Raymond Aschheim, Laurenţiu Bubuianu, Fang Fang, Klee Irwin, Vyacheslav Ruchin, Sergiu I. Vacaru (2016)

The goal of this work on mathematical cosmology and geometric methods in modified gravity theories, MGTs, is to investigate Starobinsky-like inflation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal (QC) like structures. Such spacetime aperiodic QCs are different from those discovered and studied in solid state physics but described by similar geometric methods. We prove that an inhomogeneous and locally anisotropic gravitational and matter field effective QC mixed continuous and discrete “aether” can be modeled by exact cosmological solutions in MGTs and Einstein gravity. The coefficients of corresponding generic off-diagonal metrics and generalized connections 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 conditions for integration functions and constants, we can model various nontrivial torsion QC structures or extract cosmological Levi–Civita configurations with diagonal metrics reproducing de Sitter (inflationary) like and other types homogeneous inflation and acceleration phases. Finally, we speculate how various dark energy and dark matter effects can be modeled by off-diagonal interactions and deformations of a nontrivial QC like gravitational vacuum structure and analogous scalar matter fields.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

### Tag Cloud

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].

References

[1] Planck 2015 results. XIII. Cosmological parameteres, v3, 17 Jun 2016, arXiv: 1502.01589

[2] A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys.

Lett. B 91 (1980) 99

[3] V. F. Mukhanov and B. V. Chibisov, Quantum uctuation and nonsingurlar universes,

JETP Lett. 33 (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549, in Russian]

[4] A. H. Guth, The inationary universe: a possible solution to the horizon and atness

problems, Phys. Rev. D 23 (1981) 347

[5] A. D. Linde, A new inationary universe scenario: a possible soluton of the horizon,

atness, homogeneity, isotropy and promordial monopole problems, Phys. Lett. B 108

(1982) 389

[6] A. Albrecht and P. J. Steinhardt, Cosmology for grand unied theories with ratiatively

induces symmetry breaking, Phys. Rev. Lett. 48 (1982) 1220

[7] A. G. Riess et al. (High-z Supernova Search Team), Observational evidence from super-

novae for an accelerating universe and a cosmological constant, Astronom. J. 116 (1998)

1009

[8] S. Nojiri and S. Odintsov, Unied cosmic history in modied gravity: from F(R) theory

to Lorentz non-invariant models, Phys. Rept. 505 (2011) 59-144

[9] S. Capozziello and V. Faraoni, Beyond Einstein Gravity (Springer, Berlin, 2010)

[10] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Modied gravity and cosmology,

Phys. Repts. 512 (2012) 1-189

[11] D. S. Gorbunov and V. Rubakov, Introduction to the theory of the early universe: Cosmo-

logical perturbations and inationary theory (Hackensack, USA: World Scientic, 2011)

[12] A. Linde, Inationary cosmology after Plank 2013, in: Post-Planck Cosmology: Lecture

Notes of the Les Houches Summer School: Vol. 100, July 2013: editors C. Deayet, P.

Peter, B. Wandelt, M. Zadarraga and L. F. Cugliandolo (Oxford Scholarship Online, 2015);

arXiv: 1402.0526

[13] K. Bamba and S. D. Odintsov, Inationary cosmology in modied gravity theories, Sym-

metry 7 (2015) 220-240

24

[14] M. Wali Hossain, R. Myrzakulov, M. Sami and E. N. Saridakis, Unication of ination

and drak energy á la quintessential ination, Int. J. Mod. Phys. D 24 (2015) 1530014

[15] A. H. Chamseddine, V. Mukhanov and A. Vikman, Cosmology with mimetic matter, JCAP

1406 (2014) 017

[16] S. Nojiri and S. D. Odintsov, Mimetic F(R) gravity: Ination, dark energy and bounce,

Mod. Phys. Lett. A 29 (2014) 14502011

[17] E. Guendelman, D. Singleton and N. Yongram, A two measure model of dark energy and

dark matter, JCAP 1211 (2012) 044

[18] 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

[19] T. Gheorghiu, O. Vacaru and S. Vacaru, O-diagonal deformations of Kerr black holes

in Einstein and modied massive gravity and higher dimensions, EPJC 74 (2014) 3152;

arXiv: 1312.4844

[20] S. Vacaru and F. C. Popa, Dirac spinor waves and solitons in anisotropic Taub-NUT spaces,

Class. Quant. Grav. 22 (2001) 4921; arXiv: hep-th/0105316

[21] S. Vacaru and D. Singleton, Ellipsoidal, cylindrical, bipolar and toroidal wormholes in 5D

gravity, J. Math. Phys. 43 (2002) 2486; arXiv: hep-th/0110272

[22] S. Vacaru, Ghost-free massive f(R) theories modelled as eective Einstein spaces & cosmic

acceleration, EPJC 74 (2014) 3132; arXiv: 1401.2882

[23] S. Vacaru, Equivalent o-diagonal cosmological models and ekpyrotic scenarios in f(R)-

modied massive and Einstein gravity, EPJC 75 (2015) 176; arXiv: 1504.04346

[24] E. Elizalde and S. Vacaru, Eective Einstein cosmological sapces for non-minimal modied

gravity, Gen. Relat. Grav. 47 (2015) 64 ; arXiv: 1310.6868

[25] S. Vacaru, O-diagonal ekpyrotic scenarios and equivalence of modied, massive and/or

Einstein gravity, Phys. Lett. B 752 (2016) 27-33; arXiv: 1304.1080

[26] D. Levine and P. J. Steinhardt, Quasicrystals. i. denition and structure, Phys. Rev. B 34

(1986) 596

[27] G. Y. Onoda, P. J. Steinhardt, D. P. DiVincenzo and J. E. S. Socolar, Growing perfect

quaiscrisals. Phys. Rev. Letters 60 (1988) 2653

[28] L. Bindi, P. J. Steinhardt, N. Yao and P. Lu, Natural quaiscrystals, Science 324 (2009)

1306

[29] H. Emmrich, H. Löven, R. Wittkowski, T. Gruhn, G. I. Tóth, G. Tegze and L. Gránásy,

Adv. Phys. 61 (2012) 665

25

[30] P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, Three-dimenisonal pahes

eld quasicrystals, arXiv: 1602.05361

[31] C. Hann, J. E. S. Socolar and P. J. Steinhardt, Local growth of icoshedral quasicristalline

telings, arXiv: 1604.02479

[32] R. Penrose, The role of aesthetics in pure and applied mathematical research. Bull. Inst.

Math. Appl., 10 (1974) 266

[33] R. Penrose, Tilings and quasicrystals: A nonlocal growth problem? In: Introduction to

the Mathematics of Quasicrystals, editor: Marko Jari¢, vol 2 of Aperiodicity and Order,

chapter 4 (Academic Press, 1989)

[34] C. V. Achim, M. Schemiedeberg and H. Löwen, Growth modes of quasicrystals, Phys. Rev.

Lett. 112 (2014) 255501

[35] K. Barkan, H. Diamant and R. Lifshitz, Stability of quasicrystals composed of soft isotropic

particles, arXiv: 1005.5257

[36] 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

[37] Fang Fang and K. Irwin, An icosahedral quasicrystal as a golden modication of the

icosagrid and its connection to the E8 lattice, arXiv: 1511.07786

[38] L. Bubuianu, Fang Fang, K. Irwin, V. Ruchin and S. Vacaru, Geometric Evolution of Cos-

mological Quasicrystal Structures and Dark Energy and Dark Matter Physics [in prepara-

tion]

[39] T. Gheorghiu, V. Ruchin, O. Vacaru and S. Vacaru, Geometric ows and Perelman's

thermodynamics for black ellipsoids in R2 and Einstein gravity theories; Annals of Physics,

NY, 369 (2016) 1

[40] S. Rajpoot and S. Vacaru, On supersymmetric geometric ows and R2 ination from scale

invariant supergravity, arXiv: 1606.06884

[41] M. Baake and U. Grimm, Aperiodic Order, Volume 1. A Mathematical Invitation. Part of

Encyclopedia of Mathematics and its Applications (Cambridge University Press, 2015)

[42] M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems

(Cambrdidge University Press, Cambridge, 2009)

[43] S. Vacaru, Nonholonomic relativistic diusion and exact solutions for stochastic Einstein

spaces, The European Physical Journal Plus, 127 (2012) 32; arXiv: 1010.0647

[44] S. Vacaru, Diusion and self-organized criticality in Ricci ow evolution of Einstein and

Finsler spaces, SYMMETRY: Culture and Science, 23 (2013) 105, arXiv: 1010.2021

26

[45] R. Lifshitz and D. M. Petrich, Theoretical model for Faraday waves with multiple-frequency

forcing, Phys. Rev. Lett. 79 (1997) 1261

[46] A. M. Rucklidge, M. Silber, and A. C. Skeldon, Three-wave interactions and spatio tem-

poral chaos, Phys. Rev. Lett. 108 (2012) 074504

[47] J. B. Swift and P. C. Hohenberg, Hydronynamic uctuations at the convective instability,

Phys. Rev. A 15 (1977) 319

[48] J. Ellis, N. E. Mavromatos and D. Nanopoulos, Starobinsky-like ination in dilaton-brane

cosmology, Phys. Lett. B 32 (2014) 380

[49] J. T. Lurnardi, B.M. Pimentel and R. G. Teixeira, Interacting spin 0 elds with torsion

via Dun-Kemmer-Petiau theory, Gen. Rel. Grav. 34 (2002) 491

[50] M. E. Rodrigues, M. J. S. Houndjo, D. Saez-Gomez and F. Rahaman, Anisotropic Universe

models in f(T) gravity, Phys. Rev. D 86 (2012) 104059

[51] J. Armors, J. de Haro and S. D. Odintsov, Bouncing loop quantum cosmology from F(T)

gravity, Phys. Rev. D 87 (2013) 104037

[52] K. Bamba, C. Geng, C. Lee and L. Luo, Equation of state for dark energy in f(T) gravity,

JCAP 1101 (2011) 021

[53] K. Karami and A. Abdolmaleki, f(T) modied teleparallel gravity models as an alternative

for holopgrahic and new agegraphic dark energey models, Res. Astron. Astrophys. 13

(2013) 757

[54] S. B. Nassur, M. J. S. Houndjo, I. G. Salako and J. Tossa, Interactions of some uids with

dark energy in f(T) theory, arXiv: 1601.04538

[55] Yi-Fu Cai, S. Capozziello, M. De Laurentis and E. N. Saridakis, f(T) teleparallel gravity

and cosmology, arXiv: 1511.0758

[56] V. Gorini, A. Kamenshchik and U. Moschella, Can the Caplygin gas be a plausible model

of dark energy? Phys. Rev. D 67 (2003) 063509

[57] D. Chechtman, I. Belch, D. Gratias and J. W. Cahn, Metallic phase with long-rage orien-

tational order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951

[58] D. V. Talapin, E. V. Shevchenko, M. I. Bodnarchuk, X. Ye, J. Chen, and C. B. Murray,

Quasicrystalline order in self-assembled binary nanonparticle superlattices, Nature 461

(2009) 964

[59] C. Xiao, N. Fujita, K. Miyasaka, Y. Sakamoto and O. Terasaki, Dodecagonal tiling in

mseoporous silica, Nature 487 (2012) 349

[60] T. Dotera, Quasicrystals in soft matter, Israel Journal of Chemistry 51 (2011) 1197

27

[61] X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Dulcey, and J. K. Hobbs, Nature 428 (2004)

157

[62] 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 diraction symmetry, Proceedings

of the national academy of science 108 (2011) 1810

[63] See, for instance, webpage: www.crystalinks.com/darkmatter.html

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