Scalar-induced gravitational waves from a box-shaped curvature power spectrum
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 $[k_-,k_+]=[k_*e^{-\Delta},k_*e^{+\Delta}]$ and vanishes elsewhere. This logarithmic box interpolates between a monochromatic spectrum as $\Delta\to0$ and a locally scale-invariant plateau as $\Delta\to\infty$, 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 $\Phi_{\rm box}(\kappa,\Delta)$; this factor turns the infrared slope from $k^{3}\ln^{2}k$ into $k^{2}\ln^{2}k$ at a break $k_b=2k_*\sinh\Delta$. 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 $k^{3}\ln^{2}k$ 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, $\Delta$-independent edge functions. We also provide an integral-free closed-form surrogate for these edge functions for use in parameter scans.