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How is the solution of EOM for the action (for FRW metric):

[itex] S= \int d^{4}x \sqrt{-g} [ (\partial _{\mu} \phi)^{2} - V(\phi) ][/itex]

give solution of:

[itex] \ddot{\phi} + 3H \dot{\phi} + V'(\phi) =0 [/itex]

I don't in fact understand how the 2nd term appears.... it doesn't appear in my calculation... Under the change of [itex] \phi \rightarrow \phi + \delta \phi [/itex] I'm getting:

[itex] \frac{ \partial L } {\partial \phi} = \partial_{\mu} \frac{ \partial L} {\partial \partial_{\mu} \phi} [/itex]

[itex] -\frac{ \partial V} {\partial \phi} = \partial_{\mu} \partial^{\mu} \phi [/itex]

If I want homogeneity ([itex]∇ \phi =0 [/itex] ):

[itex] \ddot{\phi} + V'(\phi) =0 [/itex]

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# Scalar field in Expanding Universe EOM

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