Living Rev Relativ (2017) 20:2 DOI 10.1007/s41114-017-0004-1 REVIEW ARTICLE Detection methods for stochastic gravitational-wave backgrounds: a unified treatment Joseph D. Romano1 · Neil. J. Cornish2 Received: 23 August 2016 / Accepted: 17 January 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 (doi:10.1007/s41114-017-0004-1) contains supplementary material, which is available to authorized users. B Joseph D. Romano [email protected] Neil. J. Cornish [email protected] 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 Living Rev Relativ (2017) 20:2 Contents 1 Introduction ............................................... 1.1 Motivation and context ...................................... 1.1.1 Why do we care about detecting a stochastic background? ................ 1.1.2 Why is detection challenging? ............................... 1.1.3 What detection methods can one use? ........................... 1.1.4 What are the prospects for detection? ........................... 1.2 Searches across the gravitational-wave spectrum ......................... 1.2.1 Cosmic microwave background experiments ....................... 1.2.2 Pulsar timing arrays .................................... 1.2.3 Space-based interferometers ................................ 1.2.4 Other detectors ...................................... 1.3 Goal of this article ......................................... 1.4 Unification ............................................. 1.5 Outline ............................................... 2 Characterizing a stochastic gravitational-wave background ...................... 2.1 When is a gravitational-wave signal stochastic? ......................... 2.2 Plane-wave expansions ...................................... 2.2.1 Polarization basis ..................................... 2.2.2 Tensor spherical harmonic basis .............................. 2.2.3 Relating the two expansions ................................ 2.3 Statistical properties ........................................ 2.3.1 Quadratic expectation values for Gaussian-stationary backgrounds ........... 2.4 Fractional energy density spectrum ................................ 2.5 Characteristic strain ........................................ 3 Statistical inference ........................................... 3.1 Introduction to Bayesian and frequentist inference ........................ 3.2 Frequentist statistics ........................................ 3.2.1 Frequentist hypothesis testing ............................... 3.2.2 Frequentist detection probability ............................. 3.2.3 Frequentist upper limits .................................. 3.2.4 Frequentist parameter estimation ............................. 3.2.5 Unified approach for frequentist upper limits and confidence intervals .......... 3.3 Bayesian inference ........................................ 3.3.1 Bayesian parameter estimation .............................. 3.3.2 Bayesian upper limits ................................... 3.3.3 Bayesian model selection ................................. 3.4 Relating Bayesian and frequentist detection statements ..................... 3.5 Simple example comparing Bayesian and frequentist analyses .................. 3.5.1 Simulated data ....................................... 3.5.2 Frequentist analysis .................................... 3.5.3 Bayesian analysis ..................................... 3.5.4 Comparison summary ................................... 3.6 Likelihoods and priors for gravitational-wave searches ..................... 3.6.1 Calculating the likelihood ................................. 3.6.2 Choosing a prior ...................................... 4 Correlations ............................................... 4.1 Basic idea ............................................. 4.2 Relating correlations and likelihoods ............................... 4.3 Extension to multiple data samples ................................ 4.3.1 White noise and signal .................................. 4.3.2 Colored noise and signal ................................. 4.4 Maximum-likelihood detection statistic ............................. 4.5 Bayesian correlation analysis ................................... 4.6 Comparing frequentist and Bayesian cross-correlation methods ................. 4.6.1 Frequentist analysis .................................... 123 Living Rev Relativ (2017) 20:2 Page 3 of 223 2 4.6.2 Bayesian analysis ..................................... 4.7 What to do when cross-correlation methods aren’t available ................... 4.7.1 Single-detector excess power statistic ........................... 4.7.2 Null channel method ................................... 5 Geometrical factors ........................................... 5.1 Detector response ......................................... 5.1.1 Spacecraft Doppler tracking ................................ 5.1.2 Pulsar timing ........................................ 5.1.3 Laser interferometers ................................... 5.2 Calculation of response functions and antenna patterns ..................... 5.2.1 One-way tracking ..................................... 5.2.2 Two-way tracking ..................................... 5.2.3 Michelson interferometer ................................. 5.3 Overlap functions ......................................... 5.3.1 Definition ......................................... 5.3.2 Interpretation ....................................... 5.3.3 Normalization ....................................... 5.3.4 Auto-correlated response ................................. 5.4 Examples of overlap functions .................................. 5.4.1 LHO-LLO overlap function ................................ 5.4.2 Big-bang observer overlap function ............................ 5.4.3 Pulsar timing overlap function (Hellings and Downs curve) ............... 5.5 Moving detectors ......................................... 5.5.1 Monochromatic plane waves ............................... 5.5.2 Stochastic backgrounds .................................. 5.5.3 Rotational and orbital motion of Earth-based detectors .................. 6 Optimal filtering ............................................ 6.1 Optimal combination of independent measurements ....................... 6.2 Correlated measurements ..................................... 6.3 Matched filter ........................................... 6.4 Optimal filtering for a stochastic background ........................... 6.4.1 Optimal estimators for individual frequency bins ..................... 6.4.2 More general parameter estimation ............................ 7 Anisotropic backgrounds ........................................ 7.1 Preliminaries ........................................... 7.1.1 Quadratic expectation values ............................... 7.1.2 Short-term Fourier transforms ............................... 7.1.3 Cross-correlations ..................................... 7.1.4 Spherical harmonic components of γ(t; f, nˆ) ....................... 7.2 Modulations in the correlated output of two detectors ...................... 7.2.1 Time-dependent cross-correlation ............................. 7.2.2 Calculation of the optimal filter .............................. 7.2.3 Inverse problem ...................................... 7.3 Maximum-likelihood estimates of gravitational-wave power ................... 7.3.1 Likelihood function and maximum-likelihood estimators ................ 7.3.2 Extension to a network of detectors ............................ 7.3.3 Error estimates ....................................... 7.3.4 Point spread functions ................................... 7.3.5 Singular-value decomposition ............................... 7.3.6 Radiometer and spherical harmonic decomposition methods ............... 7.4 Frequentist detection statistics .................................. 7.5 Phase-coherent mapping ..................................... 7.5.1 Maximum-likelihood estimators and Fisher matrix .................... 7.5.2 Point spread functions ................................... 7.5.3 Singular value decomposition ..............................