# Cosmological attractors and anisotropies in two measure theories, effective EYMH systems, and off-diagonal inflation models

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## Abstract

Applying the anholonomic frame deformation method, we construct various classes of cosmological solutions for effective Einstein–Yang–Mills–Higgs, and two measure theories. The types of models considered are Freedman–Lemaître–Robertson–Walker, Bianchi, Kasner and models with attractor configurations. The various regimes pertaining to plateau-type inflation, quadratic inflation, Starobinsky type and Higgs type inflation are presented.

## 1 Introduction

Over time, the Cosmological Constant Problem (CCP) has evolved from the “Old Cosmological Constant Problem” [1, 2], where the concern was on why the observed vacuum energy density of the universe is exactly zero, to the present form pertaining to the evidence establishing the accelerating expansion of the universe [3, For reviews of this subject see for example]. One is therefore faced with the “New Cosmological Constant Problem” [4, 5]. In other words, the problem has shifted from the question why the CCP is exactly zero, but to why the vacuum energy density is so small. Various attempts to address the issue range from the conventional to the esoteric. Conventional field theoretic models are based on a single scalar field (quintessence) while the esoteric models involve tachyons, phantoms and K-essence. The latter may also admit multi scalar field configurations. Such models have also been supplemented further to take into consideration the recent observational data from Planck [6, 7] and BICEP2 [8]. In all these models the inflationary paradigm [9, 10] is the underlying theme; see also an opposite point of view in [11]. However, present data is insufficient to determine precisely what the initial conditions were that drove inflation. In addressing the present situation there are essentially two main approaches entertained. In one approach it is assumed that there is a basic mechanism driving to zero the vacuum energy but some “residual” interactions survive that slightly shift the vacuum energy density towards the presently observed small non-zero value. In the alternative approach it is assumed that the true vacuum energy will exactly be zero when the final state of the theory is reached and the present state pertaining to the small non-zero vacuum energy density is the result of our universe having not reached that final state yet.

^{1}Quantities under consideration and with a left label (for instance, \(\ ^{\mathbf {g}}\mathbf {D}\) ) emphasize that the geometric object (\(\mathbf {D}\)) is uniquely determined by \(\mathbf {g}\). Unless otherwise stated, Einstein’s summation convention is assumed throughout with the caveat that upper and lower labels are omitted if this does not result in ambiguities. We emphasize that \( \mathbf {D}\) contains non-trivial anholonomically induced torsion \(\mathbf {T}\) relating to the underlying nonholonomic frame structure. Such a torsion field is completely defined by the metric and the nonholonomic (equivalently, anholonomic and/or non-integrable) distortion relations,

^{2}For such a splitting, all geometric constructions can be carried out equivalently with \( \nabla \) using the so-called canonical distinguished connection (d-connection), \(\widehat{\mathbf {D}}\). Here \(\widehat{\mathbf {D}}\) is distinct from \(\mathbf {D}\). This linear connection is N-adapted, i.e., preserves under parallelism the N-connection splitting, and it is uniquely determined (together with \(\nabla \)) by the constraints

^{3}Solutions thus determined describe various geometric and physical models in modified gravity theories with non-trivial nonholonomically induced torsion, \(\widehat{\mathbf {T}}\ne 0,\) and generalized connections. As special cases, we extract LC-configurations and construct new classes of cosmological solutions in Einstein’s gravity if we constrain the set of possible generating and integration functions to satisfy the following conditions:

In general, gravitational field equations (4) constitute a sophisticated system of nonlinear partial differential equations (PDEs) as opposed to the occurrence of ordinary differential equations (ODEs) in conventional general relativity. The AFDM, on the other hand, allows us to find new classes of solutions by decoupling the PDEs. We emphasize that, in the AFDM approach advocated here, constraints of type (5) and/or (6) are to be imposed after the inhomogeneous \(\mathbf {g}_{\alpha \beta }(x^{i},y^{3},t)\) are constructed in general form. If the aforementioned constraints are imposed from the very beginning in order to transform PDEs into ODEs, a large class of generic off-diagonal and diagonal solutions will be compromised. The specific goal of this work is to apply the AFDM method and explicitly construct solutions in effective TMTs addressing attractors, acceleration, dark energy and dark matter effects in the new cosmological models.

This work is organized as follows. In Sect. 2, we provide a brief introduction to the geometry of nonholonomic deformations in Einstein gravity and modifications that lead to effective TMTs. In such theories we shown how the gravitational and matter field equations can be decoupled and solved in very general off-diagonal forms for the canonical d-connection with constraints for LC-configurations. Section 3 is devoted to off-diagonal and diagonal cosmological solutions with small vacuum density. Also constructed and analysed are the off-diagonal inhomogeneous cosmological solutions with nonholonomically induced torsion. In Sect. 4, we study the equivalence of effective TMTs with sources for nonlinear potentials and EYMH self-dual fields resulting in attractor type behaviour. In Sect. 5, we analyze in explicit form how exact cosmological solutions with locally anisotropic attractor properties can be generated by deforming FLRW type diagonal metrics and off-diagonal Bianchi type cosmological models. Conclusions are presented in Sect. 6.

## 2 Nonholonomic deformations

*h*) and vertical (

*v*) decompositions (

*h*and

*v*splitting) (2) as \((\mathbf {V},\ \mathbf {N})\). To this we associate structures of N-adapted local bases, \(\mathbf {e}_{\nu }=(\mathbf {e} _{i},e_{a}),\) and cobases, \(\mathbf {e}^{\mu }=(e^{i},\mathbf {e}^{a}),\) which are the following N-elongated partial derivatives and differentials:

The geometric objects on \(\mathbf {V}\) are defined with respect to the N-adapted frames (7), (8). These are referred to as distinguished objects or d-objects in short. A vector \(Y(u)\in T\mathbf {V}\) is parameterized as a d-vector. Explicitly, \(\mathbf {Y}=\) \(\mathbf {Y}^{\alpha }\mathbf {e}_{\alpha }=\mathbf {Y} ^{i}\mathbf {e}_{i}+\mathbf {Y}^{a}e_{a},\) or \(\mathbf {Y}=(hY,vY),\) with \(hY=\{ \mathbf {Y}^{i}\}\) and \(vY=\{\mathbf {Y}^{a}\}.\) Likewise, in this frame work, the coefficients of d-tensors, N-adapted differential forms, d-connections, and d-spinors are easily accommodated.

*h*and

*v*N-adapted coefficients:

*M*is a constant. The Einstein d-tensor \(\widehat{\mathbf {G}}_{\alpha \beta }\) is given in N-adapted form by Eq. (19). The resulting nonlinear system of PDEs can be integrated in explicit form for arbitrary parameterizations of type \({\varvec{\Upsilon }}_{~\delta }^{\beta }= diag[{\varvec{\Upsilon }}_{\alpha }]\).

^{4}

*N*term.

^{5}It is a CP violating parameter and is determined to be very small from constraints from phenomenology. The non-Riemannian configuration is determined from the canonical d-connection \( \widehat{\Gamma }_{\beta \gamma }^{\alpha }\) for \(\mathbf {g}_{\alpha \beta }. \)

*M*as an integration constant for the right hand side, the left hand side has a non-trivial transformation. In terms of the metric \(\widehat{\mathbf {g}} _{\alpha \beta },\) the equation for the scalar field becomes

*M*or add a constant to

*V*, we still satisfy the conditions \(\ ^{e}V=0\) and \(\mathrm{d}\ ^{e}V/\mathrm{d}\phi =0\) if \(V+M=0.\) Here we also note that if we consider \(N\ne 0\), it implies that an external source drives the scalar field away from such vacuum points and can be addressed in terms of instanton effects.

^{6}For convenience, the partial derivatives \(\partial _{\alpha }=\partial /\partial u^{\alpha }\) are labelled

^{7}we rewrite the equations in the form

*t*in (42) and find \(n_{i}(x^{k},t)\). In the final step we solve for \(w_{i}(x^{i},y^{a})\) by solving a system of linear algebraic equations (43). Equation (44) is necessary to accommodate a non-trivial conformal (in the vertical “subspace”) factor \(\omega (x^{i},y^{a})\) that depends on all four coordinates. For convenience, we shall use \(\Psi :=e^{\varpi }\) as our redefined generating function.

We have shown that TMT theories as determined by actions of type (27) can be formulated in nonholonomic variables as effective EYMH systems with modified Einstein field equation (4). This allows one to apply the AFDM and decouple such systems of nonlinear PDEs in very general form and write them equivalently as systems of type (40)–(44). This procedure and the resulting equations provide important results for mathematical cosmology. For instance, by considering the coordinate \(y^{4}=t \) to be time-like, one can show that TMT theories and other modified gravity models can be integrated in general forms.

## 3 Off-diagonal cosmological solutions with small vacuum density

In this section we provide a series of examples of new classes of exact solutions of modified Einstein equations with (non) homogeneous cosmological configurations constructed by applying the AFDM. We emphasize that all solutions generated in this section will be for a TMT theory with sources (22) parameterized in the form (23), when the effective nonlinear scalar potential is taken in the form (24). In a similar form, we can construct solutions with effective sources for other types of modified gravity theories like in [25, 47, 48, 49].

*t*, we get

*t*, and (43), solving a system of linear algebraic equations for \(w_{i}.\) As a result, the N-coefficients are expressed recurrently as functionals (an example of which is \([\widetilde{\Psi },\Lambda _{0},\Xi ]\)) and are as follows:

After a tedious calculation for \(g_{a}=\omega ^{2}(x^{k},y^{a})h_{a}\) that involves the vertical conformal factor \(\omega (u^{\alpha })\) depending on all spacetime coordinates, the vertical metric \(h_{a}\) (48) and the N-coefficients \( N_{i}^{a}\) (50) reveals the fact that the formulae for the Ricci d-tensor \( \widehat{\mathbf {R}}_{\alpha \beta }\) (17) are invariant if the first order PDE (44) are satisfied. For non-trivial \(\omega ,\) the solutions to the modified gravitational equation (4), parameterized as a d-metric (21), do not posses in general any Killing symmetries and contain dependencies of \(\omega \) on \([\psi ,h_{a},n_{i},w_{i}]\) with as many as six independent variables for \( \mathbf {g}_{\alpha \beta }.\)

*n*-coefficients, such conditions are satisfied if \(\ _{2}n_{k}(x^{i})=0\) and \( \partial _{i}\ _{1}n_{j}(x^{k})=\partial _{j}\ _{1}n_{i}(x^{k}).\) In N-adapted form, such coefficients do not depend on generating functions and sources but only on a corresponding class of integration functions, e.g., \(_{1}n_{j}(x^{k})=\) \(\partial _{i}n(x^{k}),\) for any \(n(x^{k}).\) It is a more difficult task to find explicit solutions for the LC-conditions (37) involving variables \(w_{i}(x^{k}).\) Such nonholonomic constraints cannot be solved in explicit form for arbitrary data \((\Psi ,{\varvec{\Upsilon }} ),\) or arbitrary \((\tilde{\Psi },\Xi ,\Lambda _{0}).\) We first use the property that \(\mathbf {e}_{i}\Psi =(\partial _{i}-w_{i}\partial _{t})\Psi \equiv 0\) for any \(\Psi \) if \(w_{i}=\partial _{i}\Psi /\partial _{t}\Psi \) (it follows from Eq. (50)). This results in the expression

## 4 Time-like parameterized off-diagonal cosmological solutions

In this section we consider a subclass of solutions pertaining to \(g_{\alpha \beta }(x^{k},y^{3},t)\) extracted from either (51), or (54) which, via frame transformations \( g_{\alpha \beta }(u)=e_{\ \alpha }^{\alpha ^{\prime }}(u)e_{\ \beta }^{\beta ^{\prime }}(u)g_{\alpha ^{\prime }\beta ^{\prime }}(t),\) result in metrics \( g_{\alpha ^{\prime }\beta ^{\prime }}(t)\) that depend only on time-like coordinate *t*. For applications in modern cosmology, we consider \(g_{\alpha ^{\prime }\beta ^{\prime }}(t)\) as certain off-diagonal deformations of the FLRW, or the Bianchi type universes [22, 25]. In explicit form, we construct physical models with \(\acute{\mathbf {g}}=\{g_{\alpha ^{\prime }\beta ^{\prime }}(t)\}\rightarrow \) \(\mathbf {\mathring{g}}=\{\mathring{g}_{i}, \mathring{h}_{a}\}\) for \(\eta _{\alpha }\rightarrow 1\) and \(\mathbf {e} ^{\alpha }\rightarrow \mathrm{d}u^{\alpha }=(\mathrm{d}x^{i},\mathrm{d}y^{a})\) in (21). The strategy is first to construct solutions for a class of generating functions and sources with spacetime dependent coordinates and then to restrict the integral varieties to configurations with dependencies only on the time-like coordinate. This procedure requires that \(\widetilde{\Psi }(x^{k},t)\rightarrow \widetilde{\acute{\Psi }}(t), \widehat{\Psi }(x^{k},t)\rightarrow \widehat{\acute{\Psi }}(t);\) \({\varvec{\Upsilon }} (x^{k},t)\rightarrow \acute{{\varvec{\Upsilon }}}(t)\) with \(\Xi [{\varvec{\Upsilon }} ,\widetilde{\Psi }]=\int \mathrm{d}t{\varvec{\Upsilon }} \partial _{t}(\widetilde{\Psi }^{2})\rightarrow \acute{\Xi }(t)=\acute{\Xi }[\acute{\varvec{\Upsilon }} (t),\widetilde{\acute{\Psi }}(t)] ; \widehat{\Xi }[{\varvec{\Upsilon }} , \widehat{\Psi }]=\int \mathrm{d}t{\varvec{\Upsilon }} \partial _{t}(\widehat{\Psi } ^{2})\rightarrow \widehat{\acute{\Xi }}(t)=\widehat{\acute{\Xi }}[\acute{{\varvec{\Upsilon }}}(t), \widehat{\acute{\Psi }}(t)];\) \(\partial _{i}\acute{\Xi }\rightarrow \acute{\digamma }_{i}(t),\) \( \partial _{i}\widehat{\acute{\Xi }}\rightarrow \widehat{\acute{\digamma }} \) and with \( \omega \rightarrow 1. \) The integration functions \(\ _{1}n_{k}(x^{i})\) and \(_{2}\widetilde{n} _{k}(x^{i})\) are considered to be constants of integration, implying \(\partial _{i}n(x^{k})\rightarrow const. \text{ and } \ \partial _{i}\check{A}(x^{k},t)\rightarrow \check{\digamma }_{i}(t).\)

### 4.1 Cosmological solutions for the effective EYMH systems and TMT

*e*and \(i^{2}=-1.\) The gauge field \(\mathbf {A}_{\mu }\) enters the covariant derivative \(D_{\mu }=\mathbf {e}_{\mu }\) \(+ie[\mathbf {A} _{\mu },]\) and the “curvature”

^{8}

*SU*(2) gauge field configurations, the diagonal ansatz for generating solutions can be written in the form

*m*(

*r*) is interpreted as the total mass within the radius

*r*for which \(m(r)=0\) defines an empty de Sitter space written in a static coordinate system with a cosmological horizon at \( r=r_{c}=\sqrt{3/\Lambda }.\) The solution of (56) associated to the quadratic metric line element (61) is defined by a single magnetic potential \(\omega (r),\)

*s*is a constant and \(\varepsilon _{\beta \mu }\) is the absolute antisymmetric tensor. The gauge field curvatures \(F_{\beta \mu },\ ^{\circ }F_{\beta \mu }\) and \(\ ^{\eta }\mathbf {F}_{\beta \mu }\) are computed by substituting (62) and (64) into (55). Any antisymmetric \(\mathbf { F}_{\beta \mu }\) (65) is a solution of \(D_{\mu }(\sqrt{|g|} F^{\mu \nu })=0,\) i.e. determines \(\ ^{\eta }F_{\beta \mu },\ ^{\eta }A_{\mu }\), for any given \(\ ^{\circ }A_{\mu },\ ^{\circ }F_{\beta \mu }.\) For nonholonomic modifications of scalar fields, we take \(\ \ ^{\circ }\phi (x^{1})\rightarrow \phi (x^{i},y^{a})=\ ^{\phi }\eta (x^{i},y^{a})\ ^{\circ }\phi (x^{1})\). It is supplemented with a polarization \(\ ^{\phi }\eta \) for which

The effective gauge field \(\mathbf {F}_{\beta \mu }\) (65) with the potential \(A_{\mu }\) (64) modified nonholonomically by \(\phi \) and subject to the conditions (66) determine exact solutions of the system (31) if the spacetime metric is chosen to be in the form (21). The energy-momentum tensor is determined to be \(\ ^FT_{\beta }^{\alpha }=-4s^{2}\delta _{\beta }^{\alpha }\) [28]. Interacting gauge and Higgs fields, with respect to N-adapted frames, result in an effective cosmological constant \(\ ^{s}\Lambda =8\pi s^{2}\), which should be added to the respective source (23).

### 4.2 Effective vacuum EYMH configurations in TMTs

The effects of off-diagonal gravitational, scalar and gauge matter fields result in driving the vacuum energy density to zero even when the effective source \(\mathbf {{\varvec{\Upsilon }} }_{\alpha }\) and cosmological constant \(\Lambda _{0}\) are non-trivial. This is possible due to the contributions of effective self-dual gauge fields. Such an effect is discussed in [16] for instantons. If \(\mathbf {{\varvec{\Upsilon }} }_{~\delta }^{\beta }=0\) in (67), one imposes further nonholonomic rescaling \( \mathbf {\Upsilon \rightarrow }\Lambda _{0}\) when \( \Lambda _{0}-4s^{2}=0.\) We can generate a very large class of solutions in TMTs with effective EMYH interactions into nonholonomic vacuum configurations of modified Einstein gravity. In this section, we analyze a subclass of generic off-diagonal EYMH interactions which can be encoded as effective vacuum Einstein manifolds of various class and lead to solutions with non-trivial cosmological constant \(\Lambda _0 {=4s^{2}}.\) In general, such solutions depend parametrically on \(\Lambda _{0}-4s^{2}\) and do not have a smooth limit from non-vacuum to vacuum models. Effects of this type exist both in commutative, noncommutative gauge gravity theories [26], Einstein gravity and its various modifications [20, 23], and TMTs. Examples are provided in the following sections.

### 4.3 Examples of (off-)diagonal nonholonomic deformations of cosmological metrics

In this section, we present details of how AFDM are employed to construct a new class of inhomogeneous and anisotropic cosmological solutions with target d-metrics \(\widehat{\mathbf {g}}\) (21), with certain well-defined limits for \(\eta _{\alpha }\rightarrow 1\), to a primed metric \(\ ^{\circ }\mathbf {g}\). These can be interpreted as conformal, frame or coordinate transformations of the well-known metrics like FLRW, Bianchi, Kasner, or another metric corresponding to a particular cosmological solution [29, 30, 31, 32].

#### 4.3.1 Off-diagonal deformations of FLRW configurations in TMTs

We show how N-anholonomic FLRW deformations can be constructed to define three classes of generic off-diagonal cosmological solutions for modified EYMH systems in TMTs. Similar models for one measure theories are presented in [22].

**FLRW metrics:**For convenience, we introduce the necessary notations to describe the primed standard FLRW metric, when written in the diagonal form,

*t*and takes the form

*p*are the proper energy density and pressure in the fluid rest frame. The Einstein equations corresponding to ansatz (82) take the form of two coupled nonlinear ODEs (the Friedmann equations),

**Off-diagonal effective EYMH cosmological solutions of type 1:**In this case the d-metric is of the type (51) with \( \partial _{t}h_{a}\ne 0,\partial _{t}\varpi \ne 0\) and \(\Upsilon -4s^{2}\ne 0,\) when

*s*for gauge fields. Such values have to be fixed in a form which is compatible with experimental/ observational data, and result in a small vacuum density. Summarizing the results for off-diagonal nonholonomic deformations of the prime metric (83), we get a quadratic line element

Solutions (89) describe general off-diagonal deformations of the FLRW metrics in TMTs encoding modified EYMH interactions. Such interactions may result in changing the topology and symmetries, and they are characterized by inhomogeneous, locally anistropic configurations or non-perturbative effect. The problem of the physical interpretation of such cosmological off-diagonal solutions is simplified to some extent if we consider small deformations with polarizations of the type \(\eta _{\alpha }\) \(\approx 1+\chi _{\alpha }\) and \(\eta _{i}^{a} \approx 0+\chi _{i}^{a}\) for small values \(|\chi _{\alpha }|\ll 1\) and \(|\chi _{i}^{a}|\ll 1,\) by which we obtain small deformations of the FLRW universes by certain generalized two measure interactions and/or modified gravity theories with effective EYMH fields. Nevertheless, even in such cases the target configuration may encode nonlinear and nonholonomic parametric effects as results of rescaling (87) of generating functions. This way we model nonlinear nonholonomic transformations of a FLRW universe into an effective and small-deformed one with small values of effective cosmological constant, nonlinear anisotropic processes and other effects of similar magnitude.

**Off-diagonal cosmological solutions of type 2 and “losing” information on effective EYMH:** This class of solutions are characterized by the condition \(\partial _{t}h_{3}=0.\) Equation (69) can be solved only if \(\varvec{\Upsilon } -4s^{2}=0,\) i.e. when the contributions from effective YM fields compensate other (effective) modified gravity and/or matter field sources. We take the function \(w_{i}(x^{k},t)\) as a solution of (71), or its equivalent (76), because the coefficients \(\beta \) and \(\alpha _{i}\) from (38) are zero. To find non-trivial values of \(n_{i}\) we integrate (70) for \(\partial _{t}h_{3}=0\) for any given \(h_{3}\) and find \(n_{i}=\ ^{1}n_{k}\left( x^{i}\right) +\ ^{2}n_{k}\left( x^{i}\right) \int h_{4}\mathrm{d}t.\) Also, we take \(g_{1}=g_{2}=e^{\psi (x^{k})},\) with \( \psi (x^{k})\) determined by (68) for a given source \((\varvec{\Upsilon } -4s^{2}).\)

*v*-conformal factors \(\ ^{s}\omega (x,z,y,t)\) subject to the constraints

*t*for N-connection coefficients \(w_{i}.\)

Off-diagonal metrics (90) result only with time-like dependence in the coefficients i.e., when \(h_{4}=h_{4}(t), \,\,\,w_{i}=w_{i}(t)\) and \(n_{i}(t)\) are determined with some constant values of \(\ ^{0}h_{4},\ \ ^{1}n_{k},\ ^{2}n_{k}.\) Such conditions are relevant for the Levi-Civita configurations if \(w_{i}=const.\) This defines solutions of the Einstein equations with nonholonomic vacuum encoding TMTs contributions and effective EYMH interactions. They transform nonholonomically a FLRW universe into certain effective vacuum Einstein configurations which in this particular case are diagonalizable by coordinate transformations.

**Off-diagonal cosmological solutions of type 3 and effective matter fields interactions:**Non-vacuum metrics with \(\partial _{t}h_{3}\ne 0\) and \(\partial _{t}h_{4}=0\) are generated by taking the ansatz

Finally, we emphasize that off-diagonal deformations of FLRW metrics in TMTs with effective EYMH interactions sources of the type \(\Upsilon -4s^{2}\) can be used for driving to zero an effective cosmological constant or for modelling parametric transforms to configurations with small effective vacuum energy.

#### 4.3.2 Nonhomogeneous EYMH effects in Bianchi cosmology in TMTs

*IX*universes and their corresponding cosmologies exist.

*t*, and \(\ ^{B}\mathbf {\mathring{g}} _{\alpha \beta }\) is parameterized as a prime metric,

*v*-conformal factor \(\ ^{B}\omega (x^{k},y^{3},t)\) is a solution of (44) with coefficients (100) when

#### 4.3.3 Kasner type metrics

In a similar manner, we construct various nonholonomic deformations of the Kasner universes of types 1–3 and/or and generalize them to solutions of type (103).

## 5 Effective TMT large field inflation with \(^{c}\alpha \)-attractors

We consider a broad class of (off-) diagonal attractor solutions that arise naturally in (modified) gravity theories and TMTs and define what we imply by natural inflationary models. In this work, we study cosmological attractors as they are considered for cosmological models in Refs. [10, 12, 13]. The use of the word attractor needs to be clarified as a similar term is widely used in the theory of dynamical systems, for certain equilibrium configurations with critical points in the phase space, i.e., critical points which are stable. Our use of the word attractor solutions is in the same spirit as Refs. [10, 12, 13]. What the authors of that work mean by cosmological attractors (see, for instance, Ref. [10]) can be stated in their own words: “Several large classes of theories have been found, all of which have the same observational predictions in the leading order in 1/N. We called these theories “cosmological attractors.” In our approach, the use of the word “attractor” is similar but in a more general context for generic off-diagonal solutions. Certain configurations in our work are determined by solutions, in general, with nonholonomically induced torsion and can be restricted to LC-configurations. Under such assumptions, these configurations appear again in other models under consideration by us. We group all such models as having “cosmological attractor configurations” since the configurations are common to these class of models. It is implicit that such solutions satisfy the conditions for “standard” cosmological attractor configurations (in the sense of Linde et al.) only for certain subclasses of nonholonomic constraints when the models are determined by imposing constraints on the corresponding generating and integration functions and integration constants. For general nonholonomic constraints, such configurations do not define cosmological attractor configurations in the sense of the above mentioned original work [10, 12, 13] but positively can be considered to possess similar properties for small off-diagonal deformations (perturbations) of the metrics. The important point is that such models have the same observational predictions in the leading order of 1 / *N*. In this section, we shall define and study cosmological attractor configurations for modified gravity theories in terms of a parameter \(^{c}\alpha \) that determines the curvature and cut-off. Henceforth, in order to make our manuscript more transparent, wherever we use the word “attractor”, it will simply imply that certain classes of theories and respective off-diagonal cosmological solutions are generated which, under specific conditions on the parameter space, lead to the same observational predictions.

### 5.1 Nonholonomic conformal transforms and cosmological attractors

*q*and study the model with the lagrangian

^{9}Attractor models are usually constructed in terms of two fields. In addition to \(\phi (u^{\mu })\) we consider a second field \(\chi (u^{\mu }).\) The fields \((\phi ,\chi )\) are subject to additional nonholonomic constraints involving the generating function \(\Psi =e^{\varpi }\) (39), some possible re-definitions (49) of effective matter field sources \(\Upsilon \) and the effective cosmological constant \(\Lambda _{0}.\)

*SO*(1, 1) symmetry which is deformed by the term \(q^{2}(\phi /\chi ).\) In turn, the Lagrange density \(_{q}^{1}L\) may restore the

*SO*(1, 1) symmetry at a critical point because for large \(\phi \) there exist asymptotic limits, \( \tanh \phi \rightarrow \pm 1\) and \(q^{2}(\tanh \phi )\rightarrow const\). The terms proportional to \(q^{2}\) can be transformed into effective sources and cosmological constant via eventual rescaling of generating functions. Employing self-duality for gauge field configurations with source \(\Upsilon -4s^{2}, \) (67), and using the gauge (107) with \(\breve{\phi } =\sinh \phi \) and \(\breve{\chi }=\cosh \chi ,\) we can approximate \(_{\chi }^{1}L\) by

It is very surprising that Eqs. (106)–(109) and their physical consequences are similar to those for holonomic models considered in Refs. [10, 12, 13]. Our solutions encoding cosmological attractor configurations were derived for a class of modified theories with generalized off-diagonal metrics and nonlinear and distinguished linear connections and contributions from EYMHs for different TMT modes. It is not obvious that such nonlinear systems may have a similar cosmological attractor behaviour like in the original work with diagonal solutions. Generic off-diagonal models can be elaborated following our geometric techniques with N-adapted nonholonomic variables and splitting of corresponding systems of nonlinear PDEs. In such variables, it is possible to generate new classes of inhomogeneous and anisotropic solutions. Our goal was to find such classes of nonholonomic constraints and subclasses of generating and integration functions, and constants, when solutions with “hat” values and TMT–EYMH contributions really preserve the main physical properties of cosmological attractors. This emphasizes the general importance of the results on cosmological attractors in the cited work due to Linde et al. Our main conclusion is that, for a corresponding class of nonholonomic constraints, a cosmological attractor configuration may “survive” for very general off-diagonal and matter source deformations, in various classes of TMT theories and effective Einstein like ones encoding modified gravity theories.

### 5.2 Effective interactions and cosmological attractors

The goal of this section is to study cosmological effects of (in general, locally anisotropic and inhomogeneous) attractors parameterized by a constant \(\ ^{c}\alpha \) \(\lesssim O(1).\) Attractor configurations can be introduced in several inequivalent ways. We will generalize the constructions following [10] and analyze possible off-diagonal solutions determined by the sources \(\ ^{q}V(\phi )\) and \(\ ^{c}\alpha \) parameters.

### 5.3 Off-diagonal attractor type cosmological solutions

*q*-terms of the type \(\ ^{q}V(\phi ,\ ^{c}\alpha )\) for attractors are encoded in various classes of solutions studied in previous section. This holds for any

#### 5.3.1 Off-diagonal effective EYMH cosmological attractor solutions of type 1

*v*-metric:

Finally, we conclude that the formulae for the coefficients of the d-metric (113) depend on the type of N-adapted frame and coordinate transforms necessary to fix observational data. The conformal factor \( e^{2\ ^{c}\widehat{\sigma }(u)}\) encodes the attractor parameters in a more direct form.

#### 5.3.2 Generalized locally anisotropic Bianchi attractors

^{10}We follow the same procedure as in Sects. 3 and 4.3.2 and write in terms of the polarization functions

*v*-conformal factor \(_{c}^{B}\omega (x^{k},y^{3},t)\) (in indirect form, it also contain attractor properties) as a solution of (44) with the attractor coefficients stated above when

Applying the AFDM, we generate off-diagonal cosmological attractor solutions of types 2 and 3 for the conventional and other families of inflation potentials, for instance, when we use \(\widetilde{q}(\frac{\phi /\sqrt{\ ^{c}\alpha }}{ 1+\phi /\sqrt{\ ^{c}\alpha }})\) instead of \(q(\phi /\sqrt{\ ^{c}\alpha })\) [10]. We note that we have used a different system of notations and our approach is based on geometric methods which allows us to construct exact solutions of modified gravitational and matter field equations. For certain well-defined conditions, we reproduce the results and “diagonal” models studied in (supersymmetric) models with dark matter and dark energy effects. Nevertheless, nonlinear parametric systems of PDEs corresponding to effective EYMH interactions in (modified) TMTs contain solutions at a richer level that were not analyzed and applied to modern cosmology. Even though the off-diagonal effects at large observational scales seem to be very small, the generic nonlinear character of cosmological solutions depending on space-like coordinates result in new nonlinear physics described by rescaling via generating functions and effective sources. Attractor type configurations offer alternative solutions of crucial importance for explaining the inflation scenarios in modern cosmology.

### 5.4 Cosmological implications of TMT nonholonomic attractor type configurations

Here we concentrate on observational consequences of generic off-diagonal solutions for the effective EYMH systems with attractor properties in TMTs. We have demonstrated that Lagrangians of type \(_{q}^{1}L\) (105) and \(_{\chi }^{1}L\) (106) and their effective energy-momentum tensors are naturally included as sources (22) in action (27) with two measures, which result in a nonholonomic modification of Einstein gravity (25). Geometrically, we reproduce such effects via re-definition of generating functions (26) and fixing a cut-off constant \(\ ^{c}\alpha \) for attractor configurations, when the effective matter field interactions are modelled for a nonholonomic off-diagonal vacuum configuration with small effective cosmological constant and gravitational \(\eta \)-polarizations.

- 1.
TMTs and nonholonomic modifications of the EYMH theory contain inflationary model of the plateau-type and features of universal attractor property when \(n_{s}=1-2/N\) and \(r=12\ ^{c}\alpha /N^{2}.\)

- 2.
For \(\ ^{c}\alpha =1\) such models are related to cosmological scenarios with the Starobinsky type model and Higgs inflation [43, 44, 45, 46]; we obtain an asymptotic theory for quadratic inflation with \(n_{s}=1-2/N,r=8/N,\) for large cut-off \(\ ^{c}\alpha .\)

- 3.
Decreasing \(\ ^{c}\alpha ,\) we get a universal attractor property both for diagonal and off-diagonal configurations; there are many models which have the same values \(n_{s}\) and

*r*. This property is preserved for EYMH contributions, solitonic and/or gravitational waves for corresponding nonholonomic configurations. - 4.
In the limit of large \(\ ^{c}\alpha ,\) we have generated models of simplest chaotic inflation. We have shown that effective nonlinear potentials with a second attractor are other viable possibilities.

- 5.
For intermediate values of \(\ ^{c}\alpha ,\) the predictions interpolate between these two critical points, thus oscillating between the sweet spots of both Planck and BICEP2 [6, 7, 8].

## 6 Concluding remarks

To mention a few, the most important physical solutions in modern gravity and cosmology theories pertaining to black holes, wormhole configurations, FLRW metrics, are constructed for diagonal metrics transforming the (modified) Einstein equations into certain nonlinear systems of second (or higher) order ODEs. The solutions generally depend on integration constants. Such constants are fixed following a certain symmetry and other physical assumptions in order to explain and describe the experimental and observational data. There are also constructed more sophisticated classes of solutions, for instance, with off-diagonal rotating metrics with Killing, Lie type symmetries and solitonic hierarchies which provide important examples of nonlinear models of gravitational and matter interactions. Nevertheless, the bulk of such analytic and numerical methods of constructing exact solutions are based on certain assumptions where the corresponding nonlinear system of PDEs are transformed into ODEs. The solutions are parameterized via integration parameters, symmetry and physical constants. The main idea is to formulate an approach to simplify the equations and find solutions depending, for instance, on a radial or a time-like variable. The drawback of this approach is that a number of nonlinear parametrical solutions are lost and thus unavailable for possible applications in cosmology and astrophysics.

The AFDM is presented as a geometric method for constructing general classes of off-diagonal metrics, auxiliary connections and adapted frames of reference when gravitational and matter field equations in various modified/ generalized gravity theories, including general relativity, are decoupled. This decoupling implies that the corresponding nonlinear system of PDEs splits into certain subclasses of equations which contain partial derivative depending only on one coordinate and relates only two unknown variables and/or generating functions. As a result, we can integrate such systems in very general off-diagonal forms when various classes of solutions are determined not only by integration constants but also by generating and integration functions, symmetry parameters and anholonomy relations. The solutions depend, in general, on all spacetime coordinates and can be with Killing or non-Killing symmetries, of different smooth classes, with singularities and non-trivial topology. We can make, for instance, certain approximations on the type of generating functions and effective source at the end, after a general form of solution has been constructed. This way we generate new classes of cosmological metrics which are homogeneous or inhomogeneous, and in general, with local anisotropies, which cannot be found if one works from the very beginning with a simplified ansatz and higher symmetries. Furthermore, the possibility to re-define the generating functions and sources via nonlinear frame transformations and parametric deformations allows one to entertain new classes of solutions and study various nonlinear physical effects.

In this paper, we studied in explicit form certain classes of modified gravity theories which can be modelled as TMTs with effective EYMH interactions. Possible scalar fields and corresponding nonlinear interaction potentials were chosen to select and reproduce attractor type solutions with cut-off constants which seem to have fundamental implication in elaborating isotropic and anisotropic inflation scenarios in modern cosmology. In general, one can work with off-diagonal configurations and consider diagonal limits for minimal and/or non-minimal coupling constants. We proved that the decoupling property holds also in TMTs, which results in the possibility of constructing various classes of off-diagonal cosmological solutions with small vacuum density. Such solutions describe spacetimes with nonholonomically induced torsion. Nevertheless we formulated well-defined criteria when additional nonholonomic constraints are introduced that allow one to extract LC-configurations. We studied nonholonomic deformations of FLRW, Bianchi and Kasner type metrics encoding TMT effects and possible contributions of effective EYMH interactions.

We have shown that attractor type cosmological solutions with cut-off parameters can be derived by nonlinear re-definitions of generating functions and effective sources in TMT if a corresponding type of nonlinear scalar potential is chosen. In general, such attractor solutions are model independent and are constructed in explicit form to accommodate effective EYMH interactions. In this way various large scale inflationary models, with anisotropic expansion and parametric nonlinear processes can be realized.

For certain conditions, the gravitational and matter field equations of TMTs are expressed as effective Einstein equations with non-minimal coupling [19]. In this presentation, we proved that in nonholonomic N-adapted variables and for additional assumptions the constructions are generalized in such form that two measure configurations serve to encode massive gravity effects and nonlinear parametric off-diagonal interactions (see Eqs. (25)–(27)). In general, such a theory also has four extra degrees of freedom with the Boulware–Deser (BD) ghosts. This problem can be circumvented if one imposes additional constraints. We imposed nonholonomic constraints for constructing cosmological attractor configurations. This procedure constrains the extra dimension degrees of freedom and encodes the TMT and massive term contributions into certain subclasses of solutions for off-diagonal effective Einstein spaces (see similar constructions for ghost-free massive *f*(*R*) theories in Refs. [47, 48, 49]). We conclude that in our models the BD ghosts are absent for such special classes of nonholonomic configurations if generic off-diagonal cosmological solutions are constructed for effective Einstein equations of type (33)–(37).

There remain many open questions on how to provide viable explanations for the recent observational data from Planck and BICEP. In this work, we have shown that attractor configurations can be constructed in TMTs with effective gravitational and matter field equations. Such solutions provide a new background for investigating cosmological theories with anisotropies, inhomogeneities, dark energy and dark matter physics.

## Footnotes

- 1.
The \(2+2\) splitting is convenient for constructing exact cosmological solutions with generic off-diagonal metrics which cannot be diagonalized by coordinate transforms in a finite spacetime region. Nevertheless, realistically, we shall have to consider \(3+1\) splitting, for instance, in Sect. 4.3.1 in order to study off-diagonal deformations of FLRW configurations in TMTs, with effective fluid energy-momentum stress tensor.

- 2.
Boldface symbols will be used in order to emphasize that certain spaces and/or geometric objects are adapted to a N-connection. Here we note that, for instance, \(\ ^{h}\mathbf {V}\) is equivalent to \(h\mathbf {V}\) (in order to avoid ambiguities, we present both types of notations used in our former work and the references therein). Such a conventional decomposition (equivalently, fibred structure) can always be constructed on any 4-d metric-affine manifold. In general relativity, it is known as the diadic decomposition of tetrads. The most important outcome of our work [20, 21, 22, 23, 24] is that we proved that (modified) Einstein equations can be decoupled and solved in very general forms both for a N-adapted \(2+2\) splitting and a d-connection \(\widehat{\mathbf {D}}\) (this auxiliary connection was not considered in former work with diadic structures).

- 3.
In general, symmetric metrics of the type \(\mathbf {g}_{\alpha \beta }(x^{1},x^{2},y^{3},y^{4}=t),\) with

*t*being a time-like coordinate, contain a maximum of six independent variables since four coefficients from the ten components of the metric tensor of a 4-d spacetime can be transformed away via coordinate transforms as a result of the Bianchi identities. - 4.
We can consider other distributions which do not allow for the construction of solutions in explicit form. Our geometric approach will be applied to such N-connection splitting and frame/ coordinate transforms that parameterize the effective sources in some form and will admit the decoupling of the (modified) Einstein equations.

- 5.
- 6.
For simplicity, we shall omit “hats” on coefficients of type \( g_{i},g_{a},n_{i},\) \(w_{i}\) etc. related to \(\widehat{\mathbf {g}}\) if it will not lead to ambiguities.

- 7.
Non-trivial solutions result if such conditions are not satisfied; in such cases, we need to consider other special methods for generating solutions.

- 8.
For standard gauge field models but on nonholonomic manifolds we can follow a variational principle for a gravitating non-Abelian SU(2) gauge field \( \mathbf {A}=\mathbf {A}_{\mu }\mathbf {e}^{\mu }\) coupled to a triplet Higgs field \(\phi .\) In such cases, the value \(\phi _{[0]}\) is the vacuum expectation of the Higgs field which determines the mass \(^{H}M=\sqrt{ \lambda }\eta ,\) when \(\lambda \) is the constant of scalar field self-interaction with potential \(\mathcal {V}(\phi )=\frac{1}{4}\lambda Tr(\phi _{[0]}^{2}-\phi ^{2})^{2},\) where the trace

*Tr*is taken on internal indices. In EYMH theory, the gravitational constant \(G,\kappa =16\pi G,\) defines the Planck mass \(M_{Pl}=1/\sqrt{G}\) and it is also the mass of gauge boson, \(\ ^{W}M=ev.\) In the literature, various versions of modified gravity and TMTs are elaborated upon with different types of nonlinear scalar and gauge fields. - 9.
In this section, we use natural units \(1/\kappa =1/2\).

- 10.
It is supposed that the parameter \(\ ^{c}\alpha \) contributes to all data defining nonholonomic deformations of a primary Bianchi metric. This parameter is included into effective source when \(\Upsilon \rightarrow \ ^{c}\Upsilon \) with \(\ ^{c}\Upsilon -4s^{2}\ne 0\) and the effective cosmological constant \(\Lambda _{0}^{c}\) is chosen to satisfy the condition \(\Lambda _{0}^{c}-4s^{2}\ne 0.\)

## Notes

### Acknowledgements

SV reports certain research related to his former basic activity at UAIC; the Program IDEI, PN-II-ID-PCE-2011-3-0256; a DAAD fellowship in 2015 and support from Quantum Gravity Research, QGR-Topanga, California, USA.

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