AI 中文总结
该研究探究依赖尺度的原初非高斯性对标量诱导引力波形态的影响，采用特定核函数计算高阶贡献，对比幂律跑动与tanh权重的效应，为相关引力波观测提供诊断模板。
AI 中文摘要
标量诱导引力波（Scalar-induced gravitational waves, SIGWs）可探测宇宙微波背景尺度之外的原初曲率扰动，而原初黑洞（Primordial black holes, PBHs）则探测同一统计量的稀有尾部。本文探究SIGW的形态是否能揭示依赖尺度的原初非高斯性。我们采用内部腿可分离核函数 $\mathcal F_{\rm NL}(\mathbf{k};\mathbf q,\mathbf{k}-\mathbf q)=\bar f_{\rm NL}f(q)f(\lVert\mathbf{k}-\mathbf q\rVert)$，在二阶张量源近似下计算到 $\mathcal O(\bar f_{\rm NL}^{4})$ 的高斯、可约和连通贡献。我们对比持续幂律跑动与在高标量动量处饱和的紫外匹配tanh权重，针对窄、有限宽度、非对称及多斜率标量谱，发现跑动产生的效应不止于振幅重整化：幂律跑动会改变非高斯卷积采样的标量支撑，引发峰位移、非对称肩部及持续紫外形变；而tanh模板则产生过渡局域化的修正，并趋近于与动量无关的紫外平台。微扰层级、加权斜率、对数正态鞍点位移、峰位移及红外标度的解析估计解释了上述趋势及 $\mathcal O(\bar f_{\rm NL}^{4})$ 部分的增强灵敏度。对于具有有限加权矩的局域源，平滑跑动在对数修正下保留了主导的 $k^3$ 红外行为。由于PBH质量标度为 $M\propto k^{-2}$，在窄支撑解释中，优先高波数加权有条件地倾向于更低的PBH质量。峰位置、谱曲率、肩部及紫外斜率是脉冲星计时阵列（PTAs）、LISA及第三代观测的诊断工具，为原本不可达尺度上的原初相互作用提供有限阶模板。
英文摘要
Scalar-induced gravitational waves (SIGWs) probe primordial curvature perturbations beyond cosmic microwave background scales, while primordial black holes (PBHs) probe the rare tail of the same statistics. We ask whether SIGW morphology can reveal scale-dependent primordial non-Gaussianity. We adopt the internal-leg separable kernel, $\mathcal F_{\rm NL}(\mathbf{k};\mathbf q,\mathbf{k}-\mathbf q)=\bar f_{\rm NL}f(q)f(\lVert\mathbf{k}-\mathbf q\rVert)$, and compute Gaussian, reducible, and connected contributions through $\mathcal O(\bar f_{\rm NL}^{4})$ in the second-order tensor-source approximation. We compare persistent power-law running with a UV-matched tanh weight that saturates at high scalar momentum. Across narrow, finite-width, asymmetric, and multi-slope scalar spectra, running produces more than amplitude renormalization. Power-law running shifts the scalar support sampled by non-Gaussian convolutions, generating peak displacement, asymmetric shoulders, and persistent ultraviolet deformations. The tanh template instead produces a transition-localized modification and approaches a momentum-independent ultraviolet plateau. Analytical estimates of the perturbative hierarchy, weighted slopes, log-normal saddle displacement, peak shifts, and infrared scaling explain the trends and the enhanced sensitivity of the $\mathcal O(\bar f_{\rm NL}^{4})$ sector. For localized sources with finite weighted moments, smooth running preserves the leading $k^3$ infrared behavior up to logarithmic corrections. Since the PBH mass scales as $M\propto k^{-2}$, preferential high-wavenumber weighting conditionally favors lower PBH masses in a narrow-support interpretation. Peak position, spectral curvature, shoulders, and ultraviolet slopes are diagnostics for PTAs, LISA, and third-generation observations, providing finite-order templates for primordial interactions on otherwise inaccessible scales.
Comments47 pages, 27 figures + References
