content_image/384252.jpg content_image/384253.jpg content_image/384254.jpg content_image/384256.jpg content_image/384257.jpg content_image/384255.jpg content_image/384258.jpg content_image/384259.jpg
(6.13) are self-consistent. 

## C. Energy-momentum tensor and Klein-Gordon equation 

Here, we summarize the explicit components of the gauge-invariant parts (3.23) of the second-order pertur- 

bation of energy momentum tensor for a single scalar field in terms of gauge-invariant variables. Through Eqs. (4.4), (5.4), (6.1), the components of Eq. (3.23) are derived by the straightforward calculations. In this pa- per, we just summarize the components of $^{(2)}$T b a in the situation where the first-order Einstein equations (5.12) and (5.16) are satisfied: 

<img src='content_image/384252.jpg'>

<img src='content_image/384253.jpg'>

<img src='content_image/384254.jpg'>

More generic formulae for the components of $^{(2)}$T b a are given in Ref. [12]. 

Next, we show the gauge-invariant second-order the Klein-Gordon equation. We only consider the simple sit- uation where Eqs. (5.12) and (5.16) are satisfied. The formulae for more generic situation is given in Ref. [12]. Through Eqs. (5.4), (6.1), (4.4), the second-order per- turbation of the Klein-Gordon equation (3.29) is given by 

<img src='content_image/384256.jpg'>

where we defined 

<img src='content_image/384257.jpg'>

In Eq. (6.18), Ξ$_{(}$$_{K}$$_{)}$ is the source term which is the collection of the quadratic terms of the linear-order 

<img src='content_image/384255.jpg'>

perturbations in the second-order perturbation of the Klein-Gordon equation. If we ignore this source term, Eq. (6.18) coincide with the first-order perturbation of the Klein-Gordon equation. From this source term (6.19) of the Klein-Gordon equation, we can see that the mode- mode coupling due to the non-linear effects appear in the second-order Klein-Gordon equation. 

We cannot discuss solutions to Eq. (6.18) only through this equation, since this includes metric perturbations. To determine the behavior of the metric perturbations, we have to treat the Einstein equations simultaneously. The second-order Einstein equation is shown in Sec. VI D. 

Here, we show the all components of the second-order Einstein equation (3.33). All components of Eq. (3.33) are summarized as 

<img src='content_image/384258.jpg'>

<img src='content_image/384259.jpg'>

## D. Einstein equations 

