A new approach to the analysis of the reconstruction methods, phase space, and exact solutions of the alternative theories of gravity

Abstract

A new approach, "$\mathfrak{B}\text{-Function}$ Method", to the analysis of the reconstruction methods, phase space, and exact solutions of the alternative theories of gravity is suggested. The $\mathfrak{B}\text{-function}$ method which is constructed by suggesting eight new model-independent physical theorems, extracts the physically admissible ranges of the evolution of the parameters (i.e. the variables and constant parameters of action and those which emerged by differential equations as the constants of integration) and helps one for quick data analysis. The most significant feature and beauty of the $\mathfrak{B}\text{-function}$ method are that all conditions leading to physically admissible domains, can be expressed only by one new function, $\mathfrak{B}\text{-Function}$, which is the Lie derivative of $\ln (\mathcal{F}(x))$ along a vector field, $\mathbf{X}=x+c$, on a singleton $Q=\{x\}$.