Abstract

We study the perturbations up to the 2nd-order for a power-law inflation driven by a scalar field in synchronous coordinates. We present the 1st-order solutions, and analytically solve the 2nd-order perturbed Einstein equation and scalar field equation, give the 2nd-order solutions for all the scalar, vector, and tensor metric perturbations, as well as the perturbed scalar field. During inflation, the 1st-order tensor perturbation is a wave and is decoupled from other perturbations, the scalar metric perturbation and the perturbed scalar field are coupled waves, propagating at the speed of light, differing from those in the dust and relativistic fluid models. The 1st-order vector perturbation is not wave and just decreases during inflation. The 2nd-order perturbed Einstein equation is similar in structure to the 1st-order one, but various products of the 1st-order perturbations occur as the effective source, among which the scalar–scalar coupling is considered in this paper. The solutions of all the 2nd-order perturbations consist of a homogeneous part similar to the 1st-order solutions, and an inhomogeneous part in a form of integrations of the effective source. The 2nd-order vector perturbation is also a wave since the effective source is composed of the 1st-order waves. We perform the residual gauge transformations between synchronous coordinates up to the 2nd-order, and identify the 1st-order and 2nd-order gauge modes.

Graphic abstract

中文翻译：

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暴胀阶段标量-标量耦合产生的二阶宇宙扰动

摘要

我们研究同步坐标中标量场驱动的幂律暴涨的二阶扰动。我们给出一阶解，并解析求解二阶扰动爱因斯坦方程和标量场方程，给出所有标量、矢量和张量度量扰动以及扰动标量场的二阶解。在膨胀过程中，一阶张量扰动是波，与其他扰动解耦，标量度量扰动和扰动标量场是耦合波，以光速传播，与尘埃和相对论流体模型中的波不同。一阶矢量扰动不是波，只是在暴胀过程中减小。二阶扰动爱因斯坦方程在结构上与一阶扰动方程相似，但一阶扰动的各种乘积作为有效源出现，其中标量-标量耦合是本文考虑的。所有二阶扰动的解均由类似于一阶解的齐次部分和有效源积分形式的非齐次部分组成。二阶矢量扰动也是波，因为有效源由一阶波组成。我们在同步坐标之间执行残差规范变换直至二阶，并识别一阶和二阶规范模式。

图文摘要