Abstract
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, R2, 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.
Highlights
Modern cosmology has the very important task of providing a theoretical description of many aspects of the observable universe with exponential expansion, particle creation, and radiation
The physics community almost accepted the idea that the Einstein gravity and standard particle physics have to be modified in order to elaborate self-consistent quantum gravity theories and describe existing experimental and observational data in modern cosmology
In the present paper and partner works [26,27,28,29,30,31,32,33], we follow an orthodox point of view that inflation and accelerating cosmological models can be elaborated in the framework of effective Einstein theories via off—diagonal and diagonal solutions for nonholonomic vacuum and non–vacuum configurations determined by generating functions and integration functions and constants
Summary
Modern cosmology has the very important task of providing a theoretical description of many aspects of the observable universe with exponential expansion (inflation), particle creation, and radiation. We shall elaborate on a unified cosmological scenario for MGTs and GR with nonholonomic off-diagonal interactions when effective Finsler-like variables can be considered for a 2 + 2 splitting In such an approach, both inflation and slowly accelerated universe models are reproduced by exact solutions constructed following the AFDM. The second action term is the standard Riemannian integration measure containing a Weyl-scale symmetry, preserving R2 , or more general f ( R) terms, which, in this work, may encode modifications from massive gravity, bi-metric and biconnection theories The latter formalism and geometrization of such TMTs allows for the representation of the corresponding gravitational field equations as certain effective Einstein equations in nonholonomic variables; see various applications in modern cosmology,. Proofs and details can be found, for instance, in [24,25]
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