Detecting a Stochastic Gravitational-Wave Background:
The Overlap Reduction Function
Detection of a gravitational-wave stochastic background via ground or space-based gravitational-wave detectors requires the cross-correlation of the response of two or more independent detectors. The cross-correlation involves a frequency-dependent factor — the so-called overlap reduction function or Hellings-Downs curve— that depends on the relative geometry of each detector pair: i.e., the detector separations and the relative orientation of their antenna patterns (beams). An incorrect formulation of this geometrical factor has appeared in the literature, leading to incorrect conclusions regarding the sensitivity of proposed detectors to a stochastic gravitational-wave background. To rectify these errors and as a reference for future work we provide here a complete, first-principles derivation of the overlap reduction function and assess the nature of the errors associated with the use of the incorrect expression that has appeared in the literature. We describe the behavior of the overlap reduction function in different limiting regimes, and show how the difference between the correct and incorrect expressions can be understood physically.
pacs:04.80.Nn, 95.55.Ym, 04.30.-w
The measured response of a single gravitational-wave detector to a stationary stochastic gravitational-wave signal is indistinguishable from unidentified instrumental noise. The gravitational-wave contribution to the measured response of two or more independent detectors will, however, be correlated between detector pairs in ways that other technical noises will not. The relationship between the power in a stochastic gravitational-wave background and the cross-correlated response of a detector pair depends on the response of the individual detectors and their relative geometry: i.e., their separation and the relative orientation of their respective detector antenna patterns, or beams. In the context of ground or space-based laser interferometric detectors Lück et al. (2006); Barish and Weiss (1999); Waldman and The LIGO Scientific Collaboration (2006); Acernese et al. (2006); Merkowitz and Livas (2007); Kawamura et al. (2008); Crowder and Cornish (2005) or resonant acoustic gravitational-wave detectors Cerdonio (2003) this geometrical factor is referred to as the overlap reduction function Michelson (1987); Christensen (1992); Flanagan (1993); Allen (1997); Allen and Romano (1999); in the context of pulsar timing arrays Jenet et al. (2006) or spacecraft doppler tracking Armstrong (2006) it is called the Hellings-Downs curve Hellings and Downs (1983).
Incorrect expressions for the overlap reduction function have appeared in the recent literature Cornish and Larson (2001); Cornish (2001) and, with them, incorrect conclusions regarding the sensitivity of proposed gravitational wave detectors to stochastic gravitational waves. These errors have lead to significantly flawed appraisals of the high-frequency sensitivity of the Big Bang Observer to a stochastic gravitational wave background, including spurious nulls in the frequency-dependent detector response and a reduced estimate of the signal-to-noise ratio as a function of the gravitational-wave power. To rectify these errors and as a reference for future work we provide here a complete, first-principles derivation of the overlap reduction function and assess the nature and physical interpretation of the errors associated with the use of the incorrect expression that has appeared in the literature.
Ii The overlap reduction function
The overlap reduction function of a pair of gravitational-wave detectors is the collection of geometric factors, associated with the relative position and orientation of the detector pair, that appear in the cross-correlation of the detector pair’s response. Here we derive an expression for the overlap reduction function by deconstructing the cross-correlation, identifying those contributions that depend only on the radiation and those that depend only on the detectors, which are then identified as the overlap reduction function. This approach has the virtue of clearly illustrating the physical origins of the overlap reduction function and making less likely mistakes of the kind that may have led to the existing errors in the literature.
ii.1 Inter-detector cross-correlation
Consider two gravitational-wave detectors. A stochastic gravitational-wave “background” will manifest itself in a non-vanishing cross-correlation between the measurements made at the two detectors, calculated as a time average over the product of the measurements:
The signature of a stochastic gravitational-wave signal is just the expectation value of in the presence of . Write the measurement made at detector as the sum of a noise contribution and a signal contribution , corresponding to the detector response to 111Lower case latin indices will denote spatial components of tensors; upper case latin indices from the middle of the alphabet will label different detectors, and upper case latin indices from the beginning of the alphabet will denote different gravitational-wave polarizations.. Assume that the noise in each detector is independent and that there are no non-gravitational-wave effects that might lead to a correlation in the measurements made at each detector. Under these assumptions the expectation value of the product vanishes, implying that the expectation value of is just the expectation value of the product :
where overbar denotes expectation value and we have assumed that the expectation value of is independent of . This is equivalent to assuming that the background is stationary, as we will describe in more detail in Sec. II.3.
Turn now to the detector response. Gravitational waves are weak. Even the most sensitive detectors respond linearly to the local field . Correspondingly we write the detector response as a convolution, in time and space, of an impulse response function with the field :
where is the spatial location of detector about which its response is defined. Causality requires that vanishes outside the future light cone of . Exploiting the convolution theorem we can also write
are the field Fourier modes and detector transfer function. Note that for real and
With the above representations of the detector response, we can express in terms of the detector response and the field:
Note particularly how the detector location and the transfer function appear together in the combination . The form of this combination will be critical when we come to understand the physical character of the errors made in earlier calculations of the overlap reduction function.
ii.2 Plane-wave representation of stochastic signal
Focus attention on gravitational wave fields . These are conveniently represented as a superposition of plane waves
where is the unit vector direction of wave propagation, and are the two orthogonal polarization tensors,
Note that as a consequence of the reality of . The plane-wave field amplitudes are related to the field’s Fourier modes by
Here we have introduced separate amplitudes for the positive and negative frequency solutions to the dispersion relations for a plane wave.
Using expansion (8), we can write the detector response as
The expectation value can also be written as
Again make note of how the detector locations , are associated with the respective transfer functions and .
ii.3 Stationarity, isotropy, and polarization correlations
The statistical properties of the stochastic signal are encoded in the expectation values of products of the gravitational field
Without loss of generality, we will assume that any non-zero mean has been absorbed in the background spacetime, so that . Furthermore, for Gaussian-distributed fields, knowledge of the quadratic correlations will suffice as all higher-order moments can be constructed from these.
In our problem we expect that the gravitational wave background is effectively stationary: i.e., that depends on and only through their difference . In terms of the plane wave components , this condition becomes
Thus, the different frequency components of a stationary stochastic background are statistically independent, but they can contribute differently to the cross-correlated power through the -dependence in . Note that the factor in (16) eliminates the -dependence in , cf., Eq. 14.
If the background is isotropic — i.e., the gravitational-wave specific intensity is independent of the direction of propagation — then the most general form of the quadratic expectation value of the plane wave components is
|where depends on and only through the angle between them. If we further assume that the components corresponding to different propagation directions are statistically independent, then|
where is the covariant Dirac delta function on the two-sphere. This latter, more restrictive, condition is the definition of isotropy for gravitational-wave stochastic backgrounds typically assumed in the literature, e.g., Christensen (1992); Flanagan (1993); Allen (1997); Allen and Romano (1999).
Finally, if the background is unpolarised, by which we will mean that the different polarisation components are statistically independent and contribute equally to the cross-correlated power, then
Putting all these conditions together, we have that an unpolarised, stationary, isotropic stochastic gravitational-wave background satisfies
Here is a real-valued function proportional to the gravitational-wave energy density. This is, in turn, directly related to , the ratio of the gravitational-wave energy density to the cosmological closure density Allen and Romano (1999):
where is the Hubble expansion rate at the present epoch.
ii.4 The overlap reduction function
Combining the results of the previous subsections we find that we can express the expectation value of the inter-detector cross-correlation in the presence of an unpolarised, stationary, isotropic gravitational-wave background as
The quantity is the overlap reduction function. It is often convenient to define a normalized overlap reduction function with for two coincident and coaligned identical detectors. For identical interferometers with opening angle this leads to the normalized form
Some symmetry properties of follow from immediately from its definition: in particular,
Summarizing the results of the previous section, the overlap reduction function normalized for interferometric detectors is
Comparing this expression with our Eq. 24a we see that they differ by the substitution of for . This difference and when it is significant can be understood physically; doing so provides the occasion for a deeper discussion of the overlap reduction function.
iii.1 Detector locations
The detector location appears here in the form . Referring to Eqs. 24a and 25 for and it is clear the substitution of for is equivalent to simply exchanging the locations of detectors 1 and 2 keeping the rest of the configuration of the detectors fixed: i.e., is the overlap reduction function for the detector configuration consisting of detector 1 at location and detector 2 at location . With this understanding we now ask when that exchange is significant and when it is not.
iii.2 Radiation wavelength and detector separation
An intuitive understanding of recognizes that its behavior in different frequency regimes is governed by several independent dimensionless parameters that can be created from the radiation wavelength, the separation between the detectors, and several intrinsic properties of the detectors as they are represented in the detector impulse response functions.
Referring to Eq. 24a we note that when the exponential term may be replaced by unity. As this is the only place where the detector separation appears, in this limit the detector separation plays no role in determining the value or behavior of . Defining
we refer to as the small separation limit. In the small separation limit, then, the difference between and is negligible under all circumstances.
Now consider the case . As we have observed, the difference between and is the difference between locating detector 1 at or , and detector 2 at or . When the two detectors are identical in all other aspects, so that , this exchange leaves the physical configuration unchanged and, again, there will be no difference between and .
To understand the case when the two detectors are not identical we must consider the detector impulse response functions as they appear in Eqs. 24.
iii.3 Radiation wavelength and detector impulse response
The impulse response of a detector has finite support: i.e., for sufficiently large or . A detector doesn’t sample the field beyond its physical extent, so the support in will be on order the detector’s size . Referring to Eq. 24b it is apparent that depends on only through when :
Introducing the parameters
for the two detectors we refer to as the small antenna limit for detector .
Return now to the difference between and when and the detectors are not identical. When both and are and noting that we find
i.e., in the small antenna limit there is no distinction between and .
iii.4 Large separations and large detectors
Finally, consider the case and, without loss of generality, . In this case, the variation of the field across the spatial extent of detector is important to the detector response and, in turn to . Exchanging the detector locations changes the relationship between the spatial extent and orientation of detector 1 relative to the location of detector 2. Correspondingly, in this limit the distinction between and is significant.
An example of the case , is the anticipated sensitivity to a stochastic gravitational-wave background of the Big Bang Observer (BBO), a space-based follow-on to LISA that has been the subject of recent study Cornish and Larson (2001); Crowder and Cornish (2005); Cutler and Harms (2006); Kudoh et al. (2006); Corbin and Cornish (2006). The principal results of this study reported in the literature Cornish and Larson (2001); Cornish (2001) make use of the incorrect form of the overlap reduction function, thus mis-estimating this proposed detector’s sensitivity to a stochastic gravitational-wave signal in the higher frequency regime. Figure 1 illustrates the the physical effect of using in place of when analyzing the cross-correlation of the two BBO detectors as described in Cornish and Larson (2001). On the left are the two interferometric detectors as they are actually arranged in space; on the right are the effective location and orientation of the detectors when is used in place of : i.e., when detector 1 is translated to and detector 2 is translated to . Under this transformation the spatial extent of the two detector pairs is much greater than is actually the case; correspondingly, we expect to be a much more sensitive function of frequency than .
To calculate for comparison with , we need an explicit expression for the transfer function of the detectors. This is derived in (Cornish and Larson, 2001, Eqs. 5, 7, 11). In our notation,
Here and are unit vectors pointing in the direction of the detector arms, is the transfer frequency of the detectors (arm length ), and . Figure 2 compares (cf. (Cornish and Larson, 2001, Fig. 5)) and (Eq. 24) for this configuration. As expected decreases much more rapidly with frequency and has its nulls more closely spaced than those of . Figure 3 shows the fractional error as a function of frequency. The large amplitude spikes in the error occur where nulls of do not coincide with nulls of .
It is apparent from Fig. 2 that the use of an incorrect overlap reduction function has, in this case, led to an underestimate of BBO’s sensitivity to a stochastic gravitational wave background. Estimating the background and detector noise power spectral density as white, the magnitude of the error is just the ratio of integrated squared magnitudes and ,
i.e., the BBO estimates of Cornish and Larson (2001); Cornish (2001) underestimate the sensitivity of BBO by nearly 30% across the entire band, and substantially larger if interest is focused on the higher frequencies. Achieving BBO’s goals of detecting the stochastic gravitational-wave relics of the inflationary epoch depend on the accurate identification and subtraction of contributions owing to compact binary systems. Underestimating BBO’s response to a gravitational-wave background leads to an overestimate of the accuracy required in this identification and subtraction Cutler and Harms (2006). Recognizing and correcting the underestimate in BBO sensitivity thus relaxes the analysis problem associated with the identification of these foreground sources.
Detection of a gravitational-wave stochastic background relies on the cross-correlated response of one or more pairs of gravitational-wave detectors. The separation and relative orientation of the two detectors plays a crucial role in determining the frequency dependent sensitivity of each detector pair to the stochastic background. Recent studies Cornish and Larson (2001); Cornish (2001) of the sensitivity of the Big Bang Observer and related future generation gravitational-wave detectors have used an incorrect expression for this geometrical factor. The errors committed may be physically interpreted as an exchange in space of the two detectors, leaving their absolute orientations fixed. In the case of the Big Bang Observer, this error leads to an approximately 30% underestimate in its sensitivity to relic gravitational waves associated with, e.g., the inflationary epoch. Since achieving BBOÕs goals of detecting this background requires the accurate identification and subtraction of gravitational-wave foreground contributions from compact binary systems Cutler and Harms (2006), this underestimate has lead to commensurate overestimate of the difficulty of this analysis problem. Recognizing and correcting this error thus improves, in two ways, the prospects for the BBO missions main goal as a Big-Bang Observer.
Acknowledgements.We most gratefully acknowledge the hospitality of the Aspen Center for Physics and the 2008 Workshop on Gravitational-Wave Astronomy, where most of the work reported here was done. JDR also acknowledges B. Allen, W. Anderson, A. Lazzarini, and J.T. Whelan for discussions. Some of the results in this paper were performed using the HEALPix Górski et al. (2005) package (http://healpix.jpl.nasa.gov). The research was supported in part by NSF grant PHY-0555842 awarded to the University of Texas at Brownsville, NSF grant PHY-0653462 and NASA grant NNG05GF71G to The Pennsylvania State University, and the Center for Gravitational Wave Physics, which was supported by the NSF under cooperative agreement PHY-014375.
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