= \frac{2}{\sqrt{3}\kappa}\ln[1 + 2\alpha R] - <\Phi>
 \label{scalar and curvature}$$ Recall that $<\Phi> =\frac{2}{\sqrt{3}\kappa} \ln[\sqrt{9 - 40\kappa^2\alpha\Lambda} - 2]$.\
From the above expression, we can relate the boundary values of the scalar field (i.e $\xi(0)=v_h$ and $\xi(\pi r_c)=v_v$) with the Ricci scalar as, $$v_h = \frac{2}{\sqrt{3}\kappa} \ln\bigg[\frac{1 + 2\alpha R(0)}{\sqrt{9 - 40\kappa^2\alpha\Lambda} - 2}\bigg]
 \label{relation1}$$ and $$v_v = \frac{2}{\sqrt{3}\kappa} \ln\bigg[\frac{1 + 2\alpha R(\pi r_c)}{\sqrt{9 - 40\kappa^2\alpha\Lambda} - 2}\bigg]
 \label{relation2}$$ where $R(0)$ and $R(\pi)$ are the values of the curvature on Planck and TeV brane respectively. Later on, we derive the expression of the bulk scalar curvature which in this scenario becomes dependent on the bulk coordinate $y$. Thus the parameters that are used in the scalar-tensor theory are actually related to the parameters of the original $F(R)$ theory.\
It deserves mentioning that the solutions obtained in eqn. (\[grav.sol1.ST\]) and eqn.(\[sol.scalar.field\]) are based on the conditions that the bulk scalar potential is retained up to quadratic term (see eqn.(\[action2ST\])) and the backreaction of the bulk scalar field is neglected on five dimensional spacetime. Both these conditions are followed from the assumption that $\kappa v_h< 1$. Relaxation of this assumption is crucial to check the status of higher order self interaction terms 