The first search for scalar-induced gravitational waves via pulsar parameter drifts yields f_PBH < 10^{-10} (95% CL) for PBH masses 0.3 to 4e4 solar masses, strongly disfavoring a primordial black hole origin for LVK binary black holes.
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Scalar induced gravitational waves review
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abstract
We provide a review on the state-of-the-art of gravitational waves induced by primordial fluctuations, so-called induced gravitational waves. We present the intuitive physics behind induced gravitational waves and we revisit and unify the general analytical formulation. We then present general formulas in a compact form, ready to be applied. This review places emphasis on the open possibility that the primordial universe experienced a different expansion history than the often assumed radiation dominated cosmology. We hope that anyone interested in the topic will become aware of current advances in the cosmology of induced gravitational waves, as well as becoming familiar with the calculations behind.
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representative citing papers
Hubble-flow models of ultra-slow-roll inflation introduce spurious UV artefacts in the potential that Fourier filtering removes, restoring Wands duality.
Accounting for the minimal mass spread of primordial black holes from gravitational collapse suppresses the Poltergeist GW background to the level of generic scalar-induced signals and reopens ultra-light PBH parameter space.
For infinitely differentiable effective potentials describing the post-inflation transition, the regularized power spectrum of primary gravitational waves exhibits exponential suppression at small scales.
Dissipation of small-scale primordial perturbations after neutrino decoupling cools relic neutrinos and reduces their abundance, enabling PTOLEMY to constrain the primordial curvature power spectrum to O(0.1) on scales k ≲ 3×10^5 Mpc^{-1}.
The conventional truncation in stochastic inflation is inconsistent because quadratic-noise contributions are the same perturbative order as the deterministic non-Markovian corrections.
A quenched-disorder approach with Schwinger-Keldysh path integrals produces an averaged density matrix for gravitational waves that separates phase-suppressing exponential terms from oscillatory corrections to coherent propagation.
The work shows that free-streaming dark radiation isocurvature produces a qualitatively different gravitational wave spectrum than cold dark matter isocurvature and derives constraints on isocurvature power spectra around 10^6 Mpc^{-1} from NANOGrav data.
Lattice simulations directly calculate SIGW spectra with non-Gaussianity to all orders and show that modest non-Gaussianity alters ultraviolet spectral behavior.
In pure axion inflation, detectable gravitational wave signals arise only in parameter regions with strong backreaction that violate the upper bound on ΔN_eff.
An open quantum system treatment of curvature perturbations during Ultra-Slow-Roll inflation shows that environmental decoherence erases the interference dip, modifies the growth slope, and induces oscillatory features in the scalar power spectrum and scalar-induced gravitational waves.
Periodic warm inflation imprints one log-periodic feature on the curvature spectrum that saturates asteroid-mass PBHs, generates dual-band GW backgrounds, and offsets the bispectrum phase by a quarter cycle fixed by spectral running.
Hybrid inflation's waterfall tachyonic instability grows isocurvature modes that convert to curvature perturbations at the field-space turn, yielding a k^{3}-peaked spectrum with always-positive f_NL that enhances PBH formation.
Curvaton self-interactions in non-quadratic potentials produce a local non-Gaussian map that enables supermassive primordial black hole formation at peak amplitudes of order 10^{-5} while remaining consistent with μ-distortion bounds.
Residual anomalous coherence in scalar perturbations imprints on scalar-induced tensor modes, producing a correlated gravitational-wave background with nontrivial covariance and phase structure instead of a simple spectral shift.
Scalar-induced tensor modes inherit anomalous coherence from a decohered two-mode Gaussian scalar state, generating nonzero tensor discord and connected power covariance as probes of primordial quantum memory.
Incorporating the general-relativity mass tail df_PBH/d ln M ∝ M^3.78 smooths PBH evaporation, suppresses the scalar-induced GW signal by orders of magnitude, and reopens the ultra-light PBH window for the hot Big Bang.
Domain wall annihilation imprints a two-peaked spectrum on induced gravitational waves via an early matter-dominated phase and entropy dilution.
A phenomenological dipole anisotropy in primordial perturbations induces dipolar and quadrupolar anisotropies in SIGW energy density spectra, producing frequency-dependent PTA overlap reduction functions that depend on pulsar sky distribution, but NANOGrav 15-year data yields no significant evidence
Derives stochastic equations from Schwinger-Keldysh formalism that include quantum diffusion and classical metric perturbations for non-perturbative ultra-slow-roll inflation, validated on Starobinsky and critical Higgs models.
Multi-phase inflation with chiral vector interactions generates amplified primordial magnetic fields that induce a detectable circularly polarized gravitational-wave background.
Induced gravitational waves during inflation obey a universal tensor spectral index formula that yields near scale-invariance for slow-roll expansion regardless of the source field's original spectrum.
Extends diagrammatic approach for scalar-induced gravitational waves to arbitrary-order local PNG, deriving semi-analytic spectra for energy density, anisotropies, bispectrum and trispectrum up to quartic terms.
Covariant analysis of curvature perturbations from first-order phase transitions reveals gauge-dependent overestimation of primordial black holes and gravitational waves in prior non-covariant calculations, leading to strong suppression of both signals.
citing papers explorer
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Constraints on the Primordial Black Hole Abundance using Pulsar Parameter Drifts
The first search for scalar-induced gravitational waves via pulsar parameter drifts yields f_PBH < 10^{-10} (95% CL) for PBH masses 0.3 to 4e4 solar masses, strongly disfavoring a primordial black hole origin for LVK binary black holes.
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UV artefacts in ultra-slow-roll models of inflation
Hubble-flow models of ultra-slow-roll inflation introduce spurious UV artefacts in the potential that Fourier filtering removes, restoring Wands duality.
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Gravitational Waves from Black Hole Reheating: The Scalar-Induced Component
Accounting for the minimal mass spread of primordial black holes from gravitational collapse suppresses the Poltergeist GW background to the level of generic scalar-induced signals and reopens ultra-light PBH parameter space.
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Primary gravitational waves at high frequencies II: Emergence of the exponential cut-off in the power spectrum
For infinitely differentiable effective potentials describing the post-inflation transition, the regularized power spectrum of primary gravitational waves exhibits exponential suppression at small scales.
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Probing the small-scale primordial power spectrum via relic neutrinos and acoustic reheating
Dissipation of small-scale primordial perturbations after neutrino decoupling cools relic neutrinos and reduces their abundance, enabling PTOLEMY to constrain the primordial curvature power spectrum to O(0.1) on scales k ≲ 3×10^5 Mpc^{-1}.
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A consistent formulation of stochastic inflation I: Non-Markovian effects and issues beyond linear perturbations
The conventional truncation in stochastic inflation is inconsistent because quadratic-noise contributions are the same perturbative order as the deterministic non-Markovian corrections.
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Gravitational-wave lensing beyond rays: a disordered-system approach
A quenched-disorder approach with Schwinger-Keldysh path integrals produces an averaged density matrix for gravitational waves that separates phase-suppressing exponential terms from oscillatory corrections to coherent propagation.
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Isocurvature Induced Gravitational Waves at Pulsar Timing Arrays
The work shows that free-streaming dark radiation isocurvature produces a qualitatively different gravitational wave spectrum than cold dark matter isocurvature and derives constraints on isocurvature power spectra around 10^6 Mpc^{-1} from NANOGrav data.
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Scalar-induced gravitational waves with non-Gaussianity up to all orders
Lattice simulations directly calculate SIGW spectra with non-Gaussianity to all orders and show that modest non-Gaussianity alters ultraviolet spectral behavior.
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Gravitational waves from axion inflation in the gradient expansion formalism. Part I. Pure axion inflation
In pure axion inflation, detectable gravitational wave signals arise only in parameter regions with strong backreaction that violate the upper bound on ΔN_eff.
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When the Environment Speaks: Quantum Signatures in Non-Attractor Inflation
An open quantum system treatment of curvature perturbations during Ultra-Slow-Roll inflation shows that environmental decoherence erases the interference dip, modifies the growth slope, and induces oscillatory features in the scalar power spectrum and scalar-induced gravitational waves.
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One Feature, Three Clocks: Phase-Locked Gravitational Waves, Primordial Black Holes, and Non-Gaussianity from Periodic Warm Inflation
Periodic warm inflation imprints one log-periodic feature on the curvature spectrum that saturates asteroid-mass PBHs, generates dual-band GW backgrounds, and offsets the bispectrum phase by a quarter cycle fixed by spectral running.
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Superhorizon curvature perturbations in hybrid inflation revisited
Hybrid inflation's waterfall tachyonic instability grows isocurvature modes that convert to curvature perturbations at the field-space turn, yielding a k^{3}-peaked spectrum with always-positive f_NL that enhances PBH formation.
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Memoirs of the curvaton: non-perturbative non-Gaussianity and supermassive primordial black holes
Curvaton self-interactions in non-quadratic potentials produce a local non-Gaussian map that enables supermassive primordial black hole formation at peak amplitudes of order 10^{-5} while remaining consistent with μ-distortion bounds.
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Entanglement, Discord, and Residual Coherence in Scalar-Induced Gravitational Waves
Residual anomalous coherence in scalar perturbations imprints on scalar-induced tensor modes, producing a correlated gravitational-wave background with nontrivial covariance and phase structure instead of a simple spectral shift.
-
Quantum Memory in Scalar-Induced Gravitational Waves
Scalar-induced tensor modes inherit anomalous coherence from a decohered two-mode Gaussian scalar state, generating nonzero tensor discord and connected power covariance as probes of primordial quantum memory.
-
Opening the Window of Ultra-Light PBHs by Exorcising the Poltergeist
Incorporating the general-relativity mass tail df_PBH/d ln M ∝ M^3.78 smooths PBH evaporation, suppresses the scalar-induced GW signal by orders of magnitude, and reopens the ultra-light PBH window for the hot Big Bang.
-
Imprint of domain wall annihilation on induced gravitational waves
Domain wall annihilation imprints a two-peaked spectrum on induced gravitational waves via an early matter-dominated phase and entropy dilution.
-
Exploring the statistical anisotropy of primordial curvature perturbations with pulsar timing arrays
A phenomenological dipole anisotropy in primordial perturbations induces dipolar and quadrupolar anisotropies in SIGW energy density spectra, producing frequency-dependent PTA overlap reduction functions that depend on pulsar sky distribution, but NANOGrav 15-year data yields no significant evidence
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Nonperturbative stochastic inflation in perturbative dynamical background
Derives stochastic equations from Schwinger-Keldysh formalism that include quantum diffusion and classical metric perturbations for non-perturbative ultra-slow-roll inflation, validated on Starobinsky and critical Higgs models.
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Chiral gravitational waves from multi-phase magnetogenesis
Multi-phase inflation with chiral vector interactions generates amplified primordial magnetic fields that induce a detectable circularly polarized gravitational-wave background.
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A universal scaling law for gravitational waves induced during inflation
Induced gravitational waves during inflation obey a universal tensor spectral index formula that yields near scale-invariance for slow-roll expansion regardless of the source field's original spectrum.
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Isotropy, anisotropies and non-Gaussianity in the scalar-induced gravitational-wave background: diagrammatic approach for primordial non-Gaussianity up to arbitrary order
Extends diagrammatic approach for scalar-induced gravitational waves to arbitrary-order local PNG, deriving semi-analytic spectra for energy density, anisotropies, bispectrum and trispectrum up to quartic terms.
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Curvature Perturbations from First-Order Phase Transitions: Implications to Black Holes and Gravitational Waves
Covariant analysis of curvature perturbations from first-order phase transitions reveals gauge-dependent overestimation of primordial black holes and gravitational waves in prior non-covariant calculations, leading to strong suppression of both signals.
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Nanohertz gravitational waves from domain walls nucleated during inflation
Finite-duration domain-wall nucleation during inflation produces a curvature power spectrum peaked at nanohertz frequencies whose scalar-induced gravitational waves match NANOGrav and EPTA data in a two-field model.
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Universal Suppression of Gravitational Waves from Black Hole Evaporation Dynamics
Black hole populations with finite-width mass distributions exhibit universal late-time evaporation-driven evolution that produces characteristic power-law suppression of induced gravitational waves, directly linking the asymptotic GW spectrum to the evaporation law.
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Matching second-order classical and 1-loop quantum tensor power spectra in de Sitter spacetime
Classical part of 1-loop tensor power spectrum in de Sitter is IR divergent but cancels with vacuum part, enabling non-perturbative renormalization to extract unaffected physical information.
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Transient Parity Violation during Inflation: Implications for PTA Gravitational Waves
A transient parity-violating phase during inflation generates a robust blue-tilted (n_T ≃ 2) primordial gravitational wave spectrum at small scales with nearly maximal helicity coherence and linear polarization, offering a cosmological template for PTA data distinct from astrophysical sources.
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Purely quadratic non-Gaussianity from tachyonic instability: Primordial black holes and scalar-induced gravitational waves
For tachyonic models with ζ=A(φ²−σ²), the PBH abundance is exponentially set by the correlation ρ between field and radial gradient; narrow spectra drive ρ→−1 and suppress PBH formation without killing the induced GW signal.
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Hunting Dark Matter with the Einstein Telescope
Clustered primordial black holes may constitute all dark matter and produce a flat stochastic gravitational wave background detectable by the Einstein Telescope.
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Gravitational Waves from Matter Perturbations of Spectator Scalar Fields
A spectator scalar field with strong portal coupling to the inflaton sources a stochastic gravitational wave background reaching Ω_GW h² ∼ 10^{-11} at frequencies 10^7-10^8 Hz for benchmark parameters σ/λ ≃ 10^4 and T_reh = 2×10^{14} GeV.
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Primary gravitational waves at high frequencies I: Origin of suppression in the power spectrum
Adiabatic regularization combined with smoothed transitions suppresses the high-frequency oscillations in the power spectrum of primary gravitational waves about a zero mean.
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Stochastic Gravitational Waves from Modulated Reheating
A spectator scalar in modulated reheating with large Higgs-like couplings generates detectable scalar-induced stochastic gravitational waves for BBO and DECIGO, but only outside perturbative low-energy extrapolations.
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Relic gravitational waves from primordial gravitational collapses
Sound shell collisions from Hubble-scale primordial density perturbations generate a stochastic GW background whose peak frequency and amplitude scale with the Hubble horizon and shell abundance.
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Gravitational Waves from Primordial Black Holes: Connecting Low-Frequency Scalar-Induced Signatures to High-Frequency Binary Mergers
For monochromatic primordial black holes, the low-frequency scalar-induced gravitational-wave peak and the high-frequency binary-merger ISCO frequency are linked by fISCO ≈ 3.4×10^20 Hz × (fSIGW/Hz)^2.
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Exploring Gravitational Wave Signatures Due to Primordial Non-gaussianity and Large Scale Structure Using SKAO
Explores SKAO detection of scalar-induced GW backgrounds as probes of primordial non-Gaussianity and parity violation, with LSS cross-correlation to improve SNR.
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Multi-peaked high-frequency gravitational waves from PBH-assisted leptogenesis
PBH-assisted leptogenesis produces a multi-peaked GW spectrum in the MHz-EHz range from RHN graviton bremsstrahlung plus PBH-related sources.
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Inflationary interpretation of the gravitational-wave signal in the European Pulsar Timing Array DR2 with constraints
Constrains inflationary tensor parameters to fit the EPTA DR2 signal under CMB, BBN and LVK bounds, favoring radiation-era horizon re-entry but requiring low reheating temperatures.
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Gravitational Waves from Post-Inflationary Magnetism: Direct and Scalar-Induced Contributions
In a post-inflationary magnetogenesis scenario with time-dependent gauge couplings, magnetic anisotropic stress dominates peak GW amplitude while scalar-induced terms matter on larger scales, both showing f^3 infrared scaling for blue spectra and potentially reaching PTA frequencies.
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Precision Analysis for $\boldsymbol{H_0}$ Using Upcoming Multi-band Gravitational Wave Observations
Multi-band GW observations of PBHs can reduce H0 uncertainty to ≲2 km/s/Mpc (conservative) or O(0.1) km/s/Mpc (optimistic) via Fisher forecasts on M_PBH and f_PBH.
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Probing non-Gaussianity during reheating with SIGW in the LISA band
Non-standard reheating imprints detectable features on SIGW spectra via non-Gaussianity, with dynamics that can suppress or boost the signal amplitude for LISA.
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Probing the scalar-induced gravitational waves with the Five-hundred-meter Aperture Spherical radio Telescope and the Square Kilometer Array
Forecasts show that adding FAST/SKA upper or lower limits on scalar-induced GWs to CMB+BAO data shifts the scalar spectral index ns by several sigma in LambdaCDM+r and extended models, offering a potential detection indicator.
- Primordial Black Holes: A Review of Formation and Evolution