1 Introduction

The “ekpyrotic” and “new ekpyrotic” mechanisms and cyclic models have been elaborated as alternatives to standard big bang inflationary cosmology [2–5]. Another alternative came from the idea that the graviton may have a nontrivial mass as was proposed by Fierz and Pauli [1]. For recent reviews and related \(f(R)\) modifications and cosmological models, in general, with non-minimal coupling and dilaton–brane cosmology, local anisotropies and/or effective modeling of massive gravity, see, respectively, [6–12] and [13–18]. The modern version of a ghost-free (bimetric) massive gravity theory was made in a series of papers (recent reviews can be found in [19]): developing generic nonlinear versions of the Fierz–Pauli theory, the so-called vDVZ discontinuity problem was solved via the Vainshtein mechanism [20–22] (avoiding the discontinuity by going beyond the linear theory). There are also more recent approaches based on the DGP model [23–26]. During a long time, no solution was found for another problem with ghosts because at nonlinear order in massive gravity there appears a sixth scalar degree of freedom as a ghost. This problem was considered in a paper by Boulware and Deser [27, 28] together with similar issues related to the effective field theory approach.

A considerable amount of work has been made in order to understand the implications and find possible applications of such ghost-free models. The most substantial progress was made when de Rham et al. showed how to eliminate the scalar mode and Hassan and Rosen provided a complete proof for a class of bigravity/bimetric gravity theories; see [29–33]. In such approaches, the second metric describes an effective exotic matter related to massive gravitons and does not suffer from a ghost instability to all orders in a perturbation theory and away from the decoupling limit.

It is important to mention that the first black hole solutions in a nonlinear massive theory were found in the context of high-energy physics and for off-diagonal and/or higher dimensions generalizations [34, 35]. A general decoupling of the generalized Einstein equations can be proven following the anholonomic frame deformation methods, AFDM, [36–44]. Here we also note that the possibility that the graviton has a nonzero mass \( \mathring{\mu }\) results not only in fundamental theoretical implications but gives rise to straightforward phenomenological and cosmological consequences. For instance, a gravitational potential of Yukawa form, \(\sim \!\! e^{-\mathring{ \mu }r}/r\), results in the decay of gravitational interactions at scales \(r\ge \mathring{\mu }^{-1}\), and such an effect may result in the accelerated expansion of the Universe. In this way, a massive gravity theory, MGT, provides alternatives to dark energy and, via effective polarizations of the fundamental physical constants, may explain certain dark matter effects. We can treat this as a result of generic off-diagonal nonlinear interactions. Recently, various cosmological models derived for ghost-free (modified) massive gravity and bigravity theories have been elaborated and studied intensively [6–10, 45–51].

The goal of this work is to construct generic off-diagonal cosmological solutions in MGT and state the conditions when such configurations are modeled equivalently in general relativity (GR). It extends the later variant of [52, 53] with new results and more details of the proofs of the results, constructing exact solutions and providing a further analysis of results and speculation on the reconstruction mechanism.

As a first step, we consider off-diagonal deformations of a “prime” cosmological solution taken in a general Painlevé–Gullstrand (PG) form, when the Friedman–Lamaître–Robertson–Walker (FLRW) can be recast for well-defined geometric conditions. Such constructions are performed in Sect. 2.

At the second step, the “target” metrics will be generated so as to possess one Killing symmetry (or other non-Killing symmetries) and depend on timelike and certain (or all) spacelike coordinates. In general, such off-diagonal solutions come with a 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. We consider the method of constructing generic off-diagonal cosmological solutions in Sect. 3.

Then (the third step), we shall emphasize and speculate on the importance of off-diagonal nonlinear gravitational interactions for elaborating cosmological scenarios when dark matter and dark energy effects can be explained by anisotropic polarizations of vacuum and/or de Sitter like configurations. We provide examples of off-diagonal solutions with solitonic configurations; see Sect. 4.

A reconstruction mechanism for off-diagonal cosmological solutions with modified gravity and/or massive graviton effects is elaborated in Sect. 5. Finally, we conclude the paper in Sect. 6.

2 Equivalent modeling of \(f\)-modified and massive gravity theories

We shall work with MGTs modeled on a pseudo-Riemannian spacetime \(\mathbf {V}\) with physical metric \(\mathbf {g=\{g}_{\mu \nu }\}\) and fiducial metrics. On massive gravity, see for a review [19] and on geometric methods in gravity and constructing exact solutions, see Refs. [36–44]. In addition to well-known approaches with diadic and tetradic (vierbein) variables, we shall work with nonholonomic manifolds where certain classes of frame transforms can be adapted to preserve a chosen splitting of the nonlinear and linear connection structures into some standard components (for instance, defining the Levi-Civita, LC, connection), and we shall deal with distortion tensors which can be fixed to be zero if additional constraints are imposed.

Using different nonholonomic frame variables, the action for our model can be written in two forms:

$$\begin{aligned} S= & {} \frac{1}{16\pi }\int \delta u^{4}\sqrt{|\mathbf {g}_{\alpha \beta }|}\left[ \widehat{f}(\widehat{R})-\frac{\mathring{\mu }^{2}}{4}\mathcal {U}(\mathbf {g} _{\mu \nu },\mathbf {K}_{\alpha \beta })+\ ^{m}L\right] \nonumber \\ \end{aligned}$$
(1)
$$\begin{aligned}= & {} \frac{1}{16\pi }\int \delta u^{4}\sqrt{|\mathbf {g}_{\alpha \beta }|} [f(R)+\ ^{m}L]. \end{aligned}$$
(2)

In the above formulas, the physical/geometrical objects with a hat and/or written in boldface form are considered for a conventional \(2 \, + \,2\) splitting when the two dimensional horizontal, h, and the two dimensional vertical, v, coordinates are labeled (respectively) in the form \(u^{\alpha }=(x^{i},y^{a}),\) or \(u=(x,y),\) with indices \(i,j,k,\ldots =1,2\) and \(a,b,\ldots =3,4. \) The scalar curvature \(R\) is for the LC-connection \(\nabla \).

We write \(\widehat{R}\) for the scalar curvature for an auxiliary (canonical) connection \(\widehat{\mathbf {D}}\) uniquely determined by two conditions:

  1. 1.

    it is metric compatible, \(\widehat{\mathbf {D}}\mathbf {g}=0, \) and

  2. 2.

    the \(h\)- and \(v\)-torsions are zero (but there are nonzero \(h-v\) components of the torsion \(\widehat{\mathcal {T}}\) completely determined by \( \mathbf {g)}\) for a conventional splitting

    $$\begin{aligned} \mathbf {N}:\ T\mathbf {V}=h\mathbf {V}\oplus v\mathbf {V}, \end{aligned}$$

    with local coefficients of the so-called nonlinear connection structure, the \(N\)-connection structure, denoted in the form \(\mathbf {N}= \{N_{i}^{b}\}.\)

General frame transforms can be parameterized in the form \(e_{\alpha }=A_{\ \alpha }^{\alpha ^{\prime }}(u)\partial _{\alpha ^{\prime }},\) where the matrix \(A_{\ \alpha }^{\alpha ^{\prime }}\) is non-degenerate in a finite, or infinite region of \(\mathbf {V}\) and \(\partial _{\alpha ^{\prime }}=\partial /\partial u^{\alpha ^{\prime }}.\) Using such \(A_{\ \alpha }^{\alpha ^{\prime }},\) we can always re-define the geometric and physical object with respect to a class of \(N\)-adapted (dual) bases,

$$\begin{aligned} \mathbf {e}_{\alpha }= & {} (\mathbf {e}_{i}=\partial _{i}-N_{i}^{b}\partial _{b},e_{a}=\partial _{a}=\partial /\partial y^{a})\quad \text{ and } \nonumber \\ \mathbf {e}^{\beta }= & {} (e^{j}=dx^{i},\mathbf {e}^{b}=dy^{b}+N_{c}^{b}dy^{c}), \end{aligned}$$
(3)

which are nonholonomic (equivalently, anholonomic) because, in general, relations of type

$$\begin{aligned} \mathbf {e}_{\alpha }\mathbf {e}_{\beta }-\mathbf {e}_{\beta }\mathbf {e} _{\alpha }=W_{\alpha \beta }^{\gamma }\mathbf {e}_{\gamma } \end{aligned}$$

are satisfied for certain nontrivial anholonomy coefficients \(W_{\alpha \beta }^{\gamma }(u).\) The Einstein summation rule on repeated indices will be applied if the contrary is not stated.

The connection \(\widehat{\mathbf {D}}\) allows us to decouple the field equations in various gravity theories and construct exact solutions in very general forms. Here we note that the distortion relation from the LC-connection,

$$\begin{aligned} \widehat{\mathbf {D}}=\nabla +\widehat{\mathbf {Z}}[\widehat{\mathcal {T}}], \end{aligned}$$

is uniquely determined by a distorting tensor \(\widehat{\mathbf {Z}}\) completely defined by \(\widehat{\mathcal {T}}\) and (as a consequence for such models) by \((\mathbf {g},\mathbf {N}).\) The main idea of the AFDM [36–44] is to use \(\widehat{\mathbf {D}}\) as an auxiliary [for the (pseudo-) Riemannian spacetimes] and/or (with nonholonomically induced torsion) connection, which definitely allows us to decouple gravitational field equations for very general conditions with respect to \(N\)-adapted frames; see (3). We cannot perform such a decoupling in general form if we work from the very beginning with the data \((\nabla ,\partial _{\alpha ^{\prime }})\), but there are proofs that this is possible for \((\widehat{\mathbf {D}},\mathbf {e}_{\alpha };\mathbf {g},\mathbf {N}).\) Having constructed integral varieties (for instance, locally anisotropic and/or inhomogeneous cosmological ones), we can impose additional nonholonomic (non-integrable constraints) when \(\widehat{\mathbf {D}}_{\mid \widehat{\mathcal {T}} =0}\rightarrow \nabla \) and \(\widehat{R}\rightarrow R,\) where \(R\) is the scalar curvature of \(\nabla ,\) and it is possible to extract exact solutions in GR.Footnote 1

The MGTs with actions of type (1) generalize the so-called modified \(f(R)\) gravity and the ghost-free massive gravity [29–33]. We shall follow some conventions from [51]. Units will be used where \(\hbar =c=1\) and the Planck mass \(M_{Pl}\) is defined via \(M_{Pl}^{2}=1/8\pi G,\) with 4D Newton constant \(G.\) We write \(\delta u^{4}\) instead of \(d^{4}u\) because \(N\)-elongated differentials are used as in (3). A model can be specified by the corresponding constants, dynamical physical equations, and their solutions in the corresponding variables. It will be assumed that \(\mathring{\mu }=\mathrm{const}\) is the mass of the graviton. For LC-configurations, we can fix conditions of type

$$\begin{aligned}&\widehat{f}(\widehat{R})-\frac{\mathring{\mu }^{2}}{4}\mathcal {U}(\mathbf {g} _{\mu \nu },\mathbf {K}_{\alpha \beta })=f(\widehat{R}),\nonumber \\&\quad \text{ or } \widehat{f}(\widehat{R})= f(R), \text{ or } \widehat{f}(\widehat{R})=R, \end{aligned}$$
(4)

which depend on the type of models we elaborate and on what classes of solutions we want to construct. Such conditions can be very general ones for arbitrary frame transforms and \(N\)-connection deformations. We emphasize that it is possible to find solutions in explicit form if we choose the coefficients \(\{N_{i}^{a}\}\) and the local frames for \(\widehat{\mathbf {D}}\) when \(\widehat{R}=\mathrm{const}\) in such forms that \(\partial _{\alpha }\widehat{f}( \widehat{R})=(\partial _{\widehat{R}}\widehat{f})\times \partial _{\alpha } \widehat{R}=0\), but, in general, \(\partial _{\alpha }f(R)\ne 0.\)

The equations of motion for our nonholonomically modified massive gravity theory can be written

$$\begin{aligned} (\partial _{\widehat{R}}\widehat{f})\widehat{\mathbf {R}}_{\mu \nu }-\frac{1}{ 2}\widehat{f}(\widehat{R})\mathbf {g}_{\mu \nu }+\mathring{\mu }^{2}\mathbf {X} _{\mu \nu }=M_{Pl}^{-2}\mathbf {T}_{\mu \nu }, \end{aligned}$$
(5)

where \(M_{Pl}\) is the Planck mass, \(\widehat{\mathbf {R}}_{\mu \nu }\) is the Einstein tensor for a pseudo-Riemannian metric \(\mathbf {g}_{\mu \nu }\) and \( \widehat{\mathbf {D}},\) \(\mathbf {T}_{\mu \nu }\) is the standard matter energy-momentum tensor. We note that in the above formulas, for \(\widehat{ \mathbf {D}}\rightarrow \nabla ,\) we get \(\widehat{\mathbf {R}}_{\mu \nu }\rightarrow R_{\mu \nu }\) with a standard Ricci tensor \(R_{\mu \nu }\) for \( \nabla .\) Such limits give rise to the original modified massive gravity models, but it is important to find some generalized solutions and only at the end to consider additional assumptions on the limits and nonholonomic constraints. It should be emphasized that the effective energy-momentum tensor \(\mathbf {X} _{\mu \nu }\) in (5) is defined by the potential of the graviton \( \mathcal {U}=\mathcal {U}_{2}+\alpha _{3}\mathcal {U}_{3}+\alpha _{4}\mathcal {U} _{4},\) where \(\alpha _{3}\) and \(\alpha _{4}\) are free parameters. The quantities \(\mathcal {U}_{2},\mathcal {U}_{3}\) and \(\mathcal {U}_{4}\) are certain polynomials entailing the traces of some other polynomials of a matrix \(\mathcal {K} _{\mu }^{\nu }=\delta _{\mu }^{\nu }-\left( \sqrt{g^{-1}\Sigma }\right) _{\mu }^{\nu }\) for a tensor determined by the four Stückelberg fields \(\phi ^{\underline{\mu }}\) as

$$\begin{aligned} \Sigma _{\mu \nu }=\partial _{\mu }\phi ^{\underline{\mu }}\partial _{\nu }\phi ^{\underline{\nu }}\eta _{\underline{\mu }\underline{\nu }}, \end{aligned}$$
(6)

when \(\eta _{\underline{\mu }\underline{\nu }}=(1,1,1,-1).\) A series of arguments presented in [51] (geometrically, we can consider corresponding nonholonomic frame transforms and nonholonomic variables) prove that the parameter choice \(\alpha _{3}=(\alpha -1)/3,\alpha _{4}=(\alpha ^{2}-\alpha +1)/12\) is useful for avoiding potential ghost instabilities. By frame transforms, we can fix

$$\begin{aligned} \mathbf {X}_{\mu \nu }=\alpha ^{-1}\mathbf {g}_{\mu \nu }. \end{aligned}$$
(7)

By explicit computations, we can prove that for the configurations (7), de Sitter solutions with an effective cosmological constant are possible, for instance, for an ansatz of PG type,

$$\begin{aligned} ds^{2}= & {} U^{2}(r,t)\big [dr+\epsilon \sqrt{f(r,t)}dt\big ]^{2}+\widetilde{\alpha } ^{2}r^{2}(d\theta ^{2}\nonumber \\&+\sin ^{2}\theta d\varphi ^{2})-V^{2}(r,t)dt^{2}. \end{aligned}$$
(8)

In the above formula, spherical coordinates are used labeled in the form \( u^{\beta }=(x^{1}=r,x^{2}=\theta ,y^{3}=\varphi ,y^{4}=t),\) where the function \(f\) takes non-negative values and the constant \(\widetilde{\alpha }=\alpha /(\alpha +1)\) and \(\epsilon =\pm 1.\) We can consider bimetric configurations [determined as solutions of the system (5), (6), and (7)] with Stückelberg fields parameterized in the unitary gauge as \(\phi ^{\underline{4}}=t\) and \(\phi ^{\underline{1}}=r \widehat{n}^{\underline{1}},\phi ^{\underline{2}}=r\widehat{n}^{\underline{2} },\phi ^{\underline{3}}=r\widehat{n}^{\underline{3}},\) where a three dimensional (3D) unit vector is defined as \(\widehat{n}=(\widehat{n}^{ \underline{1}}=\sin \theta \cos \varphi ,\widehat{n}^{\underline{2}}=\sin \theta \sin \varphi ,\widehat{n}^{\underline{3}}=\cos \theta ).\)

First we note that any PG metric of type (8) defines solutions both in GR and in MGT. For instance, we can extract the de Sitter solution, in the absence of matter, and obtain standard cosmological equations with a FLRW metric, for a perfect fluid source

$$\begin{aligned} T_{\mu \nu }=\left[ \rho (t)+p(t)\right] u_{\mu }u_{\nu }+p(t)g_{\mu \nu }, \end{aligned}$$
(9)

where \(u_{\mu }=(0,0,0,-V)\) can be reproduced for the effective cosmological constant \(\ ^\mathrm{eff}\lambda =\mathring{\mu }^{2}/\alpha .\) Secondly, it is also possible to express metrics of type (8) in a familiar cosmological FLRW form (see formulas (23), (24), and (27) in [51]).

Finally, in this section, we note that off-diagonal deformations of such solutions can be constructed for different classes of zero graviton mass and/or massive theories and various \(f\)-modifications; see examples in [36–44] and [34, 35].

3 Generating off-diagonal cosmological solutions

We now make the crucial assumption that our Universe can be described by inhomogeneous cosmological metrics with Killing symmetry on \(\partial _{3}=\partial _{\varphi }\) and, in general, cannot be diagonalized by coordinate transforms. As a matter of principle, we can consider dependencies on all spacetime coordinates, but this requires more cumbersome computations; see the examples in Refs. [36–44]. Inhomogeneities and anisotropies can be very small, but it is important to find certain classes of general solutions for generic nonlinear systems and impose at the end certain homogeneity and high symmetry conditions by selecting the corresponding subclasses of generation and integration functions. Up to general classes of frame transforms, we can consider the ansatz

$$\begin{aligned} ds^{2}= & {} \eta _{1}(r,\theta )\mathring{g}_{1}(r)dr^{2}+\eta _{2}(r,\theta ) \mathring{g}_{2}(r)d\theta ^{2}\nonumber \\&+\, \omega ^{2}(r,\theta ,\varphi ,t)\bigl \{\eta _{3}(r,\theta ,t) \nonumber \\&\times \, \mathring{h}_{3}(r,\theta )[d\varphi +n_{i}(r,\theta )dx^{i}]^{2}+\eta _{4}(r,\theta ,t)\mathring{h}_{4}(r,\theta ,t)\nonumber \\&\times \, [dt+(w_{i}(x^{k},t)+\mathring{w }_{i}(x^{k}))dx^{i}]^{2}\bigr \}. \end{aligned}$$
(10)

The values \(\eta _{\alpha }\) are called “polarization” functions, where \( \omega \) is the so-called “vertical”, v, conformal factor.

For metrics (10), we can consider off-diagonal \(N\)-coefficients labeled \(N_{i}^{a}(x^{k},y^{4}),\) where (for parameterizations corresponding to this class of ansatz)

$$\begin{aligned} N_{i}^{3}=n_{i}(r,\theta )\quad \text {and}\quad N_{i}^{4}=w_{i}(x^{k},t)+\mathring{w} _{i}(x^{k}). \end{aligned}$$

The data for the “primary” metric are

$$\begin{aligned} \mathring{g}_{1}(r)= & {} U^{2}-\mathring{h}_{4}(\mathring{w}_{1})^{2}, \mathring{g}_{2}(r)=\widetilde{\alpha }^{2}r^{2},\nonumber \\\mathring{h}_{3}= & {} \widetilde{ \alpha }^{2}r^{2}\sin ^{2}\theta ,\mathring{h}_{4}=\sqrt{|fU^{2}-V^{2}|}, \nonumber \\ \mathring{w}_{1}= & {} \epsilon \sqrt{f}U^{2}/\mathring{h}_{4},\mathring{w} _{2}=0,\mathring{n}_{i}=0, \end{aligned}$$
(11)

when the coordinate system is fixed in such a way that the values \(f,U,V\) in (8) result in a coefficient \(\mathring{g}_{1}\) depending only on \(r\).

Using nonholonomic frame transforms, we can parameterize the energy momentum sources (9) and the effective equation (7) in the form

$$\begin{aligned} \Upsilon _{\beta }^{\alpha }= & {} \frac{1}{M_{Pl}^{2}(\partial _{\widehat{R}} \widehat{f})}(\mathbf {T}_{\beta }^{\alpha }+\alpha ^{-1}\mathbf {X}_{\beta }^{\alpha })\nonumber \\= & {} \frac{1}{M_{Pl}^{2}(\partial _{\widehat{R}}\widehat{f})}(\ ^{m}T+\alpha ^{-1})\delta _{\beta }^{\alpha }=(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\delta _{\beta }^{\alpha }, \end{aligned}$$
(12)

for constant values \(\ ^{m}\Upsilon :=M_{Pl}^{-2}(\partial _{\widehat{R}} \widehat{f})^{-1}\ ^{m}T\) and \(\ ^{\alpha }\Upsilon =M_{Pl}^{-2}(\partial _{ \widehat{R}}\widehat{f})^{-1}\alpha ^{-1},\) with respect to \(N\)-adapted frames (3). In general, such sources are not diagonal and may depend on all spacetime coordinates. Our assumption is that we prescribe a distribution with one Killing symmetry in a moment of time and then find the further evolution with respect to certain classes of nonholonomic frames. We emphasize that fixing such \(N\)-adapted parameterizations we can decouple the gravitational field equations in MGTs and construct exact solutions in explicit form.

Let us outline in brief the decoupling property of the gravitational and matter field equations in GR and various generalizations/modifications studied in detail in Refs. [36–44]. That anholonomic frame deformation method (AFDM) can be applied for the decoupling, and for constructing solutions of the MGT field equations (5) with any effective source parameterized in the form (12); see details in Refs. [36–44]).

We label the target off-diagonal metrics as \(\mathbf {g}=(g_{i}=\eta _{i} \mathring{g}_{i},h_{a}=\eta _{a}\mathring{h}_{a},N_{j}^{a})\) with coefficients determined by the ansatz (10). In these formulas, there is no summation on the repeating indices. We shall use short-hand notations for the partial derivatives: \(\partial _{1}\psi =\psi ^{\bullet },\partial _{2}\psi =\psi ^{\prime },\partial _{3}\psi =\psi ^{\diamond }\), and \(\partial _{4}\psi =\psi ^{*}.\) Computing the \(N\)-adapted coefficients of the Ricci tensors, when \(\widehat{R}_{1}^{1}=\widehat{R} _{2}^{2},\widehat{R}_{3}^{3}=\widehat{R}_{4}^{4},\widehat{R}_{3k}\), and \( \widehat{R}_{4k}\) are not trivial, we write (5) as a system of nonlinear partial differential equations (PDEs):

$$\begin{aligned} \psi ^{\bullet \bullet }+\psi ^{\prime \prime }= & {} 2(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon ), \\ \phi ^{*}h_{3}^{*}= & {} 2h_{3}h_{4}(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon ), \nonumber \\ n_{i}^{**}+\gamma n_{i}^{*}= & {} 0,\beta w_{i}-\alpha _{i}=0,\ \nonumber \end{aligned}$$
(13)
$$\begin{aligned} \partial _{k}\omega= & {} n_{k}\omega ^{\diamond }+w_{k}\omega ^{*}, \end{aligned}$$
(14)

for

$$\begin{aligned}&\phi =\ln \left| \frac{h_{3}^{*}}{\sqrt{|h_{3}h_{4}|}}\right| ,\quad \gamma :=\left( \ln \frac{|h_{3}|^{3/2}}{|h_{4}|}\right) ^{*},\nonumber \\&\alpha _{i}=\frac{ h_{3}^{*}}{2h_{3}}\partial _{i}\phi ,\ \beta =\frac{h_{3}^{*}}{2h_{3} }\phi ^{*}. \end{aligned}$$
(15)

In the above formulas, we take the system of coordinates and the polarization functions to be fixed for configurations with \(g_{1}=g_{2}=e^{\psi (x^{k})}\) and nonzero values \(\phi ^{*}\) and \(h_{a}^{*}.\)

We can extract solutions for the LC-configurations with zero torsion if the coefficients of metrics are subjected to additional conditions:

$$\begin{aligned} w_{i}^{*}= & {} \mathbf {e}_{i}\ln \sqrt{|\ h_{4}|},\mathbf {e}_{i}\ln \sqrt{|\ h_{3}|}=0,\nonumber \\ \partial _{i}w_{j}= & {} \partial _{j}w_{i} \text{ and } n_{i}^{*}=0. \end{aligned}$$
(16)

Step by step, the system of nonlinear PDEs (13)–(16) can be integrated in a general form for any \(\omega \) constrained by a system of linear first order equations (14); see details in [36–44]. The explicit solutions are given by quadratic elements:

$$\begin{aligned}&ds^{2}=e^{\psi (x^{k})}[(dx^{1})^{2}+(dx^{2})^{2}]+\frac{\Phi ^{2}\omega ^{2} }{4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )}\nonumber \\&\mathring{h}_{3}[d\varphi +\left( \partial _{k}\ n\right) dx^{k}]^{2}-\frac{(\Phi ^{*})^{2}\omega ^{2}}{(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\Phi ^{2}}\mathring{h} _{4}[dt+(\partial _{i}\ \widetilde{A})dx^{i}]^{2}, \nonumber \\ \end{aligned}$$
(17)

for any

$$\begin{aligned} \Phi =\check{\Phi },(\partial _{i}\check{\Phi })^{*}=\partial _{i}\check{ \Phi }^{*},w_{i}+\mathring{w}_{i}=\partial _{i}\check{\Phi }/\check{\Phi } ^{*}=\partial _{i}\ \widetilde{A}. \end{aligned}$$

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 certain sources in non-explicit form. So, the AFDM can be applied to generate both off-diagonal metrics and nonholonomically induced torsions. There are various physical arguments for what type of generating/ integration functions and sources we have to choose in order to construct realistic scenarios for acceleration of the Universe and observable dark energy/dark matter effects.

Coming back to the properties of general solutions, we note that we can generate new classes of solutions for arbitrary nontrivial sources, \(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon \ne 0,\) and generating functions, \(\Phi (x^{k},t):=e^{\phi }\) and \(n_{k}=\partial _{k}n(x^{i}).\) The resulting target metrics are generic off-diagonal and cannot be diagonalized via coordinate transforms in a finite spacetime region because, in general, the anholonomy coefficients \(W_{\alpha \beta }^{\gamma } \) for (3) are not zero (we can check this by explicit computations). The polarization \(\eta \)-functions from (17) are

$$\begin{aligned} \eta _{1}= & {} e^{\psi }/\mathring{g}_{1},\eta _{2}=e^{\psi }/\mathring{g} _{2},\eta _{3}=\Phi ^{2}/4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon ),\nonumber \\ \eta _{4}= & {} (\Phi ^{*})^{2}/(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\Phi ^{2}. \end{aligned}$$
(18)

We conclude that prescribing any generating functions \(\check{\Phi }(r,\theta ,t),\) \(n(r,\theta ),\) \(\omega (r,\theta ,\varphi ,t)\) and sources \(\ ^{m}\Upsilon ,\ ^{\alpha }\Upsilon \) and then computing \(\check{A}(r,\theta ,t),\) we can transform any PG (and, similarly, FLRW) metric \(\mathbf { \mathring{g}}=(\mathring{g}_{i},\mathring{h}_{a},\mathring{w}_{i},\mathring{n}_{i})\) 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 cosmological constants determined by \(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon .\) With respect to \(N\)-adapted frames (3) the coefficients of the metric encode contributions from massive gravity, determined by \(\ ^{\alpha }\Upsilon ,\) and matter fields, included in \(\ ^{m}\Upsilon .\)

We also note that it is possible to provide an “alternative” treatment of (17) as exact solutions in MGT. In such a case, we have to define and analyze the properties of fiducial Stückelberg fields \(\phi ^{\underline{\mu }}\) and the corresponding bimetric structure resulting in target solutions \(\mathbf {g}=(g_{i},h_{a},N_{j}^{a}).\)

Let us analyze the primary configurations related to

$$\begin{aligned} \mathring{\phi }^{\underline{\mu }}=(\mathring{\phi }^{\underline{i}}=a(\tau )\rho \widetilde{\alpha }^{-1}\widehat{n}^{\underline{i}},\mathring{\phi }^{ \underline{3}}=a(\tau )\rho \widetilde{\alpha }^{-1}\widehat{n}^{\underline{3}},\mathring{\phi }^{\underline{4}}=\tau \kappa ^{-1}), \end{aligned}$$

when the corresponding prime PG-metric \(\mathbf {\mathring{g}}\) is taken in FLRW form,

$$\begin{aligned} ds^{2}=a^{2}(d\rho ^{2}/(1-K\rho ^{2})+\rho ^{2}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}))-d\tau ^{2}. \end{aligned}$$

A corresponding fiducial tensor (6) is computed,

$$\begin{aligned} \mathring{\Sigma }_{\underline{\mu }\underline{\nu }}du^{\underline{\mu }}du^{ \underline{\nu }}= & {} \frac{a^{2}}{\tilde{\alpha }^{2}}\left[ d\rho ^{2}+\rho ^{2}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2})\right. \nonumber \\&+\left. 2H\rho d\rho d\tau -\left( \frac{ \tilde{\alpha }^{2}}{\kappa ^{2}a^{2}}-H^{2}\rho ^{2}\right) d\tau ^{2}\right] , \end{aligned}$$

where the coefficients and coordinates are re-defined in the form \(r\rightarrow \rho =\widetilde{\alpha }r/a(\tau )\) and \(t\rightarrow \tau =\kappa t,\) for \(K=0,\pm 1;\ \kappa \) is an integration constant; \(H:=d\ln a/d\tau \) and the local coordinates are parameterized in the form \(x^{\underline{1}}=\rho ,x^{\underline{2}}=\theta ,y^{\underline{3}}=\varphi ,y^{\underline{4}}=\tau .\)

In the next step, for a target metric \(\mathbf {g}={\mathbf {g}_{\alpha \beta } }\) and frames \(\mathbf {e}_{\alpha }^{\ }=\mathbf {e}_{\alpha }^{\ \underline{\alpha }}\partial _{\underline{\alpha }},\) we write

$$\begin{aligned} \mathbf {g}_{\alpha \beta }= & {} \mathbf {e}_{\alpha }^{\ \underline{\alpha }} \mathbf {e}_{\beta }^{\ \underline{\beta }}\eta _{\underline{\alpha } \underline{\beta }}=\left[ \begin{array}{l@{\quad }l} g_{ij}+N_{i}^{a}N_{j}^{b}g_{ab} &{} N_{i}^{a}h_{ab} \\ N_{i}^{a}h_{ab} &{} h_{ab} \end{array} \right] ,\\&\text{ for } \mathbf {e}_{\alpha }^{\ \underline{\alpha }}=\left[ \begin{array}{l@{\quad }l} \mathbf {e}_{i}^{\ \underline{i}} &{} N_{i}^{b}\mathbf {e}_{b}^{\ \underline{a}} \\ 0 &{} \mathbf {e}_{a}^{\ \underline{a}} \end{array} \right] . \end{aligned}$$

The values

$$\begin{aligned} g_{ij}= & {} \mathbf {e}_{i}^{\ \underline{\alpha }}\mathbf {e}_{j}^{\ \underline{\beta }}\eta _{\underline{\alpha }\underline{\beta }}=e^{\psi }\delta _{ij}=\mathrm{diag}[\eta _{i}\mathring{g}_{i}],h_{ab}\\= & {} \mathbf {e}_{i}^{\ \underline{\alpha }}\mathbf {e}_{j}^{\ \underline{\beta }}\eta _{\underline{\alpha } \underline{\beta }}=\mathrm{diag}[\eta _{a}\mathring{h}_{a}],N_{i}^{3}=\partial _{k}\ n,N_{i}^{4}=\partial _{k}\ \widetilde{A} \end{aligned}$$

are related algebraically to the data (18) resulting in off-diagonal solutions (17). Then, to work out the “target” Stückelberg fields, we compute \(\phi ^{\mu ^{\prime }}=\mathbf {e}_{\ \underline{\mu }}^{\mu ^{\prime }\ }\phi ^{\underline{\mu }}\) with \(\mathbf {e}_{\ \underline{\mu }}^{\mu ^{\prime }\ }\), these being inverse to \(\mathbf {e}_{\alpha }^{\ \underline{\alpha }},\) and the fiducial tensor

$$\begin{aligned} \Sigma _{\alpha \beta }=(\mathbf {e}_{\alpha }^{\ }\phi ^{\underline{\mu }})( \mathbf {e}_{\beta }^{\ }\phi ^{\underline{\nu }})\eta _{\underline{\mu } \underline{\nu }}=\mathbf {e}_{\alpha }^{\ \underline{\alpha }}\mathbf {e} _{\beta }^{\ \underline{\beta }}\Sigma _{\underline{\alpha }\underline{\beta }}. \end{aligned}$$

If the prime value \(\mathring{\Sigma }_{\underline{\mu }\underline{\nu }}\) carries information as regards the two constants \(\kappa \) and \(\widetilde{\alpha },\) a target tensor \(\Sigma _{\mu \nu }\) is associated to off-diagonal solutions and encodes data about generating and integration functions and via superpositions on possible Killing symmetries, on various integration constants. Similar constructions were elaborated for holonomic and nonholonomic configurations in GR; see [54–56].

In the framework of MGT, the two cosmological solutions \(\mathbf {\mathring{g}}\) and \(\mathbf {g}\) related by nonholonomic deformationsFootnote 2 are characterized, respectively, by the two invariants

$$\begin{aligned} \mathring{I}^{\underline{\alpha }\underline{\beta }}=\mathring{g}^{\alpha \beta }\partial _{\alpha }\mathring{\phi }^{\underline{\alpha }}\partial _{\beta }\mathring{\phi }^{\underline{\beta }}\quad \text {and}\quad \mathbf {I}^{\alpha \beta }=\mathbf {g}^{\alpha \beta }\mathbf {e}_{\alpha }\phi ^{\underline{ \alpha }}\mathbf {e}_{\beta }\phi ^{\underline{\beta }}. \end{aligned}$$

The tensor \(\mathring{I}^{\underline{\alpha }\underline{\beta }}\) does not contain singularities because there are no coordinate singularities on the horizon for the PG metrics. It should be emphasized that the symmetry of \(\Sigma _{\mu \nu }\) is not the same as that of \(\mathring{\Sigma }_{\underline{\mu } \underline{\nu }}\) and the singular behavior of \(\mathbf {I}^{\alpha \beta }\) depends on the class of generating and integration functions we chose for constructing a target solution \(\mathbf {g}\).

In GR, MGTs, and/or Einstein–Finsler gravity theories [36–44], off-cosmological solutions of type (17) were found to generalize various models of Biachi, Kasner, Gödel, and other universes. There are known locally anisotropic black holes and wormholes, in general, with solitonic background solutions; see [34, 35, 57–59]. For instance, Bianchi type anisotropic cosmological metrics are generated if we impose the corresponding Lie algebra symmetries on the metrics. It was emphasized in [51] 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.

Finally, 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 the details in [60, 61]. This special choice also allows us to have a structure \(X_{\mu \nu }\sim g_{\mu \nu }\) at least 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.

4 Examples of off-diagonal solutions with solitonic configurations

We now consider three examples of off-diagonal cosmological solutions 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 and can be characterized by additional parametric dependencies and solitonic symmetries. Moving solitonic configurations can mimic various types of modified gravity dark energy and dark matter effects with a nontrivial gravitational vacuum, polarization of constants and additional nonlinear diagonal and off-diagonal interactions of the gravitational and matter fields.

4.1 One soliton solutions

We shall for simplicity work with a nonlinear radial (solitonic, with a left \(s \)-label) generating function,

$$\begin{aligned} \Phi =\ ^{s}\check{\Phi }(r,t)=4\arctan e^{q\sigma (r-vt)+q_{0}}, \end{aligned}$$
(19)

and with \(\omega =1,\) we construct a metric

$$\begin{aligned} \mathbf {ds}^{2}= & {} e^{\psi (r,\theta )}(dr^{2}+\ d\theta ^{2})+~\frac{\ ^{s} \check{\Phi }^{2}}{4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )}\mathring{h} _{3}(r,\theta )d\varphi ^{2}\nonumber \\&-\frac{(\partial _{t}\ ^{s}\check{\Phi })^{2}}{ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\ ^{s}\check{\Phi }^{2}}\mathring{h} _{4}(r,t)[dt+(\partial _{r}\ \widetilde{A})dr]^{2}. \end{aligned}$$
(20)

In this metric, for simplicity, we fixed \(n(r,\theta )=0\) and consider that \(\widetilde{A}(r,t)\) is defined as a solution of \(\ ^{s}\check{\Phi } ^{\bullet }/\ ^{s}\check{\Phi }^{*}=\partial _{r}\ \widetilde{A}\) and \( \mathring{h}_{a}\) are given by PG-data (11). The generating function (19), where \(\sigma ^{2}=(1-v^{2})^{-1}\) for constants \( q,q_{0},v,\) is just a 1-soliton solution of the sine–Gordon equation

$$\begin{aligned} \ ^{s}\check{\Phi }^{**}-\ ^{s}\check{\Phi }^{\bullet \bullet }+\sin \ ^{s}\check{\Phi }=0. \end{aligned}$$

For any class of small polarizations with \(\eta _{a}\sim 1),\) we can assume that the source \((\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\) is polarized by \(\ ^{s}\check{\Phi }^{-2}\) when \(h_{3}\sim \mathring{h}_{3}\) and \(h_{4}\sim \mathring{h}_{4}(^{s}\check{\Phi }^{*})^{2}/\ ^{s}\check{\Phi }^{-4}\) with an off-diagonal term \(\partial _{r}\ \widetilde{A}\), resulting in a stationary solitonic universe. If we consider \((\partial _{ \widehat{R}}\widehat{f})^{-1}=\ ^{s}\check{\Phi }^{-2}\) in (12), we can model \(\widehat{f}\)-interactions of type (1) via off-diagonal interactions and “gravitational polarizations”.

In the absence of matter, \(\ ^{m}\Upsilon =0,\) the off-diagonal cosmology is completely determined by \(\ ^{\alpha }\Upsilon \) when \(\ ^{s}\check{\Phi }\) transforms \(\mathring{\mu }\) into an anisotropically polarized/variable mass of solitonic waves. Such configurations can be modeled if \(\ ^{m}\Upsilon \ll \ ^{\alpha }\Upsilon .\) If \(\ ^{m}\Upsilon \gg \ ^{\alpha }\Upsilon ,\) we generate cosmological models determined by a distribution off matter fields when contributions from massive gravity come with a small anisotropic polarization. For a class of nonholonomic constraints on \(\Phi \) and \(\psi \) (which may be not of solitonic type), when the solutions (20) are of type (10) with \(\eta _{\alpha }\sim 1\) and \(n_{i},w_{i}\sim 0,\) we approximate PG-metrics of type (8).

Hence, by an appropriate choice of generating functions and sources, we can model equivalently modified gravity effects, massive gravity contributions or matter field configurations in GR and MGT interactions. For well-defined conditions, such configurations can be studied in the framework of some classes of off-diagonal solutions in Einstein gravity with effective cosmological constants.

4.2 Three dimensional solitonic anistoropic waves

More sophisticated nonlinear gravitational and matter field interactions can be modeled both in MGTs and GR if we consider more general classes of solutions of effective Einstein equations.

For instance, off-diagonal solitonic metrics with one Killing symmetry can be generated, for instance, if we take instead of (19) a generating function \(\ ^{s}\check{\Phi }(r,\theta ,t)\) which is a solution of the Kadomtsev–Petivashvili, KdP, equations [62],

$$\begin{aligned} \pm \ ^{s}\check{\Phi }^{\prime \prime }+(\ ^{s}\check{\Phi }^{*}+\ ^{s} \check{\Phi }\ \ ^{s}\check{\Phi }^{\bullet }+\epsilon \ ^{s}\check{\Phi } ^{\bullet \bullet \bullet })^{\bullet }=0, \end{aligned}$$

when solutions induce a certain anisotropy on \(\theta .\) Footnote 3 In the dispersionless limit, \( \epsilon \rightarrow 0,\) we can assume that the solutions are independent on \(\theta \) and determined by Burgers’ equation,

$$\begin{aligned} \ ^{s}\check{\Phi }^{*}+\ ^{s}\check{\Phi }\ \ ^{s}\check{\Phi }^{\bullet }=0. \end{aligned}$$

Such solutions can be parameterized and treated similarly to (20) but with, in general, a nontrivial term \((\partial _{\theta }\ \widetilde{A })d\theta \) after \(\mathring{h}_{4},\) where \(\ ^{s}\check{\Phi }^{\bullet }/\ ^{s}\check{\Phi }^{*}=\ \widetilde{A}^{\bullet }\) and \(\ ^{s}\check{\Phi } ^{\prime }/\ ^{s}\check{\Phi }^{*}=\ \widetilde{A}^{\prime }.\) Similar metrics were constructed for Dirac spinor waves and solitons in anisotropic Taub-NUT spaces and in five dimensional brane gravity which can be encoded and classified by the corresponding solitonic hierarchies and geometric invariants; see [57–59]. Here we proved that AFDM can be extended to generate inhomogeneous off-diagonal cosmological solitonic solutions in various MGTs.

4.3 Solitonic waves for a nontrivial vertical conformal \(v\)-factor

The cosmological solutions we look for also come with three dimensional solitons but for a \(v\)-factor as in (10). For instance, we consider solitons of KdP type, when \(\omega =\check{\omega }(r,\varphi ,t),\) when \( x^{1}=r, x^{2}=\theta ,y^{3}=\varphi ,y^{4}=t,\) for

$$\begin{aligned} \pm \check{\omega }^{\diamond \diamond }+(\partial _{t}\check{\omega }+\check{ \omega }\ \check{\omega }^{\bullet }+\epsilon \check{\omega }^{\bullet \bullet \bullet })^{\bullet }=0. \end{aligned}$$
(21)

In the dispersionless limit, \(\epsilon \rightarrow 0,\) we can take the solutions to be independent on the angle \(\varphi \) and to be determined by Burgers’ equation,

$$\begin{aligned} \check{\omega }^{*}+\check{\omega }\ \check{\omega }^{\bullet }=0. \end{aligned}$$

The conditions (14) impose the additional constraint

$$\begin{aligned} \mathbf {e}_{1}\check{\omega }=\check{\omega }^{\bullet }+w_{1}(r,\theta ,\varphi )\check{\omega }^{*}+n_{1}(r,\theta )\check{\omega }^{\diamond }=0. \end{aligned}$$

In the system of coordinates when \(\check{\omega }^{\prime }=0,\) we can fix \( w_{2}=0\) and \(n_{2}=0.\) For any arbitrary generating function with LC-configuration, \(\check{\Phi }(r,\theta ,t),\) we construct exact solutions,

$$\begin{aligned} \mathbf {ds}^{2}= & {} e^{\psi (r,\theta )}(dr^{2}+\ d\theta ^{2})+~\frac{\check{ \Phi }^{2}\check{\omega }^{2}}{4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )} \mathring{h}_{3}(r,\theta )d\varphi ^{2}\nonumber \\&-\frac{(\partial _{t}\ \check{\Phi } )^{2}\check{\omega }^{2}}{(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\ \check{\Phi }^{2}}\mathring{h}_{4}(r,t)[dt+(\partial _{r}\ \widetilde{A})dr]^{2}, \end{aligned}$$
(22)

which are generic off-diagonal and depend on all spacetime coordinates. Such stationary cosmological solutions come with polarizations on the two angles \( \theta \) and \(\varphi .\) Nevertheless, the character of the anisotropies is different for metrics of type (20) and (22). In the third class of metrics, we obtain a Killing symmetry on \(\partial _{\varphi } \) only in the limit \(\check{\omega }\rightarrow 1,\) but in the first two such a symmetry exists generically. For (22), the value \( \check{\Phi }\) is not necessarily a solitonic one which can be used for additional off-diagonal modifications of solutions and various types of polarizations.

We can provide a physical interpretation of (22) which is similar to (20) if the generating and integration functions are chosen to satisfy the conditions \(\eta _{\alpha }\sim 1\) and \(n_{i},w_{i}\sim 0.\) We approximate the PG-metrics of type (8).

In a particular case, we can use a conformal \(v\)-factor which is a 1-solitonic one, i.e.

$$\begin{aligned} \check{\omega }\rightarrow \omega (r,t)=4\arctan e^{q\sigma (r-vt)+q_{0}}, \end{aligned}$$

where \(\sigma ^{2}=(1-v^{2})^{-1}\) and constants \(q,q_{0},v,\) define a 1-soliton solution of the sine–Gordon equation,

$$\begin{aligned} \omega ^{**}-\omega ^{\bullet \bullet }+\sin \omega =0. \end{aligned}$$

Such a soliton propagates in time along the radial coordinate.

We conclude that solitonic waves may mimic both particle type configurations as dark matter and encode certain hidden dark energy and off-diagonal gravitational and matter field interactions.

5 Reconstruction mechanism for off-cosmological solutions

We now consider a reconstruction mechanism with distinguished off-diagonal cosmological effects [36–44] by generalizing some methods elaborated for \(f(R)\) gravity in [6–10]. The main idea was to present the MGT actions (with zero or nonzero gravitational mass) as sums of actions in GR and certain effective ideal fluid contributions with parameters defined by nontrivial \(\mathring{\mu }\) and \(f\)-deformations. The reconstruction method was developed for such formulations as lead to a cosmology with cyclic evolution. Then it was proven that the ekpyrotic scenario may also be realized for MGTs and that it is possible to reconstruct models of \(f(R)\) gravity which induce a little rip cosmology.

In our approach, we work with cosmological generic off-diagonal metrics and generalized connections. We can always choose such generating functions and parameters of solutions so that off-diagonal contributions are small, and the torsion is constrained to be zero, beginning at a certain fixed moment of time. Nevertheless, we cannot neglect for such classes of solutions possible nonlinear effects determined by generating functions and effective sources resulting in diagonalized modifications. MGTs were elaborated as realistic alternatives for a unified description of inflation with dark energy when cosmological scenarios encode information on various massive and zero-mass gravitational modes. A crucial question is if and how such constructions have to be elaborated for generic off-diagonal cosmological spaces. This is not only a geometric problem for generalizing the reconstruction formalism for inhomogeneous and locally anisotropic cosmological theories. It is connected to a very important question of equivalent modeling of cosmological scenarios for different MGTs in the framework of GR with nonholonomic and off-diagonal nonlinear interactions.

Any cosmological solution in massive, MGT, and/or GR parameterized in the form (10) [in particular, as (20) and (22)] can be encoded into an effective functional (4) when

$$\begin{aligned} \widehat{f}-\frac{\mathring{\mu }^{2}}{4}\mathcal {U}=f(\widehat{R}),\quad \widehat{R }_{\mid \widehat{\mathbf {D}}\rightarrow \nabla }=R. \end{aligned}$$

This allows us to work as in MGT; the conditions \(\partial _{\alpha }f( \widehat{R})=0\) if \(\widehat{R}=\mathrm{const}\) simplify the computations substantially. Nevertheless, the contributions of the parameter \(\mathring{\mu }\) and effective potential \(\mathcal {U}\) are included as functional dependencies of \(f\) and \(\widehat{R}.\) For \(\widehat{\mathbf {D}}\rightarrow \nabla ,\) we get nonlinear modifications both in diagonal and off-diagonal terms of the cosmological metrics.

The starting point of our approach is to consider a prime flat FLRW like metric

$$\begin{aligned} ds^{2}=a^{2}(t)[(dx^{1})^{2}+(dx^{2})^{2}+(dy^{3})^{2}]-dt^{2}, \end{aligned}$$

where \(t\) is the cosmological time. In order to extract a monotonically expanding and periodic cosmological scenario, we parameterize \(\ln |a(t)|=H_{0}t+\tilde{a}(t)\) for the periodic function \(\tilde{a}(t+\tau )=\ ^{1}a\cos (2\pi t/\tau ),\) where \(0<\ ^{1}a<H_{0}.\) 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 (3) for a generalized Hubble function \(H,\)

$$\begin{aligned} 3H^{2}=8\pi \rho \quad \text{ and } \quad 3H^{2}+2\mathbf {e}_{4}H=-8\pi p. \end{aligned}$$

Using variables with \(\partial _{\alpha }f(\widehat{R})_{|\widehat{R} =const}=0,\) we can consider a function \(H(t)\) when \(\mathbf {e}_{4}H=\partial _{t}H=H^{*}.\) It should be noted that the approximation \(\mathbf {e} _{4}\rightarrow \partial _{t}\) is considered “at the end” (after a class of off-diagonal solutions was found for a necessary type connection, which allowed one to decouple the equations) and the generating functions and effective sources are constrained to depend only on \(t.\) The energy-density and pressure of an effective perfect fluid are computed thus:

$$\begin{aligned} \rho= & {} (8\pi )^{-1}\left[ (\partial _{R}f)^{-1}\left( \frac{1}{2}f(R)+3H\mathbf {e} _{4}(\partial _{R}f)\right) -3\mathbf {e}_{4}H\right] \nonumber \\= & {} (8\pi )^{-1}\left[ \partial _{\widehat{R}}\ln \sqrt{|\widehat{f}|}-3H^{*}\right] \nonumber \\= & {} (8\pi )^{-1}\left[ \partial _{\widehat{R}}\ln \sqrt{|\frac{\mathring{\mu }^{2}}{4}\mathcal {U}+f(\widehat{R})|}-3H^{*}\right] , \nonumber \\ p= & {} -(8\pi )^{-1}\left[ (\partial _{R}f)^{-1}\left( \frac{1}{2}f(R)+2H\mathbf {e} _{4}(\partial _{R}f)+\mathbf {e}_{4}\mathbf {e}_{4}(\partial _{R}f)\right) \!+\!\mathbf {e} _{4}H\right] \nonumber \\= & {} (8\pi )^{-1}\left[ \partial _{\widehat{R}}\ln \sqrt{|\widehat{f}|}+H^{*}\right] \nonumber \\= & {} -(8\pi )^{-1}\left[ \partial _{\widehat{R}}\ln \sqrt{|\frac{\mathring{\mu }^{2}}{ 4}\mathcal {U}+f(\widehat{R})|}+H^{*}\right] . \end{aligned}$$
(23)

We emphasize that, in general, such values are defined with respect to nonholonomic frames. The effect of nonlinear deformations encoding physical data \((\mathring{\mu },\mathcal {U},f,\widehat{R})\) is preserved also in diagonal contributions for \(\mathbf {e}_{4}\rightarrow \partial _{t}.\)

The equation of state, EoS, parameter for the effective dark fluid encoding MGTs parameters is defined by

$$\begin{aligned} w=\frac{p}{\rho }=\frac{\widehat{f}+2H^{*}\partial _{\widehat{R}} \widehat{f}}{\widehat{f}-6H^{*}\partial _{\widehat{R}}\widehat{f}}=\frac{\frac{\mathring{\mu }^{2}}{4}\mathcal {U}+f(\widehat{R})+2H^{*}\partial _{ \widehat{R}}\widehat{f}}{\frac{\mathring{\mu }^{2}}{4}\mathcal {U}+f(\widehat{R })-6H^{*}\partial _{\widehat{R}}\widehat{f}}. \end{aligned}$$
(24)

The corresponding EoS is

$$\begin{aligned} p=-\rho -(2\pi )^{-1}H^{*}. \end{aligned}$$

\(\mathcal {U}(t)\) is computed, for simplicity, for a configuration of “target” Stückelberg fields \(\phi ^{\mu ^{\prime }}=\mathbf {e}_{\ \underline{\mu }}^{\mu ^{\prime }\ }\phi ^{\underline{\mu }}\), and the solution found is, finally, modeled by generating functions with dependencies on \(t. \) The components of \(\phi ^{\mu ^{\prime }}\) are computed with respect to \(N\)-adapted frames.

Taking a generating Hubble parameter \(H(t)=H_{0}t+H_{1}\sin \omega t,\) for \( \omega =2\pi /\tau ,\) we can recover the modified action for the oscillations of an off-diagonal (massive) universe (see similar details in [6–10]),

$$\begin{aligned} f(R(t))= & {} 6\omega H_{1}\int dt[\omega \sin \omega t-4\cos \omega t(H_{0}+H_{1}\sin \omega t)]\nonumber \\&\times \exp [H_{0}t+\frac{H_{1}}{\omega }\sin \omega t]. \end{aligned}$$
(25)

Here we note that we cannot analytically make an inversion to find an explicit form \(R, \) or any nonholonomic deformation to \(\widehat{R}\) with respect to general \(N\)-adapted frames. Nevertheless, we can prescribe any values of constants \( H_{0}\) and \(H_{1}\) and of \(\omega \) and compute the effective dark energy and dark matter oscillating cosmology effects for any off-diagonal solution in massive gravity and/or effective MGT, GR. To extract the contributions of \( \mathring{\mu }\) we can fix, for instance, \(\widehat{f}(\widehat{R})=\) \( \widehat{R}=R\), and using (1) and (2) we can relate \(f(R(t))\) and the 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 the 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. The main idea is to develop the reconstruction techniques for the scalar–tensor theory using the AFDM, with nonholonomic off-diagonal metric and linear connection deformations. Working with general classes of off-diagonal cosmological solutions, the problem is to state the conditions for generating and integration functions when a corresponding ekpyrotic scenario will “survive” for certain constraints and in the respective limits.

Let us consider a prime configuration with energy-density for pressureless matter \(\mathring{\rho }_{m};\) for radiation and anisotropies we take, respectively, \(\mathring{\rho }_{r}\) and \(\mathring{\rho }_{\sigma }\). \(\kappa \) is the spatial curvature of the universe and for the target effective energy-density \(\rho \) we have (23). A FLRW model can be described by

$$\begin{aligned} 3H^{2}=8\pi \left[ \frac{\mathring{\rho }_{m}}{a^{3}}+\frac{\mathring{\rho } _{r}}{a^{4}}+\frac{\mathring{\rho }_{\sigma }}{a^{6}}-\frac{\kappa }{a^{2}} +\rho \right] . \end{aligned}$$

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 rest. In such cases, the results are similar to those in the inflationary scenario. For recovering (25), the ekpyrotic scenario occurs and mimics the observable universe for \(t\sim \pi /2\omega \) in the effective EoS parameter

$$\begin{aligned} w\approx -1+\sin \omega t/3\omega H_{1}\cos ^{2}\omega t\gg 1. \end{aligned}$$

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. We can compute possible locally anisotropic and inhomogeneous small contributions for self-consistent models with nonholonomic frames.

In the diversity of off-diagonal cosmological solutions which can be constructed using the above presented methods, there are cyclic ones with singularities of the type of a big bang/big crunch behavior. This is still a largely unexplored area both as regards the geometric methods of constructing exact solutions of PDEs and recovering procedures for certain “preferred” fundamental physical objects compatible with the experimental data. Choosing necessary types of generating and integration functions, we can avoid singularities and elaborate models with a 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 [6–10, 36–44], big rip and/or little rip cosmology models. In this case the phantom energy-density is modeled by off-diagonal interactions. We note that such nonsingular models for the dark energy were proposed as alternatives to \(\Lambda \)CDM cosmologies, [63–65]. Using the corresponding classes of the generating functions, we can reproduce the scenarios with a phantom scalar modeling a little rip in the framework of AFDM and nonholonomic MGTs. We omit such considerations in this article (see a summary of such a construction and further developments in [52, 53]).

6 Conclusions

In this paper we deal with new classes of cosmological off-diagonal solutions in massive gravity with flat, open, and closed spatial geometries. These solutions can be systematically constructed for various types of modified gravity theories, MGTs, and in general relativity, GR. We applied an advanced geometric technique for decoupling the field equations and constructing exact solutions in massive and zero mass \(f(R)\) gravity, theories with nontrivial torsion, nonholonomic constraints to GR, and possible extensions on (co) tangent Lorentz bundles. The so-called anholonomic frame deformation method, AFDM, was elaborated during the last 15 years in our work [36–44, 57–59], where a number of examples of off-diagonal solutions and new applications in gravity and modern cosmology were considered.

A very important property of such generalized classes of cosmological solutions is that they depend, in general, on all spacetime coordinates via generating and integration functions and constants. They describe certain models of inhomogeneous and locally anisotropic cosmology with a physical meaning and possible physical implications that are less clear [13–18]. After some classes of solutions were constructed in the most general form, we can impose at the end additional nonholonomic constraints, cosmological approximations, extract configurations with a prescribed spacetime symmetry, and/or a dependence on certain mass parameters; and we can consider asymptotic conditions, etc. Thus, our solutions can be used for elaborating homogeneous and isotropic cosmological models with an arbitrary spatial curvature, to study generalized Killing and non-Killing symmetries, with possible nonholonomically deformed (super-) symmetries [34, 35, 54–56] and to study the “non-spherical” collapse models of the formation of cosmic structure such as stars and galaxies (see also [51]).

Another aspect of the AFDM is that if we work only with cosmological scenarios for diagonalizable metrics, there are possibilities to discriminate the massive gravity theory from the \(f\)-gravity and/or GR. For diagonal metrics depending on a time like coordinate, we can formulate mathematical cosmology problems for certain nonlinear systems of ordinary differential equations with general solutions depending on integration constants. Identifying such a constant, for instance, with a graviton mass parameter, we do not have many possibilities to mimic a number of similar effects in GR for different MGTs. Following only such a “diagonal” approach, we positively have to modify the GR theory in order to explain the observational data in modern cosmology and elaborate realistic quantum models of massive gravity.

We now discuss a new and important feature of the off-diagonal anisotropic configurations, which allows us to model cosmic accelerations and massive gravity and/or dark energy and dark matter effects as certain effective Einstein spaces. Having integrated such a system of nonlinear partial differential equations, PDEs, for a large class of such solutions, we can ask such questions as: Maybe we do not need to modify radically the GR theory but only to extend the constructions to off-diagonal solutions and nonholonomic systems and try to apply this in modern cosmology? Could we explain observational data in modern cosmology via nonlinear diagonal and/or off-diagonal interactions with non-minimal coupling for matter and/or different phases of massive and zero mass gravity? This is a quite complicated theoretical and experimental problem and the main goal of this and our recent papers cited in Refs. [36–44, 52, 53] was to analyze such constructions from the viewpoint of massive gravity theory when off-diagonal effects can be alternatively explained by other types of gravity theories.

The reconstruction procedure for cosmological models with non-minimally coupled scalar fields evolving on a flat FLRW background and in different MGTs was studied in [6–12]. In this work, we elaborated a reconstruction method for the massive gravity theory which admits an effective off-diagonal interpretation in GR and \(f\)-modified gravity with a cyclic and an ekpyrotic universe solution. We concluded that the expansion can be around the GR action even if we admit a nontrivial effective torsion. For zero torsion constraints, it is possible to construct off-diagonal cosmological models keeping the approach in the framework of the GR theory. We further investigated how our results indicate that theories with massive gravitons, with possible \(f\)-modified terms and off-diagonal interactions, may lead to more complicated scenarios of cyclic universes. Following such nonlinear (off-diagonal) approaches, the ekpyrotic (little rip) scenario can be realized without the need to introduce additional fields (or modifying gravity) but only in terms of massive gravity or GR. Another interesting construction can be related to the reconstruction of scenarios of \(f(R)\) and massive gravity theories leading to little rip universes both in locally anisotropic and isotropic variants. Finally, we note that the dark energy for little rip models present an example of non-singular phantom cosmology.