Optimal combination of signals from colocated gravitational wave interferometers for use in searches for a stochastic background

This article derives an optimal (i.e., unbiased, minimum variance) estimator for the pseudodetector strain for a pair of colocated gravitational wave interferometers (such as the pair of LIGO interferometers at its Hanford Observatory), allowing for possible instrumental correlations between the two detectors. The technique is robust and does not involve any assumptions or approximations regarding the relative strength of gravitational wave signals in the Hanford pair with respect to other sources of correlated instrumental or environmental noise. An expression is given for the effective power spectral density of the combined noise in the pseudodetector. This can then be introduced into the standard optimal Wiener filter used to cross-correlate detector data streams in order to obtain an optimal estimate of the stochastic gravitational wave background. In addition, a dual to the optimal estimate of strain is derived. This dual is constructed to contain no gravitational wave signature and can thus be used as an “off-source” measurement to test algorithms used in the “on-source” observation.

Figure 1

Strain spectral densities (i.e., absolute value) of ${\tilde{s}}_H(f)$ (gray or dotted line), ${\tilde{s}}_{H_1}(f)$ (black line), and ${\tilde{s}}_{H_2}(f)$ (dashed line), representative of the S1 data. Top panel: Overlay of the individual spectral densities with that of the strain spectral density $|{\tilde{s}}_H(f)|$ calculated with the S1 run-averaged coherence, ${\Gamma }_{H_1{H}_2}(f)$, and with ${\Gamma }_{H_1{H}_2}(f)=0$. On this scale, the left-hand panel shows no discernible difference between the spectra for ${\Gamma }_{H_1{H}_2}(f)$, and with ${\Gamma }_{H_1{H}_2}(f)=0$, suggesting that even the level of coherence seen during the S1 run might be sufficiently low to allow one to simply combine the L1-H1 and L2-H2 cross-correlation measurements under the assumption of zero cross-correlated noise. The optimality of the estimate ${\tilde{s}}_H(f)$ is visible here because it is always less than the smaller of ${\tilde{s}}_{H_1}(f)$ or ${\tilde{s}}_{H_2}(f)$. The inset shows a blowup of the region near one of the spectral features. On this scale the individual spectra can be discerned. Bottom panel: Plot of the ratio of amplitude spectra for $|{\tilde{s}}_H(f)|$ calculated with ${\Gamma }_{H_1{H}_2}(f)$ as measured during S1 and ${\Gamma }_{H_1{H}_2}(f)=0$ (i.e., assuming no coherence). The difference between the two is very small except for the very lowest frequencies and at narrow line features.

Figure 2

H1-H2 coherence averaged over the whole S1 data run. Note the substantial broadband coherence throughout the band. Low-frequency seismic noise and acoustic coupling between the input electro-optics systems are considered to be the prime sources of this cross-correlated noise [6, 7].

Figure 3

Schematic showing how the H1 and H2 signals may be represented in a three-dimensional space of noise components for the two detectors and the common gravitational wave strain: {${\widehat{n}}_{H_1},{\widehat{n}}_{H_2},\widehat{h}$}. The signals ${\tilde{s}}_{H_1}(f)$ and ${\tilde{s}}_{H_2}(f)$ are not, in general, orthogonal if the coherence between the noise, ${\tilde{n}}_{H_1}(f)$ and ${\tilde{n}}_{H_2}(f)$, is nonzero. ${\tilde{s}}_H(f)$ is the minimum variance estimate of $\tilde{h}(f)$ derived from ${\tilde{s}}_{H_1}(f)$ and ${\tilde{s}}_{H_2}(f)$. Using ${\tilde{s}}_H(f)$ as the best estimate of $\tilde{h}(f)$, this signal can be subtracted from ${\tilde{s}}_{H_1}(f)$ and ${\tilde{s}}_{H_2}(f)$ to produce the vectors ${\tilde{z}}_{H_1}(f)$, ${\tilde{z}}_{H_2}(f)$ that lie in the ${\widehat{n}}_{H_1}$-${\widehat{n}}_{H_2}$ plane. These vectors give rise to the covariance matrix $\Vert {\tilde{\mathbf{C}}}_z(f)\Vert$. ${\tilde{z}}_{H_1}(f)$ and ${\tilde{z}}_{H_2}(f)$ are collinear and thus one of the eigenvectors of $\Vert {\tilde{\mathbf{C}}}_z(f)\Vert$ will be zero. The other corresponds to the dual of ${\tilde{s}}_H(f)$, denoted ${\tilde{z}}_H(f)$, which is orthogonal to ${\tilde{s}}_H(f)$, as shown in the figure. Note that it is necessary to first subtract the contribution of $\tilde{h}(f)$ from the signals before forming the covariance matrix.

Figure 4

Same as Fig. 1, but for the null signal ${\tilde{z}}_H(f)$ instead of the optimal estimate ${\tilde{s}}_H(f)$. Strain spectral densities (i.e., absolute value) of ${\tilde{z}}_H(f)$ (gray or dotted lines), ${\tilde{s}}_{H_1}(f)$ (black line), and ${\tilde{s}}_{H_2}(f)$ (dashed line), representative of the S1 data. Top panel: Overlay of the individual amplitude spectral densities with that of the strain spectral density $|{\tilde{z}}_H(f)|$ is calculated with the S1 run-averaged coherence, ${\Gamma }_{H_1{H}_2}(f)$. On this scale, the left-hand panel shows no discernible difference between the spectra for ${\Gamma }_{H_1{H}_2}(f)$, and with ${\Gamma }_{H_1{H}_2}=0$, suggesting that even the level of coherence seen during the S1 run might be sufficiently low to allow one to simply combine the L1-H1 and L2-H2 cross-correlation measurements under the assumption of zero cross-correlated noise. The optimality of the estimate ${\tilde{z}}_H(f)$ is visible here because it is always less than the larger of ${\tilde{s}}_{H_1}(f)$ or ${\tilde{s}}_{H_2}(f)$. Bottom panel: Overlay of individual amplitude spectra with that for $|{\tilde{z}}_H(f)|$ calculated with ${\Gamma }_{H_1{H}_2}(f)=0$ (i.e., assuming no coherence). The difference between the two is very small except for the very lowest frequencies and at narrow line features.