Spectral line

Spectral density

Quantum mechanics

Linguistics

Philosophy

Pure mathematics

Mathematical physics

Statistics

Exponent

Omega

Bounded function

Scaling

Spectrum (functional analysis)

Tensor (intrinsic definition)

Physics

Degrees of freedom (physics and chemistry)

Mathematics

Quantum electrodynamics

Bilinear interpolation

Mathematical analysis

Gravitational wave

Geometry

Physical review. D/Physical review. D.

We study the general infrared behavior of the power spectrum of a stochastic gravitational wave background produced by stress tensor in the form bilinear in certain dynamical degrees of freedom. We find ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{GW}}\ensuremath{\propto}{k}^{3}$ for a very wide class of the sources which satisfy a set of reasonable conditions. Namely, the ${k}^{3}$ scaling is universally valid when the source term is bounded in both frequency and time, is effective in a radiation-dominated stage, and for $k$ smaller than all the physical scales associated with the source, like the peak frequency, peak width, and time duration, etc. We also discuss possible violations of these conditions and their physical implications.

Universal infrared scaling of gravitational wave background spectra