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Gravitational wave background

The gravitational wave background, often termed the stochastic gravitational wave background (SGWB), is a diffuse, isotropic superposition of gravitational waves emanating from a vast ensemble of unresolved sources across the universe, manifesting as a random, noise-like signal in spacetime that permeates cosmic scales, much like the cosmic microwave background radiation does for electromagnetic waves.^[1] This background arises from two primary categories of origins: astrophysical sources, such as the mergers of compact binaries like black holes and neutron stars, core-collapse supernovae, and rotating neutron stars, which dominate predictions in the millihertz to kilohertz frequency bands; and cosmological sources, including relics from the early universe such as inflationary tensor modes, cosmic strings, and first-order phase transitions during the electroweak or QCD epochs, potentially contributing at nano- to microhertz frequencies.^[1] The intensity of the SGWB is quantified by its fractional energy density per logarithmic frequency interval, denoted as \Omega_\mathrm{GW}(f) = \frac{1}{\rho_c} \frac{d\rho_\mathrm{GW}}{d\ln f}, where \rho_c is the critical density of the universe and f is the gravitational wave frequency, allowing for spectral shape analyses that distinguish between source types—for instance, a power-law spectrum with index \alpha = -2/3 is expected from inspiraling binaries.^[1] Detection of the SGWB relies on indirect methods like cross-correlating signals from multiple detectors to suppress local noise, as individual waves are too weak and overlapping to resolve spatially.^[1] Ground-based interferometers such as Advanced LIGO, Virgo, and KAGRA have set stringent upper limits in the 10–1000 Hz range, with \Omega_\mathrm{GW} \leq 2.0 \times 10^{-9} at 25 Hz from the first part of the fourth observing run (as of 2025).^[2] Pulsar timing arrays (PTAs) probe the nanohertz regime by monitoring millisecond pulsars for timing residuals induced by passing gravitational waves.^[1] A landmark advancement came in 2023, when the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) and international PTA collaborations, including the European Pulsar Timing Array (EPTA) and Parkes Pulsar Timing Array (PPTA), reported compelling evidence for a low-frequency SGWB in their datasets, characterized by a strain amplitude of A = 2.4^{+0.7}_{-0.6} \times 10^{-15} at a reference frequency of 1 yr^{-1} (approximately 30 nHz), with a spectrum consistent with the Hellings-Downs spatial correlation pattern predicted by general relativity for a quadrupolar anisotropic background from unresolved supermassive black hole binaries.^[3] This signal, spanning 2–30 nHz, marks the first evidence for a gravitational wave background and opens avenues for multimessenger astronomy, though its astrophysical interpretation remains under scrutiny against alternatives like cosmic strings or dark matter scenarios.^[3] Future observatories, including space-based detectors like the Laser Interferometer Space Antenna (LISA) slated for the 2030s, are poised to resolve components of the SGWB in the 0.1–100 mHz band, potentially unveiling cosmological signals that probe the universe's earliest moments and test extensions to general relativity.^[1] The SGWB thus represents a frontier for understanding extreme astrophysics, galaxy evolution, and fundamental physics, with ongoing PTA efforts from international collaborations like the International Pulsar Timing Array (IPTA) refining the evidence and constraining source populations.^[3]

Fundamentals

Definition and Characteristics

The gravitational wave background, often referred to as the stochastic gravitational wave background (SGWB), is defined as the incoherent superposition of gravitational waves from a large number of unresolved sources, resulting in a random, noise-like signal that permeates space. This background arises from the cumulative effect of many independent gravitational wave emissions, each too weak or numerous to detect individually, creating a diffuse field analogous to the cosmic microwave background in electromagnetism but in the tensor gravitational domain. Unlike deterministic gravitational wave signals from resolvable single events, such as binary black hole mergers, the SGWB features random phases and amplitudes, making it statistically characterized rather than waveform-matched. Key characteristics of the SGWB include its frequency spectrum, which spans a broad range from nanohertz frequencies probed by pulsar timing arrays to kilohertz frequencies accessible by ground-based interferometers. The strength of the background is quantified by the dimensionless energy density parameter per logarithmic frequency interval,

\Omega_{\rm GW}(f) = \frac{1}{\rho_c} \frac{d\rho_{\rm GW}}{d \ln f},

where \rho_c is the critical energy density of the universe and \rho_{\rm GW} is the gravitational wave energy density; this parameter captures the fractional contribution of gravitational waves to the total energy density at frequency f. For primordial backgrounds, typically associated with early-universe processes, the SGWB is assumed to be isotropic—exhibiting uniform intensity across all sky directions—and Gaussian distributed, reflecting a statistical homogeneity from quantum fluctuations or inflationary origins. The concept of the gravitational wave background was first proposed in the 1960s and 1970s by theorists including Charles Misner, William H. Press, and Kip S. Thorne, who envisioned it as a fundamental probe of cosmic phenomena within the framework of general relativity.

Theoretical Framework

The theoretical framework for gravitational wave backgrounds is rooted in the linearized approximation to general relativity, where weak gravitational fields are described by small perturbations to the flat Minkowski metric. Consider the metric g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, with |h_{\mu\nu}| \ll 1, where \eta_{\mu\nu} is the Minkowski metric and h_{\mu\nu} represents the perturbation. In the weak-field limit, the Einstein field equations G_{\mu\nu} = 8\pi G T_{\mu\nu} reduce to the linearized form, where the Einstein tensor G_{\mu\nu} is expanded to first order in h. For vacuum regions (T_{\mu\nu} = 0), this yields the equation \square \bar{h}_{\mu\nu} = 0, with \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h (trace-reversed perturbation) and \square = -\partial_t^2 + \nabla^2 the flat-space d'Alembertian. To describe propagating waves, the harmonic gauge condition \partial^\mu \bar{h}_{\mu\nu} = 0 is imposed, which simplifies the equations and allows further specialization to the transverse-traceless (TT) gauge for far-field, plane-wave solutions. In the TT gauge, for a wave propagating in the z-direction, the components satisfy h_{0\mu} = 0, \partial^i h_{ij} = 0, and h^i_i = 0, leaving only the spatial transverse components h_{ij}^{TT} (with i,j = x,y) that are traceless and divergence-free. This gauge choice ensures that the perturbation represents physical gravitational radiation without coordinate artifacts.^[4] In vacuum, the TT components obey the wave equation \square h_{ij}^{TT} = 0, implying that gravitational waves propagate at the speed of light c, transverse to the direction of propagation, with two independent polarization states (plus and cross). The general solution for a monochromatic plane wave is h_{ij}^{TT}(t, \mathbf{x}) = \mathrm{Re} \left[ \tilde{h}_{ij}^{TT}(f) e^{i (2\pi f (t - \hat{n} \cdot \mathbf{x}/c))} \right], where f is the frequency and \hat{n} the propagation direction. These waves carry energy and momentum, described by an effective stress-energy pseudotensor in the high-frequency (geometric optics) limit. For a high-frequency wave packet, the averaged energy-momentum tensor is given by

\langle T_{\mu\nu} \rangle = \frac{c^4}{32\pi G} \left\langle \partial_\mu h_{ij}^{TT} \partial_\nu h^{TT ij} \right\rangle,

where the angle brackets denote time averaging over several wavelengths, and the indices are raised with \eta^{\mu\nu}. This expression captures the pseudo-localized energy flux of the gravitational radiation, with the energy density scaling as \langle T_{00} \rangle \propto (\dot{h}_{+}^2 + \dot{h}_{\times}^2), confirming that gravitational waves behave as null fluids in the linear regime.^[5] For stochastic gravitational wave backgrounds, which arise from incoherent superpositions of many unresolved sources, a statistical description is necessary rather than individual waveforms. The background is characterized by the two-point correlation function in frequency space,

\left\langle h_{ij}(\mathbf{f}) h_{kl}(\mathbf{f}') \right\rangle = \delta^{(3)}(\mathbf{f} - \mathbf{f}') P_{ijkl}(\hat{f}) S_h(f),

where \mathbf{f} is the wave vector with |\mathbf{f}| = f, S_h(f) is the one-sided power spectral density, and P_{ijkl}(\hat{f}) is the TT polarization tensor projecting onto the transverse, traceless modes orthogonal to \hat{f}. For an unpolarized, isotropic background, P_{ijkl} averages to \frac{1}{8} (P_{ik} P_{jl} + P_{il} P_{jk} - P_{ij} P_{kl}), with P_{ij} = \delta_{ij} - \hat{f}_i \hat{f}_j. This formalism treats the background as a Gaussian random field, enabling the computation of ensemble averages for detection statistics. The high-frequency approximation justifies averaging over rapid oscillations, yielding an effective stress-energy tensor for the background as in the single-wave case but with correlators replacing individual terms.^[5]^[6] In cosmological contexts, gravitational wave backgrounds are analyzed as tensor perturbations to the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, ds^2 = a(\tau)^2 \left[ -d\tau^2 + (\delta_{ij} + h_{ij}) dx^i dx^j \right], where a(\tau) is the scale factor, \tau the conformal time, and h_{ij} satisfies the TT conditions h^i_i = 0, \partial^i h_{ij} = 0. The evolution equation for these modes in the absence of anisotropic stress is

h_{ij}'' + 2 \frac{a'}{a} h_{ij}' - \nabla^2 h_{ij} = 0,

where primes denote derivatives with respect to \tau. This is analogous to the flat-space wave equation but includes a friction term from cosmic expansion, leading to redshift of the amplitude h_{ij} \propto 1/a during radiation or matter domination. Sources such as quantum fluctuations during inflation or anisotropic stresses from early-universe phase transitions can excite these modes, imprinting a primordial spectrum that evolves with the universe's expansion. This framework connects local gravitational wave propagation to global cosmological dynamics, allowing backgrounds to probe early-universe physics through their relic signatures.^[7]

Detection and Analysis

Observational Methods

Ground-based interferometers, such as the Laser Interferometer Gravitational-wave Observatory (LIGO), Virgo, and KAGRA, operate in the frequency band of approximately 10 to 1000 Hz and utilize laser Michelson interferometers to detect minute changes in spacetime caused by passing gravitational waves. LIGO features two detectors with 4 km arm lengths, achieving a strain sensitivity of around $10^{-23}/\sqrt{\rm Hz} near 100 Hz, while Virgo and KAGRA each have 3 km arms with comparable sensitivities in this band.^[8]^[9] These instruments measure differential arm length variations induced by gravitational waves, enabling searches for stochastic backgrounds from astrophysical sources like compact binary mergers. Pulsar timing arrays (PTAs) provide sensitivity to gravitational wave backgrounds in the nanohertz regime by monitoring millisecond pulsars as stable celestial clocks and analyzing timing residuals for induced perturbations.^[10] Major collaborations include the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), the European Pulsar Timing Array (EPTA), and the Parkes Pulsar Timing Array (PPTA), which collectively observe dozens of pulsars over years to detect correlated signals.^[11] International efforts, such as the International Pulsar Timing Array (IPTA), combine data from these collaborations to enhance sensitivity, with 2024-2025 analyses providing stronger evidence for the background. For an isotropic stochastic background, these arrays search for the characteristic quadrupolar spatial correlation pattern predicted by general relativity, known as the Hellings-Downs curve, which depends on the angular separation between pulsar pairs. Space-based detectors like the Laser Interferometer Space Antenna (LISA), planned for launch around 2035, target the millihertz frequency band (0.1 to 100 mHz) using a constellation of three spacecraft forming a triangular interferometer with 2.5 million km arm lengths.^[12]^[13] LISA will measure gravitational wave-induced displacements of free-floating test masses via laser interferometry, offering enhanced sensitivity to low-frequency backgrounds from supermassive black hole binaries compared to ground-based systems.^[12] Resonant detectors and emerging atom interferometer technologies address gaps in the mid-frequency range (roughly 0.01 to 3 Hz), where ground-based systems are limited by seismic noise and space-based ones by other constraints. Concepts like the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100), under development at Fermilab with a 100 m vertical baseline, use atom interferometry to detect phase shifts from gravitational waves, potentially achieving unprecedented sensitivity in this band as a precursor to longer-baseline experiments.^[14] Multi-messenger synergies enhance the detection of gravitational wave backgrounds by integrating electromagnetic observations to identify and subtract astrophysical foregrounds, such as those from known binary systems, thereby isolating stochastic signals.^[15] This approach leverages counterpart emissions in radio, optical, or X-ray bands to model and remove contaminating contributions, improving the signal-to-noise ratio for background searches across detectors.^[15]

Data Analysis Techniques

Cross-correlation methods form the cornerstone of detecting stochastic gravitational wave backgrounds in data from pulsar timing arrays (PTAs) and ground-based interferometers, leveraging the spatial separation of detectors to suppress uncorrelated noise. The optimal statistic, a frequentist approach, maximizes the signal-to-noise ratio (SNR) by filtering the data to enhance the common gravitational wave signal while minimizing instrumental noise contributions. This statistic is computed from the cross-power spectral density C(f) between detector pairs, with the expected SNR scaling as \mathrm{S/N} = \sqrt{T} \int df \, |C(f)|^2 / \sigma^2(f), where T is the observation time and \sigma(f) represents the noise variance in the cross-spectrum.^[16] For PTAs, this method adapts to timing residuals across multiple pulsars, incorporating the Hellings-Downs correlation pattern expected from an isotropic background.^[17] Bayesian inference provides a robust framework for parameter estimation of the gravitational wave energy density spectrum \Omega_\mathrm{GW}(f), particularly in PTA analyses where noise models and astrophysical foregrounds must be jointly modeled. Markov chain Monte Carlo (MCMC) sampling explores the posterior distribution of parameters such as the amplitude and spectral index of \Omega_\mathrm{GW}(f), incorporating priors on noise characteristics like red spin noise in pulsar timing residuals and potential clock errors in interferometers. This approach enables quantification of uncertainties and model comparison, distinguishing gravitational wave signals from correlated noise processes. In PTA searches, Bayesian methods have been extended to handle anisotropic components by parameterizing sky maps in spherical harmonics.^[18] Machine learning techniques, including neural networks, are increasingly applied to identify anomalies indicative of gravitational wave backgrounds in noisy time-series residuals or spectrograms from detectors. Autoencoder-based architectures, for instance, learn to reconstruct noise-dominated data and flag deviations as potential signals, enabling unsupervised detection of subtle stochastic features without assuming specific waveforms. Convolutional neural networks have shown promise in processing LIGO-Virgo strain spectrograms to separate background signals from glitches and instrumental artifacts, achieving faster inference than traditional matched filtering. In PTA contexts, these methods aid in preprocessing residuals to isolate low-frequency power excesses.^[19] Search pipelines for gravitational wave backgrounds incorporate specialized algorithms like radiometer analyses to probe anisotropic components, constructing intensity maps of the sky by cross-correlating detector pairs with direction-dependent response functions. These pipelines generate all-sky maps for broadband or narrowband searches, enabling localization of potential hotspots from unresolved sources. Map-making techniques further refine these outputs by inverting the data to estimate the background's angular power spectrum, facilitating sky localization with angular resolutions down to degrees for high-frequency bands.^[20] In PTAs, similar pipelines extend to hemispherical asymmetries in pulsar correlations.^[21] Upper limits on the gravitational wave background are derived using both frequentist and Bayesian frameworks to establish confidence intervals on \Omega_\mathrm{GW}. Frequentist methods, such as the optimal statistic, provide one-sided bounds based on null-hypothesis testing against noise-only data, while Bayesian approaches yield posterior credible intervals incorporating full parameter uncertainties. For example, the NANOGrav collaboration's 2023 analysis of its 15-year dataset reported compelling evidence for a stochastic gravitational wave background, consistent with \Omega_\mathrm{GW} h^2 \approx 10^{-8} at a frequency of 3 nHz, with subsequent international PTA efforts strengthening this evidence as of 2025.^[3] These constraints tighten with longer observations and improved noise modeling, guiding interpretations of the signal.

Implications

Cosmological Insights

The detection of a primordial gravitational wave background offers a unique probe into the inflationary epoch of the early universe, allowing tests of slow-roll inflation through measurements of the tensor-to-scalar ratio r and the tensor spectral tilt n_t. These observables directly constrain the slow-roll parameters, such as the first-order parameter \epsilon, which relates to the Hubble rate's variation, and the second-order parameter \eta, which governs the potential's curvature. For instance, current cosmic microwave background (CMB) constraints from BICEP/Keck and Planck imply r < 0.036 at 95% confidence, which under slow-roll relations bounds n_t > -0.005, providing limits on inflationary models where r \approx 16\epsilon and n_t \approx -2\epsilon, thus limiting the energy scale of inflation to below approximately $10^{16} GeV.^[22] A stochastic gravitational wave background arising from first-order phase transitions in the early universe encodes information about bubble nucleation rates and the shape of the Higgs potential, offering insights into physics beyond the Standard Model. The peak frequency and amplitude of the resulting GW spectrum depend on the phase transition's strength, characterized by the inverse duration parameter \beta/H, and the bubble wall velocity, which together determine the energy release during the transition. These signatures can reveal extensions involving new particles, such as singlet scalars or supersymmetric fields, that alter the potential barrier and nucleation dynamics, with detectable signals potentially peaking in the millihertz range for electroweak-scale transitions.^[23]^[24] Gravitational waves produced during reheating after inflation inform the relic abundance of particles and the post-inflation energy scales, particularly in scenarios involving superpartners. The GW spectrum from preheating instabilities, driven by parametric resonance of the inflaton field, depends on the reheating temperature T_{\rm RH} and the decay rates, which set the thermalization process and particle production yields.^[25] In multimessenger cosmology, the gravitational wave background can potentially complement CMB anisotropies and big bang nucleosynthesis (BBN) data to help resolve tensions in the Hubble constant H_0, by providing independent constraints on the early-universe expansion history. Joint analyses of the GW energy density parameter \Omega_{\rm GW} with CMB power spectra and BBN light element abundances (e.g., ^4He and D/H ratios) can break degeneracies in late-time measurements, offering a pathway to mitigate the ~5 km/s/Mpc discrepancy observed as of 2025. This approach leverages the GW background's sensitivity to relativistic degrees of freedom during radiation domination, offering a model-independent test of \LambdaCDM.^[26] Connections between the gravitational wave background and dark matter or dark energy emerge through propagation effects, such as those induced by axion-like particles or modified gravity theories. Axion dark matter fields can cause oscillatory modifications to GW waveforms via scalar-tensor couplings, leading to phase shifts or amplitude variations detectable in pulsar timing arrays or interferometers, with constraints tightening for axion masses around $10^{-22} eV. Similarly, in modified gravity frameworks like Horndeski theories, GW propagation speeds deviating from the speed of light by |\Delta c| < 10^{-15} imprint dispersion relations on the background, potentially linking to dark energy dynamics without altering cosmological distances.^[27]

Astrophysical Applications

The stochastic gravitational wave background (SGWB) in the 10-100 Hz band, primarily from unresolved binary black hole mergers detected by ground-based observatories like LIGO and Virgo, enables inference of the black hole mass function through analysis of merger rates and chirp mass distributions.^[28] The characteristic strain spectrum of the SGWB, modeled as h_c^2(f) \propto f^{-2/3}, encodes the population's chirp mass distribution, where the chirp mass \mathcal{M} = M \eta^{3/5} (with total mass M and symmetric mass ratio \eta) peaks around 14-35 M_\odot, revealing a mixture of power-law and Gaussian components in the mass function.^[28] This approach constrains the low-mass end of the mass function. At nanohertz frequencies, the SGWB from supermassive black hole binaries traces the galaxy merger history, reflecting hierarchical structure formation in the local universe.^[29] These binaries, relics of galaxy mergers, evolve through dynamical friction in galactic centers and clusters, hardening via stellar scattering and gas interactions before gravitational wave domination at milliparsec separations.^[30] The SGWB amplitude and spectrum (h_c(f) \propto f^{-2/3}) provide independent estimates of merger rates, consistent with galaxy pair counts from surveys, and probe delay times between mergers and binary coalescence, typically on gigayear scales.^[29] This mapping illuminates dynamical friction processes in dense environments, where friction timescales influence the binary hardening rate and overall background intensity.^[30] Contributions to the kHz SGWB from neutron star oscillations, such as r-modes and f-modes, offer tests of extreme physics, particularly the neutron star equation of state (EOS).^[31] R-modes, unstable via the Chandrasekhar-Friedman-Schutz mechanism in rapidly rotating stars, emit gravitational waves at frequencies around (4/3) f_{\rm rot} (with f_{\rm rot} the rotation frequency), where the growth rate versus viscous damping depends on the EOS's stiffness, constraining nuclear matter properties like superfluidity.^[32] F-modes, fundamental pressure-supported oscillations excited by glitches or flares, resonate at 1-2 kHz, with frequencies and damping times sensitive to stellar compactness and the pressure-density relation in the EOS.^[31] Upper limits on these signals from targeted searches (e.g., on pulsars like PSR J0537-6910) already bound EOS models, with potential detections enabling measurements of radius and moment of inertia to reduce EOS uncertainties by factors of several.^[32] Deviations in the SGWB spectral shape distinguish binary formation channels, revealing population demographics such as field binaries versus those in globular clusters.^[33] In the field, isolated binaries evolve circularly, producing a smooth f^{-2/3} spectrum, while globular cluster dynamics yield highly eccentric orbits (e > 0.9), generating burst-like signals that steepen the spectrum at millihertz frequencies due to fly-bys and three-body interactions.^[33] Merger rates from clusters (e.g., \gamma_{\rm GC} \approx 3.6 \times 10^{-7} yr^{-1} in the Milky Way) dominate local contributions, with eccentricity distributions imprinting channel-specific features like enhanced power at higher harmonics.^[33] These spectral signatures, separable via waveform modeling, quantify the fraction of dynamically formed binaries (up to 10-20% in some populations), informing isolation versus dense-environment demographics.^[33] Cross-analysis of the SGWB with galaxy catalogs from surveys like DESI and LSST enables spatial mapping of source populations, correlating gravitational wave overdensities with large-scale structure.^[34] Angular power spectra between binary merger localizations and galaxy positions constrain clustering biases, achieving ~50% precision with current LIGO data and DESI bright galaxy sample, improving to ~2.5% with future detectors like the Einstein Telescope and LSST over five years.^[34] At redshifts 1 < z < 2.5, these correlations trace host galaxy distributions, isolating astrophysical sources from contaminants and mapping merger environments without individual event redshifts.^[35] This method, using Fisher forecasts on tomographic bins, reveals spatial anisotropies tied to galaxy evolution, with magnification lensing effects constrained to ~3% precision.^[34]

Current Status and Future Prospects

Recent Results

In June 2023, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration analyzed their 15-year pulsar timing dataset, comprising observations of 68 pulsars, and reported compelling evidence for a stochastic gravitational-wave background at nanohertz frequencies. This evidence includes a common red noise process correlated across the pulsars, with the spatial correlation pattern matching the Hellings-Downs curve predicted by general relativity at a significance of 2–3σ. The signal is consistent with a power-law spectrum characteristic of a nanoHz-frequency background, potentially arising from supermassive black hole binaries, with an inferred energy density parameter in the range Ω_GW ≈ 10^{-10} to 10^{-8} around a reference frequency of a few nanohertz.^[3] The LIGO-Virgo-KAGRA (LVK) collaboration's fourth observing run (O4), which began in May 2023 and continued through 2025, has provided stringent upper limits on the isotropic stochastic gravitational-wave background in the 10–1000 Hz band. Using data from the first half of O4 (O4a), the search yielded upper limits on the energy density of Ω_GW < 1.3 × 10^{-9} (95% confidence) at a pivot frequency of 25 Hz, assuming a flat spectrum; for the astrophysical background expected from unresolved compact binary mergers, the limits are similarly tight, excluding contributions that would dominate over cosmic microwave background signals at these frequencies. These results build on prior O3 constraints and highlight the absence of detectable stochastic signals amid over 200 individual gravitational-wave detections during O4.^[2] Collaborative efforts by the International Pulsar Timing Array (IPTA), combining datasets from NANOGrav, the European Pulsar Timing Array (EPTA), Parkes Pulsar Timing Array (PPTA), and others, have further refined nanohertz constraints through joint analyses up to 2024. The IPTA's second data release and subsequent comparisons exclude high-amplitude gravitational-wave backgrounds with a f^{-2/3} power-law spectrum—typical of supermassive black hole binary populations—at levels above A ≈ 10^{-14} (characteristic strain amplitude) across frequencies of 2–30 nHz, tightening prior limits by factors of 2–3 via improved noise modeling and multi-array coherence. These combined results reinforce the NANOGrav findings while ruling out louder backgrounds that could mimic alternative astrophysical or cosmological sources.^[36]^[37] No confirmed detections of a primordial gravitational-wave background have emerged, with constraints primarily from cosmic microwave background (CMB) polarization analyses. The Planck 2018 dataset, combined with BICEP/Keck observations up to 2018, limits the tensor-to-scalar ratio to r < 0.056 (95% confidence), corresponding to negligible primordial contributions at Ω_GW < 10^{-15} on large scales; updated joint analyses through 2021 further tighten this to r < 0.036, excluding significant inflationary gravitational waves without primordial B-mode detections in ongoing ground-based searches like those from the BICEP3 and Keck Array experiments.^[38]^[39] Advancements in preparation for future detectors include refinements from the LISA mission's preparatory phase, following its formal adoption by the European Space Agency in January 2024, which leverages legacy data from the 2015–2017 LISA Pathfinder mission to validate noise reduction techniques for millihertz stochastic background sensitivity. Construction of the LISA mission began in June 2025 following an agreement with OHB System AG.^[40] Similarly, the Einstein Telescope project entered pre-construction in 2024, with 2025 design studies incorporating initial geotechnical data from proposed underground sites to optimize third-generation ground-based detection of stochastic signals in the 1–10 Hz range.^[40]

Upcoming Experiments

The Laser Interferometer Space Antenna (LISA), a joint ESA-NASA mission scheduled for launch around 2035, will pioneer space-based detection of gravitational waves in the millihertz band, with a predicted characteristic strain sensitivity of h_c \sim 10^{-20} at 3 mHz.^[12]^[41] This sensitivity will enable LISA to resolve and characterize the stochastic gravitational wave background arising from a cosmic population of supermassive black hole binaries throughout the universe. Third-generation ground-based observatories, such as the proposed Cosmic Explorer in the United States and the Einstein Telescope in Europe, are targeted for operation in the 2030s and will feature significantly longer interferometer arms to boost sensitivity. Cosmic Explorer plans for facilities with 20 km and 40 km arms, while the Einstein Telescope will employ 10 km underground arms in a triangular configuration.^[42] These designs aim to achieve stochastic background sensitivities of \Omega_{GW} \sim 10^{-12} at 10 Hz, allowing detection of faint cosmological signals that current detectors cannot access.^[43] Enhancements to pulsar timing arrays (PTAs) represent another key advancement, with next-generation efforts building on the International Pulsar Timing Array by incorporating more precisely timed millisecond pulsars and integrating observations from the Square Kilometre Array (SKA) telescope. SKA's expanded pulsar catalog and improved timing precision at nanohertz frequencies will sharpen constraints on the stochastic gravitational wave background from supermassive black hole binaries and potentially primordial sources.^[44] Conceptual space-based detectors like the Deci-hertz Interferometer Gravitational-wave Observatory (DECIGO) and the Big Bang Observer (BBO) are proposed to fill the 0.1–10 Hz frequency gap between ground-based and LISA observatories. DECIGO, a Japanese-led pathfinder mission, targets primordial gravitational waves from cosmic inflation with enhanced sensitivity in this band.^[45] BBO, envisioned as a NASA follow-on, would further refine these measurements to probe relic backgrounds at levels below astrophysical foregrounds.^[46] Ground-based cosmic microwave background (CMB) experiments, including the Simons Observatory and CMB Stage-4 (CMB-S4), will provide indirect constraints on primordial gravitational waves through measurements of CMB B-mode polarization. The Simons Observatory, deploying in the late 2020s, will map the southern sky to achieve tensor-to-scalar ratio sensitivities of \sigma(r) \sim 0.01, while CMB-S4, operational in the early 2030s across multiple sites, aims for \sigma(r) < 0.001 to distinguish inflationary signals from lensing and foreground effects.

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ESA - LISA factsheet - European Space Agency
Name: Laser Interferometer Space Antenna (LISA) · Planned launch: ~2035 · Mission: First gravitational wave detector in space. · Status: On 20 June 2017, LISA was ...Missing: length | Show results with:length
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ESA Science & Technology - Summary - European Space Agency
There are five types of gravitational wave sources that are expected to be detectable in the frequency band from 0.1 to 100 mHz where LISA is expected to have ...
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MAGIS-100 – A next-generation atom interferometer to search for ...
This novel detector will search for ultralight dark matter, test quantum mechanics in new regimes and pave the way for future gravitational wave detectors. The ...Missing: mid- frequency
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Using electromagnetic observations to aid gravitational-wave ...
We present a first-stage study of the effect of using knowledge from electromagnetic (EM) observations in the gravitational wave (GW) data analysis of Galactic ...
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Detection methods for stochastic gravitational-wave backgrounds
Apr 4, 2017 · Cross-correlation methods can be used whenever one has multiple detectors that respond to the common gravitational-wave background. For single ...
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Analytic distribution of the optimal cross-correlation statistic for ...
I. INTRODUCTION. Pulsar timing array (PTA) collaborations have recently reported evidence for a stochastic background of nanohertz gravitational wavesMissing: interferometers | Show results with:interferometers
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Bayesian parameter estimation for targeted anisotropic gravitational ...
In this paper, we introduce an alternative Bayesian formalism to search for such stochastic signals with a particular distribution of anisotropies on the sky.
[31]
A semisupervised machine learning search for never-seen ...
We apply deep learning and outlier detection algorithms to labelled spectrograms of GW strain data, and then search for spectrograms with anomalous patterns ...
[32]
Very fast stochastic gravitational wave background map making ...
Jul 2, 2018 · Radiometric techniques are used to make sky maps of anisotropies in the SGWB by cross-correlating data from pairs of detectors. The conventional ...
[33]
Search for anisotropic gravitational-wave backgrounds using data ...
Jul 27, 2021 · The Virgo detector has been used in the anisotropic GWB analysis, here we provide sensitivity maps for all the three baselines for comparison.
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[1201.6641] Limits on the Stochastic Gravitational Wave Background ...
Jan 31, 2012 · We present an analysis of high-precision pulsar timing data taken as part of the North American Nanohertz Observatory for Gravitational waves (NANOGrav) ...
[35]
Gravitational-Wave Cosmology across 29 Decades in Frequency
Mar 31, 2016 · Observations of the cosmic microwave background alone limit the gravitational-wave spectral index at 95% confidence to for a tensor-to-scalar ...
[36]
An accurate bound on tensor-to-scalar ratio and the scale of inflation
Jun 19, 2013 · In this paper we provide an accurate bound on primordial gravitational waves, ie tensor-to-scalar ratio (r) for a general class of single-field models of ...Missing: background | Show results with:background
[37]
Mapping the shape of the scalar potential with gravitational waves
May 2, 2019 · We study the dependence of the observable stochastic gravitational wave background induced by a first-order phase transition on the global ...
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Gravitational waves from phase transitions at the electroweak scale ...
If there was a first-order phase transition in the early universe, there should be an associated stochastic background of gravitational waves.
[39]
Detectable gravitational waves from preheating probes nonthermal ...
Jan 22, 2024 · In this paper, we consider the scenario in which the DM relic abundance is set via the inflaton oscillations and decay, which is similar to the ...
[40]
Observational constraint on axion dark matter with gravitational waves
If dark matter in the Milky Way (MW) is composed of axions, the propagating GWs are amplified and delayed. Since axions are expected to be cold dark matter in ...
[41]
[PDF] arXiv:2408.09024v1 [astro-ph.HE] 16 Aug 2024
Aug 16, 2024 · Kamionkowski, Black hole mass function from grav- itational wave measurements, Phys. Rev. D 95, 103010 · (2017), arXiv:1611.01157 [astro-ph.CO] ...
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[PDF] arXiv:2406.17010v1 [astro-ph.CO] 24 Jun 2024
Jun 24, 2024 · The single supermassive black hole mass function ϕ at z = 0 can be estimated from local scaling relations that relate the black hole mass to the ...
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[PDF] arXiv:1808.06020v1 [astro-ph.GA] 17 Aug 2018
Aug 17, 2018 · The dominant source of these nanohertz GWs is likely to be a population of SMBH binaries that are formed naturally in post-merger galaxies. The ...
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[PDF] arXiv:2306.17821v3 [astro-ph.HE] 9 May 2024
May 9, 2024 · The SMBH binaries are under- stood to be relics of galaxy mergers which have resulted in the hierarchical growth of structure in the Universe.
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[PDF] Gravitational Waves from Neutron Stars: A Review - arXiv
Aug 26, 2015 · 1 kHz for most equations of state Cook et al. ... average of 1/τc, and Ωgw(fgw) is the energy density in the gravitational wave background as a ...
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[PDF] arXiv:2206.06447v1 [astro-ph.HE] 13 Jun 2022
Jun 13, 2022 · ... gravitational wave background emitted by a population of neutron stars with a rotational frequency distribution similar to that of known ...
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[PDF] arXiv:2403.04832v2 [astro-ph.HE] 13 Jun 2024
Jun 13, 2024 · Mandic, “On the stochastic gravitational wave background from binary black hole mergers dynamically assembled in dense star clusters,” (2024) ...<|separator|>
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[PDF] arXiv:2412.01678v1 [astro-ph.CO] 2 Dec 2024
Dec 2, 2024 · Using Equation 22 to Equation 28 we can thus construct cross- correlations between different GW experiments and galaxy surveys, and proceed to ...
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[PDF] Constraining extended cosmologies with GW×LSS cross-correlations
Dec 1, 2023 · In this work we explore which combination of gravitational wave and galaxy survey datasets are able to provide the best constraints both on ...
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[2508.20721] Upper Limits on the Isotropic Gravitational-Wave ...
Aug 28, 2025 · This paper discusses upper limits on the isotropic gravitational-wave background from the first part of LIGO, Virgo, and KAGRA's fourth ...Missing: stochastic 2023-2025
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Comparing Recent Pulsar Timing Array Results on the Nanohertz ...
Given that each PTA made different choices in modeling their data, we perform a comparison of the GWB and individual pulsar noise parameters across the results ...Missing: limits | Show results with:limits
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The second data release from the European Pulsar Timing Array
In EPTA Collaboration and InPTA Collaboration (2024) ... Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array ...
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Planck and BICEP/Keck Array 2018 constraints on primordial ... - arXiv
Aug 22, 2022 · Current and future B-mode polarization data are the most powerful observables to constrain gravitational waves from the early Universe.Missing: detection searches 2025
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Improved Constraints on Primordial Gravitational Waves using ...
Oct 4, 2021 · BICEP3 is equivalent to about eight of the Keck Array 95 GHz receivers [15] so the BK18 dataset is equivalent to about 28, 18, and 14 Keck ...
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Capturing the ripples of spacetime: LISA gets go-ahead - ESA
Jan 25, 2024 · Gravitational waves will cause tiny changes in the distances between the masses in the different spacecraft, and the mission will track these ...
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LISA – Laser Interferometer Space Antenna – NASA Home Page
The international network of ground-based gravitational wave detectors has begun its fourth observing run after a series of upgrades to enhance sensitivity.LISA Project Home · For Scientists · LISA Preparatory Science · FAQMissing: length | Show results with:length
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Cosmic Explorer
Cosmic Explorer is a next-generation observatory concept that will greatly deepen and clarify humanity's gravitational-wave view of the cosmos.Missing: experiments PTA BBO Simons CMB- S4
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[PDF] Cosmic Explorer - arXiv
Sep 20, 2021 · Cosmic Explorer's in- creased sensitivity comes primarily from scaling up the detector's length from to km. This increases the amplitude of ...
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The prospects of pulsar timing with new-generation radio telescopes ...
Apr 16, 2018 · The improved timing precision will reveal new phenomena and also allow us to make a detection of gravitational waves in the nanohertz frequency ...Missing: enhancements | Show results with:enhancements
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Current status of space gravitational wave antenna DECIGO and B ...
Jun 24, 2020 · DECIGO is the future Japanese space mission with a frequency band of 0.1 Hz to 10 Hz. DECIGO aims at the detection of primordial gravitational waves.Missing: BBO relics
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Space-based gravitational-wave detectors can determine the ...
Jun 2, 2008 · Here we argue that future space-based gravitational-wave detectors such as DECIGO [4] or the Big Bang Observer (BBO), are very useful for this ...Missing: relics | Show results with:relics