1. Introduction
In the current
CDM model of cosmology, the energy content of the universe consists of about 70% dark energy, about 25% dark matter, and about 5% visible matter. The model is based on the standard spatially homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological solutions of Einstein’s general relativity theory [
1]. The
dark features, whose nature and origin are unknown, provide the motivation to modify and extend general relativity (GR) on galactic scales and beyond in order to account for observational data purely on the basis of the gravitational physics of the extended GR without recourse to dark ingredients.
To modify the current benchmark model of cosmology, we consider nonlocal gravity theory [
2,
3,
4], a classical history-dependent generalization of GR that bears a formal resemblance to the nonlocal electrodynamics of media [
5,
6,
7,
8,
9,
10]. It is important to digress here and mention that there are indeed various other approaches to nonlocal gravitation. For the sake of brevity, we only refer to some examples and their cosmological implications. Nonlocally modified extensions of GR can be generated by the addition of functions of □, as in infinite derivative theories, or functions of
to the Einstein–Hilbert action. Here, □ is the d’Alembert-Beltrami operator. Cosmological implications of such theories have been investigated by a number of authors in connection with dynamic dark energy and accelerated expansion of the universe, for instance, see [
11,
12] and the references cited therein. Moreover, cosmological solutions of nonlocal infinite derivative theories have been studied that involve anisotropic bouncing models [
13] or an interplay between dark matter and dark energy [
14]. Quantum field theory provides the motivation for a different class of nonlocal theories of gravitation. Higher curvature nonlocal gravity theories have recently been reviewed in [
15], and a generalized nonlocal quantum gravity theory has been formulated within the framework of inflationary cosmology. On the other hand, the phenomenological approach of Deser and Woodard has been based on an effective quantum gravitational action and has been designed to explain cosmic acceleration without dark energy; see [
16] and the references cited therein. Furthermore, primordial bouncing cosmology and anisotropy have been investigated within the framework of the Deser–Woodard nonlocal gravity model in [
17].
We now return to our classical model of nonlocal gravity that is patterned after the nonlocal electrodynamics of media. Nonlocal gravity (NLG) is a tetrad theory, where the gravitational potentials are given by the 16 components of a preferred orthonormal tetrad frame field. The extended geometric framework of NLG is based on the Weitzenböck connection [
18], which renders the spacetime a parallelizable manifold. Within the framework of teleparallelism [
19,
20,
21,
22], it is possible to express GR using the Weitzenböck torsion tensor. This teleparallel equivalent of general relativity (TEGR) is a gauge theory of the Abelian group of spacetime translations [
23]. The formal similarity between TEGR and electrodynamics can be employed to introduce nonlocality into GR via constitutive kernels [
2,
3]. In NLG, the gravitational field is local, but the theory involves an average of the field over past events resulting in 16 partial integro-differential field equations [
24,
25]. No exact nontrivial solution of NLG is known at present [
26]; however, the linear regime of the theory has been extensively studied. Nonlocal gravity, in its Newtonian regime, simulates dark matter. It is therefore possible to account for the gravitational effects in the solar system as well as in nearby galaxies and clusters of galaxies [
27,
28,
29,
30,
31]. A comprehensive account of these studies is contained in [
4].
NLG is rather intricate, and to study its cosmological implications, we resort to its local limit, which is easier to analyze. In
Section 2, we present a brief account of the modified GR field equations in the local limit of NLG. For a more detailed treatment of this limiting situation, see [
32,
33], where spatially homogeneous and isotropic (FLRW) cosmological models were investigated in this modified TEGR scheme. To explore anisotropy in the Hubble flow, we present, in
Section 3, modified gravitational field equations for a Bianchi type I class of time-dependent spatially homogeneous but anisotropic spacetimes within the framework of the local limit of NLG. The field equations contain a susceptibility function
with
and
that is characteristic of the dynamic spacetime background. For
, we recover the GR field equations. We show that de Sitter and Kasner spacetimes are not solutions of the modified field equations unless
is independent of time, which is not physically reasonable. Explicit solutions of the modified field equations are studied in the next two sections. A well-known class of dynamic solutions of GR for dust with vanishing cosmological constant is extended to the local limit of NLG in
Section 4. The new solutions contain the time-dependent function
and allow for the possibility of exploring the dependence of anisotropic acceleration on
. These modified cosmological models are locally anisotropic but tend to the isotropic modified Einstein–de Sitter model at late times (
). Similarly, we study the solution of the modified field equations for a spacetime dominated by dark energy in
Section 5 and explore anisotropic cosmic acceleration in this cosmological model, which eventually becomes isotropic as well. The presence of
could be responsible for certain new “dark” features of accelerating bulk flows in the local universe.
Anisotropy of the Hubble flow would indicate a significant departure from the presumed large-scale spatial homogeneity and isotropy of the standard FLRW cosmology. On the other hand, there is recent observational evidence in support of
local anisotropic cosmic acceleration [
34,
35,
36,
37,
38]. The purpose of the present paper is to study the possible contribution of the susceptibility function
to anisotropic features of the Hubble flow.
2. Local Limit of NLG
We consider a spacetime manifold as in general relativity (GR). In an admissible system of coordinates
, the spacetime metric can be written as
Here, Greek indices run from 0 to 3, Latin indices run from 1 to 3, and the signature of the metric is +2; moreover, we employ units such that
. As in GR, the world lines of free test particles and null rays are geodesics of the spacetime manifold. We assume the existence of a preferred set of observers in this gravitational field. The observers have adapted orthonormal tetrads
,
where
is the Minkowski metric tensor. In our convention, indices without hats are normal spacetime indices, while hatted indices indicate the tetrad axes in the local tangent space.
We employ the tetrad frame field to define the curvature-free Weitzenböck connection,
Let ∇ denote covariant differentiation with respect to the Weitzenböck connection; then, , so the preferred tetrad frames are parallel throughout the gravitational field and provide a natural scaffolding for the spacetime manifold. The spacetime is thus a parallelizable manifold by the Weitzenböck connection. In this framework of teleparallelism, two distant vectors are considered parallel if they have the same local components relative to their preferred tetrad frames. Moreover, it follows from the tetrad orthonormality relation that the Weitzenböck connection is metric compatible, namely, .
The difference between two connections on the same manifold is a tensor. We define the
torsion tensor that corresponds to the Weitzenböck connection by
In the extended GR framework, we have the Weitzenböck connection as well as the symmetric Levi–Civita connection,
We use a left superscript “0” to refer to geometric quantities directly derived from the Levi–Civita connection. The
contorsion tensor is then defined by
which is related to the torsion tensor through the metric compatibility of the Weitzenböck connection. In fact,
The Levi–Civita connection given by the Christoffel symbol is the sum of the Weitzenböck connection and the contorsion tensor. One can therefore express the Einstein tensor
and the gravitational field equations of GR in terms of the teleparallelism framework, resulting in the teleparallel equivalent of GR, namely, TEGR [
4]. Indeed, we find
and Einstein’s field equations expressed in terms of torsion thus become the TEGR field equations
where
is the cosmological constant and
. Here, we define the auxiliary torsion field
by means of the auxiliary torsion tensor
, namely,
Moreover,
is the torsion vector. As in GR,
is the symmetric energy-momentum tensor of matter. In Equation (
9), we interpret
to be the traceless energy-momentum tensor of the gravitational field
This version of GR, namely, TEGR, is the gauge theory of the 4-parameter Abelian group of spacetime translations [
23]; therefore, though nonlinear, it bears a certain resemblance to Maxwell’s electrodynamics.
In analogy with the electrodynamics of media, we can consider the torsion tensor in the form
to be similar to the Faraday tensor, while the relationship between
and the torsion tensor in Equation (
10) can be viewed as the local constitutive relation of TEGR. Let us recall that in Maxwell’s electrodynamics, the constitutive relation may change, but the field equations remain the same. We adopt the same approach for the purpose of modifying Einstein’s theory. That is, we modify TEGR by introducing a tensor
that changes the constitutive relation of TEGR as follows:
To obtain the field equations of modified TEGR, we simply replace
in Equations (
9) and (
11) by
. The gravitational field equations of extended GR based on the new tensor field
now take the form
where
is the traceless energy-momentum tensor of the gravitational field. The gravitational energy-momentum tensor is modified by the presence of
; hence, we introduce a traceless tensor
that indicates this difference, namely,
where
The total energy-momentum conservation law takes the form
It is interesting to see how
modifies GR field equations; to this end, we substitute
in the Einstein tensor (
8) and employ modified TEGR field Equation (
13) to obtain
where
is a tensor defined by
Therefore, to find the field equations of modified GR, we must add
to the right-hand side of Einstein’s field equations of GR.
Finally, we have to relate
to the torsion tensor. In NLG, the components of
measured by the preferred observers of the theory with adapted tetrads
are associated with the corresponding measured components of
that are directly connected to the torsion tensor, and its expression has been discussed in detail in [
4]. That is [
24,
25],
where
Here,
is the basic causal kernel of NLG that in essence must be determined via observation [
4],
is a constant dimensionless parameter, and
is the torsion pseudovector,
where
is the Levi–Civita tensor.
Nonlocal gravity (NLG) is thus a classical extension of GR that is highly nonlinear as well. Linearized NLG has been investigated in detail [
4]. Within the Newtonian regime of NLG, it appears possible to account for the rotation curves of nearby spiral galaxies as well as for the solar system data [
27,
28,
29,
30,
31]. Beyond the linear domain, no exact solution is known except for the trivial result that in the absence of gravity we have Minkowski spacetime [
26]. On the other hand, it is possible that certain nonlinear features of NLG that belong to the strong-field regimes such as those involving black holes or cosmological models may indeed survive in the local limit of the theory. It is therefore interesting to explore this limiting case of NLG.
To come up with the local limit of NLG, let us assume that the kernel in Equation (
20) is proportional to the 4D Dirac delta function, namely,
then,
, where
is a dimensionless scalar function. Therefore,
and the constitutive relation takes the form
Here, the susceptibility function
is a characteristic of the background spacetime just as
and
are features of the medium in electrodynamics. In general, the local electric permittivity
and magnetic permeability
functions are expected to preserve significant features of the electrodynamics of media such as spatial symmetries and temporal dependence. Similarly,
is expected to preserve the characteristics of the background spacetime. Ultimately,
must be determined based on observational data [
32,
33].
For
, we recover TEGR; otherwise, we have a natural generalization of GR that contains a new function
. Indeed, Equation (
25) implies that to have GR as a limit, we must impose the requirement that
. In this local limit of nonlocal gravity, explicit deviations from locality have vanished; however, nontrivial aspects of NLG may have survived through
, which would be interesting to study. Consequently, we explore the cosmological implications of this local limit of NLG. Spatially homogeneous and isotropic (FLRW) cosmological models have been treated in [
32,
33] in connection with
tension. Therefore, we concentrate here on a class of spatially homogeneous but anisotropic spacetimes.
3. Anisotropic Models
Let us consider a Bianchi type I model with a metric of the form
where
X,
Y, and
Z are functions of time
t. This spacetime is spatially homogeneous, with three spacelike commuting Killing vector fields
,
, and
. A detailed discussion of such spacetimes is contained in Chapter 13 of [
39]; for a recent discussion within the context of teleparallelism, see [
40].
Einstein’s gravitational field equations are
where
is assumed to be due to the presence of a comoving perfect fluid with density
and pressure
,
and
is the cosmological constant. With respect to the system of coordinates
, the perfect fluid is comoving with
; hence,
Moreover, the Einstein tensor
is diagonal as well with components
It is interesting to work out the Kretschmann scalar
,
for metric (
26). The result is
Detailed discussions of the GR solutions of these models with
for dust (
) can be found, for instance, in [
41,
42], Section 5.4 of ref. [
43], and Section 12.15 of ref. [
44]. We give a brief description of these solutions in
Section 4 in connection with cosmic deceleration.
We are interested in the extended GR framework. Therefore, consider the class of observers that are spatially at rest with adapted tetrad
field given by
where the spatial axes point along the Cartesian coordinate directions. We have
We compute the Weitzenböck torsion tensor (
4) in this case, and we find
,
, and the only nonzero components can be obtained from
Similarly, we have
,
, and the only nonzero components of
can be obtained from
It follows from these results that the torsion vector is given by
while the torsion pseudovector
in this case.
The calculations of contorsion (
7) and the auxiliary torsion (
9) tensors produce similar results. That is,
,
, and the only nonzero components of
can be obtained from
Moreover,
,
, and the only nonzero components of
can be obtained from
In the local limit of NLG, the constitutive relation of modified TEGR is given by
, where the gravitational susceptibility
S is a property of the background spacetime. In the case of the homogeneous time-dependent background (
26), we assume that
S is a function of time
t. Therefore,
,
, and the only nonzero components of
can be obtained from
We can now compute
given in Equation (
14) and
given in Equation (
18). The results are that these quantities are diagonal with elements
For
, however, we find
and
Collecting everything, the modified GR field Equation (
18) can be expressed as
and
3.1. Field Equations
To express the gravitational field equations for the anisotropic models under consideration in a more tractable form, it is useful to consider
where
and note that
Let us add Equations (
50)–(
52) to obtain
Using Equation (
49), we can write
or
Another interesting result is obtained by writing Equation (
49) as
and taking the time derivative of both sides. From the relation
we find
Using Equation (
49), we finally obtain
Equations (
57) and (
60) are important consequences of the modified field equations. Furthermore, let us define a new temporal variable
by
hence, the spacetime metric in
coordinates is
where
S is now considered, by an abuse of notation, a function of
. For instance, let us suppose
; then,
, and in the above metric we have in this case
. For metric (
62), the Kretschmann scalar
given by Equation (
35) can be expressed in terms of the new temporal variable
using
and
The gravitational field equations can now be written in terms of the temporal variable
as
To solve Equations (
66)–(
68), let us subtract, for instance, Equation (
66) from Equation (
67) to obtain
which with
can be written as
Hence,
where
is a constant of integration and similar results hold for the other metric functions.
Finally, in terms of temporal variable
, Equation (
57) can be written as
3.2. Special Solutions
We now explore some cases of particular interest.
3.2.1. de Sitter
Let us first consider de Sitter’s solution with
where
is a nonzero constant. The field equations imply
Therefore,
and
S must be constant. It follows that de Sitter spacetime is not a solution of the modified TEGR since the susceptibility function is independent of time while the background spacetime is dynamic. This is in agreement with the fact that de Sitter spacetime is not a solution of NLG [
25].
3.2.2. Kasner
Let us next consider the standard Kasner metric [
45,
46]
In GR, this empty universe model is a solution of the field equations with
and
. Note that with
, say, and hence
, we recover flat spacetime. Therefore, we can assume
; that is,
It follows from field Equations (
49) and (
50) that
etc. Because
and
together with two other cyclically related terms that vanish, we find that
,
P, and
S must be constants such that
Therefore, Kasner’s spacetime with constant
S is not a solution of the modified TEGR.
3.2.3. Flat FLRW Model
We now consider
. Then, field Equations (
49) and (
50) imply
A thorough treatment of these equations is contained in a recent paper [
32], where they were employed with
in a detailed discussion of the implications of the modified Cartesian flat cosmology in connection with
tension.
It is important to emphasize that in these time-dependent solutions considered thus far, S must be dependent upon time as well; otherwise, we do not have a physically meaningful solution of the theory.
5. Solution for Dynamic Dark Energy with
Let us imagine a universe that in the absence of the cosmological constant (
) is dominated by dynamic dark energy with
. Therefore, Equation (
60) implies
where
refers to some fiducial epoch in the expansion of the universe. The pressure of dark energy is always negative,
, and we assume on the basis of Equation (
95) that
where
is a constant with the dimensions of time, and henceforth we employ
as the new dimensionless temporal variable. To explore the anisotropic acceleration of this dark energy universe model, we assume for the sake of simplicity that
. Thus, we have a cylindrical model with the
z direction as the main direction of anisotropy. Let us note here that for
our dynamic dark energy source in effect reduces to a cosmological constant.
With
, Equation (
68) reduces to
This nonlinear equation can be easily solved with
where
are integration constants. Next, Equation (
65) implies
or
where
and
Hence,
, where
is an integration constant. Therefore,
One can check that this is the general solution of the field equations in the present case. For these solutions, the Kretschmann scalar
can be written as
where
We are interested in the behavior of the acceleration of these anisotropic solutions.
For the deceleration parameters of this model, we find
At late times
,
, and these expressions reduce to
. However, for finite
there could be anomalous behavior when
or
. If
, then
, while
. On the other hand, if
, then
, while
. To give an example of the former situation, consider, for instance, the case where
with
and
. In this case,
. Then,
and
. The universe starts from a singular pancake state at
and expands with infinite acceleration in the
x and
y directions but with zero acceleration in the
z direction. Indeed, inspection of the Kretschmann scalar
given by Equation (
103) reveals that
diverges at
provided
.
For an example of the case where
, let us consider, for instance,
and
. Then, the universe starts from a cigar state at
with infinite acceleration in the
z direction but with finite acceleration
in the
x and
y directions assuming that
. The Kretschmann scalar (
103) remains finite in this case.
These results could be interesting in connection with recent observational evidence in favor of anomalous anisotropic acceleration of bulk flow in the local universe [
34,
35,
36,
37,
38].