We study the stochastic gravitational wave (GW) background induced by the primordial scalar perturbation with the spectrum having a lognormal peak of width $\Delta$ at $k=k_*$. We derive an analytical formula for the GW spectrum $\Omega_\text{GW}$ for both narrow ($\Delta\ll1$) and broad ($\Delta\gtrsim1$) 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 $\Delta$ as $f_b/f_p\approx\sqrt{3}\Delta$, 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 $\Delta/\sqrt2$. Using these derived analytic formulae, we also present expressions for the maximum values of $\Omega_\text{GW}$ for both narrow and broad cases. Our results will provide a useful tool in searching for the induced GW signals in the coming decades.

中文翻译：

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具有对数正态峰值的标量扰动引起的引力波

我们研究了由原始标量扰动引起的随机引力波 (GW) 背景，其频谱在 $k=k_*$ 处具有宽度为 $\Delta$ 的对数正态峰。我们为窄峰 ($\Delta\ll1$) 和宽峰 ($\Delta\gtrsim1$) 推导出 GW 谱 $\Omega_\text{GW}$ 的分析公式。在窄峰情况下，光谱具有双峰特征，在 $k= 2k_*/\sqrt{3}$ 处具有更尖的峰。在光谱的红外 (IR) 侧，我们发现幂律行为在幂律指数中的 $k=k_b$ 处中断，其中它从远红外侧的 $k^3$ 变为 $k^近红外侧 2 美元。我们发现中断频率与峰值频率的比率由 $\Delta$ 确定为 $f_b/f_p\approx\sqrt{3}\Delta$，其中 $f_b$ 和 $f_p$ 是中断频率和峰值频率，分别。在宽峰情况下，我们发现 GW 频谱在 $k=k_*$ 处也有一个对数正态峰，但宽度更小，$\Delta/\sqrt2$。使用这些导出的解析公式，我们还给出了狭义和广义情况下 $\Omega_\text{GW}$ 的最大值的表达式。我们的结果将提供一个有用的工具来寻找未来几十年的诱导引力波信号。