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Physics Letters B

www.elsevier.com/locate/physletb

Off-diagonal ekpyrotic scenarios and equivalence of modified, massive and/or Einstein gravity

Sergiu I. Vacarua'b,c'*'1

a University "Al. I. Cuza" Iasi, Rector's Department, 14 Alexandru Lapusneanu Street, Corpus R, UAIC, Office 323, Iasi, 700057, Romania b Max-Planck-Institute for Physics, Werner-Heisenberg-Institute, Foehringer Ring 6, München, D-80805, Germany c Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Welfengarten 1,30167 Hannover, Germany

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A R T I C L E I N F 0

A B S T R A C T

Article history:

Received 31 March 2013

Received in revised form 25 February 2015

Accepted 9 November 2015

Available online 11 November 2015

Editor: S. Dodelson

Keywords: Massive gravity Modified gravity

Off-diagonal cosmological solutions Ekpyrotic and little rip universe

Using our anholonomic frame deformation method, we show how generic off-diagonal cosmological solutions depending, in general, on all spacetime coordinates and undergoing a phase of ultra-slow contraction can be constructed in massive gravity. In this paper, there are found and studied new classes of locally anisotropic and (in)homogeneous cosmological metrics with open and closed spatial geometries. The late time acceleration is present due to effective cosmological terms induced by nonlinear offdiagonal interactions and graviton mass. The off-diagonal cosmological metrics and related Stuckelberg fields are constructed in explicit form up to nonholonomic frame transforms of the Friedmann-Lamaitre-Robertson-Walker (FLRW) coordinates. We show that the solutions include matter, graviton mass and other effective sources modeling nonlinear gravitational and matter fields interactions in modified and/or massive gravity, with polarization of physical constants and deformations of metrics, which may explain certain dark energy and dark matter effects. There are stated and analyzed the conditions when such configurations mimic interesting solutions in general relativity and modifications and recast the general Painleve-Gullstrand and FLRW metrics. Finally, we elaborate on a reconstruction procedure for a subclass of off-diagonal cosmological solutions which describe cyclic and ekpyrotic universes, with an emphasis on open issues and observable signatures.

© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The idea that graviton may have a nontrivial mass was proposed by Fierz and Pauli work [1] (for recent reviews and related f (R) modifications, see [2]). The key steps in elaborating a modern version of a ghost free (bimetric) massive gravity theory were made in a series of papers: The so-called vDVZ discontinuity problem was solved using the Vainshtein mechanism [3] (avoiding discontinuity by going beyond the linear theory), or following more recent approaches based on DGP model [4]. But none solution was found for another problem with ghosts because at nonlinear order in massive gravity appears a sixth scalar degree of freedom as a ghost (see the Boulware and Deser paper and similar issues related to the effective field theory approach in Refs. [5]). That stoppled for almost two decades the research on formulating a consistent theory of massive gravity.

* Correspondence to: University "Al. I. Cuza" Iasi, Rector's Department, 14 Alexandru Lapusneanu street, Corpus R, UAIC, office 323, Iasi, 700057, Romania. E-mail addresses: sergiu.vacaru@uaic.ro, sergiu.vacaru@gmail.com.

1 Visiting DAAD fellowship with two affiliations in Germany, October 1-December 31, 2015.

Recently, a substantial progress was made when de Rham and co-authors have shown how to eliminate the scalar mode and Hassan and Rosen established a complete proof for a class of bigrav-ity/bimetric gravity theories, see [6]. The second metric describes an effective exotic matter related to massive gravitons and does not suffer from ghost instability to all orders in a perturbation theory and away from the decoupling limit.

The possibility that the graviton has a nonzero mass /1 results not only in fundamental theoretical implications but give rise to straightforward phenomenological consequences. For instance, a gravitational potential of Yukawa form ~ e-/1 r/r results in decay of gravitational interactions at scales r > /1and this can result in the accelerated expansion of the Universe. This way, a theory of massive gravity provides alternatives to dark energy and, via effective polarizations of fundamental physical constants (in result of generic off-diagonal nonlinear interactions), may explain certain dark matter effects. Recently, various cosmological models derived for ghost free (modified) massive gravity and bigravity theories have been elaborated and studied intensively (see, for instance, Refs. [7,2,8]).

http://dx.doi.org/10.1016/j.physletb.2015.11.024

0370-2693/© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The goal of this work is to construct generic off-diagonal cos-mological solutions in massive gravity theory, MGT, and state the conditions when such configurations are modeled equiva-lently in general relativity (GR). We shall develop and apply in massive gravity theory the so-called anholonomic frame deformation method, AFDM [9]. As a first step, we consider off-diagonal deformations of a "prime" cosmological solution taken in general Painleve-Gullstrand (PG) form, when the Friedman-Lamaitre-Robertson-Walker (FLRW) can be recast for well-defined geometric conditions. At the second step, the "target" metrics will be generated to possess one Killing symmetry (or other none Killing symmetries) and depend on timelike and certain (all) spacelike coordinates. In general, such off-diagonal solutions are with local anisotropy and inhomogeneities for effective cosmological constants and polarizations of other physical constants and coefficients of cosmological metrics which can be modeled both in MGT and GR. Finally (the third step), we shall emphasize and speculate on importance of off-diagonal nonlinear gravitational interactions for elaborating cosmological scenarios with anisotropic polarization of vacuum and/or de Sitter like configurations.

We study modified massive gravity theories determined on a pseudo-Riemannian spacetime V with physical metric g = } and certain fiducial metric as we shall explain below. The action for our model is

S = ■

yf Su4J\g0i\[f (R) - ^U(g¡v, Kap) + mL] (1)

6-f Su4^\g0i\[f (R) + mL].

In this formula, R is the scalar curvature for an auxiliary (canonical) connection D uniquely determined by two conditions 1) it is metric compatible, Dg = 0, and 2) its h- and v-torsions are zero (but there are nonzero h-v components of torsion T completely determined by g) for a conventional splitting N : TV = hV ® vV, see details in [10].2 The "priority" of the connection D is that it allows to decouple the field equations in various gravity theories and construct exact solutions in very general forms. We shall work with generic off-diagonal metrics and generalized connections depending on all spacetime coordinates when, for instance, of type D = V + Z[T|. Such distortion relations from the Levi-Civita (LC) connection V are uniquely determined by a distorting tensor Z completely defined by tt and (as a consequence for such models) by (g, N). Having constructed certain general classes of solutions (for instance, locally anisotropic and/or inhomogeneous cosmologi-cal ones), we can impose additional nonholonomic (non-integrable constraints) when D|T=0 ^ V and R ^ R, where R is the scalar curvature of V, and it is possible to extract exact solutions in GR.

The theories with actions of type (1) generalize the so-called modified f (R) gravity, see reviews and original results in [2], and the ghost-free massive gravity (by de Rham, Gabadadze and Tolley, dRGT) [6]. We use the units when h = c = 1 and the Planck mass Mpi is defined via Mj¡ = 1/8^G with 4-d Newton constant G. We write Su4 instead of d4u because there are used N-elongated differentials (see below the formulas (11)) and consider /1 = const as the mass of graviton. For LC-configurations, we can fix (as particular cases) conditions of type

2 We consider a conventional 2 + 2 splitting when coordinates are labeled in the form ua = (x', ya), or u = (x, y), with indices i, j, k... = 1, 2 and a, b,... = 3, 4. There will be used boldface symbols in order to emphasize that certain geometric/physical objects and/or formulas are written with respect to N-adapted bases (11). There will be considered also left up/low indices as labels for some geometric/physical objects.

f (R) - ¡4u(g„v, Kap) = f (R), or /(R) = f (R), or /(R) = R,

which depend on the type of models we elaborate and what classes of solutions we wont to construct. The first one is necessary if we wont to encode massive gravity effects into a MGT with a generalized connection and corresponding Ricci scalar R which allows us to decouple the gravitational field equations and generate off-diagonal solutions. The second condition is necessary for extracting MGT models with Levi-Civita conditions. We can also consider the third type classes of solutions when theories with both f - and linear connection modifications are effectively modeled as certain off-diagonal solutions in GR. It will be possible to find solutions in explicit form if we fix the coefficients of N and local frames for D when R = const and daf (R) = (dff ) x daR = 0 but, in general, da f (R) = 0. The equations of motion for such modified massive gravity theory can be written3

^^ 1^^ 2 , (dff )R^v - ^f (f )g^v + A2XMv = Mpl2T^v,

where MPi is the Plank mass, R^v is the Einstein tensor for a pseudo-Riemannian metric g/JiV and D, T^v is the standard matter energy-momentum tensor. For D ^V, we get R^v ^ R^v with a standard Ricci tensor R^v for V. The effective energy-momentum tensor X^v is defined in a "sophisticate" form by the potential of graviton u = u2 + a3u3 + a4W4, where a3 and a4 are free parameters. The values U2, U3 and U4 are certain polynomials on traces of some other polynomials of a matrix = — ^ ^g-1^ for a tensor determined by four Stuckelberg fields ^ as

^^v = d^dv^n^v,

when n^v = (1, 1,1, —1). Following a series of arguments presented in [8], when the parameter choice a3 = (a — 1)/3, a4 = (a2 — a + 1 )/12 is useful for avoiding potential ghost instabilities, we can fix

X¡¿v = a 6g^v.

De Sitter solutions for an effective cosmological constant are possible, for instance, for ansatz of PG type,

ds2 = U2 (r, t)[dr + eV f (r, t)dt]2 + a2r2 (de2 + sin2 9dy2) - V2 (r, t)dt2.

In above formula, there are used spherical coordinates labeled in the form = (x1 = r, x2 = 0, y3 = y4 = t), the function f takes non-negative values and the constant a = a/(a + 1) and e = ±1. For such bimetric configurations, the Stuckelberg fields are parameterized in the unitary gauge as = t and = rn-, $2 = rn2-, 033 = rn-, where a three dimensional (3-d) unit vector is defined as in = (n1 = sin 0 cos y, n2 = sin 0 siny, n- = cos 0). Any PG metric of type (7) defines solutions both in GR and in MGT. It allows us to extract the de Sitter solution, in the absence of matter, and to obtain standard cosmological equations with FLRW metric, for a perfect fluid source

T /iv — [p(t) + P(t) ]u¡x uv + p(t)g¡v,

3 See details on action and variational methods in [6]; we shall follow some conventions from [8]; the Einstein summation rule on repeating indices will be applied if the contrary is not stated.

where u/ = (0, 0, 0, — V) can be reproduced for the effective cos-mological constant effX = /12/a. It is also possible to express metrics of type (7) in a familiar cosmological FLRW form (see formulas (23), (24) and (27) in [8]). Let us consider an ansatz

ds2 = m (r,6)g1 (r)dr2 + n2 (r,6)g2 (r)d02

+ m2(r,e, y, t){n3(r,e, t)h3(r,0)[dy + ni(r,0)dxi]2

+ n4(r,0, t)h4(r,0, t)[dt + (wi(xk, t) + wi(xk))dxi]2}, (9)

with Killing symmetry on 93 = dy, which (in general) cannot be diagonalized by coordinate transforms. The values na are called "polarization" functions; m is the so-called "vertical", v, confor-mal factor.4 The off-diagonal, N-coefficients, are labeled Nf (xk, y4), where (for this ansatz) N3 = ni(r, 0) and N4 = wi (xk, t) + w i (xk). The data for the "primary" metric are

g1 (r) = U2 — h4(w1)2, ^2(r) = a2r2,

h3 = a2r2 sin2 0, h4 = J\ fU2 — V21,

1 = €/f U2/h4, w2 = 0, ni = 0,

when the coordinate system is such way fixed that the values f, U, V in (7) result in a coefficient gi depending only on r.

We shall work with respect to a class of N-adapted (dual) bases

ea = (ei = di - Nfdb, ea = da = d/dya) and = (ej = dxi, eb = dyb + Nbcdyc ),

which are nonholonomic (equivalently, anholonomic) because, in general, there are satisfied relations of type ea ep — ep ea = W ey, for certain nontrivial anholonomy coefficients WYp(u). It should be noted that a nonholonomic 2 + 2 splitting (11) can be always defined on a pseudo-Riemannian manifold as a fibred structure. The operators ei and eb are called respectively N-elongated partial derivatives and N-elongated differentials because they elongate the usual ones with certain linear N-terms. We can re-define all geometric and physical objects of a MGT, massive, or GR, model in N-adapted form, i.e. with respect to N-adapted bases. The motivation for such constructions is that the gravitational and matter field equations for mentioned type theories written for D decouple in very general form with respect to N-adapted frames. This allows

4 Up to certain frame/coordinates transforms, any inhomogeneous and/or locally anisotropic cosmological metrics with such a Killing type space symmetry on dy can be parameterized in a form (9). This allows us to deform nonholonomically any class of "prime" cosmological metrics g, with np = 1 and N-connection coefficients ni (xk) = 0 and nontrivial wi (xk), into certain new classes of exact cosmological solutions defining "target" metrics g with nontrivial np(Xk, t) and ni(xk) and wi(xk, t). For instance, we can chose a "prime" g to define a FLRW diagonal cosmologi-cal model and analyze how possible MGT, massive gravity or other type effects would deform it into a locally anisotropic inhomogeneous "target" one g(xk,t). It is possible to generate target cosmological metrics depending only on a time like coordinate and with very small off-diagonal deformations which may explain recent observational cosmological data. Nevertheless, certain nonlinear parametric effects can be important ones determined, for instance, by a nontrivial massive gravity term. As a matter of principle, we can construct general inhomogeneous and locally anisotropic cosmological metrics depending on all spacetime coordinates, see examples in Refs. [9], which request a more advanced geometric techniques. In this work, we restrict our approach only for generating off-diagonal cosmological target metrics which can be finally transformed in "almost" diagonalized cosmological models with dependence on a time like coordinate t and study corresponding ekpy-rotic scenarios. Such nonlinear solutions cannot be constructed if we impose from the very beginning certain homogeneity and symmetry conditions. This is the property of nonlinear/parametric/nonholonomic systems of partial derivative equations (PDEs).

to generate exact solutions with off-diagonal metrics and generalized connections and (if necessary) to impose additional conditions for LC-configurations. It is not possible to decouple such generic nonlinear systems of equations and find solutions if we work, for instance, in coordinate frames and, from the very beginning, with the Levi-Civita connection.5

For simplicity, we shall consider energy momentum sources (8) and effective (6) which up to frame/coordinate transforms can be parameterized in the form

Xa] M2Pi(dRf)

(mT + a-1)&a

= Mikn+a

= ( my + aT)Sa

for constant values mT := Mn2(dff)—1mT and aT = M— x (dff)—1a—1, with respect to N-adapted frames (11).6

Let us explain an important decoupling property of the gravitational field equations in GR and various generalizations/modifications studied in details in Refs. [10]. That anholonomic frame deformation method (AFDM) can be applied for decoupling, and constructing solutions of the MGT field equations (4) with any effective source (12). We consider target off-diagonal metrics g = (gi = Цjgj, hf = Vaha, Na) [there is not summation on repeating indices in this formula] with coefficients determined by ansatz (9). For convenience, we shall use brief denotations for partial derivatives: d1 ifr = ^•, = ^', d3^ = ^'' and = ^*. Computing the N-adapted coefficients of the Ricci and Einstein tensors (see details in Refs. [10]), we transform (4) into a system of nonlinear PDEs:

R3k dk M

f •• + f " = 2( mT + aT),

= R4 0*h*3 = 2h3h4(mT + aT), =^ n33 + yn3 = 0, i?4k Pwi - ai = 0,

= nkM + WkM ,

for 0 = ln

n \h3\3/2,3

,Y := (ln^TT)3,

5 The N-connection geometry with nonholonomic 2 + 2 splitting is an example of "diadic" and/or "tetradic/vierbein" formalism which is well known from GR textbooks. It was applied for constructing exact solutions with two Killing symmetries and with respect to certain special systems of reference with rotation/axial symmetries. In our works [10,9], we proved that it is possible to decouple and solve the gravitational field equations for a large class of gravity theories in very general forms (with coefficients of off-diagonal metrics and generalized connections depending on all spacetime variables). We have shown in explicit form how to define certain classes of reference and perform N-adapted geometric constructions for an "auxiliary" canonical d-connection D, which allows to solve such sophisticate systems of nonlinear PDEs. At the end, it is possible to impose additional nonholonomic constraints resulting in Levi-Civita configurations. We cannot argue that this way we are able to find the "general solution" of the Einstein equations, or any modified/generalized versions, because (for generic nonlinear systems) it is not possible to prove uniqueness theorems and other properties of solutions which are typical for linear systems. Nevertheless, we elaborated on advanced geometric methods for constructing various classes of generic off-diagonal exact and approximate solutions depending on all spacetime coordinates via generating functions and integration functions and constants. Usually, the physical meaning and fundamental properties of such solutions are not known. In certain cases of models with prescribed symmetries/topological/singularity configurations, for "small" offdiagonal deformations with "gravitational polarizations", inhomogeneities and local anisotropies and limits to well-known physical/cosmological solutions, it is possible to speculate on nonlinear quantum and classical effects, modified/massive gravity contributions, analogous models etc.

6 In general, such sources are not diagonal and may depend on spacetime coordinates. We fix such N-adapted parameterizations which will allow to construct exact solutions in explicit form.

ai = -3-diф, ß = -3-ф

where is used in order to show that certain equations follow from respective coefficients of the Ricci tensor . In these formulas, the system of coordinates and polarization functions are fixed for configurations with g\ = g2 = '> and nonzero values ф* and h*. The equations result in solutions for the Levi-Civita configurations (with zero torsion) if the coefficients of metrics are subjected to the conditions

= ei ln ei ln vThäi = 0,diwj = djwi and nj = 0. (16)

The system of nonlinear PDE (13)-(16) can be integrated in general forms for any a constrained by a system of linear first order equations (14). The explicit solutions are given by quadratic elements

ds2 = )[(dx1)2 + (dx2)2] +

4 (my + ay)

x h3[dp + (dkn) dxk]2

(Ф*)2ю2

( my + ау)ф2

h4[dt + (di A)dx ]

for any é = é, (dié)* = d,é* and w, + wj = d,é/é* = d, A.1 To generate new solutions we can consider arbitrary nontrivial sources, mT + aT = 0, and generating functions, é(xk, t) := e< and nk = dkn(xj ). Such metrics are generic off-diagonal and cannot be diagonalized via coordinate transforms in a finite spacetime region because, in general, the anholonomy coefficients WYp for (11) are not zero (we can check by explicit computations). The polarization n-functions for (11) are computed in the form

П1 = ef/gi, П2 = e^fg2, П3 = Ф2/4 (mУ + аУ), П4 = (Ф*)2/(т тУ + аУ)Ф2.

So, prescribing any generating functions é(r, 0, t), n(r, 0), a(r, 0, p, t) and sources mT, aT and then computing A(r, 0, t), we can transform any PG (and FLRW) metric g = (g,, h a, W j, fi j ) in MGT and/or GR into new classes of generic off-diagonal exact solutions depending on all spacetime coordinates. Such metrics define Einstein manifolds in GR with effective sources m T + aT. Following such an approach, we have to study the properties of fiducial Stuckelberg fields and the corresponding bimetric structure resulting in target solutions g = (g,, ha, Na):

Let us analyze the "prime" configurations related to = (< = a(t)pa—iff, = a(T)pa—1ff3, <4 = tk—1), when the corresponding prime PG-metric g is taken in FLRW form

ds2 = a2 (dp2/(1 - Kp2) + p2(d92 + sin2 вdp2)) - dx2. The related fiducial tensor (5) is computed

t du^du- =— [dp2 + p2 (de2 + sin2 в dp2) + 2H pdpdr

- (— - H2p2)dT2], к 2a2

7 We can construct exact solutions even if such conditions are not satisfied, i.e. the zero torsion conditions are not stated or there are given in non-explicit form; this way, it is possible to generate off-diagonal metrics and nonholonomically induced torsions etc., see details in [10]. There will be presented physical arguments for what type of generating/integration functions and sources we have to chose in order to construct realistic scenarios for Universe acceleration and observable dark energy/matter effects.

where the coefficients and coordinates are re-defined in the form r ^ p = ar/a(r) and t ^ t = Kt, for K = 0, ±1; k is an integration constant; H := d lna/dt and the local coordinates are parameterized in the form x1 = p, x2 = 0, y- = y, y4 = t .

For a target metric g = gap and frames ea = eo~da, we can write

Saß = ea eß Vaß

for ea- =

gij + NaNbgab Nahab Nahab hab

b « ^

The values gij = ei ae ^Цор = ef&ij = diagfagi ], hab = ei a e ^Цор = diag[tfaha], N? = dk n and Nf = dk A are related algebraically to data (18) resulting in off-diagonal solutions (17). We can compute the "target" StOckelberg fields as ф= e^^ with е^ц, being inverse to eJr, and the fiducial tensor = (еафц)(евф^)Пц^ =

eaa. If the value lb¡¿v carries information about two constants к and a, a tensor Ьцу associated to off-diagonal solutions encodes data about generating and integration functions and via superpositions on possible Killing symmetries, on various integration constants [11]. In the framework of MGT, two cosmological solutions g and g related by nonholonomic deformations8 are characterized respectively by two invariants Iae = gавдафадрфв and Iae = gaeеаф^евфв. The tensor Iae does not contain singularities because there are not coordinate singularities on horizon for PG metrics. The symmetry of b/JiV is not the same as that of É¡v and the singular behavior of Iae depends on the class of generating/integration functions chosen for constructing a target solution g.

In GR and/or Einstein-Finsler gravity theories [10], off-diagonal cosmological solutions of type (17) were found to generalize various models of Bianchi, Kasner, Godel and other universes. For instance, Bianchi type anisotropic cosmological metrics are generated if we impose corresponding Lie algebra symmetries on metrics. It was emphasized in [8] that "any PG-type solution in general relativity (with a cosmological constant) is also a solution to massive gravity." Such a conclusion can be extended to a large class of generic off-diagonal cosmological solutions generated by effective cosmological constants but it is not true, for instance, if we consider nonholonomic deformations with nonholonomically induced torsion like in metric compatible Finsler theories.

We note that the analysis of cosmological perturbations around an off-diagonal cosmological background is not trivial because the fiducial and reference metrics do not respect the same symmetries. Nevertheless, fluctuations around de Sitter backgrounds seem to have a decoupling limit which implies that one can avoid potential ghost instabilities if the parameter choice is considered both for diagonal and off-diagonal cosmological solutions, see details in [12]. This special choice also allows us to have a structure ~ g/JiV at list in N-adapted frames when the massive gravity effects can be approximated by effective cosmological constants and exact solutions in MGT which are also solutions in GR.

Let us consider three examples of off-diagonal cosmological so-lutons with solitonic modifications in MGT and (with alternative interpretation) GR. Two and three dimensional solitonic waves are typical nonlinear wave configurations which can be used for generating spacetime metrics with Killing, or non-Killing, symmetries

8 Involving not only frame transforms but also deformation of the linear connection structure when at the end there are imposed additional constraints for zero torsion.

Ф2Ю2

characterized by additional parametric dependencies and solitonic symmetries.

Example 1. Taking a nonlinear radial (solitonic, with left s-label) generating function

$ = s$ (r, t) = 4arctan eqa(r—vt)+q° (19)

and m = 1, we construct a metric a metric of type (17),

ds2 = e^(r'e)(dr2 + d02) + (dts( )2

s (j)2

h3(r,9)dq>2

( my + ay)s( 2

4 (my + ay)

h4(r, t)[dt + (dr À)dr]2

where, for simplicity, we consider n(r, 0) = 0, A(r, t) is defined as a solution of s$'/s$* = dr A and ha are given by PG-data (10). The generating function (19), where a2 = (1 — v 2)—1 for constants q, q0, v, is just a 1-soliton solution of the sine-Gordon equation s$** — s $•• + sins $ = 0. For any class of small polarizations with na ~ 1, we can consider that the source (m Y + a Y) is polarized by s$—2 when h3 ~ h3 and h4 ~ h4(s$*)2/s4 with an off-diagonal term dr A resulting in a stationary solitonic modification of the PG universe. If we chose a generating function that (dff)—1 = s ■2 for the source (12), we can model off-diagonal gravitational interactions and "gravitational polarizations" in f -gravity of type (1).

In absence of matter, mY = 0, the off-diagonal cosmology is completely determined by aY when the nonlinear solitonic generating function s $ transform /1 into an anisotropically polarized/variable mass of solitonic waves. Such configurations can be modeled if mY < aY.

If m Y ^ aY, the above soluton describes cosmological models determined by respective distributions of matter fields when contributions from massive gravity are with small anisotropic polarization. We approximate PG-metrics of type (7) for a class of nonholonomic constraints on $ and ^ (which may be not of solitonic type) when the solutions (20) are of type (9) with na ~ 1 and ni, wi ~ 0.

Hence, by an appropriate choice of generating functions and sources, we can model equivalently modified gravity effects, massive gravity contributions or matter field configurations and/or MGT interactions. Such configurations can be modeled alternatively in the framework of some classes of off-diagonal solutions in Einstein gravity with effective cosmological constants and respective generating functions and integration constants.

Example 2. Three dimensional solitonic anisotropic waves can be generated, for instance, if we take instead of (19) a generating functions s $ (r, 0, t) which is a solution of the Kadomtsev-Petivashvili, KdP, equations [13],

±s$" + (s$* + s$ s$• + es$•••)• = 0,

when solutions induce certain anisotropy on 0.9 In the dispersion-less limit e ^ 0, we can consider that the solutions are independent on 0 and determined by Burgers' equation s$* +s$ s$• = 0. The solutions can be parameterized and treated similarly to (20) but with, in general, a nontrivial term (d0 A)d0 after h4, when s$ys$* = A and s$'/s$* = A'.

9 In a similar form, we can construct various types of vacuum gravitational 2-d and 3-d configurations characterized by solitonic hierarchies and related bi-Hamilton structures, for instance, of KdP equations with possible mixtures with solutions for 2-d and 3-d sine-Gordon equations etc., see details in Ref. [14].

Example 3. Choosing respective types of generating functions, we can construct different classes of nonlinear solitonic modifications of certain "primary" cosmological metrics with spherical symmetry which were deformed into locally anisotropic configurations by any 4>(r, 0, t) and, finally, containing 3-d propagating solitonic waves. For instance, such solitonic waves can be considered for a nontrivial vertical conformal v-factor as in (9), for instance, of KdP type, when m = M (r, p, t), when x1 = r, x2 = 0, y3 = y4 = t, for

±6:)"' + (dtù) + Ù) a)' + ea)'")' = 0.

In the dispersionless limit e ^ 0, the solutions are independent on angle p and determined by Burgers' equation M3 + M M* = 0. The conditions (14) impose an additional constraint

e1M = M' + w1 (r,0, p)M* + n1 (r,0)M" = 0.

In the system of coordinates when M' = 0, we can fix w2 = 0 and n2 = 0. For any arbitrary generating function with LC-configuration, S(r, 0, t), we construct exact solutions

ds2 = e*(r'e)(dr2 + d92) +

4 (my + ay)

fl3(r,0)dç2

(dt ()262 (my + ay) (2

h4(r, t)[dt + (dr À)dr]2

which are generic off-diagonal and depend on all spacetime coordinates. Such cosmological solutions are with polarizations on two angles 0 and y. Nevertheless, the character of anisotropies is different for metrics of type (20) and (22). In the third class of metrics, we obtain a Killing symmetry on dy only in the limit M ^ 1, but in the first two ones, such a symmetry exists generi-cally. For (22), the value $ is not obligatory a solitonic one which can be used for additional off-diagonal modifications of solutions and various types of polarizations. We can provide an interpretation similar to that in Example 1, if the generating and integration functions are chosen to satisfy the conditions na ~ 1 and ni, wi ~ 0, we approximate PG-metrics of type (7). In a particular case, we can use a conformal v-factor which is a 1-solitonic one, i.e. M ^ M(r, t) = 4 arctaneqa(r—vt^+q°, where a2 = (1 — v2)—1 and constants q, q0, v, defines a 1-soliton solution of the sine-Gordon equation m** — m•• + sin m = 0. Such a soliton propagates in time along the radial coordinate.

The solitonic waves constructed for Examples 1-3 can be characterized by corresponding velocities and effective mass and energy. They may model different dark energy and polarized dark matter distributions and nonlinear effects determined by certain source terms mY := mT/M2pldffJ and aY = \/M2pla(dff) in MGT defined by (12).

We now consider a reconstruction mechanism with distinguished off-diagonal cosmological effects [10] by generalizing some methods elaborated for f(R) gravity in [2]. Any cosmologi-cal solution in massive, MGT and/or GR parameterized in a form (9) (in particular, as (20) and (22)) can be encoded into an effective functional f — 1/2U = f (R), f \d^v = R (3). This allows us to work as in MGT; the conditions da f (R) = 0 if R = const simplify substantially the computations. The starting point is to consider a prime flat FLRW like metric

ds2 = a2(t)[(dx1)2 + (dx2)2 + (dy3)2] - dt2,

where t is the cosmological time. In order to extract a monoton-ically expanding and periodic cosmological scenario, we parameterize ln |a(t)| = H0t + a(t) for a periodic function a(t + t) = 1acos(2nt/T), where 0 < 1a < H0. Our goal is to prove that such a behavior is encoded into off-diagonal solutions of type (20)-(22).

We write FLRW like equations with respect to N-adapted (moving) frames (11) for a generalized Hubble function H,

3H2 = 8np and 3H2 + 2e4H = —8np.

Using variables with da f (R)fi=const = 0, we can consider a function H(t) when e4H = dtH = H*. The energy-density and pressure of an effective perfect fluid are computed

p = (8n) — l[(dRf ) — l( 2 f (R ) + 3 He4(dRf )) — 3e4 H ] = (8n)—l[df ln/f —3H *]

= (8жГ l[df lnj If u + f (R) I — 3 H ],

p = —(8n)—l[(dRf )—l( 1 f (R) + 2H e4(dRf ) + e4e4(dRf )) + e4 H ] = (8n)—ll[df ln/f + H *]

= —(8n)—l[df lna/ | ^ u + f (R )| + h *].

In N-adapted variables, the equation of state, EoS, parameter for the effective dark fluid is defined by

p = f + 2H*dff_ f-u + f(R) + 2H*dff p~ f — 6H f. u + f (R) — 6 h *dff,

w = — =

when the corresponding EoS is p = —p — (2n) 1H* and U(t) is computed, for simplicity, for a configuration of "target" Stückelberg fields = e^ when a found solution is finally modeled by generating functions with dependencies on t.

Taking a generating Hubble parameter H (t) = H0t + H1 sin at, for a = 2n/T, we can recover the modified action for oscillations of off-diagonal (massive) universe (see similar details in [2]),

f (R (t)) = 6mHi I dt[M sin Mt - 4cos Mt(H0 + Hi sin Mt)] H1

x exp[H0t +--sin Mt].

We cannot invert analytically to find in explicit form R. Nevertheless, we can prescribe any values of constants H0 and H1 and of m and compute effective dark energy and dark matter oscillating cosmology effects for any off-diagonal solution in massive gravity and/or effective MGT, GR. To extract contributions of /1 we can fix, for instance, f (R) = R = R and using (1) and (2) we can relate f(R(t)) and respective constants to certain observable data in cosmology.

The MGT theories studied in this work encode, for respective nonholonomic constraints, the ekpyrotic scenario which can be modeled similarly to f (R) gravity. A scalar field is introduced into usual ekpyrotic models in order to reproduce a cyclic universe and such a property exists if we consider off-diagonal solutions with massive gravity terms and/or f-modifications. Let us consider a prime configuration with energy-density for pressureless matter pm, for radiation and anisotropies we take respectively pr and pa for radiation and anisotropies, k is the spatial curvature of the universe and a target effective energy-density p (23). A FLRW model can be described by

3H2 = 8n[^ + % + % - K + p]-

a3 a4 a6 a2

We generate an off-diagonal/massive gravity cosmological cyclic scenario containing a contracting phase by solving the initial problems if w > 1, see (24). A homogeneous and isotropic spatially flat universe is obtained when the scale factor tends to zero and the effective f -terms (massive gravity and off-diagonal contributions) dominate over the rests. In such cases, the results are similar to those in the inflationary scenario. For recovering (25), the ekpyrotic scenario takes place and mimic the observable universe for t ~ п/2ы in the effective EoS parameter w & — 1 + sin wt/3wHi cos2 at ^ 1. This allows us to conclude that in massive gravity and/or using off-diagonal interactions in GR cyclic universes can be reconstructed in such forms that the initial, flatness and/or horizon problems can be solved.

In the diversity of off-diagonal cosmological solutions which can be constructed using above presented methods, there are cyclic ones with singularities of the type of big bang/crunch behavior. Choosing necessary types generating and integration functions, we can avoid singularities and elaborate models with smooth transition. Using the possibility to generate nonholonomically constrained f -models with equivalence to certain classes of solutions in massive gravity and/or off-diagonal configurations in GR, we can study in this context, following methods in [2,10], big and/or little rip cosmology models, when the phantom energy-density is modeled by off-diagonal interactions. We omit such considerations in this letter.

In summary, we have found new cosmological off-diagonal solutions in massive gravity with flat, open and closed spatial geometries. We applied a geometric techniques for decoupling the field equations and constructing exact solutions in f (R) gravity, theories with nontrivial torsion and nonholonomic constraints to GR and possible extensions on (co)tangent Lorentz bundles. A very important property of such generalized classes of solutions is that they depend, in principle, on all spacetime coordinates via generating and integration functions and constants. After some classes of solutions were constructed in general form, we can impose at the end nonholonomic constraints, cosmological approximations, extract configurations with a prescribed spacetime symmetry, consider asymptotic conditions etc. Thus, our solutions can be used not only for elaborating homogeneous and isotropic cosmological models with arbitrary spatial curvature, but also for study "non-spherical" collapse models of the formation of cosmic structure such as stars and galaxies (see also [8]).

Elaborating cosmological scenarios only for diagonalizable metrics with spherical and/or two Killing symmetries, there are limited possibilities to study nonlinear physical effects in massive gravity theory and to determine if there are differences from similar ones in f -gravity and/or GR. The solutions of corresponding nonlinear systems of ordinary differential equations are parameterized by integration constants with certain explicit values fixed in order to satisfy certain boundary/asymptotic conditions, experimental data etc. We positively have to modify the GR theory in order to explain observational data in modern cosmology with acceleration and dark energy and dark matter and to develop self-consistent quantum models of (non)massive gravity.

In a series of our works [10,9] on geometric methods and exact solutions, we proposed a new approach constructing exact solutions in commutative and noncommutative MGTs, massive gravity and GR. The fundamental field equations in such gravity models consist very sophisticate off-diagonal coupled systems of nonlinear PDEs which in the past could be solved in very special cases. Nevertheless, we proved that it is possible to decouple and solve such equations in certain general forms by considering necessary type nonholonomic transforms and "auxiliary" connections which can be constrained at the end in order to extract LC-configurations. Usually, it is not clear what physical meaning may have such

general off-diagonal inhomogeneous and locally anisotropic solutions but it is obvious that they should be important for certain cosmological scales. A new and important feature of such solutions is that the off-diagonal anisotropic configurations allow us to model scenarios of cosmic acceleration and massive gravity and/or dark energy and dark matter effects as certain nonlinear/paramet-ric/nonholonomic interactions in effective Einstein spaces. Following such an "orthodox paradigm", it is natural to put the question: Maybe it is not necessary to modify the Einstein gravity at classical level but to consider new classes of off-diagonal solutions for nonlinear and nonholonomic gravitational and matter fields interactions and try to apply this in modern cosmology? It is not possible to give a complete answer to such fundamental problems in this letter or in other articles with explicit constructions on modeling MGTs as generalized Einstein spaces with nontrivial vacuum structure and non-minimal coupling with matter fields. The goal of this paper was to construct in explicit form and study some typical examples of generic off-diagonal solutions determined by generating and integration functions when massive gravity effects are reproduced via nonlinear solitonic wave interactions in an effective Einstein theory. We concluded that it is possible to model cosmological scenarios as in f-modified gravity theories by considering another types of off-diagonal configurations on nonholonomic Einstein spaces.

We also developed a reconstruction method for the massive gravity theory which admits and effective off-diagonal interpretation in GR and f -modified gravity with cyclic and ekpyrotic universe solution. The expansion is around the GR action if we admit a nontrivial effective torsion. For zero torsion constraints, it is also possible to perform off-diagonal cosmological models keeping the constructions in the framework of the GR theory. Our results indicate that theories with massive gravitons and off-diagonal interactions may lead to more complicated cyclic universes. Following such an approach, the ekpyrotic (little rip) scenario can be realized with no need to introduce an additional field (or modifying gravity) but only in terms of massive gravity or GR. Further constructions can be related to reconstruction scenarios of f (R) and massive gravity theories leading to little rip universe. The dark energy for little rip models presents an example of non-singular phantom cosmology. Finally, we note that other types of nonsingular super-accelerating universe may be also reconstructed in f(R) gravity.

Acknowledgements

The work is partially supported by the Program IDEI, PN-II-ID-PCE-2011-3-0256. The author is grateful to organizers and participants of respective sections MG13, where some results of this paper were presented. He thanks S. Capozziello, S. Hervik, S. Hassan, E. Guendelman, P. Stavrinos, D. Singleton and S. Rajpoot for important discussions, critical remarks, or substantial support for

seminars. The paper was modified during the DAAD advanced research fellowship and the author is grateful to D. Lust and O. Lechtenfeld for hosting and support. Finally, the author thanks the referee for hard work and very important remarks which helped to improve and understand better certain results provided in this paper.

References

[1] M. Fierz, W. Pauli, Proc. R. Soc. Lond. A 173 (1939) 211.

[2] K. Hinterbichler, Rev. Mod. Phys. 84 (2012) 671;

S. Nojiri, S. Odintsov, N. Shirai, J. Cosmol. Astropart. Phys. 1305 (2013) 020; S. Nojiri, S. Odintsov, Phys. Lett. B 716 (2012) 377;

S. Nojiri, S. Odintsov, D. Saez-Gomez, AIP Conf. Proc. 1458 (2011) 207, arXiv: 1108.0768.

[3] H. van Dam, J.J.G. Veltman, Nucl. Phys. B 22 (1970) 397; V. Zakharov, JETP Lett. (Sov. Phys.) 12 (1970) 312;

A.I. Vainshtein, Phys. Lett. B 39 (1972) 393.

[4] G.R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485 (2000) 2008; C. Deffayet, G.R. Dvali, G. Babadaze, Phys. Rev. D 65 (2002) 044023; M.A. Luty, M. Porrati, R. Rattazzi, J. High Energy Phys. 0309 (2003) 029; A. Nicolis, R. Rattazzi, J. High Energy Phys. 0406 (2004) 059.

[5] D.G. Boulware, S. Deser, Phys. Rev. D 6 (1972) 3368;

N. Arkani-Hamed, H. Gergi, M.D. Schwarz, Ann. Phys. 305 (2003) 96.

[6] C. de Rham, Phys. Lett. B 681 (2009) 89;

C. de Rham, G. Gabadadze, Phys. Rev. D 82 (2010) 044020; C. de Rham, G. Gabadadze, A.J. Tolley, Phys. Rev. Lett. 106 (2011) 231101; S. Hassan, R.A. Rosen, Phys. Rev. Lett. 108 (2012) 041101; S. Hassan, R.A. Rosen, J. High Energy Phys. 1204 (2012) 2012.

[7] G. D'Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, A.J. Tolley, Phys. Rev. D 84 (2011) 124046;

A.E. Gumrukcuoglu, C. Lin, S. Mukoyama, J. Cosmol. Astropart. Phys. 1203 (2012) 006;

M. von Strauss, A. Schmidt-May, J. Enander, E. Mortsell, S.F. Hassan, J. Cosmol.

Astropart. Phys. 1203 (2012) 042;

M.S. Volkov, J. High Energy Phys. 1201 (2012) 035;

S. Koyama, G. Niz, G. Tasinato, J. High Energy Phys. 1112 (2011) 065;

P. Gratia, W. Hu, M. Wyman, Phys. Rev. D 86 (2012) 061504.

[8] T. Kobayashi, S. Siino, M. Yamaguchi, D. Yoshida, Phys. Rev. D 86 (2012) 061505.

[9] T. Gheorghiu, O. Vacaru, S. Vacaru, Class. Quantum Gravity 32 (2015) 065004; T. Gheorghiu, O. Vacaru, S. Vacaru, Eur. Phys. J. C 74 (2014) 3152;

S. Vacaru, Eur. Phys. J. C 74 (2014) 3132; S. Vacaru, Eur. Phys. J. C 74 (2014) 2781; S. Vacaru, Int. J. Geom. Methods Mod. Phys. 11 (2014) 1450032.

[10] P. Stavrinos, S. Vacaru, Class. Quantum Gravity 30 (2013) 055012; S. Vacaru, J. Phys. Conf. Ser. 543 (2013) 012021;

S. Vacaru, Gen. Relativ. Gravit. 44 (2012) 1015; S. Vacaru, Int. J. Mod. Phys. D 21 (2012) 1250072; S. Vacaru, Europhys. Lett. 96 (2011) 5001.

[11] R. Geroch, J. Math. Phys. 12 (1971) 918; R. Geroch, J. Math. Phys. 12 (1972) 394;

S. Vacaru, Int. J. Geom. Methods Mod. Phys. 4 (2008) 1285.

[12] L. Berezhiani, G. Chkareuli, C. de Rham, G. Gababadze, A.J. Tolley, Phys. Rev. D 85 (2012) 044024;

K. Koyama, G. Niz, G. Tasinato, J. High Energy Phys. 1112 (2011) 065.

[13] B.B. Kadomtsev, V.I. Petviashvili, Dokl. Akad. Nauk SSSR 192 (1970) 753;

B.B. Kadomtsev, V.I. Petviashvili, Sov. Phys. Dokl. 15 (1970 ) 539. English translation.

[14] S. Vacaru, Acta Appl. Math. 110 ( 2010 ) 73.