Abstract A new approach, “ B -Function Method”, to the analysis of the reconstruction methods, phase space, and exact solutions of the alternative theories of gravity is suggested. The B -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 B -function method are that all conditions leading to physically admissible domains, can be expressed only by one new dimensionless function, B -Function , which is the Lie derivative of ln ( F ( x ) ) along a vector field, X = x + c , on a singleton Q = { x } . Besides these, this approach may also be regarded as a new approach to the reconstruction of alternative theories based on the behavior of the scalar field(s). In general, this type of reconstruction has three modes for a single-scalar-field theory: 1- Reconstruction from a strictly decreasing scalar field; 2- Reconstruction from a strictly increasing scalar field; 3- Reconstruction from a hybrid-behavior scalar field. For a theory with n scalar fields, there are 3 n modes of reconstruction.

中文翻译：

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一种分析替代引力理论的重建方法、相空间和精确解的新方法以及一种基于标量场行为的重建新方法

摘要 提出了一种新的方法“B 函数法”，用于分析引力替代理论的重构方法、相空间和精确解。通过提出八个新的与模型无关的物理定理构建的 B 函数方法，提取参数（即作用的变量和常数参数以及通过微分方程作为积分常数出现的参数）演化的物理允许范围) 并有助于快速数据分析。B 函数方法最显着的特点和优点是所有导致物理允许域的条件只能由一个新的无量纲函数 B 函数表示，它是 ln ( F ( x ) ) 的 Lie 导数。向量场 X = x + c ，在单例 Q = { x } 上。除此之外，这种方法也可以被视为一种基于标量场行为重建替代理论的新方法。一般来说，这种类型的重建对于单标量场理论具有三种模式： 1- 从严格递减的标量场重建；2- 从严格递增的标量场重建；3- 从混合行为标量场重建。对于具有 n 个标量场的理论，有 3 n 种重构模式。3- 从混合行为标量场重建。对于具有 n 个标量场的理论，有 3 n 种重构模式。3- 从混合行为标量场重建。对于具有 n 个标量场的理论，有 3 n 种重构模式。