[Paper Review] The stochastic gravity-wave background: sources and detection

B. Allen|arXiv (Cornell University)|Apr 16, 1996

Pulsars and Gravitational Waves ResearchPhysics and Astronomy8 references101 citations

TL;DR

This paper investigates the stochastic gravitational-wave background as a distinct class of sources, analyzing its origin, statistical properties, and detectability using interferometric detectors. It derives the cross-correlation response of baseline detectors via the overlap reduction function and establishes a direct link between the observed strain power spectrum and the cosmological energy density parameter Ω_gw(f), enabling detection of a primordial background from as early as 10⁻²² seconds after the Big Bang.

ABSTRACT

A world-wide effort is now underway to build gravitational wave detectors based on highly-sensitive laser interferometers. When data from detectors at different sites is properly combined, it will permit highly-sensitive searches for a stochastic background of relic gravitational radiation. These lectures (from the Les Houches School in October 1995) review the current status of this program, and discuss the methods by which data from different detectors can be used to make measurements of, or place limits on, a stochastic background. They also review possible cosmological sources and their potential detectability.

Motivation & Objective

• To understand the nature and detectability of a stochastic gravitational-wave background arising from unresolved, incoherent sources across the sky.
• To derive the cross-correlation response of two widely separated gravitational-wave detectors to a stochastic background, accounting for detector response patterns and source distribution.
• To establish a quantitative link between the observed strain power spectrum and the cosmological energy density parameter Ω_gw(f), enabling physical interpretation of detected signals.
• To explore the potential of detecting a primordial gravitational-wave background from the early universe, potentially probing physics at 10⁻²² s after the Big Bang.

Proposed method

• Uses a plane-wave decomposition of the stochastic background in terms of polarization states + and ×, with arbitrary complex amplitudes h_A(f, Ω̂) satisfying Hermitian symmetry.
• Assumes the background is stationary and Gaussian, leading to a power spectral density defined by H(f), related to the energy density spectrum via Ω_gw(f) = (32π³ / 3H₀²) f³ H(f).
• Derives the cross-correlation of strain signals at two detectors using the overlap reduction function γ(f), which depends on the relative orientation and separation of the detectors.
• Applies the detector response formalism, expressing the strain in terms of detector-specific antenna patterns F₁ᴬ(Ω̂), F₂ᴬ(Ω̂), and integrates over the celestial sphere.
• Uses the covariant Dirac delta function on the sphere to enforce angular correlation, ensuring statistical independence of sources at different sky positions.
• Establishes that the cross-power spectral density between two detectors is proportional to H(f)γ(f), with γ(f) calculable from the detector geometry and orientation.

Experimental results

Research questions

• RQ1 What is the statistical structure of a stochastic gravitational-wave background, and how does it differ from coherent, transient, or periodic signals?
• RQ2 How can the cross-correlation of signals from two widely separated detectors be used to detect an isotropic stochastic background?
• RQ3 What is the relationship between the observed strain power spectrum and the cosmological energy density parameter Ω_gw(f)?
• RQ4 Can a primordial gravitational-wave background from the early universe be detected, and what physical timescales could it probe?

Key findings

• The cross-correlation of strain signals from two detectors is proportional to the overlap reduction function γ(f), which depends on the relative orientation and separation of the detectors.
• The cross-power spectral density between two detectors is given by 8π/5 γ(f) H(f) δ(f−f′), linking observable strain statistics to the underlying H(f) power spectrum.
• The energy density parameter Ω_gw(f) is related to H(f) by Ω_gw(f) = (32π³ / 3H₀²) f³ H(f), enabling physical interpretation of detected signals.
• A stochastic background could carry information about the universe as early as 10⁻²² seconds after the Big Bang, if detectable with current or future interferometric detectors.
• The overlap reduction function γ(f) is calculable from detector geometry and can be used to distinguish a stochastic background from instrumental noise or correlated environmental effects.
• The formalism provides a rigorous statistical framework for detecting an isotropic stochastic background using cross-correlation of data from multiple detectors.