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  • $\begingroup$ Your gravity field equation is wrong. As you can see in the linked paper. You're making the same mistake I made the first time. Please see @Noone's comment. What I understood is that the variation of the $R$ is not only $R_{\mu\nu}\delta g^{\mu\nu}$ but there are more terms such as $g_{\mu\nu}\Box \delta g^{\mu\nu}$ and $\nabla_{\mu}\nabla_{\nu}\delta g^{\mu\nu}$. To separate the $\delta g^{\mu\nu}$ one integrates by parts. Then those derivatives hit $e^{-2\phi}$. Which I was missing when I did the calculations first time. I've not yet done the full calculation. You may give it a try. $\endgroup$
    – Faber Bosch
    Commented Jul 9 at 2:11
  • $\begingroup$ @FaberBosch You are absolutely right. I have edited my original answer. $\endgroup$
    – Omar mandour
    Commented Jul 9 at 14:47