We study scalar perturbations induced by scalar perturbations through the non-linear interaction appearing at second order in perturbations. We derive analytic solutions of the induced scalar perturbations in a perfect fluid. In particular, we consider the perturbations in a radiation-dominated era and a matter-dominated era. With the analytic solutions, we also discuss how the induced scalar perturbations evolve from outside to inside horizon.

中文翻译：

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标量摄动引起的标量摄动的解析解

我们研究了由标量摄动引起的标量摄动，它是通过在摄动中出现在第二阶的非线性相互作用而产生的。我们导出在理想流体中引起的标量摄动的解析解。特别是，我们考虑了辐射主导时代和物质主导时代的扰动。借助解析解，我们还讨论了诱导的标量摄动是如何从外部视线演变为内部视线的。