϫ␦T͑ f 4ϩ f Ј͒ϩ13,24ϩ14,23. ͑5.78͒ time delta functions in each of the factors ␦T( f 1ϩ f 2) and

102001-27 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

2 ␦T( f 3ϩ f 3) in Eq. ͑5.83͒ by ordinary Dirac delta functions, and by evaluating the others at f 1ϩ f 2ϭ0 and f 3ϩ f 4ϭ0, we eliminate two of the integrations, and introduce a factor of T2:

ϱ 2 ϱ 2 H12͑ f ͒ H34͑ f Ј͒ diagonal termsϷ24T2 df dfЈ ϩ13,24ϩ14,23 ͑5.85͒ ͵ P f P f ͵ Ϫϱ 1͉͑ ͉͒ 2͉͑ ͉͒ Ϫϱ P3͉͑f Ј͉͒P4͉͑f Ј͉͒

3H2 4 ϱ ␥2 ͑ f ͒⍀2 ͑ f ͒ ϱ ␥2 ͑ f ͒⍀2 ͑ f ͒ 0 2 12 ͉ ͉ gw ͉ ͉ 34 ͉ Ј͉ gw ͉ Ј͉ ϭ 2 T df dfЈ ͩ 10␲ ͪ ͵Ϫϱ 6 ͵Ϫϱ 6 f P1͉͑f͉͒P2͉͑f͉͒ f Ј P3͉͑f Ј͉͒P4͉͑f Ј͉͒ ϩ13,24ϩ14,23 ͑5.86͒

ϭ͑12͒SNR2 ͑34͒SNR2ϩ͑13͒SNR2 ͑24͒SNR2ϩ͑14͒SNR2 ͑23͒SNR2, ͑5.87͒ where (ij)SNR2 denotes the squared signal-to-noise ratios for the optimally filtered cross-correlation signal for the ij detector pair, which we derived in Sec. III C. For the off-diagonal term ͑5.84͒, we can again approximate ␦T( f 1ϩ f 2) and ␦T( f 3ϩ f 4) by ordinary Dirac delta functions to obtain

ϱ ϱ 1 4 off-diagonal termϷ2 ͵ df1͵ df3 ͑5.88͒ Ϫϱ Ϫϱ P1͉͑f1͉͒P2͉͑f1͉͒P3͉͑f3͉͒P4͉͑f3͉͒

2 ϫH12͑ f 1͒H34͑ f 3͒H13͑ f 1͒H24͑Ϫ f 1͒␦T͑ f 1ϩ f 3͒. ͑5.89͒

By further approximating one of the finite-time delta func- detector outputs si(ti), while the cross-correlation signal val- 2 (ij) tions in ␦T( f 1ϩ f 3) by an ordinary Dirac delta function, and ues S are quadratic in the detector outputs. ͓See, e.g., Eq. by evaluating the other at f 1ϩ f 3ϭ0, we eliminate one more ͑3.45͒.͔ integration and introduce a factor of T: Given Eq. ͑5.91͒, we can now ask the question, what is the minimum value of ⍀0 required to detect a stochastic off-diagonal term gravity-wave signal 95% of the time, from data obtained via a 4-detector correlation experiment? Since ϱ H12͑ f ͒H34͑ f ͒H13͑ f ͒H24͑ f ͒ Ϸ24T ͵ df . Ϫϱ P1͉͑f͉͒P2͉͑f͉͒P3͉͑f͉͒P4͉͑f͉͒ T/2 4 ͑5.90͒ S ª d ts1͑t1͒•••s4͑t4͒Q͑t1 ,...,t4͒ ͑5.92͒ ͵ϪT/2 But since this term grows like T, while the diagonal terms grow like T2, we can ͑for large observation times͒ ignore the ϱ contribution of the off-diagonal terms to the optimal signal- 4 ˜ ˜ ˜ 40 ϭ d fs1͑f1͒•••s4͑f4͒Q*͑f1 ,...,f4͒ ͑5.93͒ to-noise ratio. The final result is thus ͵Ϫϱ

2 ͑12͒ 2 ͑34͒ 2 ͑13͒ 2 ͑24͒ 2 SNRoptimalϷ SNR SNR ϩ SNR SNR ͑14͒ 2 ͑23͒ 2 is, effectively, a sum ͑over f 1 ,...,f4) of a large number of ϩ SNR SNR . ͑5.91͒ statistically independent random variables ͓products of the Fourier amplitudes ˜s ( f ), which are correlated only when Note that the optimal signal-to-noise ratio for the 4-detector i i ͉f iϪ f j͉Ͻ1/T], the central limit theorem guarantees that S correlation is quadratic in the signal-to-noise ratios for the will be well-approximated by a Gaussian random variable. individual ij detector pairs. This is because the 4-detector Thus, the statistical analysis of Sec. IV is valid for the correlation signal S, given by Eq. ͑5.51͒, is quartic in the 4-detector correlation as well. In particular, for a false alarm rate ␣ϭ0.05 and for a detection rate ␥ϭ0.95, the minimum value of ⍀0 for the 4-detector correlation is determined by 40This expression should be compared with setting the signal-to-noise ratio after n observation periods equal to 2ϫ1.65 ͓see Eq. ͑4.36͔͒. For a stochastic back- SNR2 ϭ͑12͒SNR2ϩ͑13͒SNR2ϩ ϩ͑34͒SNR2, optimal ••• ground having a constant frequency spectrum ⍀gw( f )ϭ⍀0 and for a total observation time T nT, the squared which is the squared signal-to-noise ratio that we found in the pre- tot ª vious section for the optimal combination of data from multiple signal-to-noise ratio for the optimally filtered 4-detector cor- detector pairs. ͓See Eq. ͑5.46͒.͔ relation signal S can be written as

102001-28 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

2 4 ϱ 2 T/2 3H0 ␥12͉͑f͉͒ 2N 2 4 2 S d ts1͑t1͒ s2N͑t2N͒Q͑t1 ,...,t2N͒. SNRoptimalϷ⍀0 2 Ttot df ª ͵ ••• ͩ10␲ ͪ ͵Ϫϱ f 6P ͉͑f͉͒P ͉͑f͉͒ ϪT/2 1 2 ͑5.99͒ ϱ 2 ␥34͉͑f Ј͉͒ ϫ dfЈ ͑ii͒ The ‘‘factorization’’ property for 2N Gaussian ran- ͵ 6 41 Ϫϱ f Ј P3͉͑f Ј͉͒P4͉͑f Ј͉͒ dom variables each having zero mean is ϩ13,24ϩ14,23. ͑5.94͒ ͗x1x2•••x2N͘ϭ͗x1x2͗͘x3x4͘•••͗x2NϪ1x2N͘ Setting SNRoptimalϭ2ϫ1.65 and rearranging terms yields ϩall possible permutations. ͑5.100͒ ͑⍀95%,5% ͒Ϫ4͑2ϫ1.65͒2 0 ͉optimal ͑iii͒ The inner product in terms of which 3H2 4 ϱ ␥2 ͉͑f͉͒ 0 2 12 ˜ 2 1 2N ˜ ˜ Ϸ 2 Ttot df ␮ϭ͑A,Q͒ and ␴ ϭ͑ 2 ͒ ͑Q,Q͒ ͑5.101͒ ͩ 10␲ ͪ ͵Ϫϱ 6 f P1͉͑f͉͒P2͉͑f͉͒ is given by ϱ 2 ␥34͉͑f Ј͉͒ ϫ dfЈ ϩ13,24ϩ14,23, ͵ 6 ϱ Ϫϱ f Ј P3͉͑f Ј͉͒P4͉͑f Ј͉͒ 2N ͑A,B͒ ª ͵ d fP1͉͑f1͉͒•••P2N͉͑f2N͉͒ ͑5.95͒ Ϫϱ ϫA*͑f ,...,f ͒B͑f ,...,f ͒, which can also be written in terms of the minimum 1 2N 1 2N (ij) 95%,5% values ⍀0 for the individual ij detector pairs: ͑5.102͒

95%,5% Ϫ2 where A( f 1 ,...,f2N) and B( f 1 ,...,f2N) are any two ͑⍀0 ͉optimal͒ complex-valued functions of 2N variables. ͑12͒ 95%,5% Ϫ2 ͑34͒ 95%,5% Ϫ2 Ϸ2ϫ1.65͓͑ ⍀0 ͒ ͑ ⍀0 ͒ ͑iv͒ The optimal filter function Q(t1 ,...,t2N) is given in the frequency domain by ϩ13,24ϩ14,23͔1/2. ͑5.96͒ ˜ Q͑ f 1 ,•••,f2N͒ϭ␭A͑f1 ,...,f2N͒, ͑5.103͒ How does this minimum value of ⍀0 compare with those found in previous sections for a single 2-detector correlation where and for the optimal combination of data from multiple detec- tor pairs? First, from Eq. ͑5.95͒ we see immediately that H12͑ f 1͒ A͑ f 1 ,...,f2N͒ϭ ␦T͑ f 1ϩ f 2͒ 95%,5% Ϫ1/2 P1͉͑f 1͉͒P2͉͑f 2͉͒ ⍀0 ͉optimalϳTtot . ͑5.97͒ H34͑ f 3͒ ϫ ␦ ͑ f ϩ f ͒ This dependence on the total observation time Ttot is the P ͑ f ͒P ͑ f ͒ T 3 4 (ij) 95%,5% 3 ͉ 3͉ 4 ͉ 4͉ same as that for ⍀0 for a single detector pair and for the optimal combination of data from multiple detector pairs: H2NϪ1,2N͑ f 2NϪ1͒ ϫ •••P ͑ f ͒P ͑ f ͒ 95%,5% Ϫ2 ͑12͒ 95%,5% Ϫ2 ͑13͒ 95%,5% Ϫ2 2NϪ1 ͉ 2NϪ1͉ 2N ͉ 2N͉ ͑⍀0 ͉optimal͒ ϭ͑ ⍀0 ͒ ϩ͑ ⍀0 ͒ ϫ␦T͑ f 2NϪ1ϩ f 2N͒ ͑34͒ 95%,5% Ϫ2 ϩ•••ϩ͑ ⍀0 ͒ . ͑5.98͒ ϩall possible permutations. ͑5.104͒ ͓See Eq. ͑5.50͒.͔ Thus, by correlating 2, 4 ͑or even 2N) detectors, one does not change the general dependence of the ͑v͒ The squared signal-to-noise ratio for the optimally fil- tered 2N-detector correlation is given by minimum value of ⍀0 on the total observation time Ttot . Ϫ1/2 However, the numerical factors multiplying T differ 2 tot SNR Ϸ͑12͒SNR2 ͑34͒SNR2 ͑2NϪ1,2N͒SNR2 from one another. In fact, it is fairly easy to show that optimal ••• 95%,5% ⍀0 ͉optimal for the single 4-detector correlation is always ϩall possible permutations. ͑5.105͒ greater than that for the optimal combination of data from multiple detector pairs. Thus, in theory, it is better to opti- ͑vi͒ The minimum value of ⍀0 required to detect a sto- mally combine data from multiple detector pairs than to op- chastic gravity-wave signal 95% of the time, with a false timally filter a single 4-detector correlation. Table V in Sec. alarm rate equal to 5%, from data obtained via a 2N-detector 95%,5% 2 correlation experiment, is given by VI D lists values of ⍀0 ͉optimalh100 for the 4-detector cor- relations taken from the five major interferometers: LIGO- WA, LIGO-LA, VIRGO, GEO-600, and TAMA-300. Finally, to conclude this section, we rewrite the key equa- 41The number of terms on the RHS of the ‘‘factorization’’ equa- tions derived above for the general case of 2N detectors: tion is given by (2NϪ1)(2NϪ3)•••1. For Nϭ1,2,3,..., this ͑i͒ The 2N-detector correlation signal S is defined by corresponds to 1,3,15,... terms.

102001-29 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

95%,5% Ϫ2 ͑2NϪ2͒/N produces a correlation between the two detectors which ͑⍀0 ͉optimal͒ ϭ͑2ϫ1.65͒ mimics the effect of a stochastic gravity-wave background. ͑12͒ 95%,5% Ϫ2 ͑34͒ 95%,5% Ϫ2 ϫ͓͑ ⍀0 ͒ ͑ ⍀0 ͒ It is straightforward to determine the point at which a

͑2NϪ1,2N͒ 95%,5% Ϫ2 non-vanishing cross spectrum C( f ) will significantly inter- ϫ•••͑ ⍀0 ͒ fere with a stochastic background search. Such interference ϩall possible permutations͔1/N, will not take place if the correlation arising from the cross spectrum of detector noise is significantly less than that aris- ͑5.106͒ ing from the stochastic gravity-wave background:

(ij) 95%,5% ͗n ͑t͒n ͑tЈ͒͘Ӷ͗h ͑t͒h ͑tЈ͒͘ for ͉tϪtЈ͉Շd/c, where ⍀0 are the analogous quantities for the ij 1 2 1 2 detector pairs. ͑5.108͒ where d/c is the light travel time between the two sites. E. Correlated detector noise Making use of Eqs. ͑5.107͒ and ͑5.74͒, we see that correlated detector noise will not impair a stochastic background search We have shown how to carry out an experimental search if for a stochastic background of gravitational radiation, by cor- relating the outputs of widely separated detectors. Our analy- 1 3H2 0 Ϫ3 sis assumed that any correlation between the two outputs ͉C͑ f ͉͒Ӷ 2 ͉f͉ ⍀gw͉͑f͉͒␥͉͑f͉͒ ͑5.109͒ arises only from a stochastic gravity-wave background. In 2 20␲ this section, we address the validity of this assumption, and over the range of frequencies included in the optimal filter look at possible sources of instrumental contamination that Q( f ). For example, for the initial LIGO detectors, this range give rise to a correlated signal between the separated detec- of frequencies is from about 40 Hz to 300 Hz, and in a 4 tors. Any source of correlated environmental noise or inter- month search, the expected level of sensitivity is about ference in the separated detectors mimics the correlation aris- ⍀ h2 ϳ10Ϫ6; so the RHS is ϳ3ϫ10Ϫ49h2 sec. Thus, in ing from a stochastic background; so it is important to 0 100 100 order that correlated sources of noise do not interfere with understand the order-of-magnitude effects of any potential initial LIGO’s 4-month stochastic background search, one sources of such correlated noise. must have This subject has already been considered in some detail, both in the the published paper of Christensen ͓6͔ and, in Ϫ49 2 ͉C͑ f ͉͒Ӷ3ϫ10 h100 sec for 40 HzϽ f Ͻ300 Hz. more detail, in Chap. 7 of his Ph.D. thesis ͓24͔. In the thesis ͑5.110͒ work, the following sources of correlated detector noise are analyzed: This limit can be stated in an interesting way. If we compare ͑i͒ Seismic noise, whose effects on initial LIGO are mini- the allowable cross spectrum C( f ) with the intrinsic noise mal. power spectrum P( f )ϳ10Ϫ45 sec in each detector, we see ͑ii͒ Fluctuations in the residual gas ͑for two interferom- that they differ by about four orders of magnitude. Thus, in eters sharing a common vacuum system such as the order that correlated noise sources do not interfere with a LIGO-WA site͒. The effects on initial LIGO are minimal, 4-month long stochastic background search for initial LIGO, and of course there is no effect for separated detectors that the correlated sources of noise must not contribute more do not share a common vacuum envelope. than 1% of the motion of the test masses, in the frequency ͑iii͒ Acoustic noise, whose effects on LIGO are minimal. range from 40 Hz to 300 Hz. ͑iv͒ Cosmic ray showers, whose effects on LIGO are Essentially the same limit on correlated noise can be writ- minimal. ten in another fashion. If we make use of Eq. ͑3.52͒, the ͑v͒ Magnetic field fluctuations, whose effects on initial contribution of correlated noise to the expected mean of the LIGO might be significant. signal can be written as In this section, we derive a general formalism for calcu- lating the effects of such correlated detector noise, and illus- 1 ϱ 2 2 ˜ trate this for the case of correlated magnetic fields, which ␮correlated noiseϭ T df C͑f͒Q͑f͒ . ͑5.111͒ ͯ2 ͵Ϫϱ ͯ Christensen concluded would be the most significant source of correlated fluctuations at two widely separated sites. Requiring that this be smaller than the magnitude of the vari- Although these ideas can be generalized to multiple site ance ␴2 ͓see Eq. ͑3.68͔͒ leads immediately to the condition locations, for simplicity we present only the two-site case. that the correlated noise will not interfere with a stochastic Correlated noise in two detectors can be described by the background search over an observation time T if cross-spectral function C( f ) defined by P ͑ f ͒P ͑ f ͒ 1 ϱ 2 1 2 i2␲f͑tϪtЈ͒ ͉C͑ f ͉͒ Ӷ . ͑5.112͒ ͗n1͑t͒n2͑tЈ͒͘ϭ df e C͑f͒. ͑5.107͒ Tf 2 ͵Ϫϱ As before, the contribution of correlated noise to the motion Because the LHS is real, the cross-spectrum satisfies C( f ) of the interferometer must be smaller than the intrinsic de- ϭC*(Ϫ f ). If this cross spectrum is non-vanishing, then it tector noise motion by a factor of (Tf)Ϫ1/2. For an observa-

102001-30 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001 tion time T of 4 months ͑i.e., 107 sec) and frequencies f of the magnetic field will not correlate between widely sepa- ϳ100 Hz, this gives the same 1% bound as before. rated sites; the global part, however, is likely to be highly One can also give a precise formula showing the effects correlated. Studies in the Caltech 40-m prototype laboratory of correlated noise sources on the expected value ␮ of the ͓25͔ have shown that the spectrum of ambient magnetic signal. Making use of Eqs. ͑5.11͒ and ͑3.57͒, we can express fields is of order 10Ϫ7 G/ͱHz, but most of this field is local. the ratio of the signal mean arising from the correlated noise Christensen reports in his thesis on a number of studies that to that arising from the stochastic background as show that the global magnetic fields are well-modeled above 20 Hz by a power-law spectrum:

␮correlated noise f Ϫ0.88 ͯ␮stochastic backgroundͯ P ͑ f ͒ϭA ͑5.115͒ B ͩ 40 Hzͪ ϱ ˜ 2 df C͑f͒Q͑f͒ 10␲ ͵Ϫϱ with Aϳ1.2ϫ10Ϫ17 G2/Hz during magnetically noisy peri- ϭ . 3H2 ϱ ods such as thunderstorms and Aϳ1.8ϫ10Ϫ19 G2/Hz dur- 0 Ϫ3 ˜ ͯ͵ df͉f͉ ⍀gw͉͑f͉͒␥͉͑f͉͒Q͑ f ͒ͯ ing magnetically quiet periods. Separate measurements have Ϫϱ shown that these fields have a coherence of order rϳ1/2 in ͑5.113͒ the frequency range of interest, over widely separated ͑al- most antipodal͒ points on the Earth’s surface. These highly correlated global fields are the main concern here. This formula allows us to precisely determine the effect of In order to reduce the effects of external magnetic fields any correlated source of noise on the signal value starting on the test masses and optics, the four magnets on each optic from a model or measurement of the correlation spectrum or test mass are arranged to approximately cancel both the C( f ). In particular, if we assume that the stochastic gravity- dipole and quadrupole parts of the magnetic field. The mag- wave background has a constant frequency spectrum nitude of the resulting force on a test mass may be described ⍀ ( f )ϭ⍀ , then gw 0 by ␮B ⌬ ⌬2 ␮correlated noise ⍀correlated noise limit Fϭ ⑀ ϩ ⑀ ϩ ⑀ ϩ . ͑5.116͒ ϭ , ͑5.114͒ l 0 l 1 l2 2 ••• ͯ␮ ͯ ⍀ ͫ ͬ stochastic background 0 Here, ␮ is the magnetic dipole moment of one of the mag- nets, B is the magnitude of the ambient magnetic field, l is where ⍀correlated noise limit is ͑by definition͒ the smallest value the length scale over which the magnetic field is varying in of ⍀0 that can be observed before the effects of correlated instrument noise interfere with the measurement. the vicinity of the test masses and optics, and ⌬ is the sepa- The effects of a given source of correlated noise can be ration between the magnets on the test masses and optics. modeled more precisely. Here, we work through one ex- The quantities ⑀0 ,⑀1 , . . . are the fractional difference be- ample: the effects of correlated magnetic field fluctuations. tween the dipole, quadrupole, . . . moments of the different Christensen’s work ͓6,24͔ concludes that these are the most magnets ͑which are not perfectly matched͒. For the initial likely environmental source of correlated noise between two LIGO detectors, the preliminary design values of these quan- 2 sites. tities are approximately ␮ϭ0.11 A m /c,⑀0ϭ0.05, The LIGO interferometers use small magnets to steer and lϭ6 cm,⌬ϭ15 cm. This leads to a force push the optical elements ͑i.e., mirrors and beam splitters͒. These magnets are an integral part of the Length Sensing and Fϭ␬B ͑5.117͒ Control ͑LSC͒ system. Forces are applied to these magnets with electromagnetic coils, and they are part of the servo loop that uses modulation techniques to lock and monitor the on the test masses, with ␬ϳ0.1 dyn/G ͓25͔. This force ac- path length difference between the test masses. External celerates the test mass, producing an equivalent strain magnetic fields, present in the environment, exert forces on these magnets thus constituting one of the different sources ˜F͑ f ͒ ˜ of instrument noise. n͑ f ͒ϳ , ͑5.118͒ M͑2␲ f ͒2L The external magnetic field is of particular concern, be- cause it propagates at the speed of light, and therefore can give rise to correlations between sites on the time scale d/c. where Mϭ10 kg is the mass of the optic, and Lϭ4kmis The magnetic field in the laboratory consists of two parts: a the length of the arm. This gives rise to a cross spectrum ‘‘local’’ part and a ‘‘global’’ part. The local part is the mag- r␬2 netic field arising from the instrumentation, wiring, and C͑ f ͒ϭ P ͑ f ͒, ͑5.119͒ M 2͑2␲ f ͒4L2 B power lines within the laboratory; the global part comes from Schuman resonances of the Earth and the ionosphere, and where r is the coherence. Evaluating the integral in Eqs. lightning strikes over the surface of the Earth. The local part ͑5.113͒ and ͑5.114͒ we find that the limits on ⍀0 are

102001-31 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

FIG. 10. A log-log plot of the predicted noise power spectra for FIG. 13. A log-log plot of the predicted noise power spectrum the initial and advanced LIGO detectors. for the TAMA-300 detector.

⍀correlated noise limit 10Ϫ7 during magnetically ‘‘noisy’’ times, ϭ ͭ 1.5ϫ10Ϫ9 during magnetically ‘‘quiet’’ times. ͑5.120͒ Our conclusion is that for the initial LIGO design ͑where sensitivities are on the order of 6ϫ10Ϫ6 for 4 months of observation͒, magnetic field induced correlations are not a concern. However, for advanced LIGO ͑where sensitivities are on the order of 6ϫ10Ϫ11 for 4 months of observation͒, the magnets must be eliminated from the design or they will they will significantly constrain the measurements of ͑or lim- its placed on͒⍀0.

VI. NUMERICAL DATA This section consists of a series of graphs and tables con- FIG. 11. A log-log plot of the predicted noise power spectrum taining numerical data for the five major interferometers: ͑i͒ for the VIRGO detector. Hanford, WA LIGO detector ͑LIGO-WA͒, ͑ii͒ Livingston, LA LIGO detector ͑LIGO-LA͒, ͑iii͒ VIRGO detector

FIG. 12. A log-log plot of the predicted noise power spectrum FIG. 14. A log-log plot of the predicted noise power spectra for for the GEO-600 detector. all the major interferometers.

102001-32 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

FIG. 15. A log-log plot of the predicted noise power spectra for FIG. 17. The overlap reduction function ␥( f ) for the LIGO-LA the ‘‘enhanced’’ LIGO detectors, showing the probable evolution of detector and the other major interferometers. the detector design over the next decade.

͑VIRGO͒, ͑iv͒ GEO-600 detector ͑GEO-600͒, and ͑v͒ TAMA-300 detector ͑TAMA-300͒. These data were derived from published site location and orientation information and detector noise power spectra design goals ͓26͔, using the stochastic background data analysis routines contained in GRASP ͓27͔. ͑See Sec. VII for more information about the computer code that we wrote to perform these calculations.͒

A. Noise power spectra Figures 10–13 show the predicted noise power spectra for the initial and advanced LIGO detectors, and for the VIRGO, GEO-600, and TAMA-300 detectors. Figure 14 displays all of the noise power spectra on a single graph. Figure 15 shows the predicted noise power spectra for the ‘‘enhanced’’ FIG. 18. The overlap reduction function ␥( f ) for the VIRGO LIGO detectors, which track the projected performance of detector and the other major interferometers. Note that the VIRGO the LIGO detector design over the next decade. The data for and GEO-600 detectors are sensitive to almost orthogonal polariza- tions.

FIG. 16. The overlap reduction function ␥( f ) for the LIGO-WA FIG. 19. The overlap reduction function ␥( f ) for the GEO-600 detector and the other major interferometers. Note that the overlap detector and the other major interferometers. Note that the VIRGO reduction functions for the more distant detectors have their first and GEO-600 detectors are sensitive to almost orthogonal polariza- zero at lower frequencies than those for the more nearby detectors. tions.

102001-33 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001

FIG. 20. The overlap reduction function ␥( f ) for the TAMA- FIG. 23. The optimal filter function Q˜ ( f ) for the initial 300 detector and the other major interferometers. LIGO-WA and GEO-600 detectors, for a stochastic background having a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function is normalized to have maximum magnitude of unity.

FIG. 21. The optimal filter function Q˜ ( f ) for the initial LIGO-WA and initial LIGO-LA detectors, for a stochastic back- ground having a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The FIG. 24. The optimal filter function Q˜ ( f ) for the initial optimal filter function is normalized to have maximum magnitude LIGO-WA and TAMA-300 detectors, for a stochastic background of unity. having a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function is normalized to have maximum magnitude of unity.

FIG. 22. The optimal filter function Q˜ ( f ) for the initial FIG. 25. The optimal filter function Q˜ ( f ) for the initial LIGO-WA and VIRGO detectors, for a stochastic background hav- LIGO-LA and VIRGO detectors, for a stochastic background hav- ing a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter ing a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function is normalized to have maximum magnitude of unity. function is normalized to have maximum magnitude of unity.

102001-34 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

FIG. 26. The optimal filter function Q˜ ( f ) for the initial FIG. 29. The optimal filter function Q˜ ( f ) for the VIRGO and LIGO-LA and GEO-600 detectors, for a stochastic background hav- TAMA-300 detectors, for a stochastic background having a con- ing a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter stant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function function is normalized to have maximum magnitude of unity. is normalized to have maximum magnitude of unity.

the noise power spectra displayed in all of these figures were taken from the published design goals ͓26͔.

B. Overlap reduction functions

Figures 16–20 show the overlap reduction functions ␥( f ) for different detector pairs.

C. Optimal filter functions

Figures 21–30 show the optimal filter functions Q˜ ( f ) for different detector pairs, for a stochastic background having a constant frequency spectrum ⍀ ( f )ϭ⍀ . The optimal fil- ˜ gw 0 FIG. 27. The optimal filter function Q( f ) for the initial ter functions are normalized to have maximum magnitude of LIGO-LA and TAMA-300 detectors, for a stochastic background unity. having a constant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function is normalized to have maximum magnitude of unity.

FIG. 28. The optimal filter function Q˜ ( f ) for the VIRGO and FIG. 30. The optimal filter function Q˜ ( f ) for the GEO-600 and GEO-600 detectors, for a stochastic background having a constant TAMA-300 detectors, for a stochastic background having a con- frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function is stant frequency spectrum ⍀gw( f )ϭ⍀0 . The optimal filter function normalized to have maximum magnitude of unity. is normalized to have maximum magnitude of unity.

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TABLE I. Theoretical signal-to-noise ratios after 4 months of observation, for the optimally-filtered cross-correlation signal between different detector pairs, for a stochastic background of gravitational radiation Ϫ6 Ϫ2 having a constant frequency spectrum ⍀gw( f )ϭ⍀0ϭ6ϫ10 h100 .

LIGO-WA LIGO-LA VIRGO GEO-600 TAMA-300

LIGO-WA — 3.45 1.74 5.09ϫ10Ϫ1 6.12ϫ10Ϫ2 LIGO-LA 3.45 — 2.10 7.66ϫ10Ϫ1 9.16ϫ10Ϫ2 VIRGO 1.74 2.10 — 1.56 9.14ϫ10Ϫ2 GEO-600 5.09ϫ10Ϫ1 7.66ϫ10Ϫ1 1.56 — 1.32ϫ10Ϫ2 TAMA-300 6.12ϫ10Ϫ2 9.16ϫ10Ϫ2 9.14ϫ10Ϫ2 1.32ϫ10Ϫ2 —

D. Signal-to-noise ratios and sensitivities discrete time series and Fourier transforms by their discrete Table I contains the values of the theoretical signal-to- frequency counterparts. The discrete data can then be pro- noise ratios after 4 months ͑i.e., 107 sec) of observation, for cessed by computer code that takes the appropriate FFTs, the optimally filtered cross-correlation signal between differ- constructs the optimal filters, whitens and windows the data, ent detector pairs, for a stochastic background of gravita- etc. We have written a number of functions ͑in ANSI-C͒ to do tional radiation having a constant frequency spectrum precisely this. These functions constitute part of a general- Ϫ6 Ϫ2 purpose data analysis package for gravitational-wave detec- ⍀gw( f )ϭ⍀0ϭ6ϫ10 h100 . Table II contains the minimum 2 tion, called GRASP ͑Gravitational Radiation Analysis and values of ⍀0h100 for 4 months of observation, for a false alarm rate equal to 5%, and for a detection rate equal to 95%, Simulation Package͓͒27͔. In this section, we describe a com- for cross-correlation measurements between different detec- puter simulation ͑made up of these functions͒ that mimics tor pairs. Tables III and IV contain the minimum values the generation and detection of a simulated stochastic 2 ⍀0h100 for 4 months of observation, for a false alarm rate gravity-wave signal in the presence of simulated detector equal to 5%, and for a detection rate equal to 95%, for the noise. Documentation and further information about the code optimal combination of cross-correlation measurements be- can be found in the GRASP user’s manual. tween multiple detector pairs, taken from all possible triples and quadruples of the five major interferometers. Table V 2 contains the minimum values of ⍀0h100 for 4 months of A. Purpose observation, for a false alarm rate equal to 5%, and for a detection rate equal to 95%, for optimally filtered 4-detector The main reason for writing the computer simulation was correlations. ͓Note that the calculation of the signal-to-noise to verify many of the theoretical calculations that were de- ratios and minimum values of ⍀0 assumes that the magni- rived in the previous sections. Specifically, we wanted to see tude of the noise intrinsic to the detectors is much larger than if the theoretically predicted signal-to-noise ratio SNR, for a the stochastic gravity-wave background. This corresponds to stochastic background of gravitational radiation having a Eq. ͑3.75͒ in the text.͔ constant frequency spectrum ⍀gw( f )ϭ⍀0 , would agree with an ‘‘experimentally’’ determined signal-to-noise ratio SNR produced by the simulation. If the theoretical and experimen- VII. COMPUTER SIMULATION tal signal-to-noise ratios agreed, we could be confidentd that In Secs. III–V, we described the data analysis and optimal the theoretical calculations were correct. If they did not signal processing required for the detection of a stochastic agree, we would know that something—either a theoretical background of gravitational radiation. This analysis was in calculation or a technical issue related to the simulation terms of continuous time functions and their associated Fou- itself—needed further investigation. rier transforms. But, in reality, when one performs the actual The theoretical and experimental signal-to-noise ratios data analysis, continuous time functions will be replaced by were said to be in agreement if the relative error defined by

2 TABLE II. Minimum values of ⍀0h100 for 4 months of observation, for a false alarm rate equal to 5%, and for a detection rate equal to 95%, for cross-correlation measurements between different detector pairs.

LIGO-WA LIGO-LA VIRGO GEO-600 TAMA-300

LIGO-WA — 5.74ϫ10Ϫ6 1.14ϫ10Ϫ5 3.89ϫ10Ϫ5 3.24ϫ10Ϫ4 LIGO-LA 5.74ϫ10Ϫ6 — 9.45ϫ10Ϫ6 2.58ϫ10Ϫ5 2.16ϫ10Ϫ4 VIRGO 1.14ϫ10Ϫ5 9.45ϫ10Ϫ6 — 1.27ϫ10Ϫ5 2.17ϫ10Ϫ4 GEO-600 3.89ϫ10Ϫ5 2.58ϫ10Ϫ5 1.27ϫ10Ϫ5 — 1.50ϫ10Ϫ3 TAMA-300 3.24ϫ10Ϫ4 2.16ϫ10Ϫ4 2.17ϫ10Ϫ4 1.50ϫ10Ϫ3 —

102001-36 DETECTING A STOCHASTIC BACKGROUND OF... PHYSICAL REVIEW D 59 102001

2 2 TABLE III. Minimum values of ⍀0h100 for 4 months of obser- TABLE IV. Minimum values of ⍀0h100 for 4 months of obser- vation, for a false alarm rate equal to 5%, and for a detection rate vation, for a false alarm rate equal to 5%, and for a detection rate equal to 95%, for the optimal combination of cross-correlation mea- equal to 95%, for the optimal combination of cross-correlation mea- surements between multiple detector pairs, taken from all possible surements between multiple detector pairs, taken from all possible triples of the five major interferometers. quadruples of the five major interferometers.

95%,5% 2 95%,5% 2 Detectors ⍀0 ͉optimalh100 Detectors ⍀0 ͉optimalh100 LIGO-WA, LIGO-LA, VIRGO 4.50ϫ10Ϫ6 LIGO-WA, LIGO-LA, VIRGO, GEO-600 4.17ϫ10Ϫ6 LIGO-WA, LIGO-LA, GEO-600 5.55ϫ10Ϫ6 LIGO-WA, LIGO-LA, VIRGO, TAMA-300 4.50ϫ10Ϫ6 LIGO-WA, LIGO-LA, TAMA-300 5.74ϫ10Ϫ6 LIGO-WA, LIGO-LA, GEO-600, TAMA-300 5.54ϫ10Ϫ6 LIGO-WA, VIRGO, GEO-600 8.28ϫ10Ϫ6 LIGO-WA, VIRGO, GEO-600, TAMA-300 8.28ϫ10Ϫ6 LIGO-WA, VIRGO, TAMA-300 1.14ϫ10Ϫ5 LIGO-LA, VIRGO, GEO-600, TAMA-300 7.27ϫ10Ϫ6 LIGO-WA, GEO-600, TAMA-300 3.86ϫ10Ϫ5 LIGO-LA, VIRGO, GEO-600 7.27ϫ10Ϫ6 LIGO-LA, VIRGO, TAMA-300 9.43ϫ10Ϫ6 the beginning of the simulation program. These parameters LIGO-LA, GEO-600, TAMA-300 2.57ϫ10Ϫ5 are VIRGO, GEO-600, TAMA-300 1.27ϫ10Ϫ5 ͑i͒ the site identification numbers for the two detectors ͑ii͒ the number of time-series data points N to be used when performing the data analysis ͑i.e., FFTs, cross- SNRϪSNR correlations, etc.͒. N should equal an integral power of 2 relative error ͑7.1͒ ª ͯ SNR ͯ ͑iii͒ the sampling period ⌬t of the two detectors ͑Note d that T ª N⌬t is the duration of a single observation period.͒ was less than ͑or approximately equal to͒ the inverse of the ͑iv͒ the constant ⍀0 that defines the stochastic back- 42 theoretical signal-to-noise ratio after n observation periods. ground frequency spectrum: ⍀gw( f )ϭ⍀0 This is the error that one would expect ͑approximately 68% ͑v͒ the total number of runs n that make up the simulation of the time͒ if we approximate the sample variance ␴ˆ 2 by the ͑Note that Ttot ª nT is the duration of the total observation true variance ␴2, and use the fact that the sample mean period.͒ ͑2͒ Using the site identification numbers, obtain site loca- 1 n tion and orientation information, and information about the ␮ˆ S ͑7.2͒ noise power spectrum and whitening filter of each detector. ª n͚ i iϭ1 ͑This information is contained in input data files.͒ 3 Using the site location and orientation information, can itself be thought of as a random variable with mean ␮ ͑ ͒ 2 construct the overlap reduction function ␥( f i) for the two and variance ␴ /n. ͑Recall from Sec. IV that S1 ,S2 ,...,Sn are n statistically independent measurements of the optimally detectors. ͓Here f i ª i/(N⌬t) where iϭ0,1,...,N/2Ϫ1. By filtered cross-correlation signal S, each associated with one convention, we ignore the value of ␥( f ), or any other func- of the n observation periods.͒ Thus, tion of frequency, at or above the Nyquist critical frequency f Nyquist ª 1/(2⌬t).] ␮ˆ ␮Ϯ␴/ͱn ␮ 1 1 ͑4͒ Simulate the generation of a stochastic background of SNR ª Ϸ ϭ Ϯ ϭSNRϮ , ͑7.3͒ gravitational radiation having a constant frequency spectrum ␴ˆ ␴ ␴ ͱn ͱn ⍀gw( f )ϭ⍀0 . This can be done in the frequency domain by using a random number generator to construct ͑complex- whichd implies ˜ ˜ valued͒ Gaussian random variables h1( f i) and h2( f i) having SNRϪSNR 1 zero mean and joint expectation values: relative error ª Ϸ . ͑7.4͒ ͯ SNR ͯ ͱnSNR 3H2 d ˜ ˜ 0 Ϫ3 ͗h1*͑ f i͒h1͑ f j͒͘ϭ 2 T␦ijfi ⍀0, ͑7.5͒ This criterion was satisfied by our simulation runs for both 20␲ the initial and advanced LIGO detector pairs. 2 TABLE V. Minimum values of ⍀0h100 for 4 months of obser- B. Flow chart vation, for a false alarm rate equal to 5%, and for a detection rate equal to 95%, for optimally filtered 4-detector correlations. A ‘‘flow chart’’ for the simulation is as follows: 95%,5% 2 ͑1͒ Input the parameters defining the simulation. This can Detectors ⍀0 ͉optimalh100 be done either interactively or via ‘‘#define’’ statements at LIGO-WA, LIGO-LA, VIRGO, GEO-600 6.49ϫ10Ϫ6 LIGO-WA, LIGO-LA, VIRGO, TAMA-300 2.51ϫ10Ϫ5 LIGO-WA, LIGO-LA, GEO-600, TAMA-300 5.44ϫ10Ϫ5 42The total observation time is T nT, where T is the duration tot ª LIGO-WA, VIRGO, GEO-600, TAMA-300 4.68ϫ10Ϫ5 of a single observation period. For our simulations, T LIGO-LA, VIRGO, GEO-600, TAMA-300 3.84ϫ10Ϫ5 ϭ3.2768 sec.

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2 whitened͒ signalϩnoise data streams output by the two de- 3H0 ͗˜h*͑ f ͒˜h ͑ f ͒͘ϭ T␦ fϪ3⍀ , ͑7.6͒ tectors. ͑Note that steps ͑4͒–͑10͒ make up the signal genera- 2 i 2 j 20␲2 ij i 0 tion part of the simulation.͒ ͑11͒ Test the input data o (t ) and o (t ) to see if they 3H2 1 i 2 i ˜ ˜ 0 Ϫ3 have probability distributions consistent with that of a ͗h*͑ f i͒h2͑ f j͒͘ϭ 2 T␦ijf ⍀0␥͑fi͒. 1 20␲ i Gaussian random variable. If either set fails this test, reject ͑7.7͒ them both, and repeat steps ͑4͒–͑10͒ to obtain new input data o (t ) and o (t ). These are just the discrete frequency versions of Eq. ͑3.56͒ 1 i 2 i ͑12͒ Window the data streams o (t ) and o (t ) in the specialized to the case ( f )ϭ . Note that the Fourier 1 i 2 i ⍀gw ⍀0 time domain, using a Hann window function to reduce side- ˜ ˜ Ϫ3/2 amplitudes h1( f i) and h2( f i) fall off like f . lobe contamination of the corresponding power spectra. ͑5͒ Simulate the generation of the noise intrinsic to the ͑13͒ FFT the windowed data into the frequency domain to detectors, using the information contained in the noise power obtain the corresponding Fourier amplitudes ˜o ( f ) and spectrum data files. This can be done in the frequency do- 1 i ˜ main by using a random number generator to construct o2( f i). ͑14͒ Unwhiten the data in the frequency domain by divid- ͑complex-valued͒ Gaussian random variables ˜n ( f ) and 1 i ing ˜o ( f ) and ˜o ( f ) by the frequency components W˜ ( f ) ˜n ( f ) having zero mean and joint expectation values: 1 i 2 i 1 i 2 i ˜ and W2( f i) of the whitening filters of the two detectors: 1 ˜ ͗˜n*͑ f ͒˜n ͑ f ͒͘ϭ T␦ P ͑f ͒, ͑7.8͒ o1͑ f i͒ 1 i 1 j 2 ij 1 i ˜s ͑ f ͒ , ͑7.15͒ 1 i ª ˜ W1͑ f i͒ 1 ˜ ͗˜n*͑ f ͒˜n ͑ f ͒͘ϭ T␦ P ͑f ͒, ͑7.9͒ o2͑ f i͒ 2 i 2 j 2 ij 2 i ˜s ͑ f ͒ . ͑7.16͒ 2 i ª ˜ W2͑ f i͒ ˜ ˜ ͗n1*͑ f i͒n2͑ f j͒͘ϭ0. ͑7.10͒ ͑15͒ Shift the input data streams o1(ti) and o2(ti) forward in time by T/2, and repeats steps ͑12͒–͑14͒, obtaining an- These are just the discrete frequency versions of Eq. ͑3.64͒. ˜ ˜ other set of Fourier components s1( f i) and s2( f i). Distin- ͑6͒ Construct the Fourier amplitudes guish these two different sets of data with superscripts: (1)˜s (f ),(1)˜s (f ),(2)˜s (f ),(2)˜s (f ). ˜s ͑ f ͒ ˜h ͑ f ͒ϩ˜n ͑ f ͒, ͑7.11͒ 1 i 2 i 1 i 2 i 1 i ª 1 i 1 i (1)˜ (2)˜ (1)˜ ͑16͒ Average s1(fi) and s1(fi) ͓ s2(fi) and (2)˜ ˜ ˜ ˜ ˜ ˜s ͑ f ͒ ˜h ͑ f ͒ϩ˜n ͑ f ͒. ͑7.12͒ s2(fi)] to produce s1( f i) ͓s2( f i)͔. s1( f i) and s2( f i) are 2 i ª 2 i 2 i the Fourier components of the unwhitened time-series data ͑7͒ Whiten the data in the frequency domain by multiply- s1(ti) and s2(ti). ͓Note that the purpose of this averaging is ˜ ˜ ˜ to reduce the variance in the estimation of the spectra ˜s ( f ) ing s1( f i) and s2( f i) by the frequency components W1( f i) 1 i ˜ and ˜s ( f )]. and W2( f i) of the whitening filters of the two detectors: 2 i ˜ ͑17͒ Construct the optimal filter function Q( f i) with the ˜o f ˜s f W˜ f , 7.13 1͑ i͒ ª 1͑ i͒ 1͑ i͒ ͑ ͒ overall normalization constant ␭ chosen so that ␮ϭ⍀0T, using the noise power spectra specified by the input data ˜ ˜ ˜ o2͑ f i͒ ª s2͑ f i͒W2͑ f i͒. ͑7.14͒ files. ˜ ˜ ˜ This multiplication in the frequency domain corresponds ͑18͒ From s1( f i), s2(fi), and Q( f i) calculate the opti- mally filtered cross-correlation signal S corresponding to a to the convolution of s1(ti) and W1(ti) ͓s2(ti) and W2(ti)] in the time domain. ͑Note that the purpose of whitening the single observation period T. ͓Note that steps ͑11͒–͑18͒ make data is to reduce the dynamic range of the corresponding up the signal analysis part of the simulation.͔ power spectra.͒ ͑19͒ Repeat steps ͑4͒–͑18͒ n times, generating a set of ˜ ˜ optimally filtered cross-correlation signal values: ͑8͒ FFT o1( f i) and o2( f i) into the time domain to obtain S1 ,S2 ,...,Sn. the corresponding time series o1(ti) and o2(ti). ͑Here ti ͑20͒ From S1 ,S2 ,...,Sn construct the sample mean ª i⌬t where iϭ0,1,...,NϪ1.) ͑9͒ Repeat steps ͑4͒–͑8͒ to obtain another set of time- 1 n ␮ˆ S ͑7.17͒ series data o1(ti) and o2(ti). Distinguish these two dif- ª n͚ i (1) (1) iϭ1 ferent sets of data with superscripts: o1(ti), o2(ti), (2) (2) and sample variance o1(ti), o2(ti). (1) (2) (1) n ͑10͒ Offset o1(ti) and o1(ti) ͓ o2(ti) and 1 (2)o (t )] by T/2, and combine them with one another ␴ˆ 2 ͑S Ϫ␮ˆ ͒2. ͑7.18͒ 2 i ª nϪ1͚ i and with data left over from the previous observa- iϭ1 tion period to produce a continuous-in-time data set The sample ͑or ‘‘experimental’’͒ signal-to-noise ratio pro- o1(ti)͓o2(ti)͔. o1(ti)and o2(ti) represent the ‘‘raw’’ ͑i.e., duced by the simulation is given by

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␮ˆ These results suggest that both the theoretical signal process- SNR . ͑7.19͒ ing formulas and the implementation of these formulas in a ª ˆ ␴ computer code are correct. But we should not stop here. For ͑21͒ Calculate the theoreticald signal-to-noise ratio, using a example, there are still a number of ways that we can im- discrete frequency approximation to the integral prove the data analysis code before we use it to search for a real stochastic background in the outputs of real interferom- 2 f 2 1/2 3H0 Nyquist ␥ ͑f͒ eters. To conclude this paper, we list some of the desired SNRϭ⍀0 2ͱT 2 df 6 . 10␲ ͫ ͵0 f P1͑f͒P2͑f͒ͬ improvements below: ͑7.20͒ ͑i͒ The first change that we would like to make is to See Eq. ͑3.75͒. ͓Note that since the data are discretely calculate real-time noise power spectra for the detectors, and sampled, we should only integrate up to the Nyquist critical to use this calculated data ͑rather than the information con- tained in the input noise power spectrum data files͒ to con- frequency f Nyquist ª 1/(2⌬t).] ˜ ͑22͒ From SNR and SNR, calculate the relative error struct the optimal filter function Q( f i). ͓See step ͑17͒ in the computer simulation described in Sec. VII B.͔ Since the real- d SNRϪSNR time noise power spectra will change slightly from one mea- relative error . ͑7.21͒ ª ͯ SNR ͯ surement to the next, we could then apply the data analysis d strategy discussed in Sec. V B for nonstationary detector As mentioned in Sec. VII A, this should be compared with noise. the inverse of the theoretical signal-to-noise ratio after n ob- ͑ii͒ In order to obtain accurate real-time noise power spec- servation periods 1/(ͱnSNR). ͓Note that steps ͑20͒–͑22͒ tra for the two detectors, it will probably be necessary to use make up the statistical analysis part of the simulation.͔ more sophisticated spectral estimation techniques. Currently, Note that in order to obtain signal-to-noise ratios on the we use a Hann window to reduce side-lobe contamination, order of 10 after nϭ1600 runs, we needed to use rather and we average two overlapped data sets to reduce the vari- Ϫ3 large values of ⍀0 ͑e.g., 10 for the initial LIGO detectors ance, when forming our estimates of ˜s ( f ) and ˜s ( f ). ͓See Ϫ8 1 i 2 i and 10 for the advanced LIGO detectors͒. These large steps ͑12͒, ͑15͒, and ͑16͒ of the computer simulation.͔ This values meant that expression ͑7.20͒ for the theoretical signal- procedure can be replaced by multitaper spectral estimation to-noise ratio had to be modified to properly take into ac- methods, which use a special set of window functions— 2 count the contributions to the theoretical variance ␴ that are called Slepian tapers—to form spectral estimates of time- due to a large stochastic gravity-wave signal. ͓See Eqs. ͑5.3͒ series data.43 GRASP ͓27͔ contains a modified version of a and ͑5.4͒.͔ Without these modifications, the theoretical and public domain package by Lees and Park ͓30͔ to perform the experimental signal-to-noise ratios would be more likely to multitaper spectral estimation. In addition to providing better disagree. Thus, instead of Eq. ͑7.20͒, we used a ‘‘mixed’’ spectral estimates, multitaper methods also provide nice expression for the theoretical signal-to-noise ratio: techniques for ‘‘spectral line’’ parameter estimation and re- 2 moval. This feature will be extremely useful when analyzing f Nyquist ␥ ͑f͒ ͱ2 df data produced by a real detector. For example, one will be 2 ͵ f6P f͒P f͒ 3H0 0 1͑ 2͑ able to track contamination of a data set by the line harmonic SNRϭ⍀0 2ͱT 2 1/2 . 10␲ fNyquist ␥ ͑f͒ at 300 Hz, and remove a pendulum resonance at, say, 590 df R f ͵ 6 2 2 ͑ ͒ Hz. ͑See GRASP ͓27͔ for more information.͒ ͫ 0 f P1͑f͒P2͑f͒ ͬ ͑7.22͒ ͑iii͒ In addition to being able to identify and to remove 2 ‘‘spectral lines’’ from a real data set, one would also like to This involves Eqs. ͑5.3͒ and ͑5.4͒ for the variance ␴ , but be able to test the data to see if the distribution of sampled the large noise expression ͑3.73͒ for the optimal filter func- values is consistent with normal detector operation. For ex- tion Q˜ ( f ). ample, one might check the input data set to see if it has a probability distribution consistent with that of a Gaussian VIII. CONCLUSION random variable. If the test reveals an exceptionally large In this paper, we derived the optimal signal processing number of ‘‘outlier’’ points, then that particular data set can strategy required for stochastic background searches. We dis- be rejected. ͓See step ͑11͒ of the computer simulation.͔ The cussed signal detection, parameter estimation, and sensitivity GRASP data analysis package already contains a routine that levels from a frequentist point of view. We also discussed performs this Gaussian test. But we would also like a more the complications that arise when one considers ͑i͒ arbitrarily rigorously characterized test that compares the distribution of large stochastic backgrounds, ͑ii͒ non-stationary detector the current data with that during ‘‘normal’’ detector opera- noise, ͑iii͒ multiple detector pairs, and ͑iv͒ correlated detec- tion, which most likely is not Gaussian. tor noise. We explained how we verified some of the theo- ͑iv͒ Finally, even though it will still be a few years before retical calculations by writing a computer simulation that we can analyze real data from any one of the major interfer- mimics the generation and detection of a simulated stochas- tic gravity-wave signal in the presence of simulated detector noise. And we noted that the ‘‘experimental’’ results and 43See the original paper by Thomson ͓28͔ and the text by Percival theoretical predictions agreed to within the expected error. and Walden ͓29͔ for more details.

102001-39 BRUCE ALLEN AND JOSEPH D. ROMANO PHYSICAL REVIEW D 59 102001 ometers, real data from prototypes—like the Caltech 40-m data analysis code is working as expected, before we can interferometer—can be used in computer simulations. For trust it when searching for a real stochastic background in the instance, rather than write a computer simulation ͑like the outputs of real interferometers. one we described in Sec. VII͒ that mimics the generation and detection of a simulated stochastic gravity-wave signal in the ACKNOWLEDGMENTS presence of simulated detector noise, we can write a com- This work has been partially supported by NSF grant puter simulation that mimics the generation and detection of PHY95-07740 to the University of Wisconsin at Milwaukee a simulated stochastic gravity-wave signal in the presence of 44 and NSF grant PHY93-08728 to Northwestern University. real detector noise. The fact that the noise level of a pro- We would like to thank Luca Gammaitoni, Albrecht Rudi- totype interferometer is much larger than that of a major ger, Kenneth Strain, Masa-Katsu Fujimoto, and Rainer interferometer poses no problem; we can simply ‘‘dial-in’’ a Weiss for kindly supplying the numerical data for the noise larger stochastic background signal to be able to detect it in power spectra for the VIRGO, GEO-600, TAMA-300, and the same amount of observation time. Another nice feature ‘‘enhanced’’ LIGO detectors. We would also like to thank of this fake stochastic background and real detector noise Sam Finn for carefully proofreading Sec. IV, and for ex- simulation is that we can address all of the issues ͑i͒–͑iii͒ plaining the differences between the Bayesian and frequen- discussed above in a context where we can still compare tist approaches to data analysis, and Eric Key for bringing ‘‘experimental’’ ͑i.e., simulation͒ performance against theo- the law of the iterated logarithm to our attention. B.A. grate- retical expectations. We must be totally convinced that the fully acknowledges the LIGO visitors program for support under NSF grant PHY96-03177, and useful conversations with Kent Blackburn, Ron Dreever, Eanna Flanagan, B.S. 44The real detector noise would be provided by the prototype out- Sathyaprakash, David Shoemaker, Kip Thorne, Robbie Vogt, put. Rainer Weiss, and Stan Whitcomb.

͓1͔ A. Abramovici et al., Science 256, 325 ͑1992͒. ͓17͔ V. Kaspi, J. Taylor, and M. Ryba, Astrophys. J. 428, 713 ͓2͔ B. Caron et al.,inGravitational Wave Experiments, Proceed- ͑1994͒. ings of the Edoardo Amaldi Conference, edited by E. Cocia, G. ͓18͔ L.S. Finn,‘‘Gravitational-wave data analysis with multiple de- Pizzella, and F. Ronga ͑World Scientific, Singapore, 1995͒,p. tectors: The gravitational-wave receiver. I. Deterministic 86. sources’’ ͑in preparation͒. ͓3͔ K. Danzmann et al.,inGravitational Wave Experiments ͓2͔,p. ͓19͔ L.S. Finn, ‘‘Gravitational-wave data analysis with multiple de- 100. tectors: The gravitational-wave receiver. II. Stochastic sig- ͓4͔ K. Tsubono, in Gravitational Wave Experiments ͓2͔, p. 112. nals’’ ͑in preparation͒. ͓5͔ P.F. Michelson, Mon. Not. R. Astron. Soc. 227, 933 ͑1987͒. ͓20͔ I. Miller and J.E. Freund, Probability and Statistics for Engi- ͓6͔ N. Christensen, Phys. Rev. D 46, 5250 ͑1992͒. neers ͑Prentice-Hall, Englewood Cliffs, NJ, 1985͒. ͓7͔ E. Flanagan, Phys. Rev. D 48, 2389 ͑1993͒. ͓21͔ W.H. Beyer, CRC Standard Probability and Statistics Tables ͓8͔ B. Allen, in Proceedings of the Les Houches School on Astro- and Formulae ͑CRC, Boca Raton, 1991͒. physical Sources of Gravitational Waves, edited by J.A. Marck ͓22͔ C.W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. ͑Pergamon, Oxford, 1968͒. and J.P. Lasota ͑Cambridge University Press, Cambridge, En- ͓23͔ W. Feller, An Introduction to Probability Theory and Its Ap- gland, 1997͒, p. 373. plications ͑Wiley, New York, 1950͒, Vol. 1. ͓9͔ E.W. Kolb and M. Turner, The Early Universe, Frontiers in ͓24͔ N. Christensen, Ph.D. thesis, Massachusetts Institute of Tech- Physics ͑Addison-Wesley, Reading, MA, 1990͒. nology, 1990. ͓10͔ B. Allen and A.C. Ottewill, Phys. Rev. D 56, 545 ͑1997͒. ͓25͔ D. Coyne, LIGO project ͑private communication͒. 11 D.G. Blair, R. Burman, L. Ju, S. Woodings, M. Mulder, and ͓ ͔ ͓26͔ The data for the predicted noise power spectra for the initial M.G. Zadnik, ‘‘The supernova cosmological background of and advanced LIGO detectors were taken from ͓1͔. Those gravitational waves,’’ report, University of Western Australia, for the VIRGO detector were supplied by Luca Gammaitoni 1997; V. Ferraria, in Proceedings of the XII Italian Conference ͑E-mail address: [email protected]͒; the GEO-600 on GR and Gravitational Physics, edited by M. Bassan et al. detector by Albrecht Rudiger and Kenneth Strain ͑E-mail ͑World Scientific, Singapore, 1997͒; D.G. Blair and L. Ju, address: [email protected] and [email protected]͒; Mon. Not. R. Astron. Soc. 283, 648 ͑1996͒. and the TAMA-300 detector by Masa-Katsu Fujimoto ͑E-mail ͓12͔ C.W. Misner, K.S. Thorne, and J.A. Wheeeler, Gravitation address: [email protected]͒. The predicted ͑Freeman, San Francisco, 1973͒. noise power spectra for the ‘‘enhanced’’ LIGO detectors ͓13͔ G.F. Smoot et al., Astrophys. J. Lett. 396,L1͑1992͒. were supplied by Rainer Weiss ͑E-mail address: ͓14͔ C.L. Bennett et al., Astrophys. J. Lett. 396,L7͑1992͒. [email protected]͒. ͓15͔ C.L. Bennett et al., Astrophys. J. 436, 423 ͑1994͒. ͓27͔ B. Allen, ‘‘GRASP: a data analysis package for ͓16͔ C.L. Bennett et al., Astrophys. J. Lett. 464,L1͑1996͒. gravitational wave detection,’’ 1997. An up-to-date

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version of the users manual may be obtained at ͓29͔ D.B Percival and A.T. Walden, Spectral Analysis for Physical http://www.ligo.caltech.edu/LIGO_web/Collaboration/ Applications ͑Cambridge University Press, Cambridge, En- manual.pdf or http://www.ligo.caltech.edu/LIGO_web/ gland, 1993͒. Collaboration/lsc_interm.html. The software package is avail- ͓30͔ J.M. Lees and J. Park, Compu. Geol. 21, 199 ͑1995͒. The able upon request. multitaper spectral estimation public domain package can be ͓28͔ D.J. Thomson, Proc. IEEE 70, 1055 ͑1982͒. found at the website http://love.geology.yale.edu/mtm/.

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