We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Examples of such Finsler two-measure formulations of general relativity (GR) and MGTs are considered for Lorentz manifolds and their (co) tangent bundles and abbreviated as FTMT. Generic off-diagonal metrics solving gravitational field equations in FTMTs are determined by generating functions, effective sources and integration constants, and characterized by nonholonomic frame torsion effects. By restricting the class of integration functions, we can extract torsionless and/or diagonal configurations and model emergent cosmological theories with square scalar curvature, $R^2$, when the global Weyl-scale symmetry is broken via nonlinear dynamical interactions with nonholonomic constraints. In the physical Einstein–Finsler frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions, which allows for a unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce massive gravity effects in two-measure theories. Finally, we study a reconstructing procedure for reproducing off-diagonal FTMT and massive gravity cosmological models as effective Einstein gravity or Einstein–Finsler theories.

中文翻译：

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具有两个量度Finsler像变数的修正的Einstein重力中的破碎尺度不变性，重力质量和暗能量

我们使用修正的色散关系（MDR）和修正的（局部）洛伦兹不变性（LIV）来研究有效重力修正（MGT）中的有效爱因斯坦方程组的新型非对角线和对角线宇宙学解。此类MGT具有两个非黎曼体积形式（类似于两个度量理论（TMT））以及相关的双度量和/或双连接几何结构，用于动作和Lagrange密度。对于常规的非完整2 + 2分裂，我们总是可以用类似于Finsler的变量描述这种模型，这对于阐述构造精确和参数解的几何方法很重要。对于洛伦兹流形及其（共）切线束，考虑了广义相对论（GR）和MGT的这种Finsler二次测量公式的示例，并缩写为FTMT。FTMT中解决引力场方程的通用非对角度量是通过生成函数，有效源和积分常数来确定的，并具有非完整框架扭转效应的特征。通过限制积分函数的类别，我们可以提取无扭转和/或对角线构型，并以方形标量曲率对新兴的宇宙学理论进行建模，${\mathrm{[R}}^{\mathrm{2个}}$，当具有非完整约束的非线性动力相互作用打破了整体Weyl尺度对称性时。在爱因斯坦-芬斯勒物理框架中，构造包括：（i）生成函数和有效源的非线性重新参数化对称性；（ii）具有两个可能平坦区域的标量场的有效电势，从而可以统一描述与加速宇宙学和暗能量有关的局部各向异性和/或各向同性的早期宇宙膨胀；（iii）对于某些非完整阶段和参数宇宙学演化，导致不同的膨胀阶段，存在非对角线和对角线解所描述的“新兴宇宙”；（iv）我们可以在两个量度的理论中重现巨大的重力效应。最后，