Early-Universe Stochastic Gravitational-Wave Background

Updated 24 August 2025

Early-Universe SGWB is a composite signal from primordial processes such as inflation, cosmic string dynamics, and phase transitions, offering insights into the universe's earliest epochs.
Its spectral properties—including scale invariance, broken power-law features, and polarization—enable precise parameter inference using detectors like LISA, PTA, and aLIGO.
Advanced cross-correlation and non-Gaussianity detection techniques help separate the SGWB from astrophysical backgrounds, constraining high-energy physics beyond conventional observations.

A stochastic gravitational-wave background (SGWB) from the early universe comprises a superposition of gravitational waves produced by a variety of primordial processes, including inflationary quantum fluctuations, first-order phase transitions, cosmic strings, preheating, bubble collisions, freely decaying turbulence, and other non-standard dynamics. This background is characterized not only by its frequency spectrum and energy density, but also by its polarization, angular anisotropies, and non-Gaussian properties. As gravitational waves interact solely through gravity, the SGWB carries direct information from physical epochs well before photon decoupling, probing energy scales and epochs not accessible to electromagnetic or standard cosmological observations.

1. Primary Early-Universe SGWB Sources and Their Physical Mechanisms

The early-universe SGWB is expected to arise from several distinct mechanisms, with each imprinted in unique spectral features and parameter dependencies (Caprini, 2015, Christensen, 2018):

Inflationary Fluctuations: Tensor (gravitational wave) modes sourced by vacuum fluctuations during an inflationary phase yield a nearly scale-invariant or slightly red-tilted SGWB. The energy density is commonly parameterized by the tensor-to-scalar ratio ( $r$ ) and the tensor spectral tilt ( $n_T$ ).
Cosmic Strings: Networks of topological or quantum strings (including cosmic superstrings) formed in symmetry-breaking phase transitions produce a background via loop oscillations and reconnections. The resultant GW signal depends sensitively on the string tension, loop distribution, and reconnection probability.
First-Order Phase Transitions: Discontinuous transitions generate GW signals through nucleation, expansion, and collision of bubbles, resulting in acoustic waves, turbulence, and direct collision-induced tensor perturbations (Caprini, 2015, Zhong et al., 2021, Auclair et al., 2022). The characteristic frequency $f_c$ today is set by the transition temperature and dynamics:

$f_c \simeq 2.6 \times 10^{-5} \ \text{Hz} \ \epsilon_*^{-1} \left(\frac{T_*}{1\,\text{TeV}}\right) \left(\frac{g_*}{100}\right)^{1/6},$

where $T_*$ is the transition temperature, $\epsilon_*$ quantifies the causality/size scale, and $g_*$ is the effective number of relativistic degrees of freedom.

Preheating and Turbulence: Non-perturbative dynamics at the end of inflation can amplify field fluctuations and generate GW through rapidly evolving anisotropic stress. Freely decaying turbulence post-phase-transition also sources SGWB, with the spectrum dependent upon the kinetic energy, time-correlations of turbulence, and the eddy evolution model (Auclair et al., 2022).
Matter-Dominated Eras and Nonlinear Structure Formation: Departures from standard radiation domination, e.g., an early matter-dominated era, enable density perturbations to grow nonlinear and collapse into halos, boosting SGWB production via anisotropic stress and out-of-equilibrium dynamics (Fernandez et al., 2023).

Table 1: Correspondence of Key SGWB Sources to Detector Bands

SGWB Source	Typical Frequency (Today)	Principal Detector(s)
Inflationary GWs	$10^{-18}$ – $10^2$ Hz	CMB, LISA, PTA, aLIGO
Cosmic Strings	$10^{-10}$ – $10^3$ Hz	PTA, LISA, aLIGO, SKA
Phase Transitions (EW)	$10^{-4}$ –$1$ Hz	LISA, DECIGO, BBO
Preheating/Turbulence	$10^{-4}$ – $10^2$ Hz	LISA, DECIGO, aLIGO
PBH binaries	$10^{-9}$ – $10^3$ Hz	PTA, LISA, aLIGO/Virgo/KAGRA

2. SGWB Spectral Properties and Detector Response

The early-universe SGWB is described by its energy density per logarithmic frequency interval, normalized to the cosmic closure density:

$\Omega_{\mathrm{GW}}(f) = \frac{f}{\rho_c} \frac{d \rho_{\mathrm{GW}}}{df},$

where $\rho_c = 3H_0^2/(8\pi G)$ . Different production mechanisms yield power spectra that are scale-invariant (inflation), broken/bumpy (phase transitions, PBH mergers) or blue-tilted (stochastic spectator-field models) (Ebadi et al., 2023, Caprini, 2015, Braglia et al., 2021).

It is often necessary to model the SGWB spectrum beyond a single power law. Templates incorporating a broken power law, with distinct spectral indices above and below a characteristic break frequency $f_*$ ,

$\Omega_{\mathrm{GW}}(f) = \Omega_* \left(\frac{f}{f_*}\right)^{n_1} \text{ for } f < f_*; \quad \Omega_* \left(\frac{f}{f_*}\right)^{n_2} \text{ for } f > f_*$

are required to capture features resulting from e.g., the peak frequency of turbulence, phase transitions, or the merger cutoff of PBH binaries (Kuroyanagi et al., 2018). Fisher matrix analyses show that failing to model such breaks biases parameter inference and diminishes the discriminating power between production scenarios.

Detector networks such as aLIGO/Virgo/KAGRA, LISA, DECIGO, and PTA experiments (NANOGrav, SKA, IPTA) cover complementary frequency windows and are sensitive to different predicted features (Christensen, 2018, Li et al., 2023).

3. Polarization, Parity, and Non-Gaussian Signatures

Early-universe mechanisms can impart distinctive polarization and non-Gaussian features onto the SGWB:

Circular Polarization and Parity Violation: Parity-violating interactions (e.g., gravitational Chern-Simons terms, pseudo-scalar couplings) during inflation can generate net circular polarization, where the two GW helicities $P_R(k) \neq P_L(k)$ differ (Crowder et al., 2012, Domcke et al., 2019). The parity-odd SGWB component is traditionally inaccessible to coplanar detectors, but exploiting kinematically-induced dipoles (from detector frame boosts) or leveraging non-coplanar networks (e.g., geographically separated observatories or LISA TDI channels) permits measurement of net circular polarization.
Measurement Formalism: The stochastic background can be decomposed via correlators in the circular polarization basis:

$\langle h_R(f,\Omega) h_R^*(f',\Omega') \rangle = \frac{\delta(f-f')\delta^2(\Omega-\Omega')}{4\pi}[I(f) + V(f)],$

with $I(f)$ the intensity and $V(f)$ the circular polarization (with $\Pi(f) = V(f)/I(f)$ the polarization fraction). Cross-correlation detection techniques employ polarization-dependent overlap reduction functions (e.g., $\gamma_I$ , $\gamma_V$ ) for optimal detection.

Non-Gaussianity: The early-universe SGWB may exhibit statistical non-Gaussianity in its angular fluctuations, particularly if sourced by non-linear primordial interactions. The bispectrum (three-point function) of the GW energy density, parameterized by a primordial nonlinearity parameter $f_{\rm NL}$ , probes the degree and nature of these non-Gaussianities (Bartolo et al., 2019). The bispectrum in the squeezed configuration is sensitive to scalar-tensor-tensor (STT) primordial correlations, i.e., mode-couplings of long-wavelength scalar and short-wavelength tensor modes (Adshead et al., 2020).

4. Angular Anisotropies and Their Cosmological Significance

Angular anisotropies in the SGWB provide information about the primordial universe beyond the isotropic monopole. The phase-space distribution function $f(\eta, x^i, q, \hat{n}^i)$ of gravitons is governed by a Boltzmann (Liouville) equation, with scalar and tensor metric perturbations generating both primary and propagation-induced anisotropies (Bartolo et al., 2019, Adshead et al., 2020). Notable features include:

Frequency Dependence: Unlike the CMB, the SGWB receives angular anisotropies both at production and along the line of propagation, and the frequency dependence can be order unity—since gravitational waves are not scattered after production. This differentiates cosmological and astrophysical SGWB sources, the latter often lacking pronounced frequency signatures.
Sachs-Wolfe and Integrated Sachs-Wolfe Effects: The anisotropies include analogs to the Sachs-Wolfe (SW) effect, associated with redshift from gravitational potential at last scattering for the SGWB (in this context at the graviton decoupling era), and the Integrated Sachs-Wolfe (ISW) effect, accruing from time-varying potentials along the graviton path (Dall'Armi et al., 2020).
Imprints of Relativistic Particles: The number of decoupled relativistic species at SGWB decoupling shifts both SW and ISW contributions to the angular power spectrum, leading to characteristic suppressions or shifts which can serve as diagnostics of new particle content (Dall'Armi et al., 2020).

5. SGWB Detection Methodologies and Constraints

The detection of the early-universe SGWB, and separation from astrophysical backgrounds, exploits several key methodologies:

Cross-Correlation Techniques: Because individual stochastic signals are below detector noise, cross-correlation of geographically separated detectors with optimized filters is crucial (Collaboration et al., 2016, Christensen, 2018). The typical estimator is:

$\hat{Y}_\alpha = \int df\,\tilde{s}_1^*(f)\tilde{s}_2(f)\tilde{Q}_\alpha(f),$

with $\tilde{Q}_\alpha(f)$ an optimal frequency filter depending on the target spectral index $\alpha$ and the detectors' response functions.

Gaussianity Tests: Analysis of SGWB data considers whether the background is Gaussian (as expected for the superposition of many primordial sources) or non-Gaussian (as in the case of unresolved astrophysical bursts). Maximum-likelihood estimators for the non-Gaussian duty cycle $\xi$ quantify this (Thrane, 2013). A Gaussian early-universe signal would have $\xi \sim 0$ .
Spectral Feature Extraction: Accurate parameterization of the SGWB using broken power-law or other template shapes is necessary for unbiased source inference. Fisher matrix and $\chi^2$ analyses identify best-fit spectral indices and, when possible, interpret breaks in the spectrum as evidence for specific mechanisms (e.g., first-order phase transitions, PBH formation epochs) (Kuroyanagi et al., 2018).
Current Constraints: Advanced LIGO/Virgo O1 observations set an upper limit of $\Omega_0 < 1.7 \times 10^{-7}$ (95% confidence) for a flat spectrum in 20–86 Hz, a factor $\sim$ 33 improvement over previous measures (Collaboration et al., 2016). CMB measurements strongly constrain the largest-scale (lowest-frequency) SGWB, especially via temperature (TT) and B-mode polarization (BB) angular power spectra (Rotti et al., 2011). Upcoming detectors (LISA, DECIGO, SKA) promise order-of-magnitude improvements in the GHz–nHz windows.

6. Theoretical Extensions and Model Diagnostics

Theoretical developments expand the interpretive reach of SGWB observations:

Big Bounce Cosmologies: Analytical frameworks classify bounce cosmologies by their background evolution phases, propagating tensor modes through four consecutive periods (contracting, bouncing, expanding phases), with explicit formulae for the output SGWB spectrum (Li, 14 Jul 2024). The measured SGWB can, in principle, distinguish bounce cosmologies from standard inflation.
Primordial Black Holes: PBH formation and mergers imprint nontrivial features in the SGWB spectrum, with characteristic bumps corresponding to epochs when the equation of state drops due to particle transitions (e.g., QCD phase transition, $e^\pm$ annihilation). Multiband observations (PTAs, LISA, LIGO) are needed to disentangle details of PBH mass functions, inflationary spectrum, and the universe thermal history (Braglia et al., 2021).
Spectator Fields and Blue-Tilted Spectra: Stochastic evolution of light spectator fields during inflation can enhance small-scale power, producing blue-tilted curvature spectra and, on horizon reentry, induced gravitational waves with $h^2\Omega_{\rm GW} \sim 10^{-20}$ – $10^{-15}$ for $10^{-5}$ –$10$ Hz—prospectively detectable by future GW observatories (Ebadi et al., 2023).
Isocurvature Anisotropies and “Primordial Clocks”: In multifield inflation or models with heavy field oscillations, isocurvature modes induce unique oscillatory, scale-invariance-breaking signatures (“primordial clocks") in the multipole anisotropies of the SGWB (Bodas et al., 2022). These are potentially observable even when hidden in the CMB by cosmic variance.

The early-universe stochastic gravitational-wave background encodes a wealth of cosmological information: the energy scale and dynamics of inflation, the properties of phase transitions, the presence of new high-energy particles, nonlinear gravitational collapse, and the possible breakdown of fundamental symmetries. Ongoing and future multimessenger observations—correlating the SGWB with the cosmic microwave background, mapping its angular, polarization, and non-Gaussian properties—offer unprecedented access to the universe’s most primordial epochs (Adshead et al., 2020).