Living Rev Relativ (2017) 20:2
https://doi.org/10.1007/s41114-017-0004-1
REVIEW ARTICLE
Detection methods for stochastic gravitational-wave
backgrounds: a unified treatment
Joseph D. Romano
1
· Neil. J. Cornish
2
Received: 23 August 2016 / Accepted: 17 January 2017 / Published online: 4 April 2017
© The Author(s) 2017. This article is an open access publication
Abstract We review detection methods that are currently in use or have been pro-
posed to search for a stochastic background of gravitational radiation. We consider
both Bayesian and frequentist searches using ground-based and space-based laser
interferometers, spacecraft Doppler tracking, and pulsar timing arrays; and we allow
for anisotropy, non-Gaussianity, and non-standard polarization states. Our focus is on
relevant data analysis issues, and not on the particular astrophysical or early Universe
sources that might give rise to such backgrounds. We provide a unified treatment of
these searches at the level of detector response functions, detection sensitivity curves,
and, more generally, at the level of the likelihood function, since the choice of sig-
nal and noise models and prior probability distributions are actually what define the
search. Pedagogical examples are given whenever possible to compare and contrast
different approaches. We have tried to make the article as self-contained and compre-
hensive as possible, targeting graduate students and new researchers looking to enter
this field.
Keywords Gravitational waves · Data analysis · Stochastic backgrounds
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s41114-
017-0004-1
) contains supplementary material, which is available to authorized users.
B
Joseph D. Romano
joseph.romano@utrgv.edu
Neil. J. Cornish
cornish@physics.montana.edu
1
Department of Physics and Astronomy, University of Texas Rio Grande Valley, Brownsville, TX
78520, USA
2
Department of Physics, Montana State University, Bozeman, MT 59717, USA
123
2 Page 2 of 223 J. D. Romano, N. J. Cornish
Contents
1 Introduction ............................................. 5
1.1 Motivation and context .................................... 6
1.1.1 Why do we care about detecting a stochastic background? .............. 7
1.1.2 Why is detection challenging? ............................. 7
1.1.3 What detection methods can one use? ......................... 7
1.1.4 What are the prospects for detection? ......................... 8
1.2 Searches across the gravitational-wave spectrum ....................... 8
1.2.1 Cosmic microwave background experiments ..................... 9
1.2.2 Pulsar timing arrays .................................. 10
1.2.3 Space-based interferometers .............................. 10
1.2.4 Other detectors .................................... 11
1.3 Goal of this article ....................................... 11
1.4 Unification ........................................... 11
1.5 Outline ............................................. 12
2 Characterizing a stochastic gravitational-wave background .................... 13
2.1 When is a gravitational-wave signal stochastic? ....................... 13
2.2 Plane-wave expansions .................................... 14
2.2.1 Polarization basis ................................... 15
2.2.2 Tensor spherical harmonic basis ............................ 15
2.2.3 Relating the two expansions .............................. 17
2.3 Statistical properties ...................................... 17
2.3.1 Quadratic expectation values for Gaussian-stationary backgrounds ......... 18
2.4 Fractional energy density spectrum .............................. 19
2.5 Characteristic strain ...................................... 20
3 Statistical inference ......................................... 21
3.1 Introduction to Bayesian and frequentist inference ...................... 21
3.2 Frequentist statistics ...................................... 22
3.2.1 Frequentist hypothesis testing ............................. 24
3.2.2 Frequentist detection probability ........................... 25
3.2.3 Frequentist upper limits ................................ 26
3.2.4 Frequentist parameter estimation ........................... 27
3.2.5 Unified approach for frequentist upper limits and confidence intervals ........ 28
3.3 Bayesian inference ...................................... 29
3.3.1 Bayesian parameter estimation ............................ 30
3.3.2 Bayesian upper limits ................................. 31
3.3.3 Bayesian model selection ............................... 31
3.4 Relating Bayesian and frequentist detection statements ................... 33
3.5 Simple example comparing Bayesian and frequentist analyses ................ 34
3.5.1 Simulated data ..................................... 38
3.5.2 Frequentist analysis .................................. 38
3.5.3 Bayesian analysis ................................... 39
3.5.4 Comparison summary ................................. 41
3.6 Likelihoods and priors for gravitational-wave searches ................... 41
3.6.1 Calculating the likelihood ............................... 41
3.6.2 Choosing a prior .................................... 43
4 Correlations ............................................. 44
4.1 Basic idea ........................................... 44
4.2 Relating correlations and likelihoods ............................. 45
4.3 Extension to multiple data samples .............................. 46
4.3.1 White noise and signal ................................ 46
4.3.2 Colored noise and signal
............................... 47
4.4 Maximum-likelihood detection statistic ........................... 48
4.5 Bayesian correlation analysis ................................. 49
4.6 Comparing frequentist and Bayesian cross-correlation methods ............... 50
4.6.1 Frequentist analysis .................................. 51
123
Detection methods for stochastic gravitational-wave backgrounds Page 3 of 223 2
4.6.2 Bayesian analysis ................................... 52
4.7 What to do when cross-correlation methods aren’t available ................. 54
4.7.1 Single-detector excess power statistic ......................... 55
4.7.2 Null channel method ................................. 55
5 Geometrical factors ......................................... 57
5.1 Detector response ....................................... 58
5.1.1 Spacecraft Doppler tracking .............................. 58
5.1.2 Pulsar timing ...................................... 59
5.1.3 Laser interferometers ................................. 59
5.2 Calculation of response functions and antenna patterns ................... 61
5.2.1 One-way tracking ................................... 63
5.2.2 Two-way tracking ................................... 66
5.2.3 Michelson interferometer ............................... 67
5.3 Overlap functions ....................................... 70
5.3.1 Definition ....................................... 70
5.3.2 Interpretation ..................................... 72
5.3.3 Normalization ..................................... 73
5.3.4 Auto-correlated response ............................... 73
5.4 Examples of overlap functions ................................ 74
5.4.1 LHO-LLO overlap function .............................. 74
5.4.2 Big-bang observer overlap function .......................... 77
5.4.3 Pulsar timing overlap function (Hellings and Downs curve) ............. 78
5.5 Moving detectors ....................................... 79
5.5.1 Monochromatic plane waves ............................. 80
5.5.2 Stochastic backgrounds ................................ 81
5.5.3 Rotational and orbital motion of Earth-based detectors ................ 82
6 Optimal filtering .......................................... 84
6.1 Optimal combination of independent measurements ..................... 85
6.2 Correlated measurements ................................... 86
6.3 Matched filter ......................................... 87
6.4 Optimal filtering for a stochastic background ......................... 88
6.4.1 Optimal estimators for individual frequency bins ................... 90
6.4.2 More general parameter estimation .......................... 91
7 Anisotropic backgrounds ...................................... 91
7.1 Preliminaries ......................................... 92
7.1.1 Quadratic expectation values ............................. 92
7.1.2 Short-term Fourier transforms ............................. 93
7.1.3 Cross-correlations ................................... 93
7.1.4 Spherical harmonic components of γ(t; f, ˆn) ..................... 95
7.2 Modulations in the correlated output of two detectors .................... 97
7.2.1 Time-dependent cross-correlation ........................... 99
7.2.2 Calculation of the optimal filter ............................ 101
7.2.3 Inverse problem .................................... 101
7.3 Maximum-likelihood estimates of gravitational-wave power ................. 102
7.3.1 Likelihood function and maximum-likelihood estimators .............. 102
7.3.2 Extension to a network of detectors
.......................... 103
7.3.3 Error estimates ..................................... 104
7.3.4 Point spread functions ................................. 104
7.3.5 Singular-value decomposition ............................. 107
7.3.6 Radiometer and spherical harmonic decomposition methods ............. 108
7.4 Frequentist detection statistics ................................ 112
7.5 Phase-coherent mapping ................................... 113
7.5.1 Maximum-likelihood estimators and Fisher matrix .................. 113
7.5.2 Point spread functions ................................. 115
7.5.3 Singular value decomposition ............................. 115
7.5.4 Basis skies ....................................... 117
7.5.5 Underdetermined reconstructions ........................... 119
123
2 Page 4 of 223 J. D. Romano, N. J. Cornish
7.5.6 Pulsar timing arrays .................................. 119
7.5.7 Ground-based interferometers ............................. 121
8 Searches for other types of backgrounds/signals .......................... 122
8.1 Non-Gaussian backgrounds .................................. 123
8.1.1 Non-Gaussian search methods: overview ....................... 125
8.1.2 Likelihood function approach for non-Gaussian backgrounds ............ 125
8.1.3 Frequentist detection statistic for non-Gaussian backgrounds ............ 128
8.1.4 Bayesian model selection ............................... 129
8.1.5 Fourth-order correlation approach for non-Gaussian backgrounds .......... 132
8.2 Circular polarization ..................................... 134
8.2.1 Polarization correlation matrix ............................ 134
8.2.2 Overlap functions ................................... 136
8.2.3 Component separation: ML estimates of I and V ................... 138
8.2.4 Example: component separation for two baselines .................. 139
8.2.5 Effective overlap functions for I and V for multiple baselines ............ 140
8.3 Non-GR polarization modes: preliminaries .......................... 141
8.3.1 Transformation of the polarization tensors under a rotation about ˆn ......... 143
8.3.2 Polarization and spherical harmonic basis expansions ................ 143
8.3.3 Detector response ................................... 144
8.3.4 Searches for non-GR polarizations using different detectors ............. 145
8.4 Searches for non-GR polarizations using ground-based detectors .............. 145
8.4.1 Response functions .................................. 145
8.4.2 Overlap functions ................................... 146
8.4.3 Component separation: ML estimates of S
(T )
h
, S
(V )
h
,andS
(S)
h
............ 148
8.4.4 Effective overlap functions for multiple baselines .................. 150
8.5 Searches for non-GR polarizations using pulsar timing arrays ................ 151
8.5.1 Polarization basis response functions ......................... 152
8.5.2 Spherical harmonic basis response functions ..................... 154
8.5.3 Overlap functions ................................... 154
8.5.4 Component separation and anisotropic backgrounds ................. 157
8.6 Other searches ......................................... 160
8.6.1 Searches for long-duration unmodelled transients .................. 161
8.6.2 Searches for targeted-sources of continuous gravitational waves ........... 162
9 Real-world complications ...................................... 163
9.1 Observatory-specific challenges ............................... 164
9.1.1 Ground-based interferometers ............................. 164
9.1.2 Pulsar timing arrays .................................. 164
9.1.3 Space-based detectors ................................. 165
9.2 Non-stationary noise ..................................... 165
9.2.1 Local stationarity ................................... 165
9.2.2 Glitches ........................................ 167
9.3 Non-Gaussian noise ...................................... 168
9.4 Gaps and irregular sampling ................................. 171
9.4.1 Interferometer data .................................. 171
9.4.2 Pulsar timing data ................................... 171
9.5 Advanced noise modeling ................................... 171
9.6 Correlated noise ........................................ 173
9.6.1 Schumann resonances ................................. 174
9.7 What can one do with a single detector (e.g., LISA)? .................... 174
10 Prospects for detection ....................................... 178
10.1 Detection sensitivity curves .................................. 179
10.2 Current observational results ................................. 182
10.2.1 CMB isotropy ..................................... 182
10.2.2 Pulsar timing ...................................... 187
10.2.3 Spacecraft Doppler tracking .............................. 188
10.2.4 Interferometer bounds ................................. 188
10.2.5 Bounds on anisotropic backgrounds .......................... 189
123
Detection methods for stochastic gravitational-wave backgrounds Page 5 of 223 2
A Freedom in the choice of polarization basis tensors ......................... 192
A.1 Linear polarization ....................................... 192
A.2 Circular polarization ...................................... 193
A.3 Polarization matrix and Stokes’ parameters .......................... 194
B Some standard results for Gaussian random variables ........................ 196
C Definitions and tests for stationarity and Gaussianity ........................ 197
C.1 Definition of stationarity .................................... 198
C.2 Definition of Gaussianity .................................... 198
C.3 Tests for stationarity ...................................... 198
C.4 Tests for Gaussianity ...................................... 199
D Discretely-sampled data ....................................... 199
D.1 Discretely-sampled time-series ................................. 200
D.2 Windowing ........................................... 202
D.3 Discrete Fourier transform ................................... 204
D.4 DFTs and discretely-sampled Fourier transforms ....................... 205
D.5 Discrete power spectra ..................................... 206
D.6 Discrete and continuous probability distributions ....................... 207
E Ordinary (scalar) and spin-weighted spherical harmonics ...................... 208
F Gradient and curl rank-1 (vector) spherical harmonics ........................ 210
G Gradient and curl rank-2 (tensor) spherical harmonics ........................ 212
H Translation between ˆn and
ˆ
k conventions .............................. 213
H.1 General relationship between the response functions ..................... 213
H.2 Polarization basis response functions ............................. 214
H.3 Spherical harmonic basis response functions ......................... 215
References ................................................ 215
1 Introduction
The real voyage of discovery consists not in seeking new landscapes, but in
having new eyes. Marcel Proust
It is an exciting time for the field of gravitational-wave astronomy. The observation,
on September 14th, 2015, of gravitational waves from the inspiral and merger of a pair
of black holes (
Abbott et al. 2016e) has opened a radically new way of observing the
Universe. The event, denoted GW150914, was observed simultaneously by the two
detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) (
Aasi
et al. 2015
). [LIGO consists of two 4 km-long laser interferometers, one located in
Hanford, Washington, the other in Livingston, LA.] The merger event that produced
the gravitational waves occured in a distant galaxy roughly 1.3 billion light years from
Earth. The initial masses of the two black holes were estimated to be 36
+5
−4
M
⊙
and
29
+4
−4
M
⊙
, and that of the post-merger black hole as 62
+4
−4
M
⊙
(
Abbott et al. 2016f). The
difference between the initial and final masses corresponds to 3.0
+0.5
−0.5
M
⊙
c
2
of energy
radiated in gravitational waves, with a peak luminosity of more than ten times the
combined luminosity of all the stars in all the galaxies in the visible universe! The fact
that this event was observed only in gravitational waves—and not in electromagnetic
waves—illustrates the complementarity and potential for new discoveries that comes
with the opening of the gravitational-wave window onto the universe.
GW150914 is just the first of many gravitational-wave signals that we expect to
observe over the next several years. Indeed, roughly 3months after the detection of
GW150914, a second event, GW151226, was observed by the two LIGO detectors
123