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arXiv 2005.12306gr-qcastro-ph.COhep-th
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对数正态峰值标量扰动诱导的引力波

Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak

Shi Pi, Misao Sasaki

备注：
15 pages, 3 figures. Some figures redrawn. References added and reorganized. Accepted for JCAP
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摘要：

本文研究了原始标量微扰引起的随机引力波背景，其谱在$k=k_*$处有一个宽度为$Δ$的对数正态峰。我们推导出窄峰（$Δ\ll1$）和宽峰（$Δ\gtrsim1$）的GW谱$Ω_\text{GW}$的解析公式。在窄峰情况下，光谱具有双峰特征，在$k= 2k_*/\sqrt{3}$处具有较尖锐的峰。在光谱的红外（IR）侧，我们发现幂律行为在幂律指数的$k=k_b$处具有中断，其中它从远IR侧的$k^3$改变为近IR侧的$k^2$。我们发现，由$Δ$确定的断裂频率与峰值频率之比为$f_b/f_p\approx\sqrt{3}Δ$，其中$f_b$和$f_p$分别是断裂频率和峰值频率。在宽峰情况下，我们发现GW谱在$k=k_*$处也有一个对数正态峰，但宽度较小，为$Δ/\sqrt2$。使用这些推导出的解析公式，我们还提出了表达式的最大值的$Ω_\text{GW}$的窄和宽的情况下。我们的研究结果将在未来几十年内提供一个有用的工具，在寻找诱导GW信号。

英文摘要：

We study the stochastic gravitational wave (GW) background induced by the primordial scalar perturbation with the spectrum having a lognormal peak of width $Δ$ at $k=k_*$. We derive an analytical formula for the GW spectrum $Ω_\text{GW}$ for both narrow ($Δ\ll1$) and broad ($Δ\gtrsim1$) peaks. In the narrow-peak case, the spectrum has a double peak feature with the sharper peak at $k= 2k_*/\sqrt{3}$. On the infrared (IR) side of the spectrum, we find power-law behavior with a break at $k=k_b$ in the power-law index where it chages from $k^3$ on the far IR side to $k^2$ on the near IR side. We find the ratio of the break frequency to the peak frequency is determined by $Δ$ as $f_b/f_p\approx\sqrt{3}Δ$, where $f_b$ and $f_p$ are the break and peak frequencies, respectively. In the broad-peak case, we find the GW spectrum also has a lognormal peak at $k=k_*$ but with a smaller width of $Δ/\sqrt2$. Using these derived analytic formulae, we also present expressions for the maximum values of $Ω_\text{GW}$ for both narrow and broad cases. Our results will provide a useful tool in searching for the induced GW signals in the coming decades.

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