arXiv:1611.05295v1 [physics.gen-ph] 7 Nov 2016Anamorphic Quasiperiodic Universes in Modified and Einstein Gravity with Loop Quantum Gravity Corrections Marcelo M. Amaral Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA emails: marcelo@quantumgravityresearch.org Raymond Aschheim Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA email: raymond@quantumgravityresearch.org Laurenţiu Bubuianu TVR Iaşi, 33 Lascǎr Catargi street, 700107 Iaşi, Romania email: laurentiu.bubuianu@tvr.ro Klee Irwin Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA email: klee@quantumgravityresearch.org Sergiu I. Vacaru Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA and University "Al. I. Cuza" Iaşi, Project IDEI 18 Piaţa Voevozilor bloc A 16, Sc. A, ap. 43, 700587 Iaşi, Romania email: sergiu.vacaru@gmail.com Daniel Woolridge Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA email: dan@quantumgravityresearch.org November 7, 2016 1 Abstract The goal of this work is to elaborate on new geometric methods of constructing exact and parametric quasiperiodic solutions for anamorphic cosmology models in modified gravity theo- ries, MGTs, and general relativity, GR. There exist previously studied generic off-diagonal and diagonalizable cosmological metrics encoding gravitational and matter fields with quasicrystal like structures, QC, and holonomy corrections from loop quantum gravity, LQG. We apply the anholonomic frame deformation method, AFDM, in order to decouple the (modified) gravita- tional and matter field equations in general form. This allows us to find integral varieties of cosmological solutions determined by generating functions, effective sources, integration func- tions and constants. The coefficients of metrics and connections for such cosmological configu- rations depend, in general, on all spacetime coordinates and can be chosen to generate observ- able (quasi)-periodic/ aperiodic/ fractal / stochastic / (super) cluster / filament / polymer like (continuous, stochastic, fractal and/or discrete structures) in MGTs and/or GR. In this work, we study new classes of solutions for anamorphic cosmology with LQG holonomy corrections. Such solutions are characterized by nonlinear symmetries of generating functions for generic off–diagonal cosmological metrics and generalized connections, with possible nonholonomic con- straints to Levi–Civita configurations and diagonalizable metrics depending only on a time like coordinate. We argue that anamorphic quasiperiodic cosmological models integrate the concept of quantum discrete spacetime, with certain gravitational QC-like vacuum and nonvacuum struc- tures. And, that of a contracting universe that homogenizes, isotropizes and flattens without introducing initial conditions or multiverse problems. Keywords: Mathematical cosmology; geometry of nonholonomic spacetimes; anamorphic cosmology; post modern inflation paradigm; ekpyrotic universes; modified gravity theories; loop quantum gravity and cosmology; quasiperiodic cosmological structures. Contents 1 Introduction and Motivation 3 2 Nonholonomic Variables and Anamorphic Cosmology 5 2.1 N-adapted frames and connection deformations in MGTs . . . . . . . . . . . . . . . . 6 2.2 Anamorphic cosmology in nonholonomic variables . . . . . . . . . . . . . . . . . . . . 7 2.3 Small parametric deformations for quasi–FLRW metrics . . . . . . . . . . . . . . . . . 8 2.4 Effective FLRW geometry for nonholonomic MGTs . . . . . . . . . . . . . . . . . . . 10 2.5 LQC extensions of MGTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Nonholonomic Friedmann equations in anamorphic cosmology with LQG corrections . 11 3 Off-Diagonal Anamorphic Cosmology in MGT and LQG 12 3.1 Generating functions encoding QC like MGT and LQG corrections . . . . . . . . . . . 12 3.2 N-adapted Weyl–invariant quantities for anamorphic phases . . . . . . . . . . . . . . 15 3.3 Cosmological QCs for (effective) matter fields and LQG . . . . . . . . . . . . . . . . . 16 3.3.1 Effective QC matter fields from MGT . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.2 Nonhomogeneous QC like scalar fields . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.3 QC configurations induced by LQG corrections . . . . . . . . . . . . . . . . . 18 3.4 Anamorphic off-diagonal cosmology with QC and LQG structures . . . . . . . . . . . 19 2 4 Small Parametric Anamorphic Cosmological QC and LQG Structures 20 4.1 N-adapted ε–deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 ε–deformations to off-diagonal cosmological metrics . . . . . . . . . . . . . . . . . . . 21 4.3 Cosmological ε–deformations with anamorphic QCs and LQG . . . . . . . . . . . . . 23 5 Concluding Remarks 24 1 Introduction and Motivation It is thought that near the Planck limit any quantum gravity theory is characterized by discrete degrees of freedom, respective of quantum minimal length and quantum symmetries, and anisotropic and inhomogeneous fluctuating / random configurations. On the other hand, observations show that the accelerating Universe is flat, smooth and scale free at large-scale distances when the spectrum of primordial curvature perturbations is nearly scale-invariant, adiabatic and Gaussian [1, 2]. We cite papers [3, 4, 5, 6, 7, 8, 9, 10] for recent reviews, discussions, critique and new results on postmodern inflation scenarios developed and advocated by prominent theorists in relation to the Planck 2013 and Planck 2015 cosmological data [11, 12, 13, 14, 15, 16, 17, 18]. Here we note that for meta- galactic and galactic distances, the Planck 2015 and WMAP, ACT and SPT teams’ observation and theoretical results1 on spacetime anisotropy and topology, dark energy, and constraints on inflation and accelerating cosmology parameters. Such works conclude on the existence of mixed aperiodic and quasiperiodic structures (for gravitational, dark matter and standard matter) described as net-works for the first group- and (super) cluster-scale, strong gravitational lensing / light filaments / polymer and quasicrystal, QC, like configurations. In our partner work [19], we proved that Starobinsky-like inflation [20] and various dark energy, DE, and dark matter, DM, effects in a Universe with quasiperiodic (super) cluster and filament configurations can be determined by a nontrivial QC spacetime structure. We cite see Refs. [21, 22, 23, 26, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 39] for important works and references on the physics and mathematics of QCs in condensed matter physics but also with possible connections to cosmology. Various F–modified (for instance, F (R) = R+αR2) cosmological models2 can be with singularities and encode inhomogeneous and locally anisotropic properties. For reviews on modified gravity theories, MGTs, readers may consider [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. In papers [55, 56], a detailed analytical and numerical study of possible holonomy corrections from LQG to f(R) gravity was performed. It was shown that, as a result of such quantum corrections (and various generic off-diagonal, nonholonomic and/or QC contributions investigated in [19, 52, 53, 54]) the dynamics may change substantially and, for certain well defined conditions, one obtains better predictions for the inflationary phase as compared with current observations. Various approaches to LQG and spin network theories, see also constructions on loop quantum cosmology, LQC, are reviewed in Refs. [57, 58, 59, 60, 61, 62, 63]. In the past, certain criticism against LQG (see, for instance, [64]) was motivated in the bulk by arguments that the mathematical formalism is not 1consistency and implications for inflationary, ekpyrotic and anamorphic bouncing cosmologies, and other type cosmological models, are discussed in Refs. [4, 13, 18] 2in more general contexts, one considers various modified gravity theories, F (R, T ), F (T ), ... determined by func- tionals on Ricci scalars, energy-momentum and/or torsion tensors etc.; in various papers, such functionals are denoted also as f(R), f(T ),... 3 that which is familiar for the particle physicists working with perturbation theory, Fock spaces, background fields etc; see reply and discussion in [65]. The main objectives of this work are to study how quasiperiodic and/or aperiodic QC like struc- tures with possible holonomy LQG corrections modify inflation and acceleration cosmology scenarios in MGTs and GR; to analyse if such effects can be modeled in the framework of the Einstein gravity theory; and to show how such generic off–diagonal cosmological solutions can be constructed and treated in anamorphic cosmology. The extensions of cosmological models to spacetimes with non- trivial quasiperiodic/ aperiodic and general anistoropic structures is not a trivial task. It is necessary to elaborate on new classes of exact and/or parametric solutions of gravitational and matter field cosmological equations which, in general, depend on all spacetime variables via generating and in- tegration functions with mixed smooth and discrete degree of freedom and anisotropically polarized physical constants. We emphasize that it is not possible to describe, for instance, growth of any QC structure and compute certain cosmological effects determined by non-perturbative and nonlinear gravitational interactions if we restrict our models to only diagonal homogeneous and isotropic met- rics like the Friedmann–Lamaître-Roberstson-Worker, FLRW, one and possible generalizations with Lie group/algebroid symmetries [52, 53, 54]. In such cases, the cosmological solutions are determined by some integration and/or structure group constants, and depend only on a time like coordinate. We can not describe in a realistic form quasiperiodic / aperiodic spacetime structures, and their evo- lution, using only time-depending functions and FLRW metrics. In order to formulate and develop an unified geometric approach for all observational data on (super) cluster and extra long cosmo- logical distances, we have to work with "non-diagonalizable" metrics3 and generalized connections, and apply new numeric and analytic methods for constructing more general classes of solutions in MGTs and cosmology models with quasiperiodic structure, inhomogeneities and local anisotropies. The new classes of cosmological solutions incorporate generating functions and integration functions, with various integration constants and parameters, which allow more opportunities to compare with experimental data. Even some subclasses of solutions can be parameterized by effective diagonal metrics4, the diagonal coefficients contain various physical data of nonlinear classical and quantum interactions encoded via generating functions and effective sources. In contrast to the general purpose of unification of physical interactions and development of fundamental and geometric principles of quantization (for instance, in string theory and deformation quantization), the approaches based on LQG and spin networks were performed originally just as theories of quantum gravity combining the general relativity (GR) and quantum mechanics. The main principle was to provide a non–perturbative formulation when the background independence (the key feature of Einstein’s theory) is preserved. At the present time, LQG is supposed to have a clear conceptual and logical setup following from physical considerations and supported by a rigorous mathematical formulation. In this work, we study a toy cosmological model with LQG contributions, whilst keeping in mind that such constructions will be expanded on for spin network models and further generalizations to QC configurations. Here, in addition to the references presented above, we cite some fundamental works [66, 67, 68, 69, 70, 71] on LQG for also considering developments in loop quantum cosmology and possible extensions, for example, to deformation quantization. We emphasize that we analyze examples with a special class of holonomy corrections from LQG in order to prove that possible quantum modifications do not affect the main results on anamorphic 3which can not be diagonalized by coordinate transforms, in a local or infinite spacetime region 4for certain limits with small off–diagonal corrections and/or nonholonomically constrained configurations, for instance, incorporating anomorphic smoothing phases 4 cosmological models with QC structure. With respect to our toy LQC model, we also note that we restrict our study to quantum gravity quasiperiodic effects in anamorphic cosmology by considering a special class of holonomy corrections from LQG in order to distinguish possible non-perturbative and background independent modifica- tions. In this approach, quantization can be performed in certain forms preserving the Lorentz local invariance in the continuous limit. Here we note that if the quantization formalism is developed on (co)tangent bundles, one gets quantum corrections and respective cosmological terms violating this local symmetry [72]. In a more general context, such an approach involves reformulation of the LQG in nonholonomic variables with double 2+2 and 3+1 fibrations considered in [71, 73]. Details on the so–called ADM, i.e. Arnowitt–Deser–Misner, formalism in GR can be found, for instance, in [74, 57, 58, 59]. In order to construct new classes of cosmological solutions, we shall apply the an- holonomic frame deformation method, AFDM (see details and examples for accelerating cosmology and DE and DM physics in [75, 76, 77, 52, 54]). The paper is organized as follows: In section 2, we outline the most important formulas on nonholonomic variables, frame, linear and nonlinear connection deformations used for constructing (in general) generic off–diagonal cosmological solutions depending on all spacetime coordinates. It is shown how using such constructions we can decouple the gravitational and matter field equations in accelerating cosmology if the Einstein gravity and various f(R) modifications, with LQG corrections. In nonholonomic variables we formulate the criteria for anamorphic cosmological phases and analyze possible small parametric deformations in terms of quasi-FLRW metrics for nonholonomic Friedmann equations. Section 3 is devoted to the study of geometric properties of new classes of generic off–diagonal cos- mological solutions modeling QC like structures in MGTs with LQG sources. In this section the con- ditions on generating and integration functions and integration constants when such configurations encode quasiperiodic/ aperiodic structure of possible different origin (induced by F-modifications, gravitational like polarization of mass like constants, anamorphic phases with effective polarization of the cosmological constant, and LQC sources) are formulated. Four such classes of solutions are constructed in explicit form and the criteria for anamorphic QC phases are formulated. Here, we also provide solutions for nonlinear superpositions resulting in hierarchies with new anamorphic QC like cosmological solutions. In section 4, we consider small parametric decompositions for quasi-FLRW metrics encoding QC like structures. It is proven that in such cases the cosmological solutions with gravitationally polarized cosmological constants and the criteria for anamorphic phases can be written in certain forms similar to homogeneous cosmological configurations. In such cases, QC and LQG modified Friedmann equations can be derived in explicit form. We discuss the results in section 5. The main conclusion is that the inflationary paradigm can be modernized in order to include cosmological acceleration scenarios determined by anamorphic quasiperiodic/ aperiodic gravitational and matter field structures in MGTs and GR with possible corrections from LQG. 2 Nonholonomic Variables and Anamorphic Cosmology To be able to construct, in explicit form, exact and parametric quasiperiodic cosmological so- lutions in MGTs with quantum corrections we have to re–write the fundamental gravitational and 5 matter field equations in such nonholonomic variables when a decoupling and general integration of corresponding systems of nonlinear partial differential equations, PDEs, are possible. Readers are referred to [72, 73, 75, 76, 77, 52, 54] for details on the geometry and applications of the AFDM as a method of constructing exact solutions in gravity and Ricci flow theories. In this section, we show how such nonholonomic variables can be introduced in MGT and GR theory and formulate a geometric approach to anamorphic cosmology [1, 2, 3, 4, 5, 6]. The constructions will be used in the next section for decoupling the fundamental cosmological PDEs with matter field sources and LQG corrections parameterized as in Refs. [78, 79, 55, 56, 71]. 2.1 N-adapted frames and connection deformations in MGTs We presume that the metric properties of a four dimensional, 4d, cosmological spacetime manifold V are defined by a metric g of pseudo–Riemannian signature (+++−) which can be parameterized as a distinguished metric, d–metric, g = gαβ(u)e α ⊗ eβ = gi(xk)dxi ⊗ dxi + ga(xk, yb)ea ⊗ eb (1) = gα′β′(u)e α′ ⊗ eβ′, for gα′β′(u) = gαβeαα′eββ′ . In these formulas, we use N–adapted frames, eα = (ei, ea), and dual frames, e α = (xi, ea), ei = ∂/∂x i −Nai (u)∂/∂ya, ea = ∂a = ∂/∂ya, (2) ei = dxi, ea = dya +Nai (u γ)dxi and eα = eαα′(u)du α′. The local coordinates on V are labeled uγ = (xk, yc), or u = (x, y), when indices run corresponding values i, j, k, ... = 1, 2 and a, b, c, ... = 3, 4 (for nonholonomic 2+2 splitting, for u4 = y4 = t being a time like coordinate and uı̀ = (xi, y3) considered as spacelike coordinates endowed with indices ı̀,j̀,k̀, ... = 1, 2, 3. We note that a local basis5 eα is nonholonomic (equivalently, non-integrable, or anholonomic) if the commutators e[αeβ] := eαeβ − eβeα = Cγαβ(u)eγ contain nontrivial anholonomy coefficients Cγαβ = {Cbia = ∂aN bi , Caji = ejNai − eiNaj }. A value N = {Nai } = Nai ∂ ∂ya ⊗ dxi determined by frame coefficients in (2) defines a nonlinear connection, N-connection, structure as an N–adapted decomposition of the tangent bundle TV = hTV ⊕ vTV (3) into conventional horizontal, h, and vertical, v, subspaces. On a 4-d metric–affine manifold V, this states an equivalent fibred structure with nonholonomic 2+2 spacetime decomposition (splitting). In particular, such a h-v-splitting states a double, h and v, diadic frame structure on any (pseudo) Riemannian spacetime. We shall use boldface symbols for geometric/ physical objects on a spacetime manifold V endowed with geometric objects (g,N,D). The values D is a distinguished connection, d–connection, D = (hD, vD) defined as a linear connection, i.e. a metric–affine one, preserving the N–connection splitting (3) under parallel transports. We denote by T = {Tαβγ} the torsion of D, which can be computed in standard form, see geometric preliminaries in [71, 72, 73, 75, 76, 77, 52, 54]. 5in literature, one uses equivalent terms like frame, tetrad, vierbein systems 6 On a nonholonomic spacetime manifold V, we can work equivalently with two linear connections defined by the same metric structure g: (g,N) → { ∇ : ∇g = 0; ∇T = 0, for the Levi-Civita, LC, -connection D̂ :D̂g = 0; hT̂ = 0, vT̂ = 0, hvT̂ 6= 0, for the canonical d–connection . (4) As a result, it is possible to formulate equivalent models of pseudo-Riemannian geometry and/or Riemann–Cartan geometry with nonholonomically induced torsion6. 2.2 Anamorphic cosmology in nonholonomic variables Based on invariant criteria, authors [1, 2, 3, 4, 5, 6] attempted to develop a complete scenario explaining the smoothness and flatness of the universe on large scales with a smoothing phase that acts like a contracting universe. In this section, we develop a model of anamorphic cosmology in the framework of MGTs with quasiperiodic / aperiodic structures and LQC–corrections. The approach relies on having time-varying masses for particles and certain Weyl-invariant values that define certain aspects of contracting and/or expanding cosmological backgrounds. For off-diagonal cosmological models with nontrivial vacuum structures, the variation of masses and physical constants have a natural explanation via gravitational polarization functions [19, 52, 53, 54]. Let us denote such variations of a particle mass m →m̌(xi, t) ≃m̌(t) and of Planck mass MP →M̌P (xi, t) ≃M̌P (t), which depends on the type of generating functions we consider. The actions for particle motion and modified gravity are written respectively as pS = ∫ m̌ M̌P ds and (5) S = ∫ d4u √ |g|[F(R̂) + mL(φ)] (6) = ∫ d4u √ |g|[1 2 M̌2P (φ)R̂− 1 2 κ(φ)gαβ(eαφ)(eβφ)− JV (φ) + mL(φ)], (7) whereM̌2P (φ) := M 0 P l √ f(φ) is positive definite (we can work in a system of coordinates when M0P l = 1). Above actions are written for a d–metric gαβ (1), κ(φ) is the nonlinear kinetic coupling function andR̂ is the scalar curvature ofD̂. In our works, we use left labels in order to denote, for instance, that mL is for matter fields (for this label, m is from "mass") and JV is for the Jordan frame representation.7 Here we note that F(R̂) = F[R̂(g,D̂, φ)] is also a functional of the scalar field 6It should be emphasized that the canonical distortion relationD̂ = ∇ +Ẑ, where the distortion distinguished tensor, d-tensor,Ẑ = {Ẑαβγ [T̂αβγ ]}, is an algebraic combination of the coefficients of the corresponding torsion d- tensorT̂ = {T̂αβγ} ofD̂. The curvature tensors of both linear connections are computed in standard forms,R̂ = {R̂αβγδ} and ∇R = {Rαβγδ} (respectively, forD̂ and ∇). This allows us to introduce the corresponding Ricci tensors, R̂ic = {R̂ βγ :=R̂γαβγ} and Ric = {R βγ := R γ αβγ}. The valueR̂ic is characterized by h-v N-adapted coefficients, R̂αβ = {R̂ij :=R̂kijk,R̂ia := −R̂kika,R̂ai :=R̂baib,R̂ab :=R̂cabc}. There are also two different scalar curvatures, R := gαβRαβ andR̂ := g αβR̂αβ = g ijR̂ij + g abR̂ab. We can also consider additional constraints resulting in zero values for the canonical d-torsion,T̂ = 0, considering some limitsD̂|T̂ →0 = ∇. 7If in (6) and (7) F(R̂) =R̂2, JV (φ) = −Λ and mL = 0, we obtain a quadratic action for nonholonomic MGTs studied in [19, 76, 77], when S = ∫ d4u √ |g|[R̂2+ mL]. The equivalence of such actions to nonholonomic deformations 7 φ but we use simplified notations using the assumption thatR̂(g,D̂) are related to φ by a source term for modified Einstein equations with such a nonlinear scalar field. The gravitational field equations for MGT with functional F(R̂) in (6) can be derived by a N– adapted variational calculus, see details in [19, 52, 53, 54] and references therein. We obtain a system of nonlinear PDEs which can be represented in effective Einstein form, R̂µν = Υµν , (8) where the right effective source is parameterized Υµν = FΥµν + mΥµν + Υµν . (9) Let us explain how three terms in this source are defined. The functionals F(R̂) and 1F(R̂) := dF(R̂)/dR̂ determine an energy-momentum tensor, FΥµν = ( F 2 1F −D̂ 2 1F 1F )gµν + D̂µD̂ν 1F 1F . (10) The source for the scalar matter fields can be computed in standard form, mΥµν = 1 2M2P mTαβ , (11) and the holonomic contributions from LQG, Υµν (19), will be defined in subsection 2.5. We shall be able to find, in explicit form, exact solutions for the system (8) for any source (9), which via frame transforms Υµν = e µ′ µe ν′ νΥµ′ν′ can be parameterized into N–adapted diagonalized form as Υµν = diag[ hΥ(x i), hΥ(x i), Υ(xi, t), Υ(xi, t)]. (12) In these formulas, the generating source functions hΥ(x i) and Υ(xi, t) have to be prescribed in some forms which will generate exact solutions compatible with observational/ experimental data. 2.3 Small parametric deformations for quasi–FLRW metrics In N–adapted bases, the models of locally anisotropic and inhomogeneous anamorphic cosmology are characterized by three essential properties during the smoothing phase: of the Einstein gravity with scalar field sources can be derived from the invariance (both for ∇ andD̂) under global dilatation symmetry with a constant σ, gµν → e−2σgµν , φ → e2σφ̃. We can re-define the physical values from the Jordan to the Einstein, E, frame using φ = √ 3/2 ln |2φ̃|, when ES = ∫ d4u √ |g| ( 1 2R̂− 12eµφ eµφ− 2Λ ) . The field equations derived from ES are R̂µν − eµφ eνφ− 2Λgµν = 0 andD̂2φ = 0. To find explicit solutions we can consider Υµν ∼ diag[0, 0, Υ, Υ], where Υ(t) will be determined by scalar fields in anamorphic QC phase and possible holonomic corrections to the Hubble constant. We obtain the Einstein gravity theory if F(R̂) = R forD̂|T̂ →0 = ∇. For simplicity, we can consider matter actions mS = ∫ d4u √ |g| mL for matter field Lagrange densities mL depending only on coefficients of a metric field and do not depend on their derivatives when mTαβ := − 2 √ |gµν | δ( √ |gµν | mL) δgαβ = mLgαβ + 2δ( mL) δgαβ . 8 1. masses are polarized with a certain dependence on time and space like coordinatesm→m̌(xi, t) and/or m→m̌(t); 2. necessary type combinations of N-adapted Weyl-invariant signatures incorporating aspects of contracting and expanding locally anisotropic backgrounds; 3. using nonlinear symmetries of generic off-diagonal solutions gαβ (1) and considering nonholo- nomic deformations on a small parameter ε, we can express, via frame transforms, the cosmo- logical solutions of (8), with prescribed sources [ hΥ, Υ], in such a quasi-FLRW form 8 ds2 =â2(xk, t)eı̀e ı̀ − e4e4, (13) for N3j = nj(x k, t) and N4j = wj(x k, t), where eı̀ = (dxi, e3 = dy3 + nj(x k, t)dxj) ≃ (dxi, dy3 + εχ3j(xk, t)dxj), e4 = dt+ wj(x k, t)dxj ≃ dt+ εχ4i (xk, t)dxj. (14) The locally anisotropic scale coefficient can be considered as isotropic in certain limits (for additional assumptions on homogeneity),â2(xk, t) ≃â2(t) and computed together with effective polarization functions χ3j and χ 4 i all encoding data on possible nonlinear generic off-diagonal interactions, QC and/or LQG contributions. In next section, we shall prove how such values can be computed for certain classes of generic off-diagonal exact solutions in MGTs and GR. Using the effective scale factorâ2 from (13), we can introduce the respective effective and locally anisotropically polarized Hubble parameter, Ĥ := e4(lnâ) = ∂t(lnâ) = (lnâ) ∗. (15) Considering a new time like coordinateť, for t = t(xi,ť) and transforming √ |h4|∂t/∂ť into a scale factorâ(xi,ť), we represent (14) in the form ds2 = ǎ2(xi,ť)[ηi(x k,ť)(dxi)2 +ȟ3(x k,ť)(e3)2 − (ě4)2], (16) where ηi = ǎ−2eψ,ǎ2ȟ3 = h3, e 3 = dy3 + ∂kn dx k,ě4 = dť+ √ |h4|(∂it + wi). For a small parameter ε, with 0 ≤ ε < 1, we the off–diagonal deformations are given by effective polarization functions ηi ≃ 1 + εχi(xk,t̂), ∂kn ≃ εn̂i(xk), √ |h4| wi ≃ εŵi(xk,t̂). We can work, for convenience, with both types of nonholonomic ε–deformations of FLRW metrics (nonholonomic FLRW models). Such approximations can be considered after a generic off–diagonal cosmological solution was constructed in a general form. 8this term means that for ε→ 0 and any approximationâ2(t) a standard FLRW metric is generated 9 2.4 Effective FLRW geometry for nonholonomic MGTs Following a N-adapted variational calculus for MGTs Lagrangians resulting in respective dynam- ical equations (see similar holonomic variants in Refs. [55, 56]), we can construct various models of locally anisotropic spacetimes [19, 52, 53, 54].9 For Υµν = FΥµν in (8) and a d–metric (1) with diagonal homogeneous approximations, we obtain from (6) that in the Einstein frame FS = M2P ∫ d4u √ |g|F(R̂) → EFS =M2P ∫ d4u EFL, for EFL = a3 [ 1 2 R+ 1 2 ( ∂φ ∂t )2 − V (φ) ] , where R = 6 ∂H ∂t + 12H 2 . In these formulas, V (φ) is an effective potential and a and φ are independent variables defined correspondingly by a := √ 1F(R̂)â, dt := √ 1F(R̂)dt, ∂ ∂t := ∂; (17) φ := √ 3 2 ln | 1F(R̂)|, V (φ) = 1 2   R̂ 1F(R̂) − F(R̂) ( 1F(R̂) )2   . Using above variables for the Hamiltonian constraint EFH := ∂a∂( EFL) ∂∂a + ∂φ (∂ EFL) ∂∂φ − EFL and effective density ρ := 1 2 (∂φ)2 + V (φ), (18) we express the effective Friedmann equation (in the Einstein frame, it is a constraint) 3H 2 = ρ when the dynamics is given by the conservation law ∂ρ = −3H(∂φ)2. This dynamics is encoded also in an effective Raychaudhury equation 2∂H = −(∂φ)2, with (∂ρ)2 = 3ρ(∂φ)2. 2.5 LQC extensions of MGTs LQC corrections to MGTs have been studied in series of works [78, 79, 55, 56]. As standard variables (we follow our notations (17)), we use β := γH, where γ is the Barbero–Immirzi parameter [68, 69, 70], and the volume V := a3. For diagonal configurations, the holonomy corrections to the Friedemann equations are of type H 2 = ρ 3 (1− ρ ρc ), (19) 9In our works, we have to elaborate more ’sophisticate’ systems of notations because such geometric modeling of cosmological scenarios and methods of constructing solutions of PDEs should include various terms with h-v– splitting; discrete and continuous classical and quantum corrections, diagonal and off-diagonal terms, different types of connections which were not considered in other works by other authors. The most important conventions on our notations are that we use boldface symbols for the spaces and geometric objects endowed with N–connection structure and that left labels are abstract ones associated to some classes of geometric/ physical objects. Right Latin and Greek indices can be abstract ones or transformed into coordinate indices with possible h- and v-splitting. Unfortunately, it is not possible to simplify such a system of notations if we follow multiple purposes related to geometric methods of constructing exact solutions in gravity and cosmology theories, analysing different phases of anamoprhic cosmology with generic off-diagonal terms etc. 10 where the critical density ρc := 2/ √ 3γ3 is computed in EF (see [60] for a status report on different approaches to LCQ). This formula can be applied for small deformations with respect to N–adapted frames taking, for simplicity, a function ρ(t) determining the component Υµν in (8) and (9). In a more general context, we can consider locally anisotropic configurations with ρ(xi, t) associ- ated to any EFH[β(xi, t), V (xi, t)], with conjugated Poisson bracket {β(xi, t), V (xi, t)} = γ/2, when H(xi, t) = sin( √ 2 √ 3γβ(xi,t)) √ 2 √ 3γ , for a re-scaling in order to have a well–defined quantum theory. We note that there were formulated different models and inequivalent approaches to LQG and LQC, see a variant [71] which is compatible with deformation quantization. For simplicity, we shall add the term Υ = − ρ 2 3ρc (20) in N-adaptedΥµν = diag[Υ,Υ,Υ,Υ], see below the formula (26), as an additional LQG contribution in the right part of certain generalized Friedmann equations with a nonlinear re–definition of scalar field effective density ρ 3 → ρ 3 (1− ρ ρc ). 2.6 Nonholonomic Friedmann equations in anamorphic cosmology with LQG corrections The cosmological models with generic off-diagonal metrics parameterized in N–adapted form with respect to bases (14) are characterized by two dimensionless quantities (being Weyl-invariant if the homogeneity conditions are imposed), mΘ := (Ĥ +H +m̌ ∗ m̌ )M̌−1 P = α̌∗m α̌mM̌P forα̌m :=âm̌/M 0 P ; (21) P lΘ := (Ĥ +H + M̌∗P M̌P )M̌−1 P = α̌∗P l α̌P lM̌P l forα̌P l :=âM̌P/M 0 P for M0P being the value of the reduced Planck mass in the frame where it does not depend on time. These values distinguish respectively such cosmological models (see details in [1, 2] but for holonomic structures): anamorphosis inflation ekpyrosis mΘ (background) < 0 (contracts) > 0 (expands) < 0 (contracts) P lΘ (curvature pert.) > 0 (grow) > 0 (grow) > 0 (decay) (22) Here we note that the priority of the AFDM is that we can consider any cosmological solution in a MGT or GR and than to write it in N-adapted form with ε–deformations. This allows us to compute all physical important values like mΘ and P lΘ and analyse if and when an anamporhic phase is possible. We note that mΘ is negative, for instance, as in modified ekpyrotic models, but P lΘ is positive as in locally anisotropic inflationary models. In such theories, the effectivem̌ andM̌P are determined by certain QC and/or LQC configurations. Reproducing in N–adapted frames for d–metrics of type (13) the calculus presented in Appendix A (with Einstein and Jordan nonholonomic frame representations) of [1], we obtain respectively such 11 a version of locally anisotropic and inhomogeneous first and second Friedmann equations, 3( mΘ)2 = [ Aρ+ mρ+ Aρ/ √ ρc M̌4P − (m̌ M̌P )2 κ α̌2m + ( m̌ M̌P )6 σ2 α̌6m ] [ 1− ∂(m̌ M̌P )/∂ lnα̌m ]−2 , ( P lΘ)∗ = −( Aρ+ mρ+ Aρ/ √ ρc)/2M̌ 3 P . (23) In these formulas, K(φ) := [3 2 (f,φ) 2) + κ(φ)f(φ)]/f 2(φ) and the values (energy density, pressure) 2(M0P ) 4 ( Aρ) := K(φ)(φ∗)2 +M̌4P JV (φ)/f 2(φ), 2(M0P ) 4 ( Ap) := K(φ)(φ∗)2 −M̌4P JV (φ)/f 2(φ), are determined by coefficients in (7), for κ = (+1, 0,−1) being the spacial curvature, and the con- stant σ2 should be considered if we try to limit the background cosmology to that described by a homogeneous and anisotropic Kasner-like metric (see formula (A.5) in [1]). For simplicity, we shall consider in this work σ2 = 1 even MGTs can contain certain locally anisotropic configurations. Finally, we note that we can identify Aρ with ρ (18) for F–modified gravity theories. 3 Off-Diagonal Anamorphic Cosmology in MGT and LQG Applying the anholonomic frame deformation method, AFDM, we can construct various classes of off-diagonal and diagonal cosmological solutions of (modified) gravitational field equations (8). After the metric, frame and connection structure, and the effective sources (9), have been parameterized in N-adapted form, we can select necessary type diagonal or off-diagonal configurations, consider small parameter decompositions, and approximate the generating/integration functions to some constant values compatible with observational data. We do not repeat that geometric formalism and refer readers to Refs. [75, 72, 76, 52, 54] for details on AFDM and applications in modern cosmology. The purpose of this section is to state the conditions for the generating functions and (effective) sources and quantum corrections which describe quasiperiodic/ aperiodic quasicrystal, QC, like cosmological structures. There are used necessary type quadratic line elements for general solutions found in in the mentioned references and the partner paper [19]. 3.1 Generating functions encoding QC like MGT and LQG corrections The metrics for off-diagonal locally anisotropic and inhomogeneous cosmological spacetimes are defined as solutions, with nonholonomically induced torsion and Killing symmetry on ∂/∂y3. 10 Via nonholonomic frame transforms, such metrics can be always written in a coordinate basis, g = gαβ(x k, t)duα ⊗ duβ, and/or in N–adapted form (1), ds2 = gijdx idxj + {h3[dy3 + ( 1nk + 2nk ∫ dt (∂tΨ) 2 Υ2|h3|5/2 )dxk]2 − ( ∂tΨ 2Υ|h3|1/2 )2 [dt+ ∂iΨ ∂tΨ dxi]2}, (24) h3 = −∂t(Ψ2)/Υ2 ( h [0] 3 (x k)− ∫ dt∂t(Ψ 2)/4Υ ) . (25) 10For simplicity, in this work we do not consider more general classes of solutions with generic dependence on all spacetime coordinates and do analyze the details how Levi-Civita, LC, configurations can be extracted by solving additional nonholonomic constraints, see [75, 72, 76, 52, 54] and references therein. 12 In this formula, gij = δije ψ(xk) and 1nk(x i), 2nk(x i) and h [0] a (xk) are integration functions. The coefficient h3, or Ψ(x i, t), is the generating function11 and the generating h- and v–sources (see (9) and (12)) are given by terms of effective gravity modifications, matter field and LQG contributions, hΥ(x i) = FhΥ(x i) + mh Υ(x i) + hΥ(x i) and Υ(xi, t) = FΥ(xi, t) + mΥ(xi, t) + Υ(xi, t). (26) Off-diagonal metrics of type (24) posses an important nonlinear symmetry, which allows us to re-define the generating function and generating source (Ψ, Υ) ↔ (Φ,Λ = const), when Υ(xk, t) → Λ, for (27) Φ2 = Λ ∫ dtΥ−1∂t(Ψ 2) and Ψ2 = Λ−1 ∫ dtΥ∂t(Φ 2), by introducing an effective cosmological constant Λ as a source and the functional Φ(Λ,Ψ, Υ) as a new generating function. This property can be proven by considering the relation Λ∂t(Ψ 2) = Υ∂t(Φ 2) in above formulas for the d-metric. We can consider that nonlinear generic off-diagonal interactions on MGTs may induce an effective cosmological constant with splitting, Λ = FΛ + mΛ + Λ. The terms of this sum are determined respectively by modifications of GR resulting in FΛ; by nonlinear interactions of matter (i.e. scalar field φ) resulting in mΛ; and by an effective Λ associated to holonomy modifications from LQG. Technically, it is more convenient to work with some data (Φ,Λ) for generating solutions and then to redefine the formulas in terms of generating function and generalized source (Ψ, Υ). We can also extract torsionless cosmological configurations12. The generating functions and/or sources can be chosen in such forms that the cosmological space- time solutions encode nontrivial gravitational and/or matter field quasicrystal like, QC, configura- tions and possible additional LQG effects. We use an additional 3+1 decomposition with spacelike coordinates xı̀ (for ı̀ = 1, 2, 3), time like coordinate y4 = t, being adapted to another 2+2 decom- position with a fibration by 3-d hypersurfacesΞ̂t, see details in [73, 74]. For such configurations, we can consider a canonical nonholonomically deformed Laplace operator b∆̂ := ( bD̂)2 = bı̀j̀D̂ı̀D̂j̀ 11we note that such solutions are defined in explicit form by coefficients of (1) computed in this form: gi = e ψ(xk) is a solution of ψ•• + ψ′′ = 2 hΥ; g3 = h3 = −∂t(Ψ2)/Υ2 ( h [0] 3 (x k)− ∫ dt∂t(Ψ 2)/4Υ ) ; g4 = h [0] 4 (x k)− ∫ dt∂t(Ψ 2)/ 4Υ; N3k = nk(x i, t) = 1nk(x i) + 2nk(x i) ∫ dt(∂tΨ) 2/Υ2 ∣∣∣∣h [0] 3 (x i)− ∫ dt ∂t(Ψ 2)/4Υ ∣∣∣∣ 5 2 ; N4 i = wi(x k, t) = ∂i Ψ ∂tΨ . 12 The nonholonomically induced torsion of solutions (24) can be constrained to be zero by choosing certain subclasses of generating functions and sources. We have to consider a subclass of generating functions and sources when for Ψ =Ψ̌(xi, t), ∂t(∂iΨ̌) = ∂i(∂tΨ̌) and Υ(x i, t) = Υ[Ψ̌] =Υ̌, or Υ = const. Then, we can introduce functionsǍ(xi, t) and n(xk) subjected to the conditions that wi =w̌i = ∂iΨ̌/∂tΨ̌ = ∂iǍ and nk =ňk = ∂kn(x i). Such assumptions are considered in order to simplify the formulas for cosmological solutions (see details in Refs. [75, 72, 76, 52, 54], where the AFDM is applied for generating more general classes of solutions depending on all spacetime coordinates. We obtain a quadratic line element defining generic off-diagonal LC-configurations, ds2 = gijdx idxj + {h3[dy3 + (∂kn)dxk]2 − 1 4h3 [ ∂tΨ̌ Υ̌ ]2 [dt+ (∂iǍ)dx i]2}. 13 (determined by the 3-d part of d-metric) as a distortion of b∆ := ( b∇)2. Such a value can be defined and computed on anyΞ̂t using a d-metric (1) and respective 3-d space like projections/ restrictions ofD̂. We chose a subclass of generating functions Ψ = Π subjected to the condition that it is a solution of an evolution equation (with conserved dynamics) of type ∂Π ∂t = b∆̂ [ δF δΠ ] = − b∆̂(ΘΠ +QΠ2 − Π3). (28) Such a nonlinear PDE can be derived for a functional defining an effective free energy F [Π] = ∫ [ −1 2 ΠΘΠ− Q 3 Π3 + 1 4 Π4 ]√ bdx1dx2δy3, (29) where b = det |bı̀j̀ | is the determinant of the 3-d spacelike metric, δy3 = e3 and the operator Θ and parameter Q are defined in the partner work [19]. Different choices of Θ and Q induce different classes of quasiperiodic, aperiodic and/or QC order of corresponding classes of gravitational solutions. We note that the functional (29) is of Lyapunov type considered in quasicrystal physics, see [29, 30, 40, 39] and references therein, and for applications of geometric flows in modern cosmology and astrophysics, with generalized Lyapunov-Perelman functionals [73, 76, 77]. In this paper, we do not enter into details how certain QC structures and their quasiperiodic/ aperiodic deformations can be reproduced in explicit form but consider that such configurations can always be modelled by some evolution equations derived for a respective free energy. The generating / integration functions and parameters should be chosen in certain forms which are compatible with experimental data. Let us explain how the quadratic element (24) defines exact solutions of MGT field equations (8). We prescribe the generating function and sources with respective associated constants, i.e. certain data for Φ(xi, t); FΛ, mΛ, Λ (defining their sum Λ); FhΥ(x i), mh Υ(x i), hΥ(x i) and FΥ(xi, t), mΥ(xi, t), Υ(xi, t) (defining respective sums hΥ and Υ). Using formulas (27), we compute the para- metric functional dependence Π = Ψ[Φ; FΛ, mΛ, Λ; FΥ, mΥ, Υ] from Π2 = ( FΛ + mΛ + Λ)−1 ∫ dt( FΥ + mΥ+ Υ)∂t(Φ 2). As a result, we can find, in explicit form, the coefficients of d-metric (1) parameterized in the form (16), for the class of generic off-diagonal solutions with Killing symmetry on ∂3, gi = ǎ2ηi = e ψ(xk) is a solution of ψ•• + ψ′′ = 2 ( FhΥ+ m h Υ+ hΥ); g3 = h3(x i, t) =ǎ2ȟ3(x k,ť) = − ∂t(Π 2) ( FΥ+ mΥ + Υ)2 ( h [0] 3 (x k)− ∫ dt ∂t(Π2) 4( FΥ+ mΥ+ Υ) ) ; g4 = h4(x i, t) = −ǎ2 = h[0] 4 (x k)− ∫ dt ∂t(Π 2) 4( FΥ + mΥ+ Υ) ; (30) N3k = nk(x i, t) = 1nk(x i) + 2nk(x i) ∫ dt (∂tΠ) 2 ( FΥ+ mΥ + Υ)2|h[0] 3 (x i)− ∫ dt ∂t(Π2) 4( FΥ+ mΥ+ Υ) | 52 ; N4 i = wi(x k, t) = ∂i Π/∂tΠ. We emphasize that these formulas allow, for instance, to "switch off" the contributions from LQG if we fix Λ = 0 and Υ but consider nontrivial values for FΛ+ mΛ and FhΥ+ m h Υ. 14 The values ( hΥ,Υ) define certain nonholonomic constraints on the sources and dynamics of (effective) matter fields and quantum corrections which allows us to integrate a system of nonlinear PDEs in explicit form and with decoupled h- v–cosmological evolution in certain N–adapted systems of reference. In explicit form, we compute using coefficients ofD̂ for a class of solutions (24). At the next step, it is possible to compute FΥµν (10), mΥµν (11) and Υµν (20) for arbitrary physically motivated values of F–modifications and solutions for scalar field φ. For instance, we generate physically motivated solutions by considering ε-parametric deformations (13) of some well defined cosmological solutions in GR or other type MGT, see examples [72, 52, 54]. For such small off diagonal locally anisotropic deformations, we have to choseâ2(xk, t) and χaj (x k, t) to be compatible with experimental gravity and observation cosmology data. Other important examples with redefinition and/or prescription of the generating function and source are those when the integration functions in a class of metrics (24) are stated to be some constants and, for instance, Φ(xi, t) ≃ Φ(t), which results in some data (Π(t), Υ(t)) following formulas (27). It is also possible to work with ε-parametric data (Φ(ε, xi, t),Λ), and respective (Π(ε, xi, t), Υ(ε, xi, t)), resulting formulas (14) for quasi-FLRW metrics (13). Here, it should be emphasized that even some further diagonal approximations withâ2(xk, t) ≃â2(t) will be consid- ered, we shall generate FLRW metrics encoding partially some data on nonlinear and/or off-diagonal interactions, MGT terms and LQG corrections. Such solutions can not be found if we introduce diagonal homogeneous cosmological ansatz which transform, from the very beginning, the nonlinear systems of PDEs into some ODEs (related to gravitational and matter field equations in respective theories of gravity and cosmology). 3.2 N-adapted Weyl–invariant quantities for anamorphic phases For any generic off-diagonal solution (24), we can compute with respect to N-adapted theα̌– coefficients and values mΘ and P lΘ in (21). Re–writing such solutions in the form (16), with re-defined time like and space coordinates and scaling factorâ =ǎ. Here, we note that we can model nonlinear off-diagonal interactions of gravitational and (effective) matter field interactions in terms of conventional polarization functions of fundamental physical constants (such values are introduced by analogy with electromagnetic interactions in certain classical or quantum media). Let us denote m̌ = m0η̌(x i, t) andM̌P l =M 0 P l √ f(φ) =M0P lηP l(x i, t), (31) whereη̌(xi, t) and ηP l(x i, t) are respective polarization of a particle mass m0 and Planck constant M0P l. Theα̌–coefficients in off-diagonal backgrounds are expressedα̌m :=âη̌m0/M 0 P andα̌P l :=âηP l. The values for analyzing the conditions for anamorphic phases of (24) are computed mΘ[Π] M0P lηP l :=Ĥ +H +η̌ ∗ = (ln |âη̌|)∗ and P lΘ[Π] M0P lηP l :=Ĥ +H + η∗P l = (ln |âηP l|)∗, (32) where the Hubble functions,Ĥ (15) and H (19) are considered for (30) with h4 = −ǎ2 and ρ (18), Ĥ = (lnâ)∗ = 1 2 ( ln ∣∣∣∣h [0] 4 (x k)− ∫ dt ∂t(Π 2) 4( FΥ+ mΥ+ Υ) ∣∣∣∣ )∗ and H = √∣∣∣∣ ρ 3 (1− ρ ρc ) ∣∣∣∣. A generating function Π = Ψ[Φ; FΛ, mΛ, Λ;F Υ, mΥ, Υ] may induce anamorphic cosmological phases following the conditions (22) determined by the data for the integration function h [0] 4 (x k); 15 effective sources FΥ, mΥ, Υ and ρ contained in the sumĤ + H. The polarizationsη̌(xi, t) and ηP l(x i, t) modify mΘ[Π] and P lΘ[Π] as follow from (32). Such values can be used for characterizing locally anisotropic cosmological models, even the analogs of generalized Friedmann equations (23) for all types of generating functions.13 We compute anamorphosis inflation ekpyrosis M0P l mΘ[Π] = (ln | √ |h4[Π]|η̌|)∗/ηP l < 0 (contracts) > 0 (expands) < 0 (contracts) M0P l P lΘ[Π] = (ln | √ |h4[Π]|ηP l|)∗/ηP l > 0 (grow) > 0 (grow) > 0 (decay) Such conditions impose additional nonholonomic constraints on generating functions, sources and integration functions and constants which induce QC structures as follow from (29). 3.3 Cosmological QCs for (effective) matter fields and LQG Quasiperiodic cosmological structures can be induced by nonholonomic distributions of (effective) matter fields sources and quantum corrections. 3.3.1 Effective QC matter fields from MGT Let us consider an effective scalar field φ := √ 3 2 ln | 1F(R̂)| with nonlinear scalar potential V (φ) = 1 2 [R̂/ 1F(R̂)−F(R̂)/ ( 1F(R̂) )2 ] determined by a modification of GR, see (17). This results in an effective matter density ρ := 1 2 (∂φ)2 + V (φ) and respective EFL. Considering that V (φ) is chosen in a form that φ = qcφ (the label qc emphasises modeling a QC structure) is a solution of ∂( qcφ) ∂t = b∆̂ [ δ( qcF) δ( qcφ) ] = − b∆̂[Θ qcφ+Q( qcφ)2 − ( qcφ)3] with effective free energy qcF [ qcφ] = ∫ [ −1 2 ( qcφ)Θ( qcφ)− Q 3 ( qcφ)3 + 1 4 ( qcφ)4 ]√ bdx1dx2δy3. This induces an effective matter source of type (10), when qcΥµν = FΥµν [ qcφ] = diag( qc h Υ, qcΥ) is taken for an energy momentum tensor FTµν computed in standard form for a QC-field qcφ. We conclude that F–modifications of GR can induce QC locally anisotropic configurations via effective matter field sources if the scalar potential is determined by a corresponding class of nonlinear interactions and associated free energy qcF . Such cosmologies for QC-modified gravity are described by N–adapted coefficients gi = ǎ2ηi = e ψ(xk) is a solution of ψ•• + ψ′′ = 2 qc h Υ; (33) g3 = h3(x i, t) =ǎ2ȟ3(x k,ť) = − ∂t[Ψ 2( qcφ)] ( qcΥ)2 ( h [0] 3 (x k)− ∫ dt∂t[Ψ 2( qcφ)] 4( qcΥ) ) ; g4 = h4(x i, t) = −ǎ2 = h[0] 4 (x k)− ∫ dt ∂t[Ψ 2( qcφ)] 4( qcΥ) ; N3k = nk(x i, t) = 1nk(x i) + 2nk(x i) ∫ dt [∂tΨ( qcφ)]2 ( qcΥ)2|h[0] 3 (x i)− ∫ dt ∂t[Ψ 2( qcφ)] 4( qcΥ) | 52 ; N4 i = wi(x k, t) = ∂i Ψ[ qcφ]/∂tΨ[ qcφ]. 13We can consider a standard interpretation as in [1, 2] for small ε–deformations in Section 4. 16 A d-metric (1) with such coefficients describes a cosmological spacetime encoding "pure" modified gravity contributions. The functional Ψ2( qcφ) has to be prescribed in a form reproducing observa- tional data. Considering additional sources for matter fields and quantum corrections, we can model quasiperiodic and/or aperiodic structures of different scales and resulting from different sources. The values necessary for analyzing the conditions for anamorphic phases induced by QC matter fields from MGT as cosmological spacetimes (33) are computed mΘ[Ψ( qcφ)] M0P lηP l :=Ĥ +η̌ ∗ = (ln |âη̌|)∗ and P lΘ[Ψ( qcφ)] M0P lηP l :=Ĥ + η∗P l = (ln |âηP l|)∗ whereĤ = 1 2 ( ln |h[0] 4 (x k)− ∫ dt ∂t[Ψ 2( qcφ)] 4( qcΥ) | )∗ . A generating function Ψ[ qcφ] may induce anamorphic cosmological phases following the conditions anamorphosis inflation ekpyrosis M0P l mΘ[ qcφ] = (ln | √ |h4[ qcφ]|η̌|)∗/ηP l < 0 (contracts) > 0 (expands) < 0 (contracts) M0P l P lΘ[ qcφ] = (ln | √ |h4[ qcφ]|ηP l|)∗/ηP l > 0 (grow) > 0 (grow) > 0 (decay) Such conditions impose additional nonholonomic constraints on modifications of gravity via F– functionals and generating function Ψ[ qcφ] and source qcΥ and integration functions. We do not consider quantum contributions in generating QCs and the mass m0 is taken by a point particle. 3.3.2 Nonhomogeneous QC like scalar fields For interactions of a scalar field φ = mφ with mass m and mΥµν = (2MP ) −2 mTαβ (11) parameterized in N-adapted form, qmΥµν = mΥµν [ mφ] = diag( qm h Υ, qmΥ), we can generate QC like configurations by this class of solutions, gi = ǎ2ηi = e ψ(xk) is a solution of ψ•• + ψ′′ = 2 qm h Υ; (34) g3 = h3(x i, t) =ǎ2ȟ3(x k,ť) = − ∂t[Ψ 2( mφ)] ( qmΥ)2 ( h [0] 3 (x k)− ∫ dt∂t[Ψ 2( mφ)] 4Ψ( qmΥ) ) ; g4 = h4(x i, t) = −ǎ2 = h[0] 4 (x k)− ∫ dt ∂t[Ψ 2( mφ)] 4( qmΥ) ; N3k = nk(x i, t) = 1nk(x i) + 2nk(x i) ∫ dt [∂tΨ( mφ)]2 ( qmΥ)2|h[0] 3 (x i)− ∫ dt ∂t[Ψ 2( mφ)] 4Ψ( qmΥ) | 5 2 ; N4 i = wi(x k, t) = ∂i Ψ[ mφ]/∂tΨ[ mφ]. We can consider additional constraints for zero torsion configurations which results in cosmological solutions in GR. Such off-diagonal metrics are determined by QC like matter distributions if ∂( mφ) ∂t = b∆̂ [ δ( qmF) δ( mφ) ] = − b∆̂[Θ mφ+Q( mφ)2 − ( mφ)3] with effective free energy qmF [ mφ] = ∫ [ −1 2 ( mφ)Θ( mφ)− Q 3 ( mφ)3 + 1 4 ( mφ)4 ]√ bdx1dx2δy3. It is possible to model double QC configurations with φ = qcφ + mφ, for instance, considering mφ as a small modification of qcφ and effective F ≃ qcF [ qcφ]+ qmF [ mφ]. In general, we do not have 17 an additive law of QC free energies for nonlinear MGT and matter field interactions. The functional Ψ[ mφ] is different from Ψ[ qcφ]. The values for anamorphic phases induced by QC matter fields from MGT as cosmological space- times (34) are computed mΘ[Ψ( qcφ), qmΥ] M0P lηP l :=Ĥ +η̌∗ = (ln |âη̌|)∗ and P lΘ[Ψ( qcφ), qmΥ] M0P lηP l :=Ĥ + η∗P l = (ln |âηP l|)∗ whereĤ =Ĥ = 1 2 ( h [0] 4 (x k)− ∫ dt∂t[Ψ 2( mφ)] 4( qmΥ) )∗ . A generating function Ψ[ qcφ] may induce anamor- phic cosmological phases following the conditions anamorphosis inflation ekpyrosis M0P l mΘ[Ψ( qcφ), qmΥ] = (ln | √ |h4|η̌|)∗/ηP l < 0 (contracts) > 0 (expands) < 0 (contracts) M0P l P lΘ[Ψ( qcφ), qmΥ] = (ln | √ |h4|ηP l|)∗/ηP l > 0 (grow) > 0 (grow) > 0 (decay) These conditions impose additional nonholonomic constraints on generating function Ψ[ qcφ] and source qmΥ and integration functions. Quantum contributions are not considered and the scalar field with QC configurations is with polarization of mass m0. 3.3.3 QC configurations induced by LQG corrections Quantum corrections may also result in quasiperiodic/ aperiodic QC like structures, for instance, if LQG sources of typeΥ̌ = Υ = −ρ2[ qφ]/3ρc (20) are considered for generating cosmological solutions. This defines an effective scalar field φ = qφ (the label q emphasizes the quantum nature of such a field). For LQG and GR, such solutions are of type (see footnote 12) ds2 = gijdx idxj + {h3[dy3 + (∂kn)dxk]2 − 9(ρc) 2 4h3 [ ∂tΨ̌( qφ) ]2 [dt+ (∂iǍ)dx i]2}, h3 = −9(ρc)2∂t(Ψ̌2)/ρ4[ qφ] ( h [0] 3 (x k) + 3ρc ∫ dt∂t[Ψ̌ 2( qφ)]/4ρ2[ qφ] ) . (35) The QC structure is generated if qφ is subjected by the conditions ∂( qφ) ∂t = b∆̂ [ δ( qF) δ( qφ) ] = − b∆̂[Θ qφ+Q( qφ)2 − ( qφ)3] for effective free energy qF [ qφ] = ∫ [ −1 2 ( qφ)Θ( qφ)− Q 3 ( qφ)3 + 1 4 ( qφ)4 ]√ bdx1dx2δy3. This type of loop QC configurations can be generated from vacuum gravitational fields. The values for anamorphic phases for QC structures determined by LQG corrections of matter fields from MGT as cosmological spacetimes (35) are computed mΘ[Ψ̌( qφ)] M0P lηP l : =Ĥ +η̌∗ = (ln |âη̌|)∗ and P lΘ[Ψ̌( qφ)] M0P lηP l : =Ĥ + η∗P l = (ln |âηP l|)∗ whereĤ = lnâ = ln | 3ρc 2h3 ∂tΨ̌( qφ)| is computed for h4 = ρ 4[ qφ] ∂tΨ̌( qφ) 8Ψ̌ ( h [0] 3 (x k) + 3ρc 4 ∫ dt ∂t[Ψ̌ 2( qφ)] ρ2[ qφ] / ) . 18 Anamorphic cosmological phases are determined following the conditions anamorphosis inflation ekpyrosis M0P l mΘ[Ψ̌( qφ)] = (ln | √ |h4|η̌|)∗/ηP l < 0 (contracts) > 0 (expands) < 0 (contracts) M0P l P lΘ[Ψ̌( qφ)] = (ln | √ |h4|ηP l|)∗/ηP l > 0 (grow) > 0 (grow) > 0 (decay) These conditions impose nonholonomic constraints on generating functionΨ̌( qφ) for quantum con- tributions computed in LQG and for polarization of mass m0 of a point particle. 3.4 Anamorphic off-diagonal cosmology with QC and LQG structures Generic off-diagonal solutions (24) encoding parameterized form QC structures generated by different type sources considered in (33), (34) and (35) can be written in the form similar to (16) with redefined time coordinate and scaling factorâ =ǎ. We obtain ds2 = â2(xi,ť)[ηi(x k,ť)(dxi)2 +ȟ3(x k,ť)(e3)2 − (ě4)2], (36) where ηi = ǎ−2eψ, e3 = dy3 + ∂kn(x i) dxk,ě4 = dť+ √ |h4|(∂it+ wi), forȟ3 = −∂t(Ψ2)/ǎ2( qcΥ+ qmΥ− ρ2[ qφ]/3ρc)2(h [0] 3 (x k)− ∫ dt ∂t(Ψ 2) 4( qcΥ+ qmΥ− ρ2[ qφ]/3ρc) ) h4 = −â2(xi, t) = h[0] 4 (x k)− ∫ dt∂t(Ψ 2)/4( qcΥ+ qmΥ− ρ2[ qφ]/3ρc), wi = ∂i Ψ/∂tΨ, for a functional Ψ = Ψ[ qcφ, mφ, qφ]. For a hierarchy of coupled three QC cosmological structures, we can subject such a functional of effective sources to conditions of type ∂Ψ ∂t = b∆̂ [ δF δΨ ] = − b∆̂(ΘΨ +QΨ2 −Ψ3), with a functional for effective free energy F [Ψ] = ∫ [ −1 2 ΨΘΨ− Q 3 Ψ3 + 1 4 Ψ4 ]√ bdx1dx2δy3, written in conventional integro-functional forms. The values characterizing anamorphic phases in QC cosmological spacetimes are computed mΘ M0P lηP l :=Ĥ +H +η̌ ∗ = (ln |âη̌|)∗ and P lΘ M0P lηP l :=Ĥ +H + η∗P l = (ln |âηP l|)∗ where the polarized Hubble functions,Ĥ (15) and H (19), are taken for the quadratic element (36) Ĥ = (lnâ)∗ = 1 2 ( ln ∣∣∣∣h [0] 4 (x k)− ∫ dt ∂t(Ψ 2) 4( qcΥ + qmΥ− ρ2[ qφ]/3ρc) ∣∣∣∣ )∗ and H = √∣∣∣∣ ρ 3 (1− ρ ρc ) ∣∣∣∣. A generating function Ψ = Ψ[Φ; FΛ, mΛ, Λ;F Υ, mΥ, Υ] may induce anamorphic cosmological phases following the conditions (22). In the case of mixed 3 type QC structures, the Weyl type 19 anamorphic characteristics are determined also by the data for the integration function h [0] 4 (x k); effective sources FΥ, mΥ, Υ and ρ contained in the sumĤ +H. We compute anamorphosis inflation ekpyrosis M0P l mΘ[Ψ, qcΥ, qmΥ, ρ2] = (ln | √ |h4|η̌|)∗/ηP l < 0 (contracts) > 0 (expands) < 0 (contracts) M0P l P lΘ[Ψ, qcΥ, qmΥ, ρ2] = (ln | √ |h4|ηP l|)∗/ηP l > 0 (grow) > 0 (grow) > 0 (decay) Such conditions impose additional nonholonomic constraints on generating functions and all types of sources and integration functions and constants which induce QC structures. 4 Small Parametric Anamorphic Cosmological QC and LQG Structures The main goal of this paper is to prove that quasiperiodic and/or aperiodic (for instance, QC like) structures in MGT with LQG helicity contributions can be incorporated in a compatible way in the framework of the anamorphic cosmology [1, 2, 3, 4, 5, 6]. For the classes of cosmological solutions constructed in general form in previous section, we can consider a procedure of small ε– deformations of d-metrics of type (24) with respective N–adapted frames and connections, see details in [72, 73, 75, 76, 77, 52, 54] and subsection 2.3. In this section, we show how using ε–deformations an off–diagonal "prime" metric,g̊(xi, y3, t, ) (for applications in modern cosmolgoy, this metric can be diagonalizable under coordinate transforms14) into a "target" metric, g(xi, y3, t). 4.1 N-adapted ε–deformations We suppose that a "prime " pseudo–Riemannian cosmological metricg̊ = [̊gi,h̊a,N̊ j b ] can be parameterized in the form ds2 = g̊i(x k, t)(dxi)2 +h̊a(x k, t)(̊ea)2, (37) e̊3 = dy3 +n̊i(x k, t)dxi,e̊4 = dt+ẘi(x k, t)dxi. For instance, some data (̊gi,h̊a) may define a cosmological solution in MGT or in GR like a FLRW, metric. The target metric g = εg for an off-diagonal deformation of the metric structure, for a small parameter 0 ≤ ε ≪ 1, is parameterized by N-adapted quadratic elements ds2 = ηi(x k, t)̊gi(x k, t)(dxi)2 + ηa(x k, t)̊ga(x k, t)(ea)2 (38) = ǎ2(xi, t)[ηi(x k, t)(dxi)2 +ȟ3(x k, t)(e3)2 − (ě4)2], e3 = dy3 + nηi(x k, t)̊ni(x k, t)dxi = dy3 + ∂kn dx k, e4 = dt+ wηi(x k, t)ẘi(x k, t)dxi =ě4 = dť+ √ |h4|(∂it+ wi), with possible re-definitions of coordinatesť =ť(xk, t) forǎ2(xi, t) →â2(xi, t) and where, for instance, nηin̊idx i = nη1n̊1dx 1 + nη2n̊2dx 2. The polarization functions are ε-deformed following rules adapted 14we note that in general,g̊ (37) may not be a solution of gravitational field equations but it will be nonholonomically deformed into such solutions. 20 to (16) and (36), when ηi = η̌i(x k, t)[1 + εχi(x k, t)], ηa = 1 + εχa(x k, t) and (39) nηi = 1 + ε nχi(x k, t), wηi = 1 + ε wχi(x k, t), ηi ≃ 1 + εχi(xk,t̂), ∂kn ≃ εn̂i(xk), √ |h4| wi ≃ εŵi(xk,t̂). Such "double" N–adapted deformations are convenient for generating new classes of solutions and further physical interpretation of such solutions with limits of quasi-FLRW metrics to some homoge- nous diagonal cosmological metrics. The target generic off–diagonal cosmological metrics g = εg = ( εgi, εha, εN jb ) = (gα = ηαg̊α, nηini, wηiẘi) (38) →g̊ (37) for ε→ 0, define, for instance, cosmological QC configurations with parametric ε-dependence determined by a class of solutions (24) (or any variant of solutions (30), (33), (34), (35) and (36)). The effective ε-polarizations of constants (see (31)) are written m̌ = m0η̌(x i, t) ≃ m0(1 + εχ(xi, t)) andM̌P l =M0P l √ f(φ) =M0P lηP l(x i, t) =M0P l(1 + εχP l(x i, t)), see formulas (6), (7) and (21), where Aρ = ρ (18) in locally anisotropic and inhomogeneous first and second Friedmann equations, (23). 4.2 ε–deformations to off-diagonal cosmological metrics The deformations of h-components of the cosmological d–metrics are εgi(x k) =g̊i(x k, t)η̌i(x k, t)[1 + εχi(x k, t)] = eψ(x k) defined by a solution of the 2-d Poisson equation. Considering ψ = 0ψ(xk) + ε 1ψ(xk) and hΥ(x k) = 0hΥ(x k) + ε 1hΥ(x k) = FhΥ(x i) + mh Υ(x i) + hΥ(x i) (in particular, we can take 1hΥ), see (12), we compute the deformation polarization functions χi = e 0  ψ 1ψ/̊giη̌i 1 hΥ. (40) Let us compute ε–deformations of v–components using formulas for a sourceΥ = FΥ+ mΥ+Υ.We consider εh3 = h [0] 3 (x k)−1 4 ∫ dt (Ψ2)∗ Υ = (1+εχ3)̊g3 ; εh4 = − 1 4 ( Ψ∗)2 (Υ)2 ( h [0] 4 − 1 4 ∫ dt (Ψ2)∗ Υ )−1 = (1+ε χ4)̊g4, (41) when the generation function can also be ε–deformed, Ψ = εΨ =Ψ̊(xk, t)[1 + εχ(xk, t)]. (42) Introducing εΨ in (41), we compute χ3 = − 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ and ∫ dt (Ψ̊2)∗ Υ = 4(h [0] 3 −g̊3). (43) 21 We conclude that χ3 can be computed for any deformation χ in (42) adapted to a time like oriented family of 2-hypersurfaces t = t(xk). This family given in non-explicit form byΨ̊ =Ψ̊(xk, t) when the integration function h [0] 3 (x k),g̊3(x k) and (Ψ̊2)∗/ Υ satisfy the conditions (43). Using (42) and (41), we get χ4 = 2(χ+ Ψ̊ Ψ̊∗ χ∗)− χ3 = 2(χ+ Ψ̊ Ψ̊∗ χ∗) + 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ . As a result, we can compute χ3 for any data ( Ψ̊,g̊3, χ ) and a compatible source Υ = ±Ψ̊∗/2 √ |̊g4h[0] 3 |. Such conditions and (43) define a time oriented family of 2-d hypersurfaces, parameterized by t = t(xk) defined in non-explicit form from ∫ dtΨ̊ = ±(h[0] 3 −g̊3)/ √ |̊g4h[0] 3 |. (44) The final step consists of ε–deformations N–connection coefficients wi = ∂iΨ/Ψ ∗ for nontrivial ẘi = ∂iΨ̊/Ψ̊ ∗, which are computed following formulas (42) and (39), wχi = ∂i(χΨ̊) ∂iΨ̊ − (χΨ̊)⋄ Ψ̊⋄ . We omit similar computations of ε–deformations of n–coefficients (we omit such details which are not important if we restrict our research only to LC-configurations). Summarizing (40)-(44), we obtain the following formulas for ε–deformations of a prime cosmo- logical metric (37) into a target cosmological metric: εgi = [1 + εχi(x k, t)]̊giη̌i = [1 + εe 0ψ 1ψ/̊giη̌i 0 hΥ]̊gi solution of 2-d Poisson eqs ); εh3 = [1 + ε χ3 ]̊g3 = [ 1− ε 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ ] g̊3; εh4 = [1 + ε χ4 ]̊g4 = [ 1 + ε ( 2(χ+ Ψ̊ Ψ̊∗ χ∗) + 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ )] g̊4; (45) εni = [1 + ε nχi ]̊ni = [ 1 + εñi ∫ dt 1 Υ2 ( χ+ Ψ̊ Ψ̊∗ χ∗ + 5 8 1 g̊3 (Ψ̊2χ)∗ Υ )] n̊i; εwi = [1 + ε wχi]ẘi = [ 1 + ε( ∂i(χΨ̊) ∂iΨ̊ − (χΨ̊) ∗ Ψ̊∗ ) ] ẘi. The factorñi(x k) is a redefined integration function. The quadratic element for such inhomogeneous and locally anisotropic cosmological spaces with coefficients (45) can be written in N–adapted form ds2 = εgαβ(x k, t)duαduβ = εgi ( xk ) [(dx1)2 + (dx2)2] + (46) εh3(x k, t) [dy3 + εnidx i]2 + εh4(x k, t)[dt+ εwk (x k, t)dxk]2. Further assumptions on generating and integration functions and source can be considered in order to find solutions of type εgαβ(x k, t) ≃ εgαβ(t). 22 4.3 Cosmological ε–deformations with anamorphic QCs and LQG We apply the procedure of ε–deformations described in the previous subsection in order to gen- erate solutions of type (36). We prescribe χ(xk, t) and 0 hΥ(x k) for any compatible ( Ψ̊,g̊3 ) and source Ψ̊∗ = ±2 √ |̊g4h[0] 3 | ( qcΥ+ qmΥ− ρ2[ qφ]/3ρc). The generated d–metric with coefficients (45) is of type (46) for Υ = qcΥ + qmΥ− ρ2[ qφ]/3ρc, ds2 = [1 + εe 0ψ 1ψ/̊giη̌i 0 hΥ]̊gi[(dx 1)2 + (dx2)2] + [1− ε 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ ]̊g3 [ dy3 + [1 + εñi ∫ dt 1 Υ2 ( χ + Ψ̊ Ψ̊∗ χ∗ + 5 8 1 g̊3 (Ψ̊2χ)∗ Υ ) ]̊nidx i ]2 + [1 + ε (2(χ+ Ψ̊ Ψ̊∗ χ∗) + 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ )]̊g4 [ dt+ [1 + ε( ∂i(χΨ̊) ∂iΨ̊ − (χΨ̊) ∗ Ψ̊∗ )]ẘkdx k ]2 . Hierarchies of coupled three QC cosmological structures are generated by a functional χ = χ[ qcφ, mφ, qφ] subjected to conditions of type ∂χ ∂t = b∆̂ [ δF δΨ ] = − b∆̂(Θχ+Qχ2 − χ3), with functionals for effective free energy F [χ] = ∫ [ −1 2 χΘχ− Q 3 χ3 + 1 4 χ4 ]√ bdx1dx2δy3, written in conventional integro-functional forms. The value εh4 = − εâ2(xi, t) = [1 + ε (2(χ+ Ψ̊ Ψ̊∗ χ∗) + 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ )]̊g4, (47) withg̊4 = a(t), allows us to compute the Weyl type invariants characterizing anamporphic phases in QC cosmological spacetimes, m ε Θ M 0 P l(1 + εχPL) : =Ĥ +H + (1 + εχ)∗ = (ln | εâ(1 + εχ)|)∗ and P l ε Θ M 0 P l(1 + εχPL) : =Ĥ +H + (1 + εχ)∗ = (ln | εâ(1 + εχPL)|)∗, where the ε-polarized Hubble functions, εĤ (15) and εH (19) are respectively computed for εh4 εĤ = (ln εâ)∗ = 1 2 ( ln ∣∣∣∣∣[1 + ε (2(χ + Ψ̊ Ψ̊∗ χ∗) + 1 4̊g3 ∫ dt (Ψ̊2χ)∗ Υ )]̊g4 ∣∣∣∣∣ )∗ and εH = √∣∣∣∣ ρ 3 (1− ρ ρc ) ∣∣∣∣. The possibility to induce and preserve certain anamorphic cosmological phases following the con- ditions (22). For mixed 3 type QC structures, the Weyl type anamorphic ε–deformed characteristics are determined also by the data for the integration function h [0] 4 (x k); effective sources FΥ, mΥ, Υ and ρ contained in the sum εĤ + εH. We compute anamorphosis inflation ekpyrosis M0P l m ε Θ = (ln | √ | εh4|(1+εχ)|)∗ (1+εχPL) < 0 (contracts) > 0 (expands) < 0 (contracts) M0P l P l ε Θ = (ln | √ | εh4|(1+εχPL)|)∗ (1+εχPL) > 0 (grow) > 0 (grow) > 0 (decay) 23 In such criteria, we use the value εh4 (47) conditions imposing additional nonholonomic constraints on generating functions and all types of sources and integration functions and constants which induce QC structures. In a similar form, we can generate ε-analogs of (33), (34) and (35), (16) and analyze if respective conditions for anamorphic phases can be satisfied. 5 Concluding Remarks The Planck temperature anistoropy maps were used to probe the large-scale spacetime structure [13, 17]. The observational data were completed with respective calculus for the Baysesian likeli- hood with simulations for specific topological models (in universes with locally flat, hyperbolic and spherical geometries). All such work found no evidence for a multiply–connected spacetime topology (when the assumption on the fundamental domain is considered within the last scattering surface). No matching circles, which would result from the intersection of fundamental topological domains with the surface of last scattering, were found. It is supposed that future Planck measurements of CMB polarization may provide more definitive conclusions on anisotropic geometries and non-trivial topologies. At present, the Planck data provides certain phenomenological evidence for a Bianchi V IIh component when parameters are decoupled from standard cosmology. There is no a well de- fined set of cosmological parameters which can produce existing patterns and observed anisotropies on other scales. Following new results of Planck2015 [14, 15, 16, 17, 18] ( with the ratio of tensor perturbation amplitude r < 0.1) authors of [1, 2, 3, 4, 5, 6] concluded that such observational data seem to "virtually eliminate all the simplest textbook inflationary models". In order to solve this problem and update cosmological scenarios, theorists elaborate [7, 8, 9, 10] on three classes of cosmological theories: • There are alternative plateau-like and multi-parameter models adjusted in such ways that necessary r is reproduced. This results in new challenges like ’unlikeness’ and multiverse– unpredictiability problems with more tuning and of parameters and initial conditions. • The classic inflationary paradigm is changed into a ’postmodern’ one and a MGT that allow certain flexibility to fit any combination of observations. Even a series of conceptual problem of initial conditions and multiverse is known and unresolved for decades, many theorists still advocate this direction. • There are developed "bouncing" cosmologies, for instance, certain versions of ekpyrotic (cyclic) cosmology and, also, anamorphic cosmology. In such models, the large scale structure of the universe is set via a period of slow contraction when the big bang is replaced by a big bounce. The anamorphic approach is also considered as a different scenario with a smoothing and flattening of the universe via a contracting phase. This way, a nearly scale-ivariant spectrum of perturbations is generated. The ekpyrotic cosmology [4] fits quite well the Planck2015 data even in the simplest version with the least numbers of parameters and the least amount of tuning. It provides a mechanism for getting a smooth and flat cosmological background via a period of ultra slow contraction before the big bang. For such a model, there are not required improbable initial conditions and the multiverse problem is 24 avoided. Realistic ekpyrotic theories [4, 81, 82] involve two scalar fields when only one has a negative potential in such a form that a non-canonical kinetic coupling acts as an additional friction term for a scalar field freezing the second one. A standard stability analysis proves that diagonal cosmological solutions for such a model are scale-invariant and stable. We note that the anamorphic cosmology [1, 2, 3, 4, 5, 6] was developed as an attempt to describe the early-universe in a form combining the advantages of the "old and modern" inflationary and ekpyrotic models. The main assumption is that the effective Plank mass, MP l(t), has a different time dependence on t, compared to the mass of a massive particle m(t) in any Weyl frame during the pri- mordial genesis phase. Such cosmological models with similar, or different, variations of fundamental constants and masses of particles can be developed in the framework of various MGTs, see discussions in [1, 3]. In our works [52, 53, 54, 77], we proved that it is possible to construct exact solutions with effective polarization of constants (in general, depending on all spacetime coordinates, MP l(x i, y3, t) and m(xi, y3, t)) in GR mimicking time-like dependencies in MGTs if generic off–diagonal metrics and nonholonomically deformations of connections are considered for constructing new classes of cosmological solutions. Such exact/ parametric solutions can be constructed in general form using the anholonomic frame deformation method, AFDM, see review of results in [75, 76] and references therein. 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