Starobinsky Ination and Dark Energy and Dark Matter
Eects from Quasicrystal Like Spacetime Structures
Raymond Aschheim
Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA
email: raymond@quantumgravityresearch.org
Laurenµiu Bubuianu
TVR Ia³i, 33 Lascar Catargi street, 700107 Ia³i, Romania
email: laurentiu.bubuianu@tvr.ro
Fang Fang
Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA
email: Fang@quantumgravityresearch.org
Klee Irwin
Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA
email: klee@quantumgravityresearch.org
Vyacheslav Ruchin
Heinrich-Wieland-Str. 182, 81735 München, Germany
email: v.ruchin-software@freenet.de
Sergiu I. Vacaru
Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA
and
University "Al. I. Cuza" Ia³i, Project IDEI
18 Piaµa Voevozilor bloc A 16, Sc. A, ap. 43, 700587 Ia³i, Romania
email: sergiu.vacaru@gmail.com
1
August 2, 2016
Abstract
The goal of this work on mathematical cosmology and geometric methods in modied
gravity theories, MGTs, is to investigate Starobinsky-like ination scenarios determined by
gravitational and scalar eld congurations mimicking quasicrystal, QC, like structures.
Such spacetime aperiodic QCs are dierent from those discovered and studied in solid
state physics but described by similar geometric methods. We prove that inhomogeneous
and locally anisotropic gravitational and matter eld eective QC mixed continuous and
discrete "ether" can be modelled by exact cosmological solutions in MGTs and Einstein
gravity. The coecients of corresponding generic o-diagonal metrics and generalized con-
nections depend (in general) on all spacetime coordinates via generating and integration
functions and certain smooth and discrete parameters. Imposing additional nonholonomic
constraints, prescribing symmetries for generating functions and solving the boundary con-
ditions for integration functions and constants, we can model various nontrivial torsion
QC structures or extract cosmological LeviCivita congurations with diagonal metrics
reproducing de Sitter (inationary) like and other type homogeneous ination and accel-
eration phases. Finally, we speculate how various dark energy and dark matter eects
can be modelled by o-diagonal interactions and deformations of a nontrivial QC like
gravitational vacuum structure and analogous scalar matter elds.
Keywords: Odiagonal cosmological metrics; eective gravitational and scalar eld
aperiodic structures; Starobinsky-Like ination; dark energy and dark matter as qua-
sicrystal structures.
Contents
1 Introduction
3
2 Generating Aperiodic Cosmological Solutions in R2 Gravity
5
3 Modied Gravity with Quasicrystal Like Structures
10
3.1 Generating functions with 3d quasicristal like structure . . . . . . . . . . . . . .
10
3.2 Eective scalar elds with quasicrystal like structure
. . . . . . . . . . . . . . .
12
3.2.1
Scalar eld Nadapted to gravitational quasicrystals . . . . . . . . . . . .
13
3.2.2
Scalar and rescaled QC generating functions . . . . . . . . . . . . . . . .
13
4 Aperiodic QC Starobinsky Like Ination
14
4.1
Ination parameters determined by QC like structures
. . . . . . . . . . . . . .
14
4.2 Reconstructing cosmological quasicrystal structures . . . . . . . . . . . . . . . .
16
5 Quasicrystal Models for Dark Energy and Dark Matter
18
5.1 Encoding o-diagonal QC structures into canonical dtorsions . . . . . . . . . .
18
5.2
Interaction between DE and DM in aperiodic QC vacuum . . . . . . . . . . . .
20
2
5.3 Quasicrystal DE structures and matter sources . . . . . . . . . . . . . . . . . . .
20
5.3.1
Interaction between DE and ordinary matter in gravitational QC media .
20
5.3.2 Van der Waals uid interacting with aperiodic DM . . . . . . . . . . . .
21
5.3.3 Chaplygin gas and DE - QC congurations . . . . . . . . . . . . . . . . .
22
6 Discussion and Conclusions
22
1
Introduction
The Plank data [1] for modern cosmology prove a remarkable consistency of Starobinsky
R2ination theory [2] and a series of classical works on inational cosmology [3, 4, 5, 6]. For
reviews of results and changing of modern cosmology paradigms with dark energy and dark
matter, and on modied gravity theories, MGTs, we cite [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
Various cosmological scenarios studied in the framework of MGTs involve certain inhomoge-
neous and local anisotropic vacuum and nonvacuum gravitational congurations determined
by corresponding type (eective) ination potentials. In order to investigate such theoretical
models, there are applied advanced numeric, analytic and geometric techniques which allow us
to nd exact, parametric and approximate solutions for various classes of nonlinear systems of
partial dierential equations, PDEs, considered in mathematical gravity and cosmology. The
main goal of this paper is to elaborate on geometric methods in acceleration cosmology physics
and certain models with eective quasicrystal like gravitational and scalar eld structures.
The anholonomic frame deformation method, AFDM, (see review and applications in four
dimensional, 4-d, and extra dimension gravity in [19, 20, 21]) allows to construct in certain
general forms various classes of (non) vacuum generic odiagonal solutions in Einstein gravity
and MGTs. Such solutions may describe cosmological observation data and explain and predict
various types of gravitational and particle physics eects [22, 23, 24, 25]. Following this geo-
metric method, we dene such nonholonomic frame transforms and deformations of connections
(all determined by the same metric structure) when the gravitational and matter eld motion
equations decouple in general forms. In result, we can integrate also in general forms certain
systems of gravitational and cosmologically important PDEs when the solutions are determined
by generating and integration functions depending (in principle) on all spacetime coordinates
and various classes of integration parameters.
The AFDM is very dierent from all other former methods applied for constructing exact
solutions, when certain ansatz with higher symmetries (spherical, cylindrical, certain Lie algebra
ones ...) are used for metrics which can be diagonalized by frame/coordinate transforms. For
such "simplied" ansatz, the motion gravitational and/or cosmological equations transform
into nonlinear systems of ordinary dierential equations, ODEs, which can be solved in exact
form for some special cases. For instance, there are considered diagonal ansatz for metrics with
dependence on one radial (or time like) coordinate and respective transforms of PDEs to ODEs
in order to construct black hole (or homogeneous cosmological) solutions, when the physical
eects are computed for integration constants determined by respective physical constants.
The priority of the AFDM is that we can use various classes of generating and integration
functions in order to construct exact and parametric solutions determined by nonholonomic
3
nonlinear constraints and transforms, with generic o-diagonal metrics, generalized connections
and various eective gravitational and matter elds congurations. Having constructed a class
of general solutions, we can always impose additional constraints at the end (for instance,
zero torsion, necessary boundary and symmetry conditions, or to consider homogeneous and
isotropic congurations with nontrivial topology etc.) and search for possible limits to well
known physically important solutions. Here we note that attempting to nd exact solutions for
nonlinear systems we lose a lot of physically important classes of solutions if we make certain
approximations with "simplied" ansatz from the very beginning. Applying the AFDM, we
can generate certain very general families of exact/parametric solutions after which there are
possibilities to consider additional nonholonomic (non integrable) constraints and necessary
type symmetry/ topology / boundary / asymptotic conditions which allow us to explain and
predict certain observational and/or experimental data.
Modern acceleration cosmology data [1] show on existence of certain complex network la-
ment and aperiodic type structures (with various fractal like, diusion processes, nonlinear wave
interactions etc.) determined by dark energy of the Universe and distribution of dark matter,
with "hidden" frame support for respective metagalactic and galactic congurations. Such
congurations can be modelled by numeric and/or analytic geometric methods as deformations
of an eective quasicrystal, QC, structure for the gravitational and fundamental scalar elds.
Similar ideas were proposed many years ago in connection both with inationary cosmology and
quasicrystal physics [26, 27, 28, 6]. On early works and modern approaches to QC mathematics
and physics, we cite [29, 30, 31, 32, 33, 34, 35, 36, 37] and references therein. Here we note that
it is not possible to introduce directly a QC like locally anisotropic and inhomogeneous structure
described by solutions of (modied) Einstein equations if we restrict the geometric approach
only for cosmological models based only on the FriedmanLemaître-Robertson-Worker, FLRW,
metric. The geometric objects and physical values with homogeneous and isotropic metrics
are determined by solutions with integration constants and this does not allow to elaborate on
realistic an physically motivated cosmological congurations with QC nontrivial vacuum and
nonvacuum structures. Realistic QC like aperiodic structures can not be described only via
integration constants or certain Bianchi / Killing type and Lie group symmetries with structure
constants. Solutions with nontrivial gravitational vacuum structure and respective cosmological
scenarios can be generated by prescribing via generating functions and generating sources possi-
ble observational QC congurations following the AFDM as in [22, 23, 24, 25]. In nonholonomic
variables, we can describe formation and evolution of QG structures as generalized geometric
ow eects, see partner work [38] as recent developments and applications in physics of R.
Hamilton and G. Pereman's theory of Ricci ows [39, 40]. Here, we emphasize that Lyapunov
type functionals (for free energy) are used both in QC and Ricci ow evolution theories. For
geometric evolution theories, such generalize entropy type functionals are known as Perelman's
functionals with associated thermodynamical variables. The purpose of a planned series of pa-
pers is to study generic odiagonal cosmological solutions with aperiodic order for (modied)
Ricci solitons. In explicit form, the main goal of this article is to provide a geometric proof
that aperiodic QC structures of vacuum and nonvacuum solutions of gravitational and scalar
matter eld equations MGTs and GR result in cosmological solutions mimicking Starobinsky
like inations and dark energy and dark matter scenarios which are compatible both with the
4
accelerating cosmology paradigm and observational cosmological data. We shall follow also
certain methods for the mathematics of aperiodic order structures summarized in [41].
This paper is structured as follows. Section 2 is devoted to geometric preliminaries and
main formulas for generating generic odiagonal cosmological solutions in MGTs and GR
(for proofs and details, readers are directed to review papers [19, 22, 23, 24]). In section 3, we
elaborate on methods of constructing exact solutions with gravitational and scalar eld eective
QC structure. We study how aperiodic structures can be dened by generating functions and
matter eld sources adapted to o-diagonal gravitational and matter eld interactions and
evolution processes. Then, in section 4, we prove that Starobinsky like ination scenarios can
be determined by eective QC gravitational and adapted scalar eld congurations. We also
outline in that section a reconstructing formalism for analogous QC cosmological structure,
with dark energy and dark matter eects. In section 5, we study certain MGTs congurations
when dark energy and dark matter physics is modelled by QCs gravitational and eective scalar
eld congurations. Finally, conclusions and discussion are provided in section 6.
2 Generating Aperiodic Cosmological Solutions in R2 Grav-
ity
Let us outline a geometric approach to constructing exact solutions, (for the purposes of
this paper), with aperiodic continuous and/or discrete, in general, inhomogeneous and locally
anisotropic cosmological structures in (modied) gravity theories. We consider a general o
diagonal cosmological metrics g on a four dimensional, 4d, pseudoRiemannian manifold V
which can be parameterized via certain frame transforms as a distinguished metric, dmetric,
in the form
g = gαβ(u)e
α ⊗ eβ = gi(xk)dxi ⊗ dxi + ga(xk, yb)ea ⊗ eb
(1)
= gαβ(u)e
α ⊗ eβ, gα′β′(u) = gαβeαα′e
β
β′ ,
ea = dya +Nai (u
γ)dxi and eα = eαα′(u)du
α′ .
(2)
In these formulas, the local coordinates uγ = (xk, yc), or u = (x, y), when indices run respective
values i, j, k, ... = 1, 2 and a, b, c, ... = 3, 4 are for conventional 2+2 splitting onV of signature
(+++−), when u4 = y4 = t is a time like coordinate and uı̀ = (xi, y3) are spacelike coordinates
withı̀,j̀,k̀, ... = 1, 2, 3. The values N = {Nai } = Nai dene a Nadapted decomposition of the
tangent bundle TV = hTV⊕ vTV into conventional horizontal, h, and vertical, v, subspaces1.
Such a geometric splitting is nonholonomic because the basis eα = (xi, ea) is dual to eα = (ei, ea)
ei = ∂/∂x
i −Nai (u)∂/∂ya, ea = ∂a = ∂/∂ya,
1If it will not be a contrary statement for an explicit formula, we shall use the Einstein rule on summation
on "up-low" cross indices. Such a system of "Nadapted notations" with boldface symbols for a nontrivial
nonlinear connection, Nconnection structure determined by N and related nonholonomic dierential geometry
is explained in details in [19, 22, 23, 24, 25] and references therein. We omit such considerations in this work.
5
which is nonholonomic (equivalently, non-intergrable, or anholonomic) if the commutators
e[αeβ] := eαeβ − eβeα = Cγαβ(u)eγ
contain nontrivial anholonomy coecients Cγαβ = {Cbia = ∂aN bi , Caji = ejNai − eiNaj }. If such
coecients are not trivial, a Nadapted metric (1) can not be diagonalized in a local nite,
or innite, spacetime region with respect to coordinate frames. Such metrics are generic o-
diagonal and characterized by six independent nontrivial coecients from a set g = {gαβ(u)}.
A frame is holonomic if all corresponding anholonomy coecients are zero (for instance, the
coordinate frames).
On V, we can consider a distinguished connection, dconnection, D, structure as a metric
ane (linear) connection preserving the Nconnection splitting under parallel transports, i.e.
D = (hD, vD). We denote the torsion of D as T = {Tαβγ}, where the coecients can be
computed in standard form with respect to any (non) holonomic basis. For instance, the well
known LeviCivita, LC, connection ∇ is a linear connection but not a dconnection because it
does not preserve under general frame/coordinate transforms a h-vsplitting. Prescribing any
dmetric and Nconnection structure, we can work on V in equivalent form with two dierent
linear connections:
(g,N)→
{
∇ :
∇g = 0; ∇T = 0, for the LCconnection
D̂ :D̂g = 0; hT̂ = 0, vT̂ = 0, hvT̂
6= 0, for the canonical dconnection .
(3)
In this formula, theD̂ = hD̂ + vD̂ is completely dened by g for any prescribed Nconnection
structure N. There is a canonical distortion relation
D̂ = ∇+Ẑ.
(4)
The distortion distinguished tensor, d-tensor,Ẑ = {Ẑαβγ[T̂αβγ]}, is an algebraic combination
of the coecients of the corresponding torsion d-tensorT̂ = {T̂αβγ} ofD̂. All such values are
completely dened by data (g,N) being adapted to the Nsplitting. It should be noted thatT̂
is a nonholonomically induced torsion determined by (Cγαβ, ∂βN
a
i , gαβ). It is dierent from that
considered, for instance, in the EinsteinCartan, or string theory, where there are considered
additional eld equations for torsion elds. We can redene all geometric constructions forD̂ in
holonomic or nonholonomic variables for ∇ when the torsion vanishes in result of nonholonomic
deformations.2
2Using a standard geometric techniques, the torsions,T̂ and ∇T = 0, and curvatures,R̂ = {R̂αβγδ}
and ∇R = {Rαβγδ} (respectively, forD̂ and ∇) are dened and can be computed in coordinate free and/or
coecient forms. We can dene the corresponding Ricci tensors,R̂ic = {R̂ βγ :=R̂γαβγ} and Ric = {R βγ :=
Rγαβγ}, when the Ricci d-tensorR̂ic is characterized h-v N-adapted coecients,R̂αβ = {R̂ij :=R̂kijk,R̂ia :=
−R̂kika,R̂ai :=R̂baib,R̂ab :=R̂cabc}. We can also dene two dierent scalar curvature, R := gαβRαβ and
R̂ := gαβR̂αβ = g
ijR̂ij+g
abR̂ab. Following the two connection approach (3), the (pseudo) Riemannian geometry
can be equivalently described by two dierent geometric data (g,∇) and (g,N,D̂). Using the canonical distortion
relation (4), we can compute respective distortionsR̂ = ∇R+ ∇Z andR̂ic = Ric+Ẑic and ∇Z andẐic.
6
The action S for a quadratic gravity model withR̂2 and matter elds with Lagrange density
mL(g,N, ϕ) is postulated in the form
S = M2P
∫
d4u
√
|g|[R̂2 + mL],
(5)
where the Plank mass MP is determined by the gravitational constant. For simplicity, we
consider in this paper actions mS =
∫
d4u
√
|g| mL depending only on the coecients of a
metric eld and not on their derivatives. In Nadapted form, the energymomentum dtensor
can be computed
mTαβ := −
2√
|gµν |
δ(
√
|gµν | mL)
δgαβ
= mLgαβ + 2δ(
mL)
δgαβ
.
In next sections, we shall chose such dependencies of mL on (eective) scalar elds ϕ which
will allow to model cosmological scenarios with dark mater and dark energy in MGTs in a
compatible form nontrivial quasicrystal like gravitational and matter elds. The action S (5)
results in the eld equations
R̂µν = Υµν ,
(6)
where Υµν =
mΥµν+Υ̂µν , for
mΥαβ =
1
2M2P
mTαβ andΥ̂µν = (
1
4
R̂−̂ R̂
R̂
)gµν+
D̂µD̂νR̂
R̂
,
and̂ :=D̂2 = gµνD̂µD̂ν . ForD̂|T̂ →0 = ∇, the equations can be re-dened via conformal
transforms gµν →g̃µν = gµνe− ln |1+2ϕ̃|, for
√
2/3ϕ = ln |1 + 2ϕ̃|, which introduces a specic
Lagrange density for matter into the gravitational equations with eective scalar elds. Such a
construction was used in the Starobinsky modied cosmology model [2]. In Nadapted frames,
such a scalar eld density can be chosen
mL̂ = −1
2
eµϕ e
µϕ− ϕV (ϕ)
(7)
resulting in matter eld equations
̂ϕ+
d ϕV (ϕ)
dϕ
= 0.
In the above formula, we consider a nonlinear potential for scalar eld φ
ϕV (ϕ) =
ς2(1− e−
√
2/3ϕ)2, ς = const,
(8)
when ϕV (ϕ  0)→ ς2, ϕV (ϕ = 0) = 0, ϕV (ϕ 0) ∼ ς2e−2
√
2/3ϕ.
To apply such geometric methods in GR and MGTs is motivated by the fact that various types of gravitational
and matter eld equations rewritten in nonholonomic variables (g,N,D̂) can be decoupled and integrated in
certain general forms following the AFDM. This is not possible if we work from the very beginning with the
data (g,∇). Nevertheless, necessary type LC-congurations can be extracted from certain classes of solutions
of (modied) gravitational eld equations if additional conditions resulting in zero values for the canonical
d-torsion,T̂ = 0, are imposed (considering some limitsD̂|T̂ →0 = ∇).
7
The aim of this work, we shall study scalar elds potentials V (ϕ) modied by eective qua-
sicrystal structures, ϕ→ ϕ = ϕ0+ψ, where ψ(xi, y3, t) with crystal, or QC, like phases described
by periodic or quasi-periodic modulations. Such modications can be modelled in dynamical
phase eld crystal, PFC, like form [42]. The corresponding 3-d nonrelativistic dynamics is
determined by a Laplace like operator 34 = ( 3∇)2, with left label 3. In Nadapted frames
with 3+1 splitting the equations for a local minimum conserved dynamics,
∂tψ =
34
[
δF [ψ]
δψ
]
,
with two lenghs scales li = 2π/ki, for i = 1, 2. Such local diusion process is described by a free
energy functional
F [ψ] =
∫ √
| 3g |dx1dx2dy3[1
2
ψ{−+
∏
i=1,2
(k2i +
34)2}ψ+1
4
ψ4],
where | 3g | is the determinant of the 3-d space metric and  is a constant. For simplicity,
we restrict our constructions only for non-relativistic diusion processes, see Refs. [43, 44] for
relativistic and Nadapted generalizations.
We shall be able to integrate in explicit form the gravitational eld equations (6) and a d
metric (5) for (eective) matter eld congurations parameterized with respect to Nadapted
frames in the form
Υµν = e
µ
µ′e
ν′
ν Υ
µ′
ν′ [
mL(ϕ+ ψ),Υ̂µν ] = diag[ hΥ(xi)δij,Υ(xi, t)δab ],
(9)
for certain vielbein transforms eµµ′(u
γ) and their duals e ν
′
ν (u
γ), when eµ = eµµ′du
µ′ , and
Υµ
′
ν′ =
mΥµ
′
ν′ +Υ̂
µ′
ν′ . The values
hΥ(x
i) and Υ(xi, t) will be considered as generating
functions for (eective) matter sources imposing certain nonholonomic frame constraints on
(eective) dynamics of matter elds.
The system of modied Einstein equations (6) with sources (9) can be integrated in general
form by such an odiagonal asatz (see details in Refs. [19, 22, 23, 24, 25]):
gi = e
ψ(xk) as a solution of ψ•• + ψ′′ = 2 hΥ;
(10)
g3 = ω
2(xi, y3, t)h3(x
i, t) = −1
4
∂t(Ψ
2)
Υ2
(
h
[0]
3 (x
k)− 1
4
∫
dt
∂t(Ψ
2)
Υ
)−1
;
g4 = ω
2(xi, y3, t)h4(x
i, t) = h
[0]
4 (x
k)− 1
4
∫
dt
∂t(Ψ
2)
Υ
;
N3k = nk(x
i, t) = 1nk(x
i) + 2nk(x
i)
∫
dt
(∂tΨ)
2
Υ2
∣∣∣h[0]
3 (x
i)− 1
4
∫
dt ∂t(Ψ2)/Υ
∣∣∣5/2 ;
N4
i = wi(x
k, t) = ∂i Ψ/ ∂tΨ;
ω = ω[Ψ,Υ] is any solution of the 1st order system ekω = ∂kω + nk∂3ω + wk∂tω = 0.
8
In these formulas, Ψ(xi, t) and ω(xi, y3, t) are generating functions; hΥ(x
i) and Υ(xi, t) are
respective generating h- and vsources; 1nk(x
i), 2nk(x
i) and h
[0]
a (xk) are integration functions.
Such values can be dened in explicit form for certain symmetry / boundary / asymptotic
conditions which have to be considered in order to describe certain observational cosmological
data (see next sections). The coecients (10) generate exact and/or parametric solutions for
any nontrivial ω2 = |h3|−1. As a particular case, we can chose ω2 = 1 which allows to construct
generic odiagonal solutions with Killing symmetry on ∂3.
The quadratic elements for such general locally anisotropic and inhomogeneous cosmological
solutions with nonholonomically induced torsion are parameterized in this form:
ds2 =
gαβ(x
k, y3, t)duαduβ = e ψ[(dx1)2 + (dx2)2] +
(11)
ω2 {h3[dy3 + ( 1nk + 2nk
∫
dt
(∂tΨ)
2
Υ2|h3|5/2
)dxk]2 − 1
4h3
[
∂tΨ
Υ
]2
[dt+
∂iΨ
∂tΨ
dxi]2}.
In principle, we can consider that h3 and Υ are certain generating functions when Ψ[h3, B,Υ]
is computed for ω2 = 1 from ∂t(Ψ
2) = B(xi, t)/Υ as a solution of
Υ
(
h
[0]
3 (x
k)−
∫
dtB
)
h3(x
i, t) = −B.
This equation is equivalent to the second equation (10) up to re-denition of the integration
function h
[0]
3 (x
k). Various classes of exact solutions with nontrivial nonholonomically induced
torsion can be constructed, for instance, choosing data (Ψ,Υ) for solitonic like functions and/or
for various singular, or discrete like structures. Such generic odiagonal metrics can en-
code nontrivial vacuum and non-vacuum congurations, fractional and diusion processes, and
describe structure formation for evolving universes, eects with polarization of gravitational
and matter eld interaction constants, modied gravity scenarios etc., see examples in Refs.
[19, 22, 23, 24, 43, 44].
The class of metrics (11) denes exact solutions for the canonical dconnectionD̂ inR̂2
gravity with nonholonomically induced torsion and eective scalar eld encoded into a gravita-
tionally polarized vacuum. We can impose additional constraints on generating functions and
sources in order to extract LeviCivita congurations. This is possible for a special class of
generating functions and sources when for Ψ =Ψ̌(xi, t), ∂t(∂iΨ̌) = ∂i(∂tΨ̌) and Υ(x
i, t) = Υ[Ψ̌],
or Υ = const. For such LCsolutions, we nd some functionsǍ(xi, t) and n(xk) when
wi =w̌i = ∂iΨ̌/∂tΨ̌ = ∂iǍ and nk =ňk = ∂kn(x
i).
The corresponding quadratic line element can be written
ds2 = e ψ[(dx1)2 + (dx2)2] + ω2{h3[dy3 + (∂kn)dxk]2 −
1
4h3
[
∂tΨ̌
Υ
]2
[dt+ (∂iǍ)dx
i]2}.
(12)
Both classes of solutions (11) and/or (12) posses additional nonlinear symmetries which allow
to redene the generation function and generating source in a form determined by an eective
9
(in the v-subspace) gravitational constant. For certain special parameterizations of (Ψ̌,Υ)
and other coecients, we can reproduce Bianchi like universes, extract FLRW like metrics, or
various inhomogeneous and locally anisotropic congurations in GR. Using generic o-diagonal
gravitational and matter eld interactions, we can mimic MGTs eects, or model fractional/
diusion / crystal like structure formation. Finally, we note that, such metrics (12) can not be
localized in nite or innite space time region if there are nontrivial anholonomy coecients
Cγαβ.
3 Modied Gravity with Quasicrystal Like Structures
To introduce thermodynamical like characteristics for gravitational and scalar eld we con-
sider an additional 3+1 splitting when odiagonal metric ansatz of type (1), (11) (12) can be
re-written in the form
g = bi(x
k)dxi ⊗ dxi + b3(xk, y3, t)e3 ⊗ e3 −N̆2(xk, y3, t)e4 ⊗ e4,
(13)
e3 = dy3 + ni(x
k, t)dxi, e4 = dt+ wi(x
k, t)dxi.
In such a case, the 4d metric g is considered as an extension of a 3d metric bı̀j̀ = diag(bı̀) =
(bi, b3) on a family of 3-d hypersurfacesΞ̂t parameterized by t. We have
b3 = g3 = ω
2h3 andN̆
2(u) = −ω2h4 = −g4,
(14)
dening a lapse functionN̆(u). For such a double 2+2 and 3+1 bration,D̂ = (D̂i,D̂a) =
(D̂ı̀,D̂t) (in coordinate free form, we write (
qD̂,
tD̂)). Similar splitting can be performed
for the LC-operator, ∇ = (∇i,∇a) = (∇ı̀,∇t) = ( q∇, t∇). For simplicity, we elaborate the
constructions for solutions with Killing symmetry on ∂3.
3.1 Generating functions with 3d quasicristal like structure
Gravitational QC like structures can be dened by generic odiagonal exact solutions if
we chose a generating function Ψ = Φ as a solution of an evolution equation with conserved
dynamics of type
∂Φ
∂t
= b∆̂
[
δF
δΦ
]
= − b∆̂(ΘΦ +QΦ2 − Φ3),
(15)
where the canonically nonholonomically deformed Laplace operator b∆̂ := ( bD̂)2 = qı̀j̀D̂ı̀D̂j̀ as
a distortion of b∆ := ( b∇)2 can be dened on anyΞ̂t. Such distortions of dierential operators
can be always computed using formulas (4). The functional F in the evolution equation (15)
denes an eective free energy (it can be associate to a model of dark energy, DE)
F [Φ] =
∫ [
−1
2
ΦΘΦ− Q
3
Φ3 +
1
4
Φ4
]√
bdx1dx2δy3,
(16)
where b = det |bı̀j̀|, δy3 = e3 and the operator Θ and parameter Q will be dened below.
Such a conguration stabilized nonlinearly by the cubic term when the second order resonant
10
interactions can be varied by setting the value of Q. The average value Φ of the generating
function Φ is conserved for any xed t. This means that Φ can be considered as an eective
parameter of the system and that we can choose Φ|t=t0 = 0 since other values can be redened
and accommodated by altering Θ and Q. Further evolution can be computed for any solution
of type (11) and/or (12).
The eective free energy F [Φ] denes an analogous 3-d phase gravitational eld crystal
(APGFC) model that generates modulations with two length scales for odiagonal cosmologi-
cal congurations. This model consists a nonlinear PDE with conserved dynamics. It describes
(in general, relativistic) time evolution of Φ over diusive time scales. For instance, we can
elaborate such a APGFC model in a form including resonant interactions that may occur in
the case of icosahedral symmetry considered for standard quasicrystals in [29, 30]. In this work,
such gravitational structures will be dened by redening Φ into respective generating functions
Ψ orΨ̌. Let us explain respective geometric constructions with changing the generating data
(Ψ, Υ)↔ (Φ,Λ̃ = const) following the conditions
∂t(Ψ
2)
Υ
=
∂t(Φ
2)
Λ̃
, which is equivalent to
(17)
Φ2 =Λ̃
∫
dtΥ−1∂t(Ψ
2) and/or Ψ2 =Λ̃−1
∫
dtΥ∂t(Φ
2).
In result, we can write respective v- and hvcoecients in (10) in terms of Φ (redening the
integration functions),
h3(x
i, t) = −1
4
∂t(Φ
2)
ΥΛ̃
(
h
[0]
3 (x
k)− Φ
2
4Λ̃
)−1
=
1
Υ
∂t(Φ
2)
Φ2 − h[0]
3 (x
k)
;h4(x
i, t) = h
[0]
4 (x
k)− Φ
2
4Λ̃
;
nk =
1nk + 2nk
∫
dt
h4[Φ]
| h3[Φ]|3/2
and wi =
∂i Ψ
∂tΨ
=
∂i Ψ
2
∂t(Ψ2)
=
∂i
∫
dtΥ ∂t(Φ
2)
Υ∂t(Φ2)
. (18)
The nonlinear symmetry (17) allows us to change generate such eective sources (9) which
allow to generate QC structures in self-consistent form when
Υ(xk, t)→ Λ =
fΛ + ϕΛ,
(19)
with associated eective cosmological constants in MGT,
fΛ, and for the eective QC struc-
ture, ϕΛ. We can identifyΛ̃ with Λ, or any other value
fΛ, or ϕΛ depending on the class of
models with eective gauge interactions we consider in our work.
Let us explain how the formation and stability of gravitational congurations with icosa-
hedral quasicrystalline structures can be studied using a dynamical phase eld crystal model
with evolution equations (15). Such a 3-d QC structure is stabilized by nonlinear interactions
between density waves at two length scales [30]. Using a generating function Φ, we elaborate
a 3-d eective phase eld crystal model with two length scales as in so-called LifshitzPetrich
model [45]. The density distribution of matter mimics a "solid" or a "liquid " on the micro-
scopic length. The role of operator Θ to allow two wave marginal numbers and to introduce
possible spatio temporal chaos is discussed in [46, 30]. The eect is similar at metagalactic
11
scales when Φ has a two parametric dependence with k = 1 (the system is weakly stable) and
k = 1/τ (where, for instance, for τ = 2 cos π
5
= 1.6180 we obtain the golden ratio, when the
system is weakly unstable).
Choosing a QC type form for Φ and determining the coecients of dmetric in the form
(18), we generate a QC like structure for generic odiagonal gravitational eld interactions.
Such a structure is formed by some type ordered arrangements of galaxies (as "atoms") with
very rough rotation and translation symmetries. A more realistic picture of the observational
data for the Universe is for a non crystal structure with lack of the translational symmetry
but yet with certain discrete observations. There is certain analogy of such congurations
for quasiperiodic two and three dimensional space like congurations, for instance, in metallic
alloys, or nanoparticles, [as a review, see [46, 30, 45] and references therein] and at meta-
galactic scales when the nontrivial vacuum gravitational cosmological structure is generated as
we consider in this section.
3.2 Eective scalar elds with quasicrystal like structure
Following our system of notations, we shall put a left label "q" to the symbols for geometric/
physical object in order to emphasize that they encode an aperiodic QC geometric structure and
write, for instance,
(
qg,
qD̂,
qϕ
)
. We shall omit left labels for continuous congurations
and/or if that will simplify notations and do not result in ambiguities.
The quadratic gravity theory with action (5) is invariant (both for ∇ andD̂) under global
dilatation symmetry with a constant σ,
gµν → e−2σgµν , ϕ→ e2σϕ̃.
(20)
We can pas from the Jordan to the Einstein frame with a redenition ϕ =
√
3/2 ln |2ϕ̃| and
obtain
ΦS =
∫
d4u
√
|g|
(
1
2
R̂− 1
2
eµϕ e
µϕ− 2Λ
)
,
(21)
where the scalar potential ϕV (ϕ) in (8) is transformed into an eective cosmological constant
term Λ using (Ψ, Υ)↔ (Φ,Λ̃) (17). Such an integration constant can be positive / negative /
zero, respectively for de Sitter / anti de Sitter / at space.
The corresponding eld equations derived from ES are
R̂µν − eµϕ eνϕ− 2Λgµν = 0,
(22)
D̂2ϕ = 0.
(23)
We obtain a theory with eective scalar eld adapted to a nontrivial vacuum QC structure
encoded into gµν , eµ andD̂ as generic odiagonal cosmological solutions. At the end of
this section, we consider three examples of such QC gravitational-scalar eld congurations
as aperiodic and mixed continuous and discrete solutions of the gravitational and matter eld
equations (22) and (23).
12
3.2.1 Scalar eld Nadapted to gravitational quasicrystals
In order to generate integrable odiagonal solutions, we consider certain special conditions
for the eective scalar eld ϕ when eαϕ =
0ϕα = const in N-adapted frames. For such
congurations,D̂2ϕ = 0. We restrict our models to congurations of φ, which can be encoded
into Nconnection coecients
eiϕ = ∂iϕ−ni∂3ϕ−wi∂tϕ = 0ϕi; ∂3ϕ = 0ϕ3; ∂tϕ = 0ϕ4;
for 0ϕ1 =
0ϕ2 and
0ϕ3 =
0ϕ4.
(24)
This way we encode the contribution of scalar eld congurations into additional source
ϕΥ̃ = ϕΛ̃0 = const and
ϕΥ = ϕΛ0 = const
even the gravitational vacuum structure is a QC modeled by Φ as a solution of (15).
3.2.2 Scalar and rescaled QC generating functions
The scalar eld equations (15) can be solved if ϕ = ZΦ, for Z = const
6= 0. The conditions
(24) with 0ϕ1 =
0ϕ2 =
0ϕ3 = 0 and nontrivialΓ̂
4
44 = −∂th4/h4 transform into
∂tϕ = − b∆̂(Θϕ+Qϕ2 − ϕ3),
(25)
∂iϕ− wi∂tϕ = ∂iϕ−
∂iΦ
∂tΦ
∂tϕ ≡ 0,
D̂2ϕ = h−1
4 (1 +Γ̂
4
44)∂tϕ = 0.
(26)
For h4(x
k, t) given by (18), we obtain nontrivial solutions of (26) if 1+Γ̂444 = 0. This constraints
additionally Φ, i.e. ϕ = ZΦ, to the condition 2∂tϕ =
4h
[0]
4 (x
k)
Λ̃Z2ϕ
−ϕ. Together with (25) we obtain
that Nadapted scalar elds mimic a QC structure if
Λ̃Z2ϕ
[
ϕ− b∆̂(Θϕ+Qϕ2 − ϕ3)
]
= 2h
[0]
4 (x
k).
Using dierent scales, we can consider the energy of such QC scalar structures as hidden energies
for dark matter, DM, modeled by ϕ, determined by an eective functional
DMF [ϕ] =
∫ [
−1
2
ϕΘ̂ϕ−Q̂
3
ϕ3 +
1
4
ϕ4
]
√
bdx1dx2δy3,
(27)
where operatorsΘ̂ andQ̂ have to be chosen in some forms compatible to observational data
for the standard matter interacting with the DM. Even the QC structures for the gravitational
elds (with QC congurations for the dark energy, DE) and for the DM can be dierent, we
parameterize F and DMF in similar forms because such values are described eectively as exact
solutions of Starobisky like model with quadratic Ricci scalar term. Here we note that such a
similar ϕmodel was studied with a similar Lyapunov functional (eective free energy) DMF [ϕ]
resulting in the SwiftHohenberg equation (25), see details in Refs. [47, 45].
13
4 Aperiodic QC Starobinsky Like Ination
The Starobinsky model described an inationary de Sitter cosmological solution by pos-
tulating a quadratic on Ricci scalar action [2]. In nonholonomic variables, such MGTs were
developed in [22, 23, 24, 25].
4.1
Ination parameters determined by QC like structures
Although the Starobinsky cosmological model might appear not to involve any quasicrystal
structure as we described in previous section, it is in fact conformally equivalent to a non-
holonomic deformation of the Einstein gravity coupled to an eective QC structure that may
drive ination and acceleration scenarios. This follows from the fact that we can linearize the
R̂2term in (5) as we considered for the action (21). Let us introduce an auxiliary Lagrange
eld λ(u) for a constant ς = 8π/3M2 for a constantM of mass dimension one, with κ2 = 8πG
for the Newton's gravitational constant G = 1/M2P and Plank's mass, and perform respective
conformal transforms with dilaton symmetry (20). We obtain that the action for our MGT can
be written in three equivalent forms,
S =
1
2κ2
∫
d4u
√
|g|
{
R̂[g] + ςR̂2[g]
}
, with
{
gµν →g̃µν = [1 + 2ςλ(u)]gµν
λ(u)→ ϕ(u) :=
√
3/2 ln[1 + 2ςλ(u)]
,

1
2κ2
∫
d4u
√
|g|
{
[1 + 2ςλ(u)]R̂[g]− ςλ2(u)
}

1
2κ2
∫
d4u
√
|g̃|
{
R̂[g̃]− 1
2
g̃µνe
νϕ eµϕ− ϕV (ϕ)
}
,
(28)
with eective potential ϕV (ϕ) (8) with for a gravitationally modied QC structure ϕeld
which for (Ψ, Υ)↔ (Φ,Λ̃) (17) denes Nadapted congurations of type (24) or (25). Such non-
linear transforms are possible only for generic o-diagonal cosmological solutions constructed
using the AFDM. We shall write qV ( qϕ) for certain eective scalar like structures determined
by a nontrivial QC conguration with gµν →g̃µν and ϕ = qϕ described above.
In order to understand how actions of type (28) with eective free energy F (16), for DE,
and DMF (27), for DM, encode conditions for ination like in the Starobinsky quadratic gravity,
let us consider small odiagonal deformations of FLRW metrics to solutions of type (11) and
(12). We introduce a new time like coordinatet̂, when t = t(xi,t̂) and
√
|h4|∂t/∂t̂, and a scale
factorâ(xi,t̂) when the dmetric (1) can be represented in the form
ds2 =
â2(xi,t̂)[ηi(x
k,t̂)(dxi)2 +ĥ3(x
k,t̂)(e3)2 − (ê4)2],
(29)
where ηi =
â−2eψ,â2ĥ3 = h3, e
3 = dy3 + ∂kn dx
k,ê4 = dt̂+
√
|h4|(∂it+ wi).
Using a small parameter ε, with 0 ≤ ε < 1, we model odiagonal deformations if
ηi ' 1 + εχi(xk,t̂), ∂kn ' εn̂i(xk),
√
|h4|(∂it+ wi) ' εŵi(xk,t̂).
(30)
This correspond to a subclass of generating functions, which for ε → 0 result in Ψ(t), or
Ψ̌(t), and, correspondingly Φ(t), and generating source Υ(t) in a form compatible toâ(xi,t̂)→
14
â(t),ĥ3(x
i,t̂) →ĥ3(t̂) etc. Conditions of type (30) and homogeneous limits for generating
functions and sources have to be imposed after a locally anisotropic solution (for instance, of
type (12)), was constructed in explicit form. If we impose homogeneous conditions from the
very beginning, we transform the (modied) Einstein equations with scalar led in a nonlinear
system of ODEs which do not describe gravitational and scalar eld analogous quasicrystal
structures. Applying the AFDM with generating and integration functions we solve directly
nonlinear systems of PDEs and new classes of cosmological solutions are generated even in
diagonal limits because of generic nonlinear and nonholonomic character of odiagonal systems
in MGFT. For ε → 0 andâ(xi,t̂) →â(t), we obtain scaling factors which are very dierent
from those in the FLRW cosmology with GR solutions. Nevertheless, we can mimic such
cosmological models using redened parameters and possible small odiagonal deformations
of cosmological evolution for MGTs as we explain in details in [22, 23, 24, 25]. In this work,
we consider eective sources encoding contributions from the QC gravitational and scalar eld
structures, with
â2ĥ3 = ∂t(Φ
2)/Υ[Φ2 − h[0]
3 (x
k)],
where ∂t(Φ
2) =Λ̃ ∂t(Ψ
2)/Υ, as follows respectively from formulas (18) and (17).
Nonhomogeneous QC structures with mixed discrete parameters and continuous degrees
of freedom appear in a broader theoretical context related to quantum-gravity corrections and
from the point of view of an exact renormalisation-group analysis. We omit such considerations
in this work by note that ination in our MGTs models can be generated for 1  ς and
MMP , which corresponds to an eective quasicrystal potential with magnitude qV M4P ,
see details and similar calculations in [48].
In our approach, such values are for nontrivial
QC congurations with diagonal limits. At certain nontrivial values qϕ, when κ−1 qϕ are
large compared to the Planck scale, a potential qV = ϕV ( qϕ) (8) is eectively suciently
at to produce phenomenologically acceptable ination. In this model, the QC conguration
determined by qϕ play the role of scalar eld. This conguration determines a region with
positive-denite Starobinsky potential where the term exp[−
√
2/3 qϕ] is dominant.
In general, a nontrivial QC gravitational and eective scalar conguration may result via
generic odiagonal parametric interactions described by solutions type (11) and (12) in ef-
fective potentials qV with constants dierent from (8), with Q
6= ς2, $
6= 2 and P
6=
√
2/3,
when
qV = Q(1−$e−P qϕ + ...),
where dots represent possible higher-order terms like O(e−2P qϕ). This means that ination
can be generated by various types of eective quasicrystal structures which emphasizes the
generality of the model. Possible cosmological implications of QCs can be computed following
standard expressions in the slow-roll approximation for inationary observables (we put left
labels q in order to emphasize their eective QC origin). We have
q =
M2P
16π
(
∂ ϕV/∂ϕ
ϕV
| qϕ
)2
, qη =
M2P
8π
∂2 ϕV/∂2ϕ
ϕV
| qϕ,
qns = 1− 6 q+ 2 qη, qr = 16 q.
15
The a e-folding number for the inationary phase
qN? = −
8π
M2P
qϕ(e)
∫
qϕ(i)
dϕ
ϕV
∂ ϕV/∂ϕ
with qϕ(i) and
qϕ(e) being certain values of QC modications at the beginning and, respectively,
end of ination. At leading order, considering the small quantity e−P
qϕ, one computes qN? =
eP
qϕ/P 2$ yielding
qns = 1− 2P 2$e−P
qϕ ' 1− 2/ qN? and qr = 8P 2$2e−2P
qϕ ' 8/P 2 qN2? .
In result, we get a proof that we can elaborate Starobinsky like scenarios using generic o-
diagonal gravitational congurations (in GR and/or MGTs) determined by QC generating
functions. For qN? = 54± 6 for P =
√
2/3, we obtain characteristic predictions qns ' 0.964
and qr ' 0.0041 in a form highly consistent with the Plank data [1].
Finally, we note that for dierent QC congurations we may deviate from such characteristic
MGTs predictions but still remain in GR via o-diagonal interactions resulting in QC structures.
Such scenarios could not be involved in cosmology [6] even the authors [26, 27, 28] made
substantial contributions both to the inationary cosmology and physics of quasicrystals. The
main problem was that nonlinearities and parametric o-diagonal interactions were eliminated
from research from the very beginning in [3, 4, 5, 6] considering only of FLRW ansatz.
4.2 Reconstructing cosmological quasicrystal structures
We consider a model with Lagrange density (5) for qf(R̂) =R̂2 +M( qT), where qT is the
trace of the energymomentum tensor for an eective QC-structure determined by (gαβ,Dµ, ϕ).
Let us denote qM :=∂M/∂ qT andĤ := ∂tâ/â for a limitâ(x
i,t̂)→â(t) in (29). In general,
cosmological solutions are characterized by nonlinear symmetries (17) of generating functions
and sources when the valueâ(t) is dierent fromå(t) for a standard FLRW cosmology.
To taste the cosmological scenarios one considers the redshift 1 + z =â−1(t) for a function
qT = qT (z) and a new shift derivative when ∂ts = −(1 + z)H∂z,for instance, for a function
s(t). Following the method with nonholonomic variables elaborated in [24], we obtain for QC
structures a set of three equations
3Ĥ2 +
1
2
[ qf(z) + M(z)]− κ2ρ(z) = 0,
−3Ĥ2 + (1 + z)Ĥ(∂zĤ)−
1
2
{ qf(z) + M(z) + 3(1 + z)Ĥ2 = 0,
(31)
ρ(z) ∂z f = 0.
Using transforms of type (17) for the generating function, we x ∂z
qM(z) = 0 and ∂z f = 0
which allows nonzero densities in certain adapted frames of references. The functional M( qT)
encodes QC gravitational congurations for the evolution of the energy-density of type ρ =
16
ρ0a
−3(1+ϑ) = ρ0(1 + z)a
3(1+ϑ) for the dust matter approximation with a constant ϑ and ρ ∼
(1 + z)3.
Using (31), it is possible to elaborate reconstruction procedures for nontrivial QC congura-
tions generalizing MGTs in nonholonomic variables. We can introduce the e-folding variable
χ := ln a/a0 = − ln(1 + z) instead of the cosmological time t and compute ∂ts =Ĥ∂χs for any
function s. In N-adapted frames, we derive the nonholonomic eld equation corresponding to
the rst FLRW equation is
qf(R̂) = (Ĥ2 +Ĥ ∂χĤ)∂χ[
qf(R̂)]− 36Ĥ2
[
4Ĥ + (∂χĤ)
2 +Ĥ∂2χχĤ
]
∂2χχ
qf(R̂)]+κ2ρ.
Introducing an eective quadratic Hubble rate,κ̃(χ) :=Ĥ2(χ), where χ = χ(R̂) for certain
parameterizations, this equation transforms into
qf = −18κ̃(R̂)[∂2χχκ̃(χ) + 4∂χκ̃(χ)]
∂2 qf
∂R̂2
+ 6
[
κ̃(χ) +
1
2
∂χκ̃(χ)
]
∂ qf
∂R̂
+ 2ρ0a
−3(1+ϑ)
0
a−3(1+ϑ)χ(R̂).
(32)
O-diagonal cosmological metrics encoding QC structures are of type (29) with t → χ, and a
functional qf(R̂) used for computing the generating source Υ for prescribed generating function
Φ. Such nonlinear systems can be described eectively by the eld equations for an (nonholo-
nomic) Einstein spaceR̂αβ =Λ̃δ
α
β. The functional ∂
qf/∂R̂ and higher functional derivatives
vanish for any functional dependence f(Λ̃) because ∂χΛ̃ = 0. The recovering procedure can be
simplied substantially by using re-denitions of generating functions.
Let us consider an example with explicit reconstruction of MGT and nonholonomically
deformed Einstein spaces with QC structure when the ΛCDM era can be reproduced. We
chose anyâ(χ) andĤ(χ) for an o-diagonal (29). We obtain an analog of the FLRW equation
for ΛCDM cosmology,
3κ−2Ĥ2 = 3κ−2H20 + ρ0â
−3 = 3κ−2H20 + ρ0a
−3
0 e
−3χ,
where H0 and ρ0 are constant values. The eective quadratic Hubble rate and the modied
scalar curvature,R̂, are computed respectively,
κ̃(ζ) := H20 + κ
2ρ0a
−3
0 e
−3χ andR̂ = 3∂χκ̃(χ) + 12κ̃(χ) = 12H
2
0 + κ
2ρ0a
−3
0 e
−3χ.
The equation (32) transforms into
X(1−X)∂
2 qf
∂X2
+ [z3 − (z1 + z2 + 1)X]
∂ qf
∂X
− z1z2 qf = 0,
for constants subjected to the conditions z1 + z2 = z1z2 = −1/6 and z3 = −1/2, when 3χ =
− ln[κ−2ρ−1
0 a
3
0(R̂−12H20 )] andX := −3+R̂/3H20 . The solutions of such nonholonomic QC equa-
tions with constant coecients and for dierent types of scalar curvatures can be constructed
similarly to [24]. In terms of Gauss hypergeometric functions, qf = qF (X) := qF (z1, z2, z3;X),
we obtain
F (X) = KF (z1, z2, z3;X) +BX
1−z3F (z1 − z3 + 1, z2 − z3 + 1, 2− z3;X)
17
for some constants K and B. Such reconstructing formulas prove in explicit form that MGT
and GR theories with QC structure encode ΛCDM scenarios without the need to postulate
the existence of an eective cosmological constant. Such a constant can be stated by nonlin-
ear transforms and redenitions of the generating functions and (eective) energy momentum
source for matter elds.
5 Quasicrystal Models for Dark Energy and Dark Matter
The modern cosmological paradigm is constructed following observational evidences that
our Universe experiences an accelerating expansion [1]. Respectively, the dark energy, DE, and
dark matter, DM, are considered to be responsible for acceleration and the dynamics of spiral
galaxies. In order to solve this puzzle of gravity and particle physics and cosmology a number
of approaches and MGTs were elaborated during last 20 years, see reviews and original results
in [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In a series of our recent works [22, 23, 24, 25],
we proved that DE and DM eects can be modelled by generic odiagonal gravitational and
matter eld interactions both in GR and MGTs. For models with QC structure, we do not need
to "reconsider" the cosmological constant for gravitational eld equations. We suppose that
an eectiveΛ̃ can be induced nonlinear symmetries of the generating functions and eective
source which results in a QC Starobinsky like scenarios. In this section, we prove that QC
structures can be also responsible for Universe acceleration and DE and DM eects.
5.1 Encoding o-diagonal QC structures into canonical dtorsions
It possible to reformulate the GR with the LCconnection in terms of an equivalent telepar-
allel theory with the Weizenboock connection (see, for instance, [49, 50, 51]) and study f(T )-
theories of gravity, with T from torsion, which can be incorporated into a more general approach
for various modications of the gravitational Lagrangian R→ f(R, T, F, L...). Such models can
be integrated in very general forms for geometric variables of type (g,N,D̂). Odiagonal con-
gurations on GR and MGT with nonholonomic and aperiodic structures of QC or other type
(noncommutative, fractional, diusion etc.) ones can be encoded respectively into the torsion,
T̂ α, and curvature,R̂αβ, tensors. By denition, such values are dened and denoted respectively
qT α :=T̂ α[ qΨ] and
qRα :=R̂α[ qΨ] in order to emphasize the QC structure of generating
functions and sources. Such values can be computed in N-adapted form using the canonical
dconnection 1form qΓαβ =Γ̂
α
βγe
γ, where qD = { qΓ̂αβγ},
qT α
: = qDeα = deα + qΓαβ ∧ eβ = qTαβγeβ ∧ eγ and
qRαβ :=
qD qΓαβ = d
qΓαβ − qΓ
γ
β ∧
qΓαγ =
qRαβγδe
γ ∧ eδ.
In such formulas, we shall omit the left label "q" and write, for instance, D,Γ̂αβγ, T
α
βγ etc.
if certain continuous limits are considered for the generating functions/sources and respective
geometric objects. Hereafter we shall work with standard Nadapted canonical values of met-
rics, frames and connections which are generated by apperiodic QC ( qΨ,
qΥ) ↔ ( qΦ, qΛ̃).
18
Such congurations are with a nontrivial eective qT α induced nonholonomically. Even at the
end we can extract LCcongurations by imposing additional nonholonomic constraints and
integral sub-varieties with ( qΨ̌,
qΥ) ↔ ( qΦ̌, qΛ̃) all diagonal and o-diagonal cosmological
solutions are determined by geometric and physical data encoded in { qTαβγ}. For qD→ q∇,
the gravitational and matter eld interactions are encoded into eαα′ [g,N] like in (1). In general,
we can work in equivalent form with dierent type theories when
R⇐⇒R̂⇐⇒ f( qR)⇐⇒ f( qT )
are all completely dened by the same metric structure and data (g,N). Here we note that qT
is constructed for the canonical dconnection qD in a metricane spacetime with aperiodic
order and this should be not confused with theories of type f(R, T ), where T is for the trace
of the energy-momentum tensor.
We construct an equivalent f( qT ) theory for DE and DM congurations determined by a
QC structure in this form: Let consider respectively the contorsion and quasi-contorsion tensors
qKµν λ =
1
2
( qTµν λ −
qTνµ λ +
qT
νµ
λ
) , qS
νµ
λ =
1
2
( qKµν λ + δ
µ
λ
qTαν α − δνλ qTαµ α)
for any qT α = { qTµν λ}. Then the canonical torsion scalar is dened qT := qTαβγ qT
βγ
α
.
The nonholonomic redenition of actions and Lagrangians (5) and (28) in terms of qT
is
S =
∫
d4u
[ q
GL
2κ2
+ mL̂
]
(33)
where the Lagrange density for QC gravitational interactions is qGL = qT + f( qT ).
The equations of motion in a at FLRW universe derived for solutions o type (29) (for
simplicity, we omit small parameter o-diagonal deformations (30)) are written in the form
6H2 + 12H2f ◦( qT ) + f( qT ) = 2κ2ρ[i],
2(2∂tH + 3H
2) + f( qT ) + 4(∂tH + 3H2)f ◦( qT )− 48H2(∂tH)f ◦◦( qT ) = 2κ2p[i],
where f ◦ := df/d qT and ρ[i] and p[i] denote respectively the energy and pressure of a perfect
uid matter imbedded into a QC like gravitational and scalar matter type structure. Similar
equations have been studied in Refs. [52, 53, 54, 55]. For κ2 = 8πG, these equations can be
written respectively as constraints equations
3H2 = ρ[i] +
qρ, 2∂tH = −(ρ[i] + p[i] + qρ+ qp),
with additional eective QC type matter
qρ = −6H2f ◦ − f/2,
qp = 2∂tH(1 + f
◦ − 12H2f ◦◦) + 6H2f ◦ + f/2.
(34)
Now we can elaborate our approach with DE and DM determined by aperiodic QC congura-
tions of gravitational - scalar eld systems.3
3Above equations can be written in a standard form for fmodied cosmology with
efΩ = Ω[i] +
qΩ :=
ρ[i]
3H2
+
qρ
3H2
= 1,
with certain eective efρ = ρ[i] +
qρ, efp = p[i] +
qp and efω = efp/ efρ encoding an aperiodic QC order.
19
5.2
Interaction between DE and DM in aperiodic QC vacuum
In this section, we ignore all other forms of energy and matter and study how interact
directly aperiodically QC structured DE and DM. Respective densities of QC dark energy and
dark matter are parameterized
qρ = qDEρ+
q
DMρ and
qp = qDEp+
q
DMp,
when (34) is written in the form
2∂t(
q
DEρ+
q
DMρ) = (∂t
qT ) (f ◦ + 2 qT f ◦◦) .
For perfect two uid models elaborated in Nadapted form [22, 23, 24, 25], the interaction DE
and DM equations are written
κ2( qρ+ qp) = −2∂tH, subjected to
(35)
∂t(
q
DEρ) + 3H(
q
DEp+
q
DEρ) = −Q and ∂t(
q
DMρ) + 3H(
q
DMp+
q
DMρ) = Q.
Above equations result in such a functional equation
2 qT f ◦◦ + f ◦ + 1 = 0,
which can be integrated in trivial and nontrivial forms with certain integration constants C,C0
and C1 = 0 (this condition follows from (35)),
f( qT ) =
{
− qT + C
− qT − 2C0
√
− qT + C1
.
So, the QC structure eectively contributes to DE and DM interaction via a nontrivial non-
holonomically induced torsion structure. Such nontrivial aperiodic congurations exist via
nontrivial C and C0 even we impose the conditions
qT
in order to extract certain diago-
nal LCcongurations. We note that in both cases of solutions for f( qT ) we preserve the
conditions efΩ = 1 and efω = −1.
5.3 Quasicrystal DE structures and matter sources
We analyse how aperiodic QC structure modify DE and DM and ordinary matter OM
interactions and cosmological scenarios, see similar computations in [54, 55] but for a dierent
type of torsion (for the Weitzenböck connection).
5.3.1
Interaction between DE and ordinary matter in gravitational QC media
Now, we model conguration when aperiodic DE interacts with OM (we use label "o" from
ordinary, ( oρ+ op) for qρ = qDEρ. We obtain such equations of interactions between DE and
DM equations are written
κ2( qDEρ+
q
DEp+
oρ+ op) = −2∂tH, subjected to
(36)
∂t(
q
DEρ) + 3H(
q
DEp+
q
DEρ) = −Q and ∂t(
oρ) + 3H( oρ+ op) = Q.
20
The equation (34) transform into
∂t(
q
DEρ) = (∂t
qT )( qT f ◦◦ + 1
2
f ◦),
which together with above formulas result in
∂t(
q
DEρ+
oρ+
1
2
qT ) = 0.
For f( qT ), these formulas result in a second order functional equation
(2 qT f ◦◦ + f ◦ + 1) = −2( oρ)◦.
We can construct solutions of this equation by a splitting into two eective ODEs with a nonzero
constant Z0, when
f ◦◦ + (2 qT )−1f ◦ = −Z0 and 2 oρ+ 1 = 2Z0 qT .
Such classes of solutions are determined by integration constants C2 and C3 = 2C4 (this condi-
tion is necessary in order to solve (35)): for oρ = −C4 − qT + Z0( qT )2/2, the aperiodic QC
contribution is
f( qT ) = C3 − 2C2
√
| − qT | − Z0( qT )2/3.
We can chose H0 = 74.2 ± 3.6Km
s
Mp
c
and t0 as the present respective Hubble parameter and
cosmic time and state the current density of the dust ρ(t0) =
mρ0 = 3 × 1.5 × 10−67eV 2. For
an arbitrary constant C2, we get the gravitational action (35) and
oρ both modied by QC
contributions via
C4 =
mρ0 − 3H20 (1− 6Z0H20 ).
The eective parameters of state
efω = −( qT )−1{Z0( qT )2 + 4[1− 2∂tH Z0( qT )] + C3} and efΩ = 1
describe an universe dominated by QC dark energy interacting with ordinary matter.
5.3.2 Van der Waals uid interacting with aperiodic DM
The state equation for such a uid (with physical values labeled by w) is
wp(3− wρ) + 8 wp wρ− 3( wρ)2 = 0,
which results in the equations for interaction of the QC DE with such a van der Waals OM,
κ2( qDEρ+
q
DEp+
wρ+ wp) = −2∂tH, subjected to
∂t(
q
DEρ) + 3H(
q
DEp+
q
DEρ) = −Q and ∂t(
wρ) + 3H( wρ+ wp) = Q.
Such equations are similar to (36) but with the OM pressure and density subjected to another
state equation and modied DE interaction equations. The solutions the aperiodic QC con-
tribution can be constructed following the same procedure with two ODEs and expressed for
wρ = −C5 + Z0( qT )2/2, C7 = 2C5, as
f( qT ) = C7 + C6
√
| − qT |/2− qT − Z0( qT )2/3.
21
Taking ∂tH(t0) = 0 and
q
DEp(t0)+
q
DEρ(t0) = 0, which constraints (see above equations)
wp(t0) +
wρ(t0) = 0, and results in
C5 = 3Z0H
2
0 + |74− 96 wω|1/2 +
5
3
,
for typical values wω = 0.5 and E = 10−10.
5.3.3 Chaplygin gas and DE - QC congurations
Another important example of OM studied in modern cosmology (see, for instance, [56]) is
that of Chaplygin, ch, gas characterized by an equation of state chp = −Z1/ chρ, for a constant
Z1 > 0. The corresponding equations for interactions between DE and such an OM is given by
κ2( qDEρ+
q
DEp+
chρ+ chp) = −2∂tH, subjected to
∂t(
q
DEρ) + 3H(
q
DEp+
q
DEρ) = −Q and ∂t(
chρ) + 3H( chρ+ chp) = Q.
The solutions for this system can be written for chρ = C8 + Z0(
qT )2/2, C10 = 2C8, as
f( qT ) = C10 + C9
√
| − qT |/2− qT − Z0( qT )2/3.
Let us assume ∂tH(t0) = 0 and
q
DEp(t0)+
q
DEρ(t0) = 0, which results in
chp(t0) +
chρ(t0) = 0
and
C8 = 18Z0H
4
0 + |Z1 − 9Z0H40 (1 + 36Z0H40 )|1/2,
for typical values Z1 = 1 and E = 10
−10.
The solutions for dierent type of interactions of QC like DE and DM with OM subjected
to corresponding equations of state (for instance, of van der Waals or Chaplygin gase) prove
that aperiodic spacetime strucutres result, in general, in odiagonal cosmological scenarious
which in diagonal limits result in eects related directly to terms containing contributions
of nonholonomically induces torsion.
If the constructions are redened in coordinate type
variables, such terms transform into certain generic odiagonal coecients of metrics.
6 Discussion and Conclusions
This paper is devoted to the study of aperiodic quasicrystal, QC, like gravitational and scalar
eld structures in acceleration cosmology. It apply certain geometric methods for constructing
exact solutions in mathematical cosmology. The main conclusion is that exact solutions with
aperiodic order in modied gravity theories, MGTs, and general relativity, GR, and with generic
odiagonal metrics, conrm but also oer interesting alternatives to the original Starobinsky
model. Here we emphasize that our work concerns possible spacetime aperiodic order and QC
like discrete and continuous congurations at cosmological scales. This is dierent from the
vast majority of QCs (discovered in 1982 [57], which attracted the Nobel prize for chemistry in
2011) are made from metal alloys. There are also examples of QCs found in nanoparticles and
soft-matter systems with various examples of block copolymers etc. [58, 59, 60, 61, 62]. In a
22
complimentary way, it is of special interest to study congurations with aperiodic order present
also interest in astronomy and cosmology, when a number of observational data conrm various
type lament and deformed QC structures, see [63].
In this work, the emergence of aperiodic ordered structure in acceleration cosmology is
investigated following geometric methods of constructing exact and parametric solutions in
modied gravity theories, MGTs, and in general relativity, GR (such methods with applica-
tions in modern cosmology are presented in [19, 22, 23, 24]). For instance, QC congurations
may be determined by generating functions encoding, for instance, a "golden rotation" of
arccos(τ 2/2
√
2) ≈ 22.2388◦ (where the golden ration is given by τ = 1
2
(1 +
√
5)), see [36, 37].
The reason to use such aperiodic and discrete parameterized generating functions and eective
sources of matter is that various cosmological scales can be reproduced as certain nonholo-
nomic deformation and diusion processes from a chosen QC conguration. The priority of
our geometric methods is that we can work both with continuous and discrete type generating
functions which allows to study various non-trivial deformed networks with various bounds and
lengths re-arranging and deforming, for instance, icosahedral arrangements of tetrahedra etc.
The aperiodic QC gravity framework proves to be very useful, since many geometric and
cosmological evolution scenarios can be realized in the context of this approach.The question
is can such models be considered as viable ones in order to explain alternatively Starobinsky-
lyke scenarious and provide a physical ground for explicit models of dark energy, DE, and dark
matter, DM. Working with arbitrary generating functions and eective source it seems that with
such MGTs everything can be realized and certain lack of predictibility is characteristic. From
our point of view, the anholonomic frame deformation method, AFDM, is more than a simple
geometric methods for constructing exact solutions for certain classes of important nonlinear
systems of partial dierential equations, PDEs, in mathematical relativity. It reects new and
former un-known properties and nonlinear symmetries of (modied) Einstein equations when
generic odiagonal interactions and mixed continuous and discrete structures are considered for
vacuum and non-vacuum gravitational congurations. The AFDM is appealing in some sense
to be "economical and very ecient" because allows to treat in the same manner by including
fractional / random / noncommutative sources and respective interaction parameters. We can
speculate on existence of noncommutative and/or nonassociative QC generalized structures
in the framework of classical MGTs. Moreover, certain compatibility with the cosmological
observational data can be achieved and, in addition, there are elaborated realistic models of
geometric ows and grow of QC related to accelerating cosmology.
An interesting and novel research steam is related to the possibility to encode aperiodic
QC structures into certain nonholonomic and generic odiagonal metric congurations with
nonholonomically induced canonical torsion elds. Such alternatives to the teleparallel and
other MGTs equivalents of the GR allows to elaborate in a most "economic" way on QC models
for DE and DM and study "aperiodic dark" interactions with ordinary matter (like van der
Vaals and Chapliygin gas). A procedure which allows to reconstruct QCs is outlined following
our former nonholonomic generalizations [22, 23, 24]. However, QC structures are characterised
by Lypunov type functionals for free energy which for geometric models of gravity are related
to certain generalized Perelman's functionals studied in [39]. Following such an approach, a
new theory with aperiodic geometric originating DE and DM with generalized Ricci ows with
23
(non) holonomic/ commutative / fractional structures has to be elaborated and we defer such
issues to our future work [38].
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