massless scalar field. The action is given by [@de1992] $$\label{eq:action}
    I=\frac{1}{16\pi G_*}\int\sqrt{-g_*}\left( R_* -
    2g_*^{\mu\nu}\varphi_{,\mu}\varphi_{,\nu}
    \right)d^4x
    +I_m[\Psi_m,A^2(\varphi)g_{*\mu\nu}],$$ where $g_{*\mu\nu}$ is the “Einstein” frame metric tensor, $\Psi_m$ denotes matter fields collectively, and $G_*$ is some dimensionful constant. In this Einstein frame, the Einstein-Hilbert term is isolated from other sectors. The “Brans-Dicke” frame metric tensor $\tilde{g}_{\mu\nu}$ is related to the Einstein frame metric tensor by the following conformal transformation: $$\tilde{g}_{\mu\nu}=A^2(\varphi)g_{*\mu\nu}.
      \label{eq:conformaltransformation}$$ Because of the “universal coupling,” which is the way of the coupling of the scalar field in the matter sector seen in Eq. (\[eq:action\]), a test particle moves on the geodesic of the Brans-Dicke frame metric $\tilde{g}_{\mu\nu}$. For this reason the Brans-Dicke frame is often called a “physical” frame. The tilde denotes the physical frame quantity.

In the Einstein frame, the field equations are given by $$\begin{aligned}
      & & G_{*\mu\nu}=8\pi G_* T_{*\mu\nu}
      +2\left(\varphi_{,\mu}\varphi_{,\nu}-\frac{1}{2}g_{*\mu\nu}
      g_*^{\alpha\beta}\varphi_{,\alpha}\varphi_{,\beta}\right), 
    \label{eq:fieldeq1} \\
      & & \Box_* \varphi = - 4\pi G_* \alpha(\varphi) T_*,
      \label{eq:fieldeq2}\end{aligned}$$ while the equations of motion for matter are $$\label{eq:eom}
  \nabla_{*\nu}T_{*\mu}^{\nu}=\alpha(\varphi)T_*\nabla_{*\mu}\varphi,$$ where the energy-momentum tensor of the matter, $T_*^{\mu\nu}$, is defined and related to the physical energy-momentum tensor $\tilde{T}^{\mu\nu}$ as $$\label{eq:defofem}
      T_*^{\mu\nu} \equiv \frac{2}{\sqrt{-g_*}}
      \frac{\delta I_m[\Psi_m,A^2(\varphi)g_{*\mu\nu}]}
      {\delta g_{*\mu\nu}}
      = A^6(\varphi) \tilde{T}^{\mu\nu}.$$ $G_{*\mu\nu}$ and $\Box_*$ are 