Scalar-induced gravitational waves from a box-shaped curvature power spectrum
Abstract
We compute the stochastic background of gravitational waves (GWs) produced at second order in cosmological perturbation theory by a primordial curvature power spectrum that is flat in ln k over a finite band [k-,k+]=[k*e−Δ,k*e+Δ] and vanishes elsewhere. This logarithmic box interpolates between a monochromatic spectrum as Δ→0 and a locally scale-invariant plateau as Δ→∞, and its sharp boundaries make the geometry of the source convolution unusually transparent. Working in the radiation era, we reduce the convolution to a compact integral over the overlap between the momentum triangle and the box, and evaluate it analytically in two regimes. For a narrow box we show that, to leading order in the width, the spectrum equals the Dirac-spectrum result multiplied by a purely geometric overlap factor Φbox(κ,Δ); this factor turns the infrared slope from k3ln2k into k2ln2k at a break kb=2k*sinhΔ. For a broad box we separate the lower edge, the scale-invariant interior, and the upper edge, derive the leading behaviour in each (an infrared k3ln2k rise, a scale-invariant plateau, and a quartic cut-off at the hard endpoint k=2k+), and combine them into a single uniform formula cast as a product of two universal, Δ-independent edge functions. We also provide an integral-free closed-form surrogate for these edge functions for use in parameter scans.
Related
- PrimordialPy: A Python library for single-field inflation, primordial black holes, and scalar-induced gravitational waves
- Scalar Perturbations and Induced Gravitational Waves from First-Order Phase Transitions in Lattice Simulations
- Primordial Black Hole Abundances and Scalar Induced Gravitational Waves from Finite-Width Power Spectra in a Stiff Thermal History
- Ultra-slow-roll Inflation with Non-perturbative Non-Gaussianity and Scalar Induced Gravitational Waves
- Angular power spectrum of induced gravitational waves: breaking model and parameter degeneracies in PTA observations
BibTeX
@misc{chen2026scalarinduced,
title = {{Scalar-induced gravitational waves from a box-shaped curvature power spectrum}},
author = {Zu-Cheng Chen and Lang Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.10730},
eprint = {2607.10730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10730},
}
RIS
TY - UNPB
TI - Scalar-induced gravitational waves from a box-shaped curvature power spectrum
AU - Zu-Cheng Chen
AU - Lang Liu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2607.10730
UR - https://arxiv.org/abs/2607.10730
AB - We compute the stochastic background of gravitational waves (GWs) produced at second order in cosmological perturbation theory by a primordial curvature power spectrum that is flat in ln k over a finite band [k_-,k_+]=[k_*e^−Δ,k_*e^+Δ] and vanishes elsewhere. This logarithmic box interpolates between a monochromatic spectrum as Δ→0 and a locally scale-invariant plateau as Δ→∞, and its sharp boundaries make the geometry of the source convolution unusually transparent. Working in the radiation era, we reduce the convolution to a compact integral over the overlap between the momentum triangle and the box, and evaluate it analytically in two regimes. For a narrow box we show that, to leading order in the width, the spectrum equals the Dirac-spectrum result multiplied by a purely geometric overlap factor Φ_box(κ,Δ); this factor turns the infrared slope from k^3ln^2k into k^2ln^2k at a break k_b=2k_*sinhΔ. For a broad box we separate the lower edge, the scale-invariant interior, and the upper edge, derive the leading behaviour in each (an infrared k^3ln^2k rise, a scale-invariant plateau, and a quartic cut-off at the hard endpoint k=2k_+), and combine them into a single uniform formula cast as a product of two universal, Δ-independent edge functions. We also provide an integral-free closed-form surrogate for these edge functions for use in parameter scans.
ER -