Junction conditions in gravity theories with extra scalar degrees of freedom

In this work we present a general method to obtain the junction conditions of modified theories of gravity whose action can be written in the form $f\left(X_1,...,X_n\right)$, where $X_1$ to $X_n$ are any combination of scalar dependencies, e.g. the Ricci scalar $R$, the trace of the stress-energy tensor $T$, the Gauss-Bonnet invariant $\mathcal G$, among others. We discriminate the junction conditions into three sub-groups: the immediate conditions, arising from the imposition of regularity of the relevant quantities in the distribution formalism; the differential conditions, arising from the differential terms in the field equations; and the coupling conditions, arising from the interaction between different scalars $X_i$. Writing the modified field equations in terms of a linear combination of the different contributions of the scalars $X_i$ allows one to analyze the direct and differential junction conditions independently for each of these scalars, whereas the coupling junction conditions can be analyzed separately afterwards. We show that the coupling junction conditions induced on a scalar $X_i$ due to a coupling with a scalar $X_j$ are of the same form as the differential junction conditions of the scalar $X_j$, but applied to the analogous quantity in the framework of the scalar $X_i$. We provide a complete analysis of three different types of spacetime matching, namely smooth matching, matching with a thin-shell, and matching with double gravitational layers, and we also describe under which conditions the full sets of junction conditions might be simplified. Our results are applicable to several well-known theories of gravity e.g. $f\left(R\right)$, $f\left(T\right)$, and $f\left(\mathcal G\right)$, and can be straightforwardly extrapolated to other theories with scalar dependencies and more complicated scenarios where several dependencies are present simultaneously.