terms up to quadratic order in $\xi$ are retained for $\kappa\xi < 1$. A detailed justification of neglecting the higher order terms as well as their possible effects will be discussed in the end part of this section.\
Taking a negligible backreaction of the scalar field ($\xi$) on the background spacetime, the solution of metric $\tilde{G}_{MN}$ is exactly same as RS model i.e. $$d\tilde{s}^2 = e^{- 2 k|y|} \eta_{\mu\nu} dx^{\mu} dx^{\nu} - dy^2
 \label{grav.sol1.ST}$$ where $k = \sqrt{\frac{-\Lambda_{eff}}{24M^3}}$. With this metric, the scalar field equation of motion in the bulk is following, $$\begin{aligned}
 -\partial_y[\exp{(-4ky)}\partial_y\xi] + m_{\Phi}^2\exp{(-4ky)}\xi(y) = 0
 \label{eom.scalar.field}\end{aligned}$$ where the scalar field $\xi$ is taken as function of extra dimensional coordinate only. Considering non zero value of $\xi$ on branes, the above equation (\[eom.scalar.field\]) has the general solution, $$\xi(y) = e^{2ky} \big[Ae^{\nu ky} + Be^{-\nu ky}\big]
 \label{sol.scalar.field}$$ with $\nu = \sqrt{4 + m_{\Phi}^2/k^2}$. Moreover $A$ and $B$ are obtained from the boundary conditions, $\xi(0)=v_h$ and $\xi(\pi r_c)=v_v$ as follows : $$A = v_v e^{-(2+\nu)kr_c\pi} - v_h e^{-2\nu kr_c\pi}
 \label{A}$$ and $$B = v_h (1 + e^{-2\nu kr_c\pi}) - v_v e^{-(2+\nu)kr_c\pi}
 \label{B}$$

It may be observed that the scalar field degrees of freedom is related to the curvature as, $$\xi(y) 