Broken Scale Invariance, Gravity Mass, and Dark Energy in Modified Einstein Gravity with Two Measure Finsler Like Variables

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, encoding possible violations of (local) Lorentz invariance, LIVs. Such MGTs are constructed for Lagrange densities with two non-Riemannian volume forms (two-measure theories, TMTs) and associated bimetric/ biconnection geometric structures. For conventional 2+2 splitting, we can describe such models in Finsler like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Such formulations of general relativity, GR, and MGTs are considered for Lorentz manifolds and their (co) tangent bundles, in brief, FTMT. 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. Restricting the class of integration functions, we can extract torsionless and 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 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 solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce in two-measure theories massive gravity effects.


Introduction
Modern cosmology has a very important task to provide a theoretical description of many aspects of observable universe with exponential expansion (inflation), particle creation, and radiation. We cite books [4,5,6,7,8] on standard cosmology [1,2,3] and further developments. Then, on acceleration cosmology [9,10] and related dark energy and dark matter physics, one can be considered a series of works on modified gravity theories, MGTs, and cosmology [11,12,13,14,15,16]. Another direction of research is devoted to nonholonomic and Finsler like locally anisotropic cosmological models [17,18,19,20,21,22,23], see [24,25] for an axiomatic approach to Finsler-Lagrange-Hamilton gravity theories. The physical 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 result, a number of MGTs and cosmological scenarios have been elaborated during the last 20 years.
In a series of works [26,27,28,29,30,31,32,33], see also references therein, it was developed a geometric approach (the so-called anholonomic frame deformation method, AFDM) for constructing exact and parametric solutions in MGTs, general relativity, GR, and theory of nonholonomic geometric and classical/ quantum information flows. Such solutions are with generic off-diagonal metrics 1 and generalized connections when their coefficients depend on all spacetime coordinates via generating and integration functions, for vacuum and non-vacuum configurations. One can be considered effective and matter fields sources for possible Killing and non-Killing symmetries and various types of commutative and noncommutative parameters etc. For Finsler like modified gravity theories, FMGTs, the coefficients of geometric and physical objects depend, in general, on (co) fiber velocity (momentum) type coordinates. Following the AFDM, the geometric constructions and variational calculus are preformed with respect to certain classes of (adapted) nonholonomic frames for a formal splitting of spacetime dimension in the form 2(3)+2+...+ and a well-defined geometrically "auxiliary" linear connection which is convenient for performing, for instance, a deformation quantization procedure, or for constructing exact and/or parametric solutions. This allows to decouple the gravitational field equations in MGTs, FMGTs, and GR, and geometric/information flow equations. Such nonholonomic deformations of fundamental geometric objects determined by distortions of nonlinear and linear connection structures were not considered in other approaches with vierbeins (tetrads), 2+2 and/or 3+1 splitting, see standard textbooks on general relativity and exact solutions [34,35,36,37,38]. The methods elaborated by other authors were successful only for generating exact solutions with two and three Killing symmetries but do not provide a geometric/ analytic formalism for a general decoupling of gravitational and matter field equations. The surprising result is that such a decoupling is possible for various classes of effective / modified Einstein equations and matter fields which can be derived for certain physically motivated general assumptions in MGTs.
Let us summarize most important ideas and methods developed in Refs. [16,24,25,26,27,28,29,30,31,32,33]: a) The (modified) Einstein equations with some effective and/or matter field sources consist very sophisticate systems of nonlinear partial derivative equations, PDEs. The bulk of most known and important physical applications (of black hole, cosmological and other type solutions) were elaborated for ansatz of metrics which can be diagonalized by certain frame/coordinate transforms and when physically important systems of nonlinear PDEs can be reduced to systems of decoupled nonlinear ODE (ordinary differential equations). In such cases, the generated exact or parametric solutions (i.e. integrals, with possible non-trivial topology, singularities, of different smooth classes etc.) depend on one space, or time, like coordinate, being determined by certain imposed symmetries (for instance, spherical / axial ones, being invariant on some rotations, with Lie algebras symmetries etc). The integration constants can be found in explicit form considering certain symmetry / Cauchy / boundary / asymptotic conditions. This way, there were constructed various classes of black/worm hole and isotropic and anisotropic cosmological solutions.
b) The AFDM allows us to decouple and integrate physically important systems of nonlinear PDEs in more general forms than in a) when the integral varieties are parameterized not only by integration constants but also by generating and integration functions subjected to nonholonomic constraints and functional/ nonlinear dependence on sources and data for certain classes of 'prime metrics and connections'. The resulting 'target' off-diagonal metrics and generalized connections depend, in general, on all spacetime coordinates. It is important to note that at the end we can impose additional nonholonomic constraints and consider 'smooth' limits or various type non-trivial topology and/or parametric transitions to Levi-Civita configurations (with zero torsion) and/or diagonal metrics. This way we can reproduce well known black hole / cosmological solutions etc., or which can be with deformed horizons (for instance, with ellipsoid / toroid symmetries), anisotropic polarized physical constants and, for instance, imbedding into nontrivial gravitational vacuum configurations. Such new classes of solutions can not be constructed if we impose from the very beginning certain particular type ansatz for diagonalizable metrics, frames of references and/or sources depending only on one spacetime coordinate. This is an important property of nonlinear parametric physical systems subjected to certain nonholonomic constraints. More general solutions with geometric rich structure and various applications for a nonlinear gravitational and matter fields dynamics can be found if we succeed to solve directly certain generic nonlinear systems of PDEs not transformed into systems of ODEs. Having constructed such general classes of solutions, one might be analyzed limits to diagonal configurations and possible perturbative effects. We "loose" the bulk of generic nonlinear solutions with multi-variables if we consider from the very beginning certain "simplified" ansatz for "higher-symmetries" resulting in ODEs.
Applying the AFDM as we explained above in paragraph b) and choosing corresponding types of generating functions and integration functions and constants, it is possible to model various MGTs and accelerating cosmology effects by considering generic off-diagonal interactions and re-parameterizations of generating functions and sources in effective Einstein gravity. In the present paper, we shall elaborate on an 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. In general, such solutions are inhomogeneous and with local anisotropy. For a corresponding class of generating and integration functions and for necessary type effective sources, we can model effective scalar field potentials with anisotropy and limits to two flat regions. We shall construct and study nonlinear parametric cosmological theories generalizing the standard models based on Friedman-Lamaître-Robertson-Walker, FLRW, diagonalizable configurations derived for ODEs. The goal is to address the initial singularity problem and to explain how two periods of exponential expansion with widely different scales can be described via solutions of effective gravitational equations.
A well-known mechanism for generating accelerated expansion as a consequence of vacuum energy can be performed in the context of scalar field theory paradigm which is described by an effective potential ef U with flat regions. For such "slow roll" configurations of the vacuum field, the kinetic energy terms are small and the resulting energy-momentum tensor is of type T µν ≃ g µν ef U . If the potential ef U contains contributions of some modified gravity terms (two measures, massive gravity and other ones), we can analyse possible effects of such terms in the inflationary phase. But this is not enough for elaborating a theory of modern cosmology with acceleration and dark energy and dark matter contributions. Theoreticians developed different quintessential, k-essence and "variable gravity" inflation scenarios [39,40,41,42,43,44,45,46] and f (R) modified models, in general, with contributions from massive gravity, Finsler like theories, bi-metrics and bi-connections and/or generic off-diagonal interactions, see [47,48,14,15,17,18,24,25,30] and references therein.
The solutions with anisotropies and flat regions can be used for speculations on the phase that proceeds the inflation and may explain both a non-singular origin of universe and the early universe evolution. This is similar to the concept of "emergent universe" which was considered with the aim to solve the problem of initial singularity including the singularity theorems for inflationary cosmology driven by scalar field [49,50,51,52,53,54]. In our approach with solutions constructed following the AFDM, the universe does not start as a static Einstein universe but as a parametric effective one when the scalar field rolls with an almost constant speed for a non-singular configuration with small anisotropies.
Let us briefly motivate and explain the origins of the present work. The main ideas and methods on constructing generic off-diagonal solutions in MGTs comes from Refs. [26,27,28,29,30,31]. In articles [11,12,13,14,15,16,17,18,19,14,15], there were considered various examples when the gravitational and matter field equations in MGTs can be re-defined and solved as certain effective/ generalized Einstein equations or their Finsler like modifications. A series of papers [55,56,57,58,59] is devoted to a new class of modified-measure gravity-matter theories containing different terms in the pertinent Lagrangian action, for instance, one with a non-Riemannian integration measure and another with standard Riemannian integration measure. In brief, we shall call such models as two measure theories, TMTs, of gravity. In a more general case, there are considered two non-metric densities [60]. An important feature of such theories is that the constructions are with global Weyl-scale invariance and further dynamical breaking. In particular, the second action term is the standard Riemannian integration measure containing a Weyl-scale symmetry preserving R 2 , or more general f (R) terms, which in this work may encode modifications from massive gravity, bi-metric and bi-connection theories. The latter formalism and geometrization of such TMTs allow to represent the corresponding gravitational field equations as certain effective Einstein equations in nonholonomic variables, see various applications in modern cosmology, (super) string/ brane theories, non-Abelian confinement etc. [61,62]. The main goal of this article is to develop the AFDM for generating exact solutions in TMTs formulated in nonholonomic and Einstein-Finsler variables, see also a partner work [63], and analyze possible implications in modern cosmology and f dark energy and dark matter physics.
The work is organized as follows. Geometric preliminaries on nonholonomic Lorentz manifolds and relativistic Lagrange-Finsler spaces are provided in section 2. Then, in section 3, we formulate a geometric approach to MGT cosmology in the framework of TMT with nonholonomic variables and effective Einstein-Finsler gravity theories. We apply the AFDM for constructing generic off-diagonal cosmological solutions in various MGTs in section 4. Cosmological models with locally anisotropic effective scalar potentials and two flat regions are studied in section 5. We devote the section 6 to formulation of certain conditions when modified massive gravity can be reproduced as TMTs and effective GR theories, with nonholonomic Finsler like variables, and speculate on reconstructing procedure for such massive gravity cosmological models. Finally, we provide a discussion and conclusions in section 7.

Nonholonomic variables and (modified) Einstein and Lagrange-Finsler equations
In this section, we outline some necessary results from the geometry of four dimensional, 4-d, Lorentz manifolds with so-called canonical nonholonomic variables which can be transforms in Finsler-Lagrange like variables. The motivation to consider canonical variables is that they allow to prove certain general decoupling and integration properties of gravitational field equations in MGTs and GR, but Finsler-Lagrange like variables and associated almost symplectic structures which can be used for deformation and other type quantization procedures of gravity theories. Proofs and details can be found, for instance, in [24,25].

Geometric objects and GR and MGTs in nonholonomic variables
Let us consider a 4-d pseudo-Riemannian manifold V defined by a metric structure of signature (+, +, +−), with local coordinates u = {u γ }, where indices α, β, γ, ... run values 1, 2, 3 and (for the time like coordinate) 4. The Einstein summation rule on up/low repeating indices is applied if it will be not stated the contrary. For a corresponding causality structure combining locally the postulates of the special relativity theory, the principle of equivalence etc. (see a review of axiomatic appraoches in GR and Finsler like modified theories in [17,24,25]) such a curved spacetime is called a Lorentz manifold. In this work, we study generalizations of geometric and gravitational and cosmological models when certain nonholonomic (nonintegrable, anholonomic) distributions and related bimeasure structures, and Lagrangians for MGTs, are considered on V.
On a curved spacetime V, we can always introduce a nonholonomic 2 + 2 splitting which is determined by a non-integrable distribution N : where T V is the tangent bundle of V, the Withney sum ⊕ defines a conventional splitting into horizontal (h), hV, and vertical (v), vV, subspaces. In local cooridnates states a nonlinear connection, N-connection structure. For such a N-connection decomposition, the indices and coordinates split in the form u = (x, y), or u α = (x i , y a ), for x = {x i } and y = {y a }, with i, j, k, ... = 1, 2 and a, b, c, ... = 3, 4, which is respectively adapted to a nonholonomic 2 + 2 splitting. The data (V, N) define a nonholonomic manifold with a prescribed fibered structure described locally by fiber like coordinates y a . In our works, there are used "boldface" symbols in order to emphasize that certain geometric/ physical objects are defined for spaces enabled with a 2+2 splitting determined by a N-connection structure. On pseudo-Riemannian manifolds, to introduce a N-connection with a 2+2 splitting is equivalent to the convention that there are used certain subclasses of local (N-adapted) bases e µ = (e i , e a ) and their duals e ν = (e j , e b ), where Such frames are called nonholonomic because they satisfy, in general, the relations [e α , e β ] = e α e β − e β e α = W γ αβ e β , with nontrivial anholonomy coefficients . For zero W -coefficients, we get holonomic bases which allows to consider coordinate transforms e α → ∂ α and e β → du β .
On any manifold V and its tangent and cotangent bundle, there are also possible general vierbein (tetradic) transforms e α = e α α (u)∂/∂u α and e β = e β β (u)du β , where the coordinate indices are underlined in order to distinguish them from arbitrary abstract ones and the matrix e β β is inverse to e α α for orthonormalized bases.
We do not use boldface symbols for such transforms because an arbitrary decomposition (we can consider as particular cases certain diadic 2+2 splitting) is not adapted to a N-connection structure. With respect to N-adapted bases, we shall say that a vector, a tensor and other geometric objects are represented correspondingly as a distinguished vector (d-vector), a distinguished tensor (d-tensor) and distinguished objects (d-object). Using frame transforms g αβ = e α ′ α e β ′ β g α ′ β ′ , any metric g (1) on V can be written in N-adapted from as a distinguished metric (in brief, d-metric) In brief, such a h-v decomposition of a metric structure is parameterized in the form On nonholonomic manifolds, we can work with a subclass of linear connections D = (hD, vD), called distinguished connections, d-connections, preserving under parallelism the N-connection splitting. A d-connection is determined by its coefficients Γ α βγ = {L i jk ,Ĺ a bk ,Ć i jc , C a bc } computed with respect to a N-adapted base (3). Not adapted to N-connections linear connections structures can be also considered but they do not preserve under parallelism (2) and satisfy other types transformation laws under frame/coordinate transforms.
For any d-vectors X and Y, we can define in standard form, correspondingly, the torsion d-tensor, T , the nonmetricity d-tensor, Q, and the curvature d-tensor, R, of a D, which (in general) do not depend on g and/or N. The formulas are In N-adapted coefficients labeled by h-and v-indices, such geometric d-objects are parameterized respectively Such coefficients can be computed in explicit form by introducing X = e α and Y = e β , see (3), and coefficients of a d-connectionD = {Γ γ αβ } into formulas (5). To elaborate the AFDM is convenient to work with two "preferred" linear connections: the Levi-Civita connection ∇ and the canonical d-connection D. Both such connections are completely defined by a metric structure g following the conditions g → ∇ : ∇ Q = 0 and ∇ T = 0; where the left label ∇ is used for the geometric objects determined by the Levi-Civita, LC, connection. It should be noted here that the N-adapted coefficients of the torsion T are not zero for the case of mixed h-and vcoefficients computed with respect to N-adapted frames (conventionally, we can write this as hv T = 0, with some nontrivial N-adapted coefficients from the subset {T i ja , T a ji , T a bi }). Such a torsion T is completely determined by the coefficients of N and g (in coordinate frames, such values determine certain generic off-diagonal terms g αβ which can not be diagonalized in a finite spacetime region U ⊂ V by coordinate transforms). We can consider a distortion relation when both linear connections and the distortion tensors Z are completely defined by the geometric data (g, ∇), or (in nonholonomic variables) by (g, N, D).
Contracting the indices of a canonical Riemann d-tensor of D, R={ R α βγδ }, we construct a respective canonical Ricci d-tensor, Ric = { R αβ := R γ αβγ }. The corresponding nontrivial N-adapted coefficients are when the scalar curvature is computed It should be noted that, in general, R αβ = R βα , even such a tensor is symmetric for the LC-connection, R αβ = R βα . This a nonholonomic deformation and nonholonomic frame effect. We can introduce the Einstein d-tensor and consider an effective Lagrangian L for which the stress-energy momentum tensor, T αβ , is defined by an N-adapted (with respect to e β and e α ) variational calculus on a nonholonomic manifold (g, N, D), Following geometric principles, we can postulate the Einstein equations in GR for the data (g, D), and/or to re-write them equivalently for the data(g, ∇) if additional nonholonomic constraints for zero torsion are imposed, and T = 0, additional condition for∇.
In general, the condition D | T =0 = ∇ may not have a smooth limit and such an equation can be considered as a nonholonomic or parametric constraint. Here we note that the source is computed with the trace T := g αβ T αβ and κ should be determined by the Newton constant N ew G as in GR if we wont to study limits to the Einstein gravity theory. In this work, we shall use the units when N ew G = 1/16π and the Planck mass P l M = (8π N ew G) −1/2 = √ 2. If we do not impose the LC-conditions (10), the system of nonholonomic nonlinear PDEs (9), and similar higher dimension ones, for instance, with noncommutative and/or supersymmetric variables can be considered in various classes of MGTs, Finsler-Lagrange gravity etc.
The values R, Ric and R for the canonical d-connection D are different from the similar ones, R, Ric and R, computed for the LC-connection ∇. Nevertheless, both classes of such fundamental geometric objects are related via distorting relations derived in a unique form for a given metric structure and N-connection splitting. There are at least two priorities to work with D instead of ∇. The first one is that we can find solutions for generalized gravity theories with nontrivial torsion. The second priority is that the equations (9) decouple in very general forms with respect to certain classes of N-adapted frames. The basic idea of the AFDM is to write the Lagrange densities and the resulting fundamental gravitational and matter field equations in terms of such nonholonomic variables which allows us to decouple and solve nonlinear systems of PDEs. This can not be done if we use from the very beginning the LC-connection ∇. It is not a d-connection, does not preserve under general transforms the h-and v-splitting and the condition of zero torsion, ∇ T = 0 and does not allow to decouple the equations in general forms. Working with D, we introduce certain "flexibility" in order apply a corresponding geometric techniques for integration PDEs. In such cases, we do not make additional assumptions on particular cases for ansatz and connections transforming the fundamental field equations into nonlinear systems of ODEs. Having defined a quite general class of solutions expressed via generating functions and integration functions and constants, we can impose additional nonholonomic constraints (10) which allows to extract LC-configurations. This way, we can construct in explicit form new classes of exact solutions in GR and MGTs both in (g, ∇) and (g, N, D) variables.

Finsler-Lagrange variables in GR and MGTs
On a 4-d/ Lorentz manifold V, we can introduce always Finsler like variables considering a conventional 2+2 splitting of coordinates u α = (x i , y a ) for a nonholonmic fiber structure where y = {y a }, for a = 3, 4, are treated as effective fiber coordinates (which are analogous to velocity ones in theories on tangent bundles). This way we elaborate a toy model of relativistic Finsler-Lagrange geometry. Let us explain how such constructions provide examples of above formulated nonholonomic models of (pseudo) Riemannian geometry. A fundamental function (equivalently, generating function) V ∋ (x, y) → L(x, y) ∈ R, i.e. a real valued function (in brief, called an effective Lagrangian, or a Lagrange density) which is differentiable on V := V /{0}, for {0} being the null section of V, and continuous on the null section of π : V → hV. A relativistic 4-d model of a fibered effective Lagrange space L 3,1 = (V, L(x, y)) is determined by a prescribed regular Hessian metric (equivalently, v-metric) is non-degenerate, i.e. det | g ab | = 0, and of constant signature. Non-regular configurations can be studied as special cases. The non-Riemannian total phase space geometries are characterized by nonlinear quadratic line elements We can elaborate on geometric and physical theories with a spacetime enabled with a nonholonomic frame and metric and (non) linear connection structures determined by a nonlinear quadratic line element (12) and related v-metric (11). The geometric objects on L 3,1 will be labeled by a tilde "~" (for instance, g ab ) if they are defined canonically by an effective Lagrange generating function. We write L 3,1 with tilde in order to emphasize that V is enabled with an effective relativistic Lagrange structure and respective nondegenerate Hessian. The dynamics of a probing point particle in L 3,1 is described by Euler-Lagrange equations, These equations are equivalent to the nonlinear geodesic (semi-spray) equations where g ij is inverse to g ab (11). This way we define a canonical Lagrange N-connection structure determining an effective Lagrange N-splitting N : T V = hV ⊕ vV, similar to (2). Using N a i from (14), we define effective Lagrange N-adapted (co) frames Such N -adapted frames can be considered as results of certain vierbein (frame, for 4-d, tetradic) transforms of type e α = e α α (u)∂/∂u α and e β = e β β (u)du β . 2 We can also consider frame transforms (11), define the respective h-and v-components of a d-metric of signature (+ + +−). In result, we can construct a relativistic Sasaki type d-metric structure Using respective frame transforms g α ′ β ′ = e α α ′ e β β ′ g αβ and g α ′ β ′ = e α α ′ e β β ′ g αβ , such an effective Lagrange-Sasaki can be represented as a general d-metric (4), or equivalently, as a off-diagonal metric (1), Parameterizations of type (17) for metrics are considered in Kaluza-Klein theories but in our approach the N-coefficients are determined by a general or Lagrange N-connection structure. The Lagrange N-connections N defines an almost complex structure J. Such a linear operator J acts on e α = ( e i , e b ) using formulas J(e i ) = −e n+i and J(e n+i ) = e i , and defines globally an almost complex structure J• J = − I, where I is the unity matrix. We note that J is a (pseudo) almost complex structure only for a (pseudo) Euclidean signature. There are omitted tildes and written, for instance, J for arbitrary frame/ coordinate transforms.
A Lagrange Neijenhuis tensor field is determined by a Lagrange generating function introduced as the curvatures of a respective N-connection, for any d-vectors X, Y. Such formulas can be written without tilde values if there are considered arbitrary frame transforms. In local form, a N-connection is characterized by such coefficients of (18) (i.e. the coefficients of a N-connection curvature) : An almost complex structure J transforms into a standard complex structure for the Euclidean signature if Ω = 0.
Using the Lagrange d-metric g and d-operator J, we can define the almost symplectic structure θ := g( J·, ·). Then, we can construct canonical d-tensor fields defined by L(x, y) and N-adapted respectively to N a i (14) and e α = ( e i , e b ) (15): P = e i ⊗ e i − e a ⊗ e a almost product structure ; θ = g aj (x, y) e a ∧ e i almost symplectic structure .
We can define the Cartan-Lagrange d-connection D = (h D, v D) which by definition satisfy the conditions (compair with (6)), The geometric d-objects (16), (20) and (4) can be subjected to arbitrary frame transforms on a Lorentz nonholonomic manifold V when we can omit tilde on symbols, for instance, writing such geometric data in the form (g, J, P,), but we have to preserve the notation D in all systems of frames/coordinates because such a d-connection is different, for instance, from the LC-connection ∇.
We can elaborate on a Lorentz manifold V a relativistic 4-d model of Finsler space is an example of Lagrange space when a regular L = F 2 is defined by a fundamental (generating) Finsler function F (x, y), called also a Finsler metric, when the nonlinear quadratic element (12) is changed into and when there are satisfied the conditions: 1) F is a real positive valued function which is differential on T V and continuous on the null section of the projection π : T V → V ; 2) it is imposed a homogeneity condition F (x, λy) = |λ| F (x, y), for a nonzero real value λ; and 3) the Hessian (11) is defined by F 2 in such a form that in any point (x (0) , y (0) ) the v-metric is of signature (+−). The conditions 1-3) allow to construct various types of geometric models with homogeneity of fiber coordinates with local anisotropy distinguished on directions. Nevertheless, to extend, for instance, the GR theory in a relativistic covariant form, we need additional assumptions and physical motivations on the type of nonlinear and linear connections we involve into consideration, how to extract effective quadratic elements etc., see details and references in [24,25]. In this work, we consider that we can always prescribe on a Lorentz manifold V a Finsler, or Lagrange, type function and state a respective nonholonomic geometric modeling using canonical data ( L, N; e α , e α ; g jk , g ab ), when certain homogeneity conditions can be satisfied for Finsler configurations. For general frame transforms and modified dispersion relations, we do not consider a Lagrange of Finsler like nonholonomic variables but can preserve a conventional h-and v-splitting adapted to a N-connection structure with geometric data (V, N; e α , e α ; g jk , g ab ). To elaborate physically realistic gravity models we need further conventions on the type of linear connection structure (covariant derivative) we shall use for our geometric constructions. We can consider always distortion relations with distortion d-tensors Z, Z, and Z, and postulate the (modified) Einstein equations (9) in various forms where the (effective) matter sources are respective functionals on distortions and energy-momentum tensors for matter fields. Such systems of nonlinear PDEs are different and characterized by different types of Bianchi identities, local conservation laws and associated symmetries. Nevertheless, we can establish such classes of nonholonomic frame and distortion structures, with respective equivalence relations when the equations (9), (23) and (24) describe equivalent gravitational and matter field models. Different geometric data have their priorities in constructing in explicit form different classes of exact/ parametric / approximate solutions or for performing certain procedures of quantization and further generalizations of physical theories. If we work with a respective canonical d-connection structure D, we can prove a general decoupling property of (9) and construct exact solutions with generic off-diagonal metrics g αβ (u γ ) (17) being represented as d-metrics g α ′ β ′ (x, y) (4), when the coefficients of such metrics and associated nonlinear and linear connection structures depend, in principle, on all spacetime coordinates u γ . We can not decouple in general form the systems of nonlinear PDEs (23), in Lagrange-Finsler variables, and (24), in local coordinates and for the LC-connection. In MGTs with modifications of (23) or (24), even in GR, we are able to find exact solutions for some "special" ansatz of metrics which, for instance are diagonalizable and depend only on a radial or time like coordinate (for instance, for black hole and/or cosmological solutions). In this work, we shall apply the AFDM in order to construct cosmological locally anisotropic solutions in MGTs with (in general, generic off-diagonal) metrics of type g α ′ β ′ (x i , y 4 = t). In geometric and analytic form, this is possible if we work with nontrivial N-connection structures and certain variables which are similar to those in Lagrange-Finsler geometry but on Lorentz manifolds. The almost symplectic Lagrange-Finsler variables ( θ, P, J, D) have the priority that they allow to elaborate on deformation quantization and together with (g, N, D) allow to introduce nonholonomic and Finsler like spinors and, for instance, nonholonomic Einstein-Finsler-Dirac systems. This is not possible if the so-called Berwald-or Chern-Finsler connections are used because they are not metric compatible and it is a problem to define in a self-consistent form locally anisotropic versions of the Dirac equation.

TMTs and other MGTs in canonical nonholonomic variables
The goal of this section is to show how various classes of MGTs can be extracted from certain effective Einstein gravity theories using nonholonomic or Finsler like variables. This allows to decouple the gravitational field equations and to generate exact solutions in very general forms, with generic off-diagonal metrics and generalized connections, and with constraints to zero-torsion configurations, see details in Refs. [26,27,28,29,30,31,32,33].
In [14,15,16,26,27,28,29,30,31,32,33,63], there were analyzed different possibilities to model different MGTs by imposing corresponding nonholonomic constraints on the metric and canonical d-connection structures and source in (9). One of the main goals of this work is to prove that using corresponding type parameterizations of the effective Lagrangian L in (8) the so-called modified massive gravity theories (in general, with biconnection and bi-metric structures) can be modeled at TMTs with effective Einstein equations for D when additional constraints D | T =0 = ∇ have to be imposed in order to extract LC-configurations.
The actions for equivalent TMT, MGT and nonholonomically deformed Einstein models are postulated: where | g| = det | g αβ | for a d-metric, g αβ , constructed effectively by a conformal transform of a TMT reference one, g αβ , (see below, formula (34)); Φ L defines a class of theories with two independent non-Riemannian volume-forms 1 Φ(A) and 2 Φ(B) as in [61,62] but with a more general functional for modification, of type ǫf (Ř), than ǫR 2 if D → ∇; the Lagrange density functional f,µ L = F( R) is determined similar to a modified massive gravity by a mass-deformed scalar curvature [64,65,66,14,15], 3 where µ is the graviton's mass and q = {q αβ } is the so-called non-dynamical reference metric; m L is the Lagrangian for matter fields.

Nonholonomic ghost-free massive configurations
The term ǫf (Ř) in (26) contains possible contributions from a nontrivial graviton mass. Such a theory can be constructed to be ghost free for very special conditions [14,15], see explicit results and discussions on possible applications in modern cosmology in Refs. [64,65,66]. In this section, we show how prescribing necessary type nonholonomic configurations such a theory can be equivalently realized as a TMT one (taking equal actions (26) and (27)). For any ( g, N, D), we consider the d-tensor ( g −1 q) µ ν computed as the square root of g µρ q ρν , where We chose the functional for Lagrange density in (27) . For simplicity, we consider Lagrange densities for matter, m L, which only depend on the coefficients of a metric field and not on their derivatives. The energy-momentum d-tensor can be computed via N-adapted variational calculus, Applying such a calculus to F,µ S+ m S, with 1 F(Ř) := dF(Ř)/dŘ, see details in [64,65,66,14,15], we obtain the modified gravitational field equations The field equations for massive gravity (30) are constructed as nonholonomic deformations of the Einstein equations (9) when the source Υ βγ → F,µ Υ µν .

TMT massive configurations with (broken) global scaling invariance
Let us explain the notations and terms used in above actions chosen in such forms that a TMT (26) is equivalent to a massive MGT model (27) when both classes of such theories are encoded via corresponding nonholonomic structures into a nonholonomically deformed Einstein gravity model (25). The non-Riemannian volume-forms (integration measures on nonholonomic manifold (g, N, D)) in (26) are determined by two auxiliary 3-index antisymmetric d-tensor fields A αβγ and B αβγ , when Nevertheless, for non-triviality of the TMT model is crucial the presence of the 3d auxiliary antisymmetric d-tensor gauge field H αβγ , when Φ(H) := 1 3! ε µαβγ e µ H αβγ . In order to model in some limits two flat regions for the inflationary and accelerating universe, we consider two Lagrange densities for a scalar field with dimensional positive parameters q, 1 a, 2 a and a dimensionless one 2 b. The action (26) is invariant under global N-adapted Weyl-scale transforms with a positive scale parameter λ, g αβ → λg αβ , ϕ → ϕ + q −1 ln λ, A αβγ → λA αβγ , B αβγ → λ 2 B αβγ and H αβγ → H αβγ . For holonomic configurations and quadratic functionals on LC-scalar f (Ř) → R 2 , such a theory is equivalent to that elaborated in [55,56,57,61,62]. In a more general context, the developments in this work involve non-quadratic nonlinear and nonholonomic functionals and mass gravity deformations viaŘ (28) and generic off-diagonal interactions encoded in R.
A variational N-adapted calculus on form fields A, B, H and on d-metric g (with respect to coordinate bases and for ∇ being similar to that presented in section 2 of [61,62]) results in effective gravitational field equations where F,µ Υ µν is determined by (31) and ef Υ βγ := κ( ef T αβ − 1 2 g αβ ef T ) is computed using formulas (8) and (29) for g αβ → g αβ and L → ef L, where when the conformal factor Θ for the Weyl re-scaling of d-metric is induced by the nonlinear functional in the action and the two measure functionals 1 χ = 1 Φ(A)/ | g µν | and 2 χ = 2 Φ(B)/ | g µν |.
The variations on auxiliary anti-symmetric form fields impose certain constants The nonconstant solutions of such nonholonomic constraints allow to preserve the global Weyl-scale invariance for certain configurations. If we take constant values we select configuration with nonholonomic dynamical spontaneous breakdown of global Weyl-scale invariance when the condition preserves the scale invariance. There are certain constraints on the scale factor 1 χ = 1 Φ(A)/ | g|, which can be derived from variation of (26) on g µν in N-adapted form. The conditions (36) relate 1 χ and 2 χ, i.e. the integration measures, to traces 1,2 T := g αβ 1,2 T αβ of the energy momentum tensors 1,2 T αβ = g αβ 1,2 L − 2∂( 1,2 L)/∂g αβ of Lagrangians for scalar fields (32). 4 This follows form the N-adapted variation on g αβ of the action (26) taken for simplicity with zero m L. which results in Taking the trace of these equations and using (36), we obtain the formula 1 χ = 2 χ 2 T+2 2 M 2 1 L− 2 T−2 1 M , which does not depend on the type of f -modifications containing possible µ-terms. We conclude that above considered non-Riemannian integration measures and the interactions of scalar fields (32) can be modelled as additional distributions on nonholonomic manifold (g, N, D). They modify the conformal factor Θ (34) and allows to express the field equations (38) in Einstein like form (33), where F,µ Υ µν is added as an additional effective matter contribution the source of scalar fields 1,2 T αβ .
It should be noted that using the canonical d-connection we obtain D α T αβ = Q β = 0, when Q β [g, N] is completely defined by the d-metric and chosen N-connection structure. Considering nonholonomic distortions with ∇ = D − Z, we obtain standard relations A similar property exists in Lagrange mechanics with non-integrable constraints when the standard conservation laws do not hold true. A new class of effective variables and new types of conservation laws can be introduced and, respectively, constructed using Lagrange multiples. The main conclusion of this section is that various MGTs with two integration measures, possible deformations by mass graviton terms, bi-connection and bi-metric structures can be expressed as nonholonomic deformations of the Einstein equations in the form (9). Different theories are characterized by respective sources (in explicit form, F,µ Υ µν in (30), or ef Υ µν + F,µ Υ µν in (33)). Our next goal is to prove that such effective Einstein equations can be integrated in certain general forms for D and possible constraints (10) for LC-configurations.

Cosmological Solutions in Effective Einstein Gravity and FMGTs
We can generate in explicit form integral varieties of systems of PDEs of type (9) for d-metrics g (34) and sources Υ βγ = ef Υ µν + F,µ Υ µν as in (33) which via frame and coordinate transforms, for a time like coordinate y 4 = t (i ′ , i, k, k ′ , ... = 1, 2 and a, a ′ , b, b ′ , ... = 3, 4), can be parameterized in the form: and These ansatz for the d-metric and sources are very general one but for an assumption that there are N-adapted frames with respect to which the MGTs interactions are with Killing symmetry on ∂/∂y 3 when geometric and physical values do not depend on coordinate y 3 . 5 We use parameterizations g 1 = g 2 = e ψ(x i ) and h a (x k , t) for i, j, ... = 1, 2 and a, b, ... = 3, 4; and N-connection coefficients N 3 i = n i (x k , t) and N 4 i = w i (x k , t). Introducing brief denotations for partial derivatives like a we transform (33) into a nonlinear system of PDEs with decoupling property for the un-known functions ψ(x i ), h a (x k , t), w i (x k , t) and n i (x k , t), This system posses another very important property which allows us to re-define the generating function, Ψ ←→ Ψ, when Λ(Ψ 2 ) * = |Υ|( Ψ 2 ) * and for Ψ := exp ̟ and any prescribed values of effective (for different types of contributions ef, m, f, µ) cosmological constants in Λ = ef Λ + m Λ + f Λ + µ Λ associated respectively to For generating off-diagonal cosmological solutions depending on t, we have to consider generating functions for which Ψ * = 0. The equations (42) for ansatz (39) transform respectively into such a system of nonlinear PDEs and ̟ * ∂ i ω − ω * ∂ i ̟ = 0, for the vertical conformal factor.
We have to subject the d-metric and N-connection coefficients to additional constraints (10) in order to satisfy the torsionless conditions, which for the ansatz (39) are written We can generate exact solutions in TMT, MGT and nonholonomically deformed Einstein theories with respective actions (25), (26) and (27) using integral varieties 6 of the system of PDEs (44) which can be found in very general forms. Let us briefly explain this geometric formalism elaborated in the framework of the AFDM (see details, for instance, in Refs. [30,31,32,33]): 1. The first equation for ψ is just the 2-d Laplace/ d' Alambert equation which can be solved for any given Υ, which allows us to find g 1 = g 2 = e ψ(x k ) . (44) and (41), the coefficients h a can be expressed as functionals on (Ψ, Υ). We re-define the generating function as in (43) and consider an effective source

Using the second equation in
3. We have to integrate two times on t in order to find from the 3d subset of equations in (44) for some integration functions 1 n i (x k ) and 2 n i (x k ). (44) are algebraic ones which allows us to compute

The 4th set of equations in
5. We can satisfy the conditions for ω in the second line in (44) if we keep, for simplicity, the Killing symmetry on ∂ i and take, for instance, ω 2 = |h 4 | −1 .
Different types of inhomogeneous cosmological solutions of the system (33) are determined by corresponding classes of and effective sources generating functions: effective sources: We can generate solutions with any nontrivial ef Λ, m Λ, f Λ, µ Λ even any, or all, effective source ef Υ, m Υ, f Υ, µ Υ can be zero.

Inhomogeneous FTMT and MGT configurations with induced nonholonomic torsion
The solutions with coefficients computed above in 1-5 can be parametrized in a form to describe nonholonomic deformations, g αβ = e α ′ α e β ′ βg α ′ β ′ , of the Friedman-Lemaître-Robertson-Walker, FLRW, diagonal quadratic element 7 into a generic off-diagonal inhomogeneous cosmological metric of type (39) with g i = η i e ψ and h a = η aga with effective polarization functions η 1 = η 2 = a −2 e ψ , η 3 =å −2 h 3 , η 4 = 1 and h 3 = h 3 /a 2 |h 4 |, when The inhomogeneous scaling factor a(x k , t) in (47) is related to the generating function Ψ via formula In general, such target metrics g αβ (x k , t) determine new classes of cosmological metrics with nontrivial nonholonomically induced torsion computed for D. Such modified spacetimes can not be diagonalized by coordinate transforms if the anholonomy coefficients W γ αβ are not zero. For trivial gravitational polarizations, η α = 1, trivial N-connection coefficients, N 3 i = n i = 0 and N 4 i = w i = 0, and for a(x k , t) →å(t) we obtain torsionless FLRW metrics. We emphasize that one could not be smooth limits g αβ →g αβ for arbitrary generating function Ψ and any nontrivial effective cosmological constant ef Λ, m Λ, f Λ, or µ Λ, associated to respective mater fields.
We can generate off-diagonal cosmological configurations as "small" deformations with η α = 1+ ǫ α , n i = ǫ n i and w i = ǫ w i , with |ǫ α |, | ǫ n i |, | ǫ w i | ≪ 1. In particular, we can study only TMT models if m Ξ = f Ξ = µ Ξ = 0 and m Λ = f Λ = µ Λ = 0 but ef Υ(x k , t) = 0 and ef Λ = 0. Off-diagonal cosmological scenarios in massive and bi-metric gravity with nontrivial µ Ξ and µ Λ were studied in our recent works [14,15]. Other classes of MGTs and cosmological models with off-diagonal configurations when f -modified gravity effects are modelled in GR were studied in [26,27,28,29,30,31,32,33]. The goal of section 6 is to show how TMT gravity and cosmological models can be associated to certain nonholonomic off-diagonal de Sitter configurations with nontrivial ef Λ for an effective Einstein-Lagrange spacetime and such constructions can be generalized to reproduce MGTs and massive gravity.

Extracting Levi-Civita cosmological configurations
Let us show how we can generate in explicit form solutions of the system (45) for nonholonomic generic off-diagonal configurations with zero torsion. We have to consider certain special classes of generating and integration functions. By straightforward computations we can check that such conditions are satisfied if we state such conditions for a metric (47) that 2 n i = 0 and 1 n i = ∂ i n(x k ), for any n(x k ) Ψ =Ψ, for (∂ iΨ ) * = ∂ i (Ψ * ) and find a functionǍ(x k , t) when when ΛΨ 2 = Ψ 2 |Υ| + dt Ψ 2 |Υ| * and Ξ := dtΥ( Ψ 2 ) * are computed following formulas (43) but for Ψ( Ψ) →Ψ( Ψ) and Ψ → Ψ. For certain configurations, we can consider functional dependencies Ψ = Ψ(ln |h 3 |) and invertible functional dependencies h 3 [ Ψ [Ψ]]. In such cases, we take a h 3 (x k , t) as a generating function and consider necessary type functionalsΨ[h 3 ] with the property Putting together the conditions (48), we generate nonhomogeneous cosmological LC-configurations with quadratic linear elements The inhomogeneous scaling factorǎ(x k , t) is computed similarly to (47) but using the generating function Ψ, Having constructed a class of generic off-diagonal solutions (49), we can impose additional constraints on the generating/integration functions and constants and source in order to explain certain observational cosmological data. For instance, we can fix subclasses of functions Ψ → Ψ(t), (∂ iǍ ) → w i (t) etc. describing small deformations of a FLRW metric (46) in a nonlinear parametric way re-defined generating functions (43) and different types of effective sources in TMT, MGT and/or massive gravity models.

Locally Anisotropic Effective Scalar Potentials and Flat Regions
We study three examples of off-diagonal cosmological solutions reproducing the TMT model with two flat regions of the effective scalar potental studied in Ref. [60], than analyse how massive gravity can be modelled as a TMT theory and effective GR, and (in the last subsection) we speculate on non-singular emergent anisotropic universes. The solutions in this section will be constructed to contain nontrivial nonholonomically induced torsion as for quadratic elements (47). For certain important limits, LC-configurations of type (49) will be also examined.

Off-diagonal interactions and associated TMT models with two flat regions
We chose the nontrivial off-diagonal data in (47) for m Λ = f Λ = µ Λ = 0 and m Υ = f Υ = µ Υ = 0 resulting in m Ξ = f Ξ = µ Ξ = 0, but consider nonzero ef Λ and ef Υ is taken as a one-Killing configuration not depending on y 3 in is computed using formula (8) and (29) for g αβ → g αβ and L → ef L for two scalar densities (32) as in (34). We generate solutions of R µν [ g αβ ] = ef Υ µν (in a particular case of (33)) for g αβ (x k , t) = Θ(x k , t)g αβ (x k , t), parameterized in the form The inhomogeneous scaling factor a(x k , t) is related to the generating function Ψ via formula Choosing a function Ψ, we prescribe a corresponding dependence for Θ(x k , t) and, respectively, a(x k , t) as follow from above formulas. Let us speculate on the structure of Θ which describe off-diagonal generalizations of the model given by formulas (18)- (23) in [60] in the assumption that the relation (35) for zero graviton mass and quadratic Ricci scalar curvature has the limit In this subsection, we shall consider 1 f ≈ 1 U − 1 M for a nonhomogeneous ϕ(x k , t)≈ϕ(t) in order to construct cosmological TMT models with limits to diagonal two flat regions. We consider Θ as a conformal factor Θ in (34) not depending on y 3 written in explicit form for an Einstein N-adapted frame with effective scalar Lagrangian where we omit cumbersome formulas for A(ϕ) and B(ϕ) in the second line (see similar ones given by formulas (24) and (25) in [60]) but present For simplicity, we can construct off-diagonal configurations with h 3 ≃ 1 in (51), prescribing a value ef Λ corresponding to observational data in accelerating Universe and computing ef Ξ for ef L using formulas and constraints of type (40), Then, we can compute ef Ξ := dt ef Υ( Ψ 2 ) * . Such a problem can be also solved in inverse form for a given a(x k , t), when Ψ has to be defined from an integro-differential equation (51), a 2 = . For cosmological solutions, we can consider a(x k , t) ≃ a(t) and Ψ(x k , t) ≃ Ψ(t), when the generation function Ψ(t) is prescribed to depend only on time-like coordinate t. The observable effective scaling factor a(t) is expressed as a functional on constant ef Λ, on TMT source ef Υ(t) and generating function Ψ(t). For instance, variations of ef Υ(t) are determined by the variation of the second auxiliary 3-index antisymmetric d-tensor field B αβγ in 2 Φ(B) in the formula (37). We adapt and write a similar formula with "tilde" values in order to emphasize that the values are computed for a prescribed value a(t), There are two options to fix a constant 2 χ : the first one is to chose a function Ψ and/or to modify B in the second measure. In general, this is a nonlinear effect of re-definition of generation functions (43) which holds for generic off-diagonal configurations. We can prescribe finally some small off-diagonal corrections but the diagonal values will be re-scalled (we shall keep "tilde " in order to distinguish such values from similar ones computed from the very beginning using diagonalized equations). The main conclusion of this subsection is that working with generic off-diagonal solutions for effective Einstein equations (33), see equivalent formulas (38), we can chose such generating functions and effective source that we reproduce in generalized forms the properties of TMT gravity theories determined by action (26) and scalar Lagrangians (32). In the next subsection, we prove that such models may be generated to have limits to diagonal two flat regions reproducing accelerating cosmology scenarios.

Limits to diagonal two flat regions
Let us consider in ef L (52) the approximation with Ψ(t) and a(t) resulting in diagonal cosmological solutions with effective FLRW metrics. We approximate the effective potential ef U (53) for a prescribed constant 2 χ by a relation (54), for ϕ → +∞ .
For such diagonal approximations, the A-and B-functions can be computed in explicit form Such values reproduce the results of section 3 in [60] with two flat regions of the effective potential ef U but in our approach the effective diagonalized metric is of type (49) withǎ ≃ a(t) for η α ≃ 1. This class of diagonalized solutions but determined by generating functions contain in explicit form solutions with effective scalar field evolving on the first flat region for large negative ϕ and describing non-singular "emergent universes" [49,50,51,52,53,54].

Reproducing Modified Massive Gravity as TMTs and Effective GR
The goal of this section is to study solution of effective Einstein equations (33) when the source (40) is taken for m Υ = f Υ = 0 and m Υ = f Υ = 0, i.e.
with a left label "eµ" emphasizing that such sources are considered for TMT configurations with a nontrivial mass term µ but zero matter field configurations and for a possible quadratic ǫR 2 cosmological term. We shall chose such N-adapted frames of reference and generating functions when the TMT gravity model will describe modifications my µ 2 terms for nonholonomic ghost-free configurations and corrections to scalar curvature (28) is determined by the graviton's mass µ and q = {q αβ } is the so-called non-dynamical reference metric. For simplicity, we make the assumption that such values can be re-defined to be constant for certain choices of the generating functions, effective sources ef Υ(x k , t), µ Υ(x k , t) and, respective, nontrivial constants eµ Λ = ef Λ + µ Λ.

Massive gravity modifications of flat regions
We can integrate in generic off-diagonal form such TMT systems as subclasses of solutions (47) when We write Ψ → Ψ, when the generating function is chosen to satisfy the conditions In general, such nonhomogeneous locally anisotropic configurations are with nontrivial nonholonomically induced canonical d-torsion which can be constrained to be zero for corresponding subclasses of generating functions and sources.
We study off-cosmological solutions depending only on time like coordinate when a(x k , t) ≃ a(t) and Ψ(x k , t) ≃ Ψ(t) and the generation function Ψ(t). The formula relating variations of eµ Υ(t) to the variation of the second auxiliary 3-index antisymmetric d-tensor field B αβγ in 2 Φ(B), a particular case of (37) is given by where the constant µ χ is zero for µ = 0 and | µ χ| ≪ | 2 χ|. Another assumption is that we can formulate a TMT theory corresponding to "pure" µ-deformations of GR even ǫ = 0. The formula (55) has to be generalized for nontrivial µ, when is a version of generalized Starobinsky relation (35), formulas (36) and (28) and approximation of typeR ≃ R + µ 2 . The resulting formulas for effective potential (53) contain additional µ-terms for ϕ → +∞ .
The A-and B-functions can also contain contributions of µ-terms,

when [−]
A is not modified. We conclude that solutions with nontrivial generating functions for nontrivial massive gravity terms modelled as effective TMT theories may also describe non-singular "emergent universes" [49,50,51,52,53,54] with corresponding modifications.

Reconstructing off-diagonal TMT and massive gravity cosmological models
For the class of solutions (56), we show how we can perform a reconstruction procedure. We introduce a new time coordinate t for t = t(x i , t) and |h 4 |∂t/∂ t, and re-defined the scale factor, a → a(x i , t), representing the quadratic elements in the form for η i = a −2 e ψ , a 2 h 3 = h 3 , e 3 = dy 3 + ∂ k n dx k , e 4 = d t + |h 4 |(∂ i t + w i ).
To model small off-diagonal deformations we use a small parameter ε, 0 ≤ ε < 1, when and there are subclasses of generating functions and sources for which a(x i , t) → a(t), h 3 (x i , t) → h 3 ( t) etc. , see details for such a procedure from section 5 of [67] (see also references therein). The analogous TMT massive gravity theory is taken with a source µ Υ µν (31) and parametrization f (Ř) = R + S( µ T), for any N-adapted value Introducing values 1 S := dS/d µ T and H := a * / a for a limit a(x i , t) → a(t) with N a i = {n i , w i (t)} and effective polarizations η α (t).
In order to test cosmological scenarios, we consider a redshift 1 + z = a −1 (t) for µ T = µ T (z) by introducing a new "shift" derivative. For instance, for a function s(t)) s * = −(1 + z)H∂ z . We can derive TMT massive modified off-diagonal deformed FLRW equations using formulas (63) and (64) in [67], when for ρ(z) ∂ z f = 0. We can fix the condition ∂ z 1 S(z) = 0, re-scale the generating function in order to satisfy the condition ∂ z f = 0. We have nonzero densities in certain adapted frames of references. Here we note that the functional S( µ T) encodes effects of mass gravity for the evolution of the energy-density when ρ = ρ 0 a −3(1+̟) = ρ 0 (1 + z)a 3(1+̟) , when for the dust matter approximation ̟ and ρ ∼ (1 + z) 3 . Any FLRW cosmology can be realized in a corresponding class of f -gravity models, which can be re-encoded as a TMT theories using actions of type (25)- (27). Let us introduce ζ := ln a/a 0 = − ln(1 + z) as the "e-folding" variable to be used instead of the cosmological time t and consider with dependencies on (x i , ζ) of generating functions ∂ ζ = ∂/∂ζ with q * = H∂ ζ q for any function q.
Repeating all computations leading to Eqs. (2)-(7) in [68], in our approach for f (Ř), we construct a FLRW like cosmological model with nonholonomic field equation corresponding to the first FLRW equation We consider an effective quadratic Hubble rate,κ(ζ) := H 2 (ζ), where ζ = ζ(Ř), we write this equation in the form For any off-diagonal cosmological models with quadratic metric elements of type (57) for redefined t → ζ when a functional f (Ř) is used for computing Υ, the generating function and respective d-metric and N-connection coefficients as solutions of certain effective Einstein spaces for auxiliary connections and effective cosmological constant eµ Λ. The value df /dŘ and higher derivatives vanish for any functional dependence f ( eµ Λ) because ∂ ζ eµ Λ = 0. We conclude that the recovering procedure simplifies substantially even in TMT theories by using re-scaling of generating function and sources following formulas of type (43). Now we speculate how we can reproduce the ΛCDM era. Using values a(ζ) and H(ζ) determined by an off-diagonal quadratic element (57) and write analogs of the FLRW equations for ΛCDM cosmology in the form for fixed constant values H 0 and ρ 0 . The second term in this formula describes, in general, an inhomogeneous distribution of cold dark mater (CDM). This allows to compute the effective quadratic Hubble rate and the modified scalar curvature,Ř, in the formsκ(ζ) := H 2 0 + κ 2 ρ 0 a −3 0 e −3ζ anď The solutions of (60) can be found following [68] and [67] as Gauss hypergeometric functions. We might denote f = F (X) := F (χ 1 , χ 2 , χ 3 ; X), where for some constants A and B, This is the proof that MGTs and various TMT models can indeed describe Λ CDM scenarios without the need of an effective cosmological constant because we have effective sources and this follows from the re-scaling property (43) of generic off-diagonal configurations. The equation (60) transforms into for certain constants, for which χ 1 + χ 2 = χ 1 χ 2 = −1/6 and ] and X := −3 +Ř/3H 2 0 . Finally, we note that the reconstruction procedure can be performed in similar form for any MGTs and TMT ones which can modeled, for well-defined conditions, by effective nonholonomic Einstein spaces.

Modified gravity and cosmology theories with metric Finsler connections on (co) tangent Lorentz bundles or for nonholonmic Einstein manifolds
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. Fixing respective classes of such functions and constants, we can extract various types of modified gravity-matter theories defined in terms of non-Riemannian volume-forms (for instance, in a manifestly globally Weyl-scale invariant form) and with certain modified Lagrange densities of type f (Ř) including contributions from the Einstein-Hilbert term R, its square R 2 , possible massive gravity µ parametric terms, nonholonomic deformations etc. The principal results are as follows: 1. We defined nonholonomic geometric variables for which various classes of modified gravity theories, MGTs, (in general, with nontrivial gravitational mass) can be modelled equivalently as respective two measure (TMT) [55,56,57,60,61,62], bi-connection and/or bi-metric theories. For well defined nonholonomic constraint conditions, the corresponding gravitational and matter field equations are equivalent to certain classes of generalized Einstein equations with nonminimal connection to effective matter sources and nontrivial nonholonomic vacuum configurations.
2. We stated the conditions when nonholonomic TMT models encode ghost-free massive configurations with (broken) scale invariance and such interactions can modelled by generic off-diagonal metrics in effective general relativity (GR) and generalizations with induced torsion. Such a nonholonomic geometric techniques was elaborated in Finsler geometry in gravity theories and for a corresponding 2+2 splitting we can consider Finsler like variables and work with so-called FTMT models.
3. We developed the anholonomic frame deformation method [30,31,32,33], AFDM, in order to generate off-diagonal, in general, inhomogeneous and locally anisotropic cosmological solutions in TMT snd MGTs. It was proved that the effective Einstein equations for such gravity and cosmological models can be decoupled in general form which allow to construct various classes of exact solutions depending on generating functions and integration functions and constants.
4. We analysed a very important re-scaling property of generating functions with association of effective cosmological constants for different types of modified gravity and matter field interactions which allow to define nonholonomic variables for which the associated systems of nonlinear partial differential equations, PDEs, can be integrated in explicit form when the coefficients of generic off-diagonal metrics and (generalized) nonlinear and linear connections depend on all space-time coordinates.
5. There were stated conditions on generating functions and effective sources when zero torsion (Levi-Civita, LC) configurations can be extracted in general form with possible nontrivial limits to diagonal configurations in ΛCDM cosmological scenarios, encoding dark energy and dark matter effects, possible nontrivial zero mass contributions, effective cosmological constants induced by off-diagonal interactions but finally constrained nonholonomically to result in nonlinear diagonal effects.
6. A special attention was devoted to subclasses of generic off-diagonal cosmological solution with effective scalar potentials and two flat regions and studied limits to diagonal cosmological TMT scenarios investigated in [61,62].
7. We studied possible massive gravity modifications of flat regions and speculated on reconstructing offdiagonal TMT and massive gravity cosmological models. Via corresponding frame transforms and redefinition of generating functions and nonholonomic variables, we proved that the same geometric techniques is applicable in all such MGTs.
Let us explain why it is important to study in different MGTs exact solutions for off-diagonal and nonlinear gravitational interactions depending on 2-4 spacetime coordinates and consider possible implications in modern cosmology. The gravitational and matter field equations in such theories consist very sophisticate systems of nonlinear PDEs. It was possible to construct physically important, for instance, black hole and cosmological solutions for certain diagonal ansatz depending on one space/time like variable modelling (generalized) Einstein spacetimes with two and three Killing symmetries or other type high symmetry and asymptotic conditions. There were two kinds of motivations for such assumptions: The technical one was that for diagonalizable ansatz the systems of nonlinear PDEs transform in "more simple" systems of nonlinear ordinary differential equations, ODEs, which can be integrated in general form. The physical interpretation of such solutions determined by integration constants is more intuitive and natural. Nevertheless, a series of problems arisen in modern acceleration cosmology with dark energy and dark matter effects. It became clear that standard diagonal cosmological solutions in GR together with standard scenarios from particle physics and former elaborated cosmological models can not be applied in order to explain observational cosmological data. A number of MGTs and new cosmological theories have been proposed and developed.
Haven chosen mathematically some special diagonalizable ansatz with prescribed symmetries, we eliminate from consideration another more general classes of solutions which seem to be important for explaining nonlinear parametric and nonholonomic off-diagonal interactions. This can be related to a new nonlinear physics in gravity and particle filed theory which have not been yet investigated. In the past, there were a number of technical restrictions to construct such solutions and study their applications but at present there are accessible advanced numerical, analytic and geometric methods. In this work, we follow a geometric approach developed in [14,15,16,26,27,28,29,30,31,32,33,63], which allow to construct exact solutions in different classes of gravity and cosmology theories. Even observational data in modern cosmology can be explained by almost diagonal and homogeneous models, when possible off-diagonal effects and anisotropies are very small, this does not constrain us to study only solutions of associated systems ODEs. For nonlinear gravitational and matter field systems, a well-defined mathematical approach is to generate (if possible, exact) solutions in the most general form and then to impose additional constraints for diagonal configurations. In result, a number of effects of MGTs and accelerating cosmology can be explained as standard but off-diagonal nonlinear ones in effective GR. Alternative interpretations in the framework of TMT and other type theories are also possible.

Alternative Finsler gravity theories with metric non-compatible connections
The referee of this work requested "minimal modification" in order to cite and discuss papers [69,70,71,72,73,74] where some alternative Finsler gravity and geometry models are considered. This is a good opportunity for authors which allows them to explain in a more detailed form their approach, geometric methods, and new results on constructing new classes of generic off-diagonal cosmological solutions and elaborating applications in non-standard particle physics and modified gravity. To comment and compare key ideas and constructions in authors' works with similar ones from the mentioned alternative geometric and cosmological theories we have to cite additionally the papers [75,76,77,78,79], and references therein. We note that in introduction (readers should pay attention to footnote 2) and conclusion sections, and Appendix B, to [24] there are provided a number of historical remarks and a review of last 80 years research activity main achievements on Finsler-Lagrange-Hamilton geometry and applications in modern physics, gravity, cosmology, mechanics, information theories. The axiomatic part was published in [25]. In just mentioned works, it is included a study of evolution of main research groups on "Finsler geometry and physics" in different countries, and formed international collaborations. There were reviewed the results and bibliography of for conventional 20 directions and more than 100 sub-directions of research and publications, of present and other authors, related to Finsler geometry and applications. We also cite as a brief critical review the paper [76] and the monograph [79], (for a collection of works on (non) commutative metric-affine generalized Finsler geometries and nonholonomic supergravity and string theories, locally anisotropic kinetic and diffusion processes, Finsler spinors etc.), and articles [77,78].
Here we summarize and discuss such issues: 1. In the abstract and introduction, see also subsection 2.2, of this article, it is emphasized that we do not elaborate a typical work on Finsler gravity and cosmology but rather provide a cosmological work on Einstein gravity and MGTs, TMTs ones, with two measures/ two connections and/or bi-metrics, mass terms, etc., when the constructions are modelled on a Lorentz manifold V of signature (+++-) with conventional nonholonomic 2+2 splitting. For such theories, the spacetime metrics g αβ (x i , y a ) (with i, j, ... = 1, 2 and a, b, ... = 3, 4) are generic off-diagonal and together with the coefficients of other fundamental geometric objects, depend generically on all spacetime conventional fibred coordinates. Lagrange-Finsler like variables are introduced on V for "toy" models, when y a are treated similarly to (co) fiber coordinates on a (co) tangent manifold (T * V ) T V, for a prescribed a fundamental Lagrange, L(x, y) (or Finsler, for certain homogeneity conditions F (x, βy) = βF (x, y), x = {x i } etc., for a real constant β > 0, when L = F 2 ). This states on V a canonical Finsler like N-connection and nonholonomic (co) frames structures, which can be also described in coordinate bases, extracting by additional constraints the LC-connection or distorting to other linear connections determined by the same metric structures. In dual form, we can consider momentum like p a -dependencies in g αβ (x i , p a ), for a conventional Hamiltonian H(x, p), which can be related to a L via corresponding Legendre transforms. The reason to introduce such Finsler like and other type nonholonomic variables on a manifold V, or on a tangent bundle T V , is that in so-called nonholonomic canonical variables (with hats on geometric objects) the modified Einstein equations (9) can be decoupled and integrated in vary general forms. We have to consider some additional nonholonomic constraints (10) in order to extract LC-configurations. This is the main idea of the AFDM [30,31,32,33] which was applied in a series of works for constructing locally anisotropic black hole and cosmological solutions defied by generic off-diagonal metrics and (generalized) connections in Lagrange-Finsler-Hamilton gravity in various limits of (non) commutative/ supersymmetric string/ brain theories, massive gravity, TMT models etc. as we consider in partner works [26,27,28,29,30,31,32,33].
2. One of the formal difficulties in modern Finsler geometry and gravity is that some authors (usually mathematicians) use a different terminology comparing to that elaborated by physicists in GR, MGTs, TMTs etc. For instance, a theory of "standard static Finsler spaces", with a time like Killing field and/or for static solutions of a type of filed equations in Finsler gravity is elaborated in [69,70,71]. Of course, it is possible always to prescribe a class of static and corresponding smooth class of Finsler generating functions, F (x, y), when semi-spray, N-connections and d-connections, and certain Finsler-Ricci generalized tensors etc. can be computed for static configurations embedded in locally anisotropic backgrounds. Such constructions can be chosen to be with spherical symmetry. But introducing and computing corresponding "standard static" Sasaki type metrics of type (16), and their off-diagonal coordinate base equivalents, involving N-coefficients (see the total (phase) spacetime metric (17)), we can check that such geometric d-objects (and corresponding canonical d-connection, or LC-connection) do not solve the (modified) Einstein equations (9) if the data are general ones considered in [69,70,71]. If the d-metric coefficients g αβ (x i , y a ) are generic off-diagonal with nontrivial N-connection coefficients, such metrics can be only quasi-stationary following the standard terminology in mathematical relativity and MGTs (when coefficients do not depend on time like variable, i.e. ∂ t is a Killing symmetry d-vector), but there are nontrivial off-diagonal metric terms because of rotation, N-connections etc. Stationary metrics of type (16) and/or (17) can be prescribed to describe, for instance, black ellipsoids, which are different from the solutions for Kerr black holes, BHs, because of a more general Finsler local anisotropy. Static configurations with diagonal metrics of Schwarzschild type BHs can be introduced for some trivial N-connection structures (but in Finsler geometry this is a cornerstone geometric object). For Finsler like gravity theories, there are not proofs of BH uniqueness theorems, and it is not clear if such static configurations (for instance, with spherical symmetry) can be stable. Such proofs are sketched for black ellipsoids, see details references in [26,27,28,29,30,31,32,33]. So, the existing concepts, definitions, and proofs of "standard" static/ stationary/ cosmological / stable / nonlinear evolution models etc. depend on the type of postulated principles for respective concepts and theories of Finsler spacetime.
3. In [72,73,75], certain attempts to elaborate models of Finsler spacetime geometry and gravity are considered for some types of N-connections and chosen classes of Finsler metric compatible and noncompatible d-connections. In many cases, it is considered the Berwald-Finsler d-connection, which (in general) is noncompatible but can be subjected to certain metrization procedures. Different geometric constructions with non-fixed signature for Hessians and sophisticate causality conditions via semi-sprays and generalized nonlinear geodesic configurations have been proposed and analyzed. In such approaches, there are a series of fundamental unsolved physical and geometric problems for developing such Finsler theories in a some self-consistent and viable physical forms. We point here only some most important issues (for details, critics, discussions, and motivation on Finsler gravity principles we cite [24,25,76,79,17]): • For theories with metric noncompatible connections, for instance, of Chern or Berwald type, there are not unique and simple possibilities to define spinors, conservation laws of type D i T jk , elaborate on supersymmetric and/or noncommutative/ nonassociative generalizations, to consider generalized type classical and quantum symmetries, considering only Finsler type d-connections proposed by some prominent geometers like E. Cartan, S. Chern, B. Berwald etc., and physically un-motivated (effective) energy-momentum tensors with local anisotropy.
• Physical principles and nonlinear causality schemes being elaborated on a base manifold with undetermined lifts, without geometric and physical motivations, on total bundles depend on the type of Finsler generating functions, Hessians and nonlinear and linear connections are chosen for elaborating geometric and physical models. A Finsler geometry is not a (pseudo/ semi-) Riemannian geometry where all constructions are determined by the metric and LC-connection structures. For instance, certain constructions with cosmological kinetic/ statistical Finsler spacetime in [73,75] are subjected to very complex type conservation laws and nonlinear kinetic/ diffusion equations. Those authors have not cited and did not applied in their works more early locally anisotropic generalized Finsler kinetic/ diffusion / statistical constructions performed for metric compatible connections studied in [77,78,79] (N. Voicu were at S. Vacaru seminars in Brashov in 2012, on Finsler kinetics, diffusion and applications in modern physics and information theory, see also [33], but together with her coauthors do not cite, discuss, or apply such locally anisotropic metric compatible and exactly solvable geometric flow, kinetic, geometric thermodynamic theories).
• Various variational principles, certain versions of Finsler modified Einstein equations were proposed and developed in [72,73,75] but such theories have been not elaborated on total bundle spaces, for certain metric compatible Finsler connections. Usually, there were used metric non-compatible Finsler connections, when it is not possible to elaborate on certain general methods for constructing exact and parametric solutions of such nonlinear systems of PDEs, for instance, describint locally anisortopic interactions of modified Finsler-Einstein-Dirac-Yang-Mills-Higgs systems. In the S. Vacaru and co-authors axiomatic approach to relativistic Finsler-Lagrange-Hamilton theories [24,25,76,17], such generalized systems can be studied, for instance, on (co) tangent Lorentz bundles (and on Lorentz manifolds with conventional nonhlonomic fibred splitting), when the AFDM was applied for generating exact and parametric solutions and certain deformation quantization, gauge like etc. schemes were developed. 4. In result, authors of [74] concluded their work in such a pessimistic form: "Finsler geometry is a very natural generalisation of pseudo-Riemannian geometry and there are good physical motivations for considering Finsler spacetime theories. We have mentioned the Ehlers-Pirani-Schild axiomatic and also the fact that a Finsler modification of GR might serve as an effective theory of gravity that captures some aspects of a (yet unknown) theory of Quantum Gravity. We have addressed the somewhat embarrassing fact that there is not yet a general consensus on fundamental Finsler equations, in particular on Finslerian generalisations of the Dirac equation and of the Einstein equation, and not even on the question of which precise mathematical definition of a Finsler spacetime is most appropriate in view of physics. We have seen that the observational bounds on Finsler deviations at the laboratory scale are quite tight. By contrast, at the moment we do not have so strong limits on Finsler deviations at astronomical or cosmological scales." In that work, there were not discussed and analyzed the approach developed for Lorentz-Finsler-Lagrange-Hamilton and nonholonomic manifolds developed by authors of this paper beginning 1994 and published in more than 150 papers in prestigious mathematical and physical journals and summarized in 3 monographs (for reviews, see [24,25,79]).
S. Vacaru's research group was more optimistic (than the authors cited above point 4) on obtained results and perspectives of Finsler geometry in physics. Having obtained by 10 NATO, CERN, DAAD research grants, the group elaborated an axiomatic approach to Finsler-Lagrange-Hamilton gravity theories using constructions on nonholonomic (co) tangent Lorentz bundles and Lorentz manifolds, with N-connection structure and Finsler like metric compatible connections. They began their activity almost 40 years ago, see historical remarks, summaries of results and discussions in [24,25,76,79,17], with recent developments in [26,27,28,29,30,31,32,33]. P. Stavrinos (with more than 40 years research experience on Finsler geometry and applications) and his coauthors also published a series of works on modified Finsler gravity and cosmology theorise involving tangent Lorentz bundles [18,22,79]. For such classes of modified Finsler geometric flow and gravity theories, there exists a general geometric method for constructing exact and parametric solutions, the AFDM, with self-consistent extensions to noncommutative and nonassociative, supestring and supergravity, Clifford-Finsler etc. theories. Together with papers on deformation and other type quantum Finsler-Einstein-gauge gravity theories, which were elaborated and developed in more than 20 directions of research on Finsler geometry and applications, this article belongs to an axiomatized and self-consistent direction of mathematical and acceleration cosmology, dark matter and dark energy physics, involving Finsler geometry methods.