Detecting stochastic gravitational waves

We make the following assumptions about the characteristics of a stochastic signal:

• stationary

• stochastic, zero mean, broadband, and Gaussian

• much smaller than the noise in the detectors.

To describe a stochastic source of gravitational waves, we treat the complex strain amplitudehA(f,Ω)ˆ (for ˆΩ≡(Right Ascension, Declination)) as a random variable with zero mean, where the wave vec- tor forhA(f,Ω) isˆ ~k= 2πfΩ/c. Then for stochastic, stationary gravitational waves the expectationˆ value of the fields is

hh^∗_A(f,Ω)hˆ ⁰_A(f⁰,Ωˆ⁰)i=HA(f)P( ˆΩ)δ²( ˆΩ,Ωˆ⁰)δAA⁰δ(f −f⁰), (10.1) whereP( ˆΩ) is the power distribution of the source andHA(f) is the two-sided power spectral density in polarization A, and the delta functions have their usual interpretation. We have assumed that the source power spectrum is separable in frequency and sky location. We simultaneously drop the polarization label and move to a one sided spectrum H(f), which is then the signal strength assuming an unpolarized source:

H(f) = 4HA(f) for A= +,×. (10.2)

One factor of two is from the move from two-sided to one-sided, and the other is from the sum over polarizations. We can convert this to an energy flux by [106],

FGW =c³π 4G

Z fmax

fmin

H(f)f²df. (10.3)

It is the power spectrumH(f) that we want to detect. Because the signal is random, broadband, and smaller than the noise, it is indistinguishable from detector noise, unless it is present in two detectors.

This suggests an analysis strategy that involves cross-correlating the outputs of two detectors (whose noises must be assumed to be uncorrelated) over long time periods. We first describe the strategy for an isotropic background (P( ˆΩ) = 1), which is interesting for cosmological reasons ([107], [108], [98]), and which is easily extended to the directional analysis. Isotropic analyses are typically quoted as Ωgw(f), which is the fractional energy density of gravitational waves, relative to the critical energy density of the universe, per unit logarithmic frequency interval. For an isotropic background, Ωgw(f) is related toH(f) by [109]

Ωgw(f) = 8π³

3H₀²f³H(f), (10.4)

whereH0is the Hubble constant.

10.2.1 Isotropic strategy

We definesi(t) as the output of detectoriat timet, which can be expressed as a sum of the strain sensed by the detector and any noise in the detector:

si(t) =hi(t) +ni(t). (10.5)

We make the reasonable assumptions that the noise in the two detectors is uncorrelated, and the noise in each detector is uncorrelated with the signal in the other detector. We can enforce that the noise is also zero mean by high-pass filtering. Then, working in the frequency domain (so ˜si(f) is the Fourier transform ofsi(t) ), we can write down the cross-correlation,

Y = Z ∞

−∞

df⁰ Z ∞

−∞

dfs˜^∗₁(f) ˜s2(f) ˜Q(f)δT(f⁰−f), (10.6)

where δT(f⁰−f) is a finite time (T) delta function approximation, and T is the observation time.

We have defined the optimal filter,

Q(f˜ )≡ 1 N

γ(f)H(f)

P1(f)P2(f), (10.7)

in which Pi(f) is the power-spectral density of the output (assumed to be noise dominated) from detector i, H(f) is the power spectrum of the signal (equation (E.7)), N is a normalization, and γ(f) is an overlap reduction function which describes the geometric configuration of the detectors:

γ(f) =1 2

X

A

Z

S²

dΩeˆ ^{i2πfΩ·∆~ˆ x/c}F₁^A( ˆΩ)F₂^A( ˆΩ), (10.8)

where ∆~x=x~1−x~2is the separation vector between the two detector sites, andF_i^A( ˆΩ) is the response of the i detector to waves of polarizationA= (+,×) coming from direction ˆΩ. The integration is

over the two-sphereS²(the surface of the sky). Usually this overlap reduction function is normalized (with a factor of _4π⁵) so thatγ(f) = 1 for co-aligned and co-located identical detectors; we omit this normalization here, in anticipation of the directional search in section10.2.3.

The filter ˜Q(f) is optimal in the sense that it maximizes the SNR for source spectrumH(f); since we do not know in advance whatH(f) is (in principle it can be predicted for a given cosmological model [99]), the usual strategy is to define several versions of H(f) (templates) and repeat the analysis for each one. In LIGO, we useH(f)∝f^β,

H(f) =Hβ

f 100 Hz

β

, (10.9)

with attention focused on β = 0 (a constant strain spectrum) and β = −3 (a constant Ωgw).

Following [99], we define a scalar product, (A, B) =

Z ∞

−∞

A^∗(f)B(f)P1(f)P2(f)df. (10.10)

Then, with the assumptions outlined so far (stationary Gaussian noise uncorrelated between the detectors, and uncorrelated with and larger than the signal), the expectation value of Y (over random instantiations of the random noise and the random signal) is [99, 109]

hYi=T(Q, γH P1P2

), (10.11)

and its variance is

σ_Y² ≡ hY²i − hYi²≈T

4(Q, Q). (10.12)

We choose the normalizationN so that the expectation value ofY is

hYi=Hβ. (10.13)

Y thus has units of strain squared per Hertz. The signal-to-noise ratio (SNR) is SNR = Y

σ, (10.14)

which we note grows with the integration timeT as√ T.

10.2.2 Segmenting Data and Optimal Combination

The are several wrinkles to the analysis described above, which include the non-stationarity of the detector noise spectra Pi(f), computational resource limits, and the bias resulting from using the same data to determine a result and its variance. A strategy to address the first two concerns

was also presented in [99]. This strategy centers around breaking the data streams into time-based segments (usually 60 seconds). An estimate forYiandσiis formed for each segment (indexed byi), and these estimates are then combined optimally. The optimal combination is given by a weighted average, with the weights chosen to maximize the signal-to-noise ratio. In [99] it is shown that the optimal weights are the reciprocals of the variance estimates:

Yˆ = 1 N

X

i

Yiσ⁻_i², (10.15a)

N=X

i

σ⁻_i ², (10.15b)

σ⁻²=N =X

i

σ_i⁻². (10.15c)

The same strategy of segmenting the data can also help with the third concern: a bias can arise when the same data is used to estimate a cross-power spectrum and a power spectrum from the same data [110], and would thus appear in the result from equation (10.15), where the cross-correlation estimator and the variance of that estimator are computed from the same data. To circumvent this, theσiterms in equation (10.15) can be replaced with quantities calculated for the neighboring time segments, as illustrated in figure10.1, to yield ˆσi. This ˆσi is called thetheoretical sigma, and is computed by averaging the power spectra from neighboring segments, calculating the optimal filter, and then using equation (10.12) to estimate the variance. This does not entirely eliminate the bias, but it does reduce it to one that depends solely on the signal strength. If the assumptions in section10.2hold, then this residual bias can be ignored. Thenaive sigmas, calculated with the data from the current segment, are also recorded for use in a data analysis veto (in section10.4.3.2). There is a further wrinkle—we also window the data (usually with a Hann window), and thus effectively discard half the data. Because of this, equation (10.15) is not the exact one used in the analysis.

The segments are instead combined in an overlapping manner to recover the sensitivity lost due to

Y ˆ _i Y ˆ _i+1 Y ˆ _{i − 1}

ˆ σ _i

σ i+1

σ _{i − 1} σ _i

Figure 10.1: The estimate of the variance for each data segment is calculated using data from neighboring time segments to avoid biasing the estimator.

windowing. The details of this procedure are in [111].

10.2.3 Gravitational Wave Radiometry

The discussion in section10.2describes the optimal strategy for averaging the contributions to the gravitational wave signal at the detectors from all sources within the measurement bandwidth, using an optimal filtering strategy. One can form an alternative filter that specifically (and optimally) targets certain portions of the sky from which stochastic gravitational waves may originate. Such a technique would not be limited to signals of cosmological origin (or other isotropic signals), but would be useful for any signals which can be described by the traits in section 10.2: constant, random, broadband, and weak.

We now want to find a source withP( ˆΩ)6= 1, and so we generalize equation (10.11),

hYi=T Q,4π 5

R

S²dΩγˆ ΩˆP( ˆΩ)H P1P2

!

, (10.16)

where we have a new direction dependent overlap reduction function which is just the integrand of equation (10.8),

γΩˆ(t, f) = 1 2

X

A

e^{i2πfΩˆ·∆~x/c}F₁^A( ˆΩ)F₂^A( ˆΩ). (10.17) This overlap reduction is also sidereal time dependent through both ∆~x/candF_i^A( ˆΩ).

For the case of a point source P( ˆΩ) =δ²( ˆΩ,Ωˆ⁰) we have a ˆΩ and time dependent optimal filter for a point source at sky direction ˆΩ,

Q˜Ωˆ(t, f) = 1 N

γΩˆ(t, f)H(f)

P1(f)P2(f) , (10.18)

wheret is the sidereal time. The direction dependent strain power estimate is hYΩˆi=T

Q˜Ωˆ, γΩˆH P1P2

, (10.19)

and the variance is

σ_Y²_ˆ

Ω ≈T

4( ˜QΩˆ,Q˜Ωˆ). (10.20)

The optimal filter is thus calculated individually for each segment i and each direction ˆΩ. It is only calculated once per segment, which introduces an error because the value must change over the course of the segment. The timet chosen is mid-segment; the remaining error is second order and is of size

Yerr(Tseg)

Y 'O

2πf d c

Tseg

1 day 2!

, (10.21)

where dis the detector separation. For the values used in the search (Tseg = 60 sec), this error is less than 1%.

10.2.4 The Point Spread Function

The radiometer search described in section 10.2.3 is optimal for point sources, where optimality again means that it yields the maximum SNR for a given source strainH(f). This does not mean that it maximally avoids contamination with signal from other directions; there is significant co- variance between different sky locations. This can be seen in the point spread function (PSF), which is the result of this covariance. Figure 10.2 is an example of the point spread function, with five injected sources at different locations to illustrate how the PSF changes with sky location. This is computed by injecting a strong source at a chosen location into simulated data (noise), and calculating the radiometer response across the whole sky. The detailed structure of the PSF will vary with sky position and frequency of the source. Because the radiometer algorithm depends on sidereal variations in the antenna acceptance to provide direction discrimination, in theory the PSF should not vary with right ascension of the source, although it will vary with declination. Its angular size will in general be smaller for higher frequency sources. A lower limit can be placed on the size of the point spread function by assuming diffraction limited detection of the gravitational waves, which limits the resolving power of an instrument by the familiar formula,

θ≈ λ

D. (10.22)

For gravitational waves wherec=λf, whenfGW = 500 Hz andD = 3000 km, this yieldsθ ≈12^◦. This means the resolution of the radiometer is limited to something of order ∼ 400 independent points in the sky, a number which has been estimated in [71].