来自盒形曲率功率谱的标量诱导引力波
Scalar-induced gravitational waves from a box-shaped curvature power spectrum
AI总结：
研究来自盒形曲率功率谱的标量诱导引力波，通过在辐射时代简化卷积并解析评估，给出窄盒和宽盒情况下引力波频谱公式及相关函数，为参数扫描提供无积分封闭形式替代。
AI中文摘要：
我们计算了宇宙学微扰理论中二阶产生的引力波随机背景，其原初曲率功率谱在有限频段$[k_-,k_+]=[k_*e^{-\Delta},k_*e^{+\Delta}]$内$\ln k$平坦，其他地方为零。这种对数盒在$\Delta\to0$时插值到单色谱，$\Delta\to\infty$时为局部尺度不变平台。在辐射时代，将卷积简化为动量三角形与盒重叠部分的紧凑积分，并在两种情况下进行解析评估。对于窄盒，频谱等于狄拉克谱结果乘以纯几何重叠因子$\Phi_{\rm box}(\kappa,\Delta)$；对于宽盒，分离出下边缘、尺度不变内部和上边缘，推导各自主导行为并组合成统一公式，还提供了无积分封闭形式替代用于参数扫描。
英文摘要：
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 $Δ\to0$ and a locally scale-invariant plateau as $Δ\to\infty$, 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 $Φ_{\rm box}(κ,Δ)$; this factor turns the infrared slope from $k^{3}\ln^{2}k$ into $k^{2}\ln^{2}k$ 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 $k^{3}\ln^{2}k$ 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, $Δ$-independent edge functions. We also provide an integral-free closed-form surrogate for these edge functions for use in parameter scans.
