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I have came across a text where an interaction term in a classical Lagrangian is presented that couples a matter density $\rho$ and a scalar field $\phi$ as

\begin{equation} \mathcal{L}_{\text{int}} = \Lambda \phi \rho, \end{equation}

where $\Lambda $ is some coupling constant. The text then claims that (in a static profile) the $\phi$-mediated acceleration from this coupling is

\begin{equation} \vec{a}_{\phi} = \Lambda \vec{\nabla}\phi. \end{equation}

I have unfortunately not worked very much with classical field theory, so I don't see how they arrive at this statement.

The text in question is https://arxiv.org/abs/1711.05748, pages 4-5, and equations (4) and (7).

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