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. Following this geometric method, we perform such nonholonomic deformations of the coef-
ficients of frames, generic off-diagonal metrics and (generalized) connections when the (generalized)
Einstein equations can be decoupled in general forms and integrated for various classes of metrics
gαβ(x
i, y3, t).
Finally, we note that noholonomic anamorphic scenarios allow us to preserve the paradigm of
Einstein’s GR theory and to produce cosmological (expanding for certain phases and contracting
in other cases) inflation and acceleration, if generic off-diagonal gravitational interactions model
equivalently modifications of diagonal configurations in MGTs. This is possible if more general
classes of cosmological solutions encoding QC structures and LQG corrections are considered.
References
[1] A. Ijjas and P. J. Steinhardt, The anamorphic universe, JCAP 10 (2015) 001
[2] A. Ijjas, P. J. Steinhardt and A. Loeb, Scale-free primordial cosmology, Phys. Rev. D89 (2014)
023525
[3] A. Ijjas, P. J. Steinhardt and A. Loeb, Inflationary paradigm in trouble after Plank2013, Phys.
Lett. B723 (2013) 261-266
[4] A. Ijjas and P. J. Steinhardt, Implications of Planck2015 for inflationary, ekpyrotic and anamor-
phic bouncing cosmologies, Class. Quant. Grav. 33 (2016) 044001
[5] A. Ijjas, J. Ripley and P. J. Steinhardt, NEC violation in mimetic cosmology revisited, Phys.
Lett. B760 (2016) 132-138
[6] A. Ijjas, Cyclic anamorphic cosmology, arXiv: 1610.02752
[7] A. H. Guth, D. I. Kaiser and Y. Nomura, Inflationary paradigm after Planck 2013, Phys. Lett.
B733 (2014) 112-119
25
[8] R. Kallosh and A. Linde, Cosmological attractors and asymptotic freedom of the inflation field,
JCAP 1606 (2016) 047
[9] A. H. Chamseddine, V. Mukhanov and A. Vikman, Cosmology with mimetic matter, JCAP
1406 (2014) 017
[10] A. A. Starobinsky, S. V. Sushkov and M. S. Volkov, The screening Horndeski cosmologies, JCAP
1606 (2016) 007
[11] Planck 2013 results. XVI. Cosmological parameters, Astron. Astrophysics, 571 (2014) A16
[12] Planck 2013 results. XXII. Constraints on inflation, Astron. Astrophysics, 571 (2014) A22
[13] Planck 2013 results. XXIII. Isotropy and statistics of the CMB, Astron. Astrophysics, 571 (2014)
A23
[14] Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophysics, 594 (2016) A13
[15] Planck 2015 results. XIV. Dark energy and modified gravity, Astron. Astrophysics, 594 (2016)
A14
[16] Planck 2015 results. XX. Constraints on inflation, Astron. Astrophysics, 594 (2016) A20
[17] Planck 2015 results. XXVI. Background geometry and topology of the Universe, Astron. Astro-
physics, 594 (2016) A26
[18] Planck 2015 results. XXXI. Consistency of the Planck data, Astron. Astrophysics, 594 (2016)
A31
[19] R. Aschheim, L. Bubuianu, F. Fang, K. Irwin, V. Ruchin and S. Vacaru, Starobinsky Inflation
and Dark Energy and Dark Matter Effects from Quasicrystal Like Spacetime Structures [under
preparation]
[20] A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett.
B 91 (1980) 99
[21] D. Levine and P. J. Steinhardt, Quasicrystals. i. definition and structure, Phys. Rev. B 34 (1986)
596
[22] G. Y. Onoda, P. J. Steinhardt, D. P. DiVincenzo and J. E. S. Socolar, Growing perfect quais-
crisals. Phys. Rev. Letters 60 (1988) 2653
[23] L. Bindi, P. J. Steinhardt, N. Yao and P. Lu, Natural quaiscrystals, Science 324 (2009) 1306
[24] H. Emmrich, H. Löven, R. Wittkowski, T. Gruhn, G. I. Tóth, G. Tegze and L. Gránásy, Phase-
field-crystal models for condensed matter dynamics of atomic elngth and diffusive time scales:
an overview, Adv. Phys. 61 (2012) 665
[25] P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, Three-dimenisonal pahes field
quasicrystals, arXiv: 1602.05361
26
[26] C. Hann, J. E. S. Socolar and P. J. Steinhardt, Local growth of icoshedral quasicristalline telings,
arXiv: 1604.02479
[27] R. Penrose, The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math.
Appl., 10 (1974) 266
[28] R. Penrose, Tilings and quasicrystals: A nonlocal growth problem?
In: Introduction to the
Mathematics of Quasicrystals, editor: Marko Jarić, vol 2 of Aperiodicity and Order, chapter 4
(Academic Press, 1989)
[29] C. V. Achim, M. Schemiedeberg and H. Löwen, Growth modes of quasicrystals, Phys. Rev. Lett.
112 (2014) 255501
[30] K. Barkan, H. Diamant and R. Lifshitz, Stability of quasicrystals composed of soft isotropic
particles, arXiv: 1005.5257
[31] F. Fang, J. Kovacs, G. Sadler and K. Irwin, An icosahedral quasicrystal as a packing of regular
tetrahedra, Acta Phys. Polonica A. 126 (2014) 458
[32] Fang Fang and K. Irwin, An icosahedral quasicrystal as a golden modification of the icosagrid
and its connection to the E8 lattice, arXiv: 1511.07786
[33] D. Chechtman, I. Belch, D. Gratias and J. W. Cahn, Metallic phase with long-rage orientational
order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951
[34] D. V. Talapin, E. V. Shevchenko, M. I. Bodnarchuk, X. Ye, J. Chen, and C. B. Murray, Qua-
sicrystalline order in self-assembled binary nanonparticle superlattices, Nature 461 (2009) 964
[35] C. Xiao, N. Fujita, K. Miyasaka, Y. Sakamoto and O. Terasaki, Dodecagonal tiling in mseoporous
silica, Nature 487 (2012) 349
[36] T. Dotera, Quasicrystals in soft matter, Israel Journal of Chemistry 51 (2011) 1197
[37] X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Dulcey, and J. K. Hobbs, Supramolecular dendritic
liquid quasicrystals, Nature 428 (2004) 157
[38] S. Fischer, A. Exner, K. Zielske, J. Perlich, S. Deloudi, W. Steurer, P. Lindner, and S. Förster,
Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry, Proceedings of the national
academy of science 108 (2011) 1810
[39] L. Boyle and P. Steinhardt, Coxeter pairs, Ammann patterns and Penrose-like tilngs, arXiv:
1608.08215
[40] See, for instance, webpage: www.crystalinks.com/darkmatter.html
[41] S. Capozziello and V. Faraoni, Beyond Einstein Gravity (Springer, Berlin, 2010)
[42] S. Nojiri and S. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to
Lorentz non-invariant models, Phys. Rept. 505 (2011) 59-144
27
[43] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys.
Repts. 512 (2012) 1-189
[44] D. S. Gorbunov and V. Rubakov, Introduction to the theory of the early universe: Cosmological
perturbations and inflationary theory (Hackensack, USA: World Scientific, 2011)
[45] A. Linde, Inflationary cosmology after Plank 2013, in: Post-Planck Cosmology: Lecture Notes of
the Les Houches Summer School: Vol. 100, July 2013: editors C. Deffayet, P. Peter, B. Wandelt,
M. Zadarraga and L. F. Cugliandolo (Oxford Scholarship Online, 2015); arXiv: 1402.0526
[46] K. Bamba and S. D. Odintsov, Inflationary cosmology in modified gravity theories, Symmetry
7 (2015) 220-240
[47] M. Wali Hossain, R. Myrzakulov, M. Sami and E. N. Saridakis, Unification of inflation and drak
energy á la quintessential inflation, Int. J. Mod. Phys. D 24 (2015) 1530014
[48] A. H. Chamseddine, V. Mukhanov and A. Vikman, Cosmology with mimetic matter, JCAP
1406 (2014) 017
[49] S. Nojiri and S. D. Odintsov, Mimetic F(R) gravity: Inflation, dark energy and bounce, Mod.
Phys. Lett. A 29 (2014) 14502011
[50] E. Guendelman, D. Singleton and N. Yongram, A two measure model of dark energy and dark
matter, JCAP 1211 (2012) 044
[51] E. Guendelman, H. Nishino and S. Rajpoot, Scale symmetry breaking from total derivative
densities and the cosmological constant problem, Phys.Lett. B 732 (2014) 156-160
[52] S. Vacaru, Equivalent off-diagonal cosmological models and ekpyrotic scenarios in f(R)-modified
massive and Einstein gravity, EPJC 75 (2015) 176; arXiv: 1504.04346
[53] E. Elizalde and S. Vacaru, Effective Einstein cosmological spaces for non-minimal modified
gravity, Gen. Relat. Grav. 47 (2015) 64 ; arXiv: 1310.6868
[54] S. Vacaru, Off-diagonal ekpyrotic scenarios and equivalence of modified, massive and/or Einstein
gravity, Phys. Lett. B 752 (2016) 27-33; arXiv: 1304.1080
[55] J. de Haro, How can holonomy corrections be introduced in f(R) gravity? Eurphys. Lett. 107
(2014) 29001; arXiv: 1403.4529
[56] S. D. Odintsov and V. K. Oikonomou, Matter bounce loop quantum cosmology for F(R) gravity,
Phys. Rev. D90 (2014) 124083
[57] C. Rovelli, Quantum Gravity (Cambridge Un. Press, Cambridge, 2004)
[58] T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press,
Cambridge, 2006), gr–qc/0110034
[59] A. Ashtekar and J. Lewandowski, Background independent quantum gravity: a status report.
Class. Quant. Grav. 21 (2004) R53–R152
28
[60] A. Ashtekar and P. Singh, Loop quantum cosmology: a status report, Class. Quant. Grav. 28
(2011) 213001, arXiv: 1108.0893
[61] M. Bojowald, Loop quantum cosmology, Living Rev. 11 (2008) 4
[62] M. M. Amaral and M. Bojowald, A path-intergral approach to the problem of time, arXiv:
1601.07477
[63] L. Smolin, An invitation to loop quantum gravity, hep–th/0408048
[64] H. Nicolai and K. Peeters and M. Zamaklar, Loop quantum gravity: an outside view. Class.
Quant. Grav. 22 (2005), R193–R247
[65] T. Thiemann, Loop quantum gravity: an inside view, Lect. Notes Phys. 721:
(2007) 185–263,
hep-th/0608210
[66] A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244–
2247
[67] A. Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D36 (1987) 1587–
1602
[68] F. Barbero, Reality conditions and Ashtekar variables: a different perspective, Phys. Rev. D51
(1995) 5498–5506
[69] F. Barbero, Real Ashtekar variables for Lorentzian singature space times, Phys. Rev.D51 (1995)
5507–5510
[70] G. Immirzi, Quantum gravity and Regge calculus, Nucl. Phys. Proc. Suppl. 57 (1997) 65–72
[71] S. Vacaru, Loop quantum gravity in Ashtekar and Lagrange-Finsler variables and Fedosov quan-
tization of general relativity, (The Icfai University Journal of Physics) The IUP Journal of
Physics, Vol. II, No. 4. (2009) 15-58; arXiv: 0801.4942
[72] P. Stavrinos and S. Vacaru, Cyclic and ekpyrotic universes in modified Finsler osculating gravity
on tangent Lorentz bundles, Class. Quant. Grav. 30 (2013) 055012
[73] O. Vacaru and S. Vacaru, Perelman’s W-entropy and Statistical and Relativistic Thermodynamic
Description of Gravitational Fields, arXiv: 1312.2580
[74] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, 1973)
[75] T. Gheorghiu, O. Vacaru and S. Vacaru, Off-diagonal deformations of Kerr black holes in Ein-
stein and modified massive gravity and higher dimensions, EPJC 74 (2014) 3152
[76] T. Gheorghiu, V. Ruchin, O. Vacaru and S. Vacaru, Geometric flows and Perelman’s thermo-
dynamics for black ellipsoids in R2 and Einstein gravity theories, Annals of Physics, NY, 369
(2016) 1
[77] S. Rajpoot and S. Vacaru, On supersymmetric geometric flows and R2 inflation from scale
invariant supergravity, arXiv: 1606.06884
29
[78] X. Zhang and Y. Ma, Extension of loop quantum gravity to f(R) theories, Phys. Rev. D84
(2011) 064040
[79] X. Zhang and Y. Ma, Loop quantum modified gravity and its cosmological application, Front.
Phys. 8 (2013) 80
[80] M. Li, Note on the production of scale-invariant entropy perturbation in the Ekpyrotic universe,
Phys. Lett. B724 (2013) 192
[81] A. Fertig, J. -L. Lehners, and E. Mallwitz, Ekpyrotic perturbations with small non-Gaussian
corrections, Phys. Rev. D89 (2014) 103537
[82] A. Ijjas, J. -L. Lehners, and P. J. Stenhardt, A general mechanism for producing scale-invariant
perturbations and small non–Gaussianity in ekpyrotic models, Phys. Rev. D89 (2014) 123520
[83] A. M. Levy, A. Ijjas, and P. J. Steinhardt, Scale-invariant perturbations in ekpyrotic cosmologies
without fine-tuning of initial conditions, Phys. Rev. D92 (2015) 063524
[84] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, From big crunch to big
bang, Phys. Rev. D65 (2002) 086007
30