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Recently, I have been researching about modified gravity theories and one of the theories has been the theory of the graviton. If one starts with the metric tensor $g_{\mu\nu}$ and then performs the perturbation $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ one then obtains the perturbation for a linearized gravitational wave, which can also represent a spin-2 particle, the graviton. The Riemann tensor is then simplified in a local inertial frame to be:

$R^\alpha_{\beta\mu\nu}=\partial _\mu \Gamma^a_{\beta\nu}-\partial _\nu \Gamma^a_{\beta\mu}$

The other curvature tensors can then be calculated and the result could construct the Einstein Hilbert Action for the graviton field. Now the question that I have is the following: since the tensor $h_{\mu\nu}$=$kH_{\mu\nu}$, where k is the gravitational constant commonly seen in the Einstein Field Equation, could one instead of the k plug in a scalar field $\phi$ like in Brans-Dicke Theory? Would this yield a result similar to the Brans-Dicke action?

I tried to calculate the Ricci scalar by plugging-in the scalar field into the graviton tensor and got:

$R$=$\partial_{\mu}\partial_{\nu}h^{\mu\nu}-\square h$

$R$=$\partial_{\mu}\partial_{\nu}(\phi H^{\mu\nu})$-$\square (\phi H^\mu_\mu)$=$(\partial_{\mu}\partial_{\nu} \phi)H^{\mu\nu}$+$\partial _\mu H^{\mu\nu}\partial _\nu \phi$+$\partial _\nu H^{\mu\nu}\partial _\mu \phi$+$(\partial _\mu \partial_\nu H^{\mu\nu} )\phi$$-...$

To get:

$\phi R$+$(\partial_{\mu}\partial_{\nu} \phi)H^{\mu\nu}$+$\partial _\mu H^{\mu\nu}\partial _\nu \phi$+$\partial _\nu H^{\mu\nu}\partial _\mu \phi$$-...$

Could one also use the metric tensor instead of the graviton tensor? I am relatively new to this level of physics, so forgive me if I make obvious mistakes.

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Well, you'd have to linearize $\phi$ as well, so you'd assume that $\phi = \phi_{0} + (c){}^{1}\phi + (c^{2})\,{}^{2}\phi\, + ...$, along with assuming that $g_{ab} = \eta_{ab} + (c) {}^{1}h_{ab} + (c^{2}){}^{2}h_{ab} + ...$, and then you would drop all of the terms in the Brans-Dicke EOM that contain terms where the exponent of $c$ is greater than 1.

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