Local Limit of Nonlocal Gravity: A Teleparallel Extension of General Relativity

We describe a general constitutive framework for a teleparallel extension of the general theory of relativity. This approach goes beyond the teleparallel equivalent of general relativity (TEGR) by broadening the analogy with the electrodynamics of media. In particular, the main purpose of this paper is to investigate in detail a local constitutive extension of TEGR that is the local limit of nonlocal gravity (NLG). Within this framework, we study the modified FLRW cosmological models. Of these, the most cogent turns out to be the modified Cartesian flat model which is shown to be inconsistent with the existence of a positive cosmological constant. Moreover, dynamic dark energy and other components of the modified Cartesian flat model evolve differently with the expansion of the universe as compared to the standard flat cosmological model. The observational consequences of the modified Cartesian flat model are briefly explored and it is shown that the model is capable of resolving the H_0 tension.


I. INTRODUCTION
To explain current astronomical observations, dark matter is apparently necessary to describe the dynamics of galaxies, clusters of galaxies, and structure formation in cosmology.
Similarly, dark energy seems necessary to explain the accelerated expansion of the universe.
In the benchmark model of cosmology, the energy content of the universe comprises about 70% dark energy, about 25% dark matter and about 5% visible matter.The physical nature of dark matter and dark energy is unknown at present.One or both dark aspects could conceivably be characteristic of the gravitational interaction.It therefore seems reasonable to attempt to modify Einstein's general relativity (GR) on the scales of galaxies and beyond in order to account for the observational data without any need for the dark content of the universe.To this end, many modified gravity theories have been proposed [1].This paper is about a certain constitutive extension of teleparallel gravity.The resulting theory can be described as the local limit of nonlocal gravity.Nonlocal gravity (NLG) is a classical nonlocal generalization of GR patterned after the nonlocal electrodynamics of media.To motivate this approach to the modification of GR, let us note that a postulate of locality runs through the special and general theories of relativity.In special relativity, Lorentz transformations are applied point by point along the world line of an accelerated observer in Minkowski spacetime in order to determine what the observer measures [2].However, to measure the properties of electromagnetic waves, one must take their intrinsic nonlocal nature into account in accordance with the Huygens principle.Moreover, Bohr and Rosenfeld [3] have pointed out that electromagnetic fields cannot be measured at a spacetime event; instead, a certain spacetime averaging procedure is necessary for this purpose.To extend the postulate of locality to the measurement of wave phenomena in Minkowski spacetime, one must consider the past history of the accelerated observer and this leads to nonlocal special relativity theory in which the observer's memory of its past acceleration is properly taken into account [4].
The locality postulate plays an essential role in Einstein's local principle of equivalence in rendering observers pointwise inertial in a gravitational field.The gravitational field equations in general relativity are thus partial differential equations.The intimate connection between inertia and gravitation, clearly revealed via Einstein's development of the general theory of relativity, implies that the universal gravitational interaction could be nonlocal as well.That is, the gravitational field equations could allow for the gravitational memory of past events in such a way that the local gravitational field would then satisfy partial integro-differential field equations.
Einstein's GR, as a field theory of gravitation, was modeled after Maxwell's electrodynamics.A nonlocal extension of general relativity theory has been developed that is modeled after the nonlocal electrodynamics of media [5,6].The nonlocal constitutive kernel in the electrodynamics of media has its origin in atomic physics [7][8][9]; however, no analogous atomic medium exists in the gravitational case.Therefore, the nonlocal kernel in the gravitational case must be determined from observation.A comprehensive account of the resulting nonlocal gravity (NLG) theory is contained in Ref. [10].A significant observational consequence of this classical nonlocal generalization of Einstein's theory of gravitation is that the nonlocal aspect of gravity in the Newtonian regime of the theory appears to simulate dark matter [11][12][13][14][15].
The classical nonlocal extension of GR can be accomplished through the framework of teleparallelism.Briefly, we start with GR and consider a gravitational field where events are characterized by an admissible system of spacetime coordinates x µ with metric ds 2 = g µν dx µ dx ν . ( Free test particles and null rays follow geodesics in this spacetime manifold.Here, Greek indices run from 0 to 3, while Latin indices run from 1 to 3; moreover, the signature of the metric is +2.We use units such that c = 1, unless specified otherwise. In this spacetime, we choose a preferred set of observers with adapted tetrads e µ α(x) that are orthonormal; that is, In our convention, hatted indices enumerate the tetrad axes in the local tangent space, while indices without hats are normal spacetime indices; moreover, η αβ is the Minkowski metric tensor given by diag(−1, 1, 1, 1).
This curvature-free connection is such that ∇ ν e µ α = 0, where ∇ denotes covariant differentiation with respect to the Weitzenböck connection.Our preferred tetrad frames are thus parallel throughout spacetime; that is, the Weitzenböck connection renders spacetime a parallelizable manifold.In this teleparallelism framework [17][18][19], distant vectors can be considered parallel if they have the same local components with respect to the preferred frame field.Furthermore, the Weitzenböck connection is metric compatible, ∇ ν g αβ = 0, since the metric can be defined through the tetrad orthonormality relation.
In our extended pseudo-Riemannian structure of GR, we have both the Levi-Civita and Weitzenböck connections that are compatible with the same spacetime metric g µν .Curvature and torsion are basic tensors associated with a given connection.The symmetric Levi-Civita connection is given by the Christoffel symbols This connection is torsion free, but has Riemannian curvature 0 R αβγδ that represents the gravitational field in GR.In our convention, a left superscript "0" is used to refer to geometric quantities directly related to the Levi-Civita connection.In particular, the gravitational field equations of GR can be expressed as [2] 0 G µν + Λ g µν = κ T µν , where 0 G µν is the Einstein tensor We denote the symmetric energy-momentum tensor of matter by T µν ; moreover, Λ is the cosmological constant and κ := 8πG/c 4 .In Eq. ( 6), the Ricci tensor 0 R µν = 0 R α µαν is the trace of the Riemann tensor and the scalar curvature 0 R = 0 R µ µ is the trace of the Ricci tensor.
The spacetime torsion tensor associated with the Weitzenböck connection is given by Furthermore, the difference between two connections on the same manifold is always a tensor.
Hence, we define the contorsion tensor The metric compatibility of the Weitzenböck connection means that contorsion is related to torsion; that is, The contorsion tensor is antisymmetric in its last two indices; in contrast, the torsion tensor is antisymmetric in its first two indices.It turns out that the curvature of the Levi-Civita connection 0 R µνρσ and the torsion of the Weitzenböck connection C µνρ are complementary aspects of the gravitational field in this extended framework [10].
It is important to point out a certain natural connection between the gravitational field strength C µνρ and the electromagnetic field strength Writing the torsion tensor as we note that for each α = 0, 1, 2, 3, we have here an analog of the electromagnetic field tensor defined in terms of the vector potential The traditional construction of the GR field equations is based on the Levi-Civita connection.On the other hand, the Levi-Civita connection is the sum of the Weitzenböck connection and the contorsion tensor in accordance with Eq. ( 8).Hence, GR field equations can be expressed in terms of the torsion tensor.This formulation of Einstein's theory is naturally analogous to Maxwell's electrodynamics.Indeed, the result is the teleparallel equivalent of general relativity (TEGR) to which we now turn.
It is possible to express the Einstein tensor as [5,6,10] where the auxiliary torsion field H µνρ is defined by The auxiliary torsion tensor C αβγ , where C µ := C α µα = −C µ α α is the torsion vector, is antisymmetric in its first two indices just like the torsion tensor.Moreover, as in GR, g := det(g µν ) and √ −g = det(e µ α).Einstein's field equations (5) expressed in terms of torsion thus become the TEGR field equations where T µν is the traceless energy-momentum tensor of the gravitational field and is defined by The antisymmetry of H µν α in its first two indices leads to the law of conservation of total energy-momentum tensor, namely, which follows from taking partial derivative ∂/∂x µ of Eq. ( 13).Finally, it is important to note that while GR is based on the metric tensor g µν , TEGR is based on the orthonormal tetrad frame field e µ α(x) that is globally parallel via the Weitzenböck connection.The teleparallel equivalent of general relativity (TEGR) is the gauge theory of the Abelian group of spacetime translations [20][21][22].Though nonlinear, TEGR is therefore in a certain sense analogous to Maxwell's equations in a medium with a simple constitutive relation [9].
That is, C µν α is, as noted before, similar to the electromagnetic field F µν , where (E, B) → F µν , while, H µν α is similar to the electromagnetic excitation H µν , where (D, H) → H µν .Moreover, we can look upon Eq. ( 12), namely, as the local constitutive relation of TEGR, since it connects H αβγ to C αβγ .
When we study the electrodynamics of media, we keep the fundamental field equations intact and change only the constitutive relation appropriate for the medium at hand.We adopt this approach in extending TEGR; that is, we simply modify Eq. ( 16) in the rest of this paper.This approach differs from other extensions of TEGR that involve, for instance, the introduction of scalar fields into the theory.That is, scalar-torsion theories of gravity, which are analogues of scalar-tensor theories that extend GR, have been studied by a number of authors; for recent reviews, see [23,24].
It is important to recognize that we could have arrived at TEGR using any other smooth orthonormal tetrad frame field λ µ α(x).This circumstance is natural, since GR only depends on the metric tensor g µν .At each event, the two tetrad frame fields λ µ α(x) and e µ α(x) are related by a six-parameter element of the local Lorentz group involving three boosts and three rotations; that is, λ µ α(x) = L α β (x) e µ β (x).This pointwise 6-fold degeneracy is generally removed when we modify the constitutive relation of TEGR.In our teleparallel extension of GR, the modified theory is then invariant only under the global Lorentz group.

II. CONSTITUTIVE EXTENSION OF TEGR
To maintain the analogy with electrodynamics, we retain the gravitational field equations of TEGR, while the constitutive relation is modified.This means, in effect, that we replace H in Eqs. ( 13) and ( 14) by H given by where N µνρ = −N νµρ is a tensor that is related to the torsion tensor C µνρ .For the moment, let us find the new extension of GR based on the new tensor field N µνρ .The gravitational field equations take the form where T µν is the traceless energy-momentum tensor of the gravitational field in this case.
We have where Q µν is a traceless tensor defined by The total energy-momentum conservation law can now be expressed as To find the modified GR field equations, let us start with Eq. ( 17) and substitute in the Einstein tensor (11) to get where we have used Eq.(18).Here, N µν is a tensor defined by It is natural to split the modified GR field equations into its symmetric and antisymmetric parts; that is, we have the modified Einstein equations and the constraint equations We thus have 16 field equations for the 16 components of the tetrad frame field.Of the 16 components of the fundamental tetrad e µ α, 10 fix the components of the metric tensor g µν via the orthonormality condition, while the other 6 are local Lorentz degrees of freedom (i.e., boosts and rotations).Similarly, the 16 field equations of modified GR for the 16 components of the fundamental tetrad e µ α naturally split into 10 modified Einstein equations plus 6 integral constraint equations for the new tensor N µνρ .
We have here a general framework for the teleparallel extension of GR.It remains to specify the exact connection between N µνρ and C µνρ .A nonlocal relation has led to nonlocal gravity(NLG) theory [5,6,10].The local limit of this nonlocal relation is the main focus of the present investigation.

A. Nonlocal Gravity
In nonlocal gravity (NLG), we assume, in close analogy with the nonlocal electrodynamics of media, that the components of N µνρ , as measured by the fundamental observers of the theory with adapted tetrads e µ α, must be physically related to the corresponding measured components of X µνρ that is directly connected to the torsion tensor [10,22,25].That is, where K(x, x ) is the basic causal kernel of NLG and Here, p = 0 is a constant dimensionless parameter and Čµ is the torsion pseudovector defined via the Levi-Civita tensor E αβγδ by It is important to remark that the only known exact solution of NLG is the trivial solution, namely, we recover Minkowski spacetime in the absence of the gravitational field.Thus far, it has only been possible to show that de Sitter solution is not an exact solution of NLG [25].
Linearized NLG has been treated in [10]; moreover, nonlocal Newtonian cosmology has been treated within the Newtonian regime of NLG [26,27].
There are indeed many other nonlocal models of gravity and cosmology; see, for instance [28][29][30][31] and the references cited therein.

B. Local Limit of NLG
The local limit of Eq. ( 27) can be obtained by assuming that the kernel is proportional to the 4D Dirac delta function, namely, where S(x) is a dimensionless scalar function that must be determined on the basis of observational data.In this case, the nonlocal constitutive relation (27) reduces to Moreover, Eq. ( 17) takes the form If S(x) = 0, we recover TEGR; otherwise, we have a generalization of GR.Of course, the local and linear relation (31) can be generalized; that is, Such six-index gravitational constitutive tensors as χ µνρ αβγ have been studied and classified in [17].
The new local constitutive relation enlarges TEGR, which is equivalent to a pure spin-2 theory, namely, GR, by the addition of a scalar function.It is clear from Eq. ( 32) that 1 + S > 0; otherwise, the new theory will not have GR as a limit.There is no field equation for S(x); however, we may be able to select S(x) on the basis of certain consistency conditions in analogy with the electrodynamics of media.Eventually, we have to determine S(x) from observation.In nonlocal gravity, we must ultimately determine the fundamental nonlocal kernel of the theory from the comparison of NLG with observational data [11][12][13][14].

C. Analogy with the Electrodynamics of Media
In the electrodynamics of media, especially magnetic media, the phenomena associated with hysteresis cannot be ignored; therefore, the constitutive relations are in general nonlocal [8].However, in most applications of the electrodynamics of media, one uses the simple relations D = (x)E and B = µ(x)H.Presumably, these local limits of nonlocal constitutive relations of linear media capture some important aspects of the general nonlocal problem.
The same is expected to hold in the gravitational case.We can partially compensate for the lack of exact solutions of NLG in the case of strong gravitational fields by searching for solutions of the new local theory in the areas of cosmology and black hole physics.
The local quantities (x) and µ(x) are characteristics of the medium in electrodynamics; similarly, S(x) is characteristic of the background spacetime in the new local theory.The functional form of S(x) must therefore be consistent with the nature of the background spacetime.In analogy with electrodynamics, we call S(x) the susceptibility function and tentatively call the new theory "Modified TEGR", since we simply add an extra term to the constitutive relation of TEGR.It must be mentioned that other modified teleparallel theories of gravity have been considered by a number of authors; see, for instance [32][33][34][35] and the references cited therein.
Modified TEGR has not been derived from a stationary action principle.Indeed, many important physical systems have dynamics that cannot be derived from stationary action principles.Dissipative processes generally lack Lagrangian descriptions.An example of this situation in fluid dynamics is the Navier-Stokes system [36].In the electrodynamics of media, averaging procedures are necessary to determine the contributions of the polarization and magnetization of the medium to the electromagnetic field.As a consequence, there is no classical field theory Lagrangian in the literature for the electrodynamics of media.This circumstance extends to nonlocal electrodynamics of media as well.In the gravitational case, the field equations of nonlocal gravity (NLG) include a certain average of the gravitational field over past events; in essence, it is this nonlocal aspect of the theory that simulates effective dark matter.The analogy with the electrodynamics of media suggests that theories that modify general relativity in this way do not have Lagrangian descriptions.Previous work in this direction implies that an action principle for NLG would be in conflict with causality [6].In the local limit of nonlocal gravity considered in the present work, the averaging is replaced by a local weight function S(x) that is reminiscent of (x) and µ(x) of the local electrodynamics of media.Can the field equations for such a modified theory of gravitation be derived from an action principle?This seems rather unlikely based on the analogy with the electrodynamics of media, but a definitive answer is not known at present.

III. MODIFIED TEGR (MTEGR)
To gain insight into the nature of this theory, we must solve its field equations which are obtained from the substitution of the auxiliary torsion field (32) in Eq. ( 18).The resulting field equations are indeed satisfied in the case of Minkowski spacetime with Cartesian coordinates x µ and preferred tetrad frame field e µ α(x) = δ µ α in the absence of any sources (T µν = 0 and Λ = 0); here, S(x) = 0 is a function of the background Cartesian coordinates.
Let us therefore look for an approximate solution of modified TEGR field equations with S(x) = 0 and Λ = 0 that is a first-order perturbation about Minkowski spacetime.The purpose of this section is to develop and study the resulting general linear weak-field solution of modified TEGR.

A. Linearization
We begin by linearizing the theory about Minkowski spacetime; that is, we assume the fundamental frame field of the theory is given by [10,37] In the linear perturbing field ψ µν (x), the distinction between spacetime and tetrad indices can be ignored at this level of approximation.Here, the 16 components of ψ µν represent the gravitational potentials of a finite source of mass-energy that is at rest in a compact region of space.We thus break the invariance of the theory under the global Lorentz group by fixing the background inertial frame of reference to be the rest frame of the source.We define the symmetric and antisymmetric components of ψ µν by The tetrad orthonormality condition then implies As in GR, we introduce the the trace-reversed potentials where h = −h and The gravitational potentials are now in suitable form to calculate the gravitational field quantities to first order in the perturbation.The linearized torsion tensor is and the torsion vector and pseudovector are given by Similarly, the auxiliary torsion tensor is and the Einstein tensor can be written as where 2 := η αβ ∂ α ∂ β and ∂ ν 0 G µν = 0, since the auxiliary torsion tensor is antisymmetric in its first two indices.
Let us recall that in general the linearized form of modified GR field equations (25) and ( 26) are given by [10] and respectively.These imply, just as in GR, the energy-momentum conservation law for massenergy, namely, ∂ ν T µν = 0. Writing Eq. (31) as the modified GR field equations become and With X µσν given by Eq. ( 28) and p = 0, we find that in the linear regime, The torsion pseudovector Čσ is the dual of C and To have regular second-order field equations, we must have 1 + S(x) > 0.
Let us briefly digress here and consider the possibility that 1 + S = 0.It is simple to see that source-free (i.e., T µν = 0 and Λ = 0) linearized field equations ( 49) and ( 50) are satisfied for φ µν = 0 and S = −1.Extending this result to the nonlinear case is not so simple; that is, the general source-free field equations are satisfied for S = −1 provided Čµ = 0 and It is very possible that no such solution exists.The general field equations of modified TEGR for S = −1 are first-order partial differential equations that have a peculiar form.It seems that there is no reasonable spacetime that could satisfy these conditions.
The gravitational potentials ψ µν are gauge dependent.Under an infinitesimal coordinate transformation, Therefore, and h = h − 2 α ,α .On the other hand, the gravitational field tensors C µνρ and C µνρ as well as the gravitational field equations are gauge invariant, as expected.
In general, we can impose the transverse gauge condition which does not completely fix the gauge.We can still use four functions µ such that µ = 0.
With the transverse gauge condition, we have which simplifies Eq. ( 49).

B. Newtonian Limit
In this limit, we consider weak fields and slow motions; moreover, we can formally let c → ∞.As in Ref. [37], we assume the transverse gauge condition holds and φ µν = 0.
Furthermore, we assume S = S 0 is constant.A detailed examination of the field equations ( 49) and ( 50) reveals that only the µ = ν = 0 case is important with h00 = −4Φ/c 2 and T 00 = ρ c 2 , where Φ is the Newtonian gravitational potential and ρ is the density of matter [10,37,38].The background is the Newtonian space and time; hence, S = S 0 must be a constant.We find from Eq. ( 49) that and the second field equation disappears.
Let us assume that Then, we can write the modified Poisson Eq. ( 54) as where ρ D has the interpretation of the density of dark matter in this framework.For a point mass m, say, the dark matter associated with the point mass m is located at the point mass and has magnitude Q 0 m.In the Newtonian regime of nonlocal gravity (NLG), the density of dark matter is the convolution of the density of matter with a spherically symmetric reciprocal kernel.In the local limit of NLG in the Newtonian regime, the folding with the reciprocal kernel reduces to multiplication by a constant, namely, ρ D = Q 0 ρ.For a point mass, the static spherical cocoon of effective dark matter associated with a point mass in NLG [13,14] settles on the point particle in the local limit of NLG.
In this Newtonian framework, the force of gravity on a particle of mass m is given by F = −m ∇Φ.Thus the gravitational force between two point masses has the Newtonian form except that it is augmented by a constant factor of (1 In contrast to the Newtonian regime of NLG [11,14], the local limit of NLG is not capable of explaining the rotation curves of spiral galaxies; however, we are mainly interested in the cosmological aspects of this locally modified TEGR.

C. Free Gravitational Waves
In linearized nonlocal gravity, free gravitational waves satisfy the nonlocal gravitational wave equation [10] hij (x) + W (x − y) hij,0 (y) where W is a certain kernel of NLG.This result follows from the source-free linearized field equations once the gauge conditions hµν ,ν = 0, h0µ = 0 and φ µν = 0 are imposed.The nonlocal wave Eq. ( 57) is reminiscent of a damped oscillator whose velocity would be represented by ∂ hij /∂t.The wave amplitude decays as the wave propagates due to fading memory in NLG.
In the case of source-free field Eqs. ( 49) and (50), we can simply impose the transverse gauge condition and φ µν = 0. To enforce the additional gauge conditions h0µ = 0, we must require that S is only a function of time, i.e. S = S(t).With these assumptions, the field equations then become which are consistent with h0µ = 0. Here, we have used and For hij , Eq. ( 58) where The transverse gauge condition reduces to hij ,j = 0, which is consistent with the propagation Eq. (62).
To solve Eq. ( 62), we assume that each component of the wave function hij has the form where H satisfies a harmonic oscillator equation with time-dependent damping.That is, where k = | k|.For a positive constant γ, we have the standard damped harmonic oscillator.
On the other hand, the solution will always be damped if γ(t) > 0, since the energy associated with the harmonic oscillator constantly decays, namely, As in NLG [10,39,40], free gravitational waves in our modified TEGR theory are indeed damped provided dS/dt > 0, since 1 + S > 0. A comment is in order here regarding the fact that in a dynamic time-dependent spacetime, dS/dt > 0 appears to be a natural requirement in order to maintain 1 + S > 0; that is, dS/dt > 0 ensures that an initially positive 1 + S will remain positive for all time.The WKB treatment of wave Eq. ( 62) is contained in Appendix A.
Beyond linearized NLG, no exact nonlinear solution of nonlocal gravity is known at present; in fact, some of the difficulties have been discussed in [41].In the local limit of NLG, we have the prospect of finding exact solutions that explore strong field regimes involving cosmological models and black holes.Furthermore, parity-violating solutions may exist in which the torsion pseudovector Č that appears in the constitutive relation (32) of modified TEGR may be nonzero.Modified theories that exhibit gravitational parity violation have been of current interest [42,43].
The rest of this paper is devoted to finding the simplest exact cosmological models of modified TEGR.The current benchmark model of cosmology is the flat FLRW solution; therefore, we focus on the way the FLRW model is modified in our approach.We begin with the simplest modified conformally flat spacetimes.

IV. MTEGR: CONFORMALLY FLAT SPACETIMES
We are interested in exact solutions of modified TEGR that have a conformally flat metric where x µ = (η, x i ) and U (x) is a scalar.We use conformal time x 0 = η in this section to agree with standard usage in cosmology and choose preferred observers that are at rest in space and their adapted tetrad axes point along the coordinate directions We seek an exact solution of the field Eqs. ( 25) and ( 26) in this case.
The torsion tensor can be simply computed and is given by where U µ := ∂ µ U in our convention.Similarly, the torsion vector, the contorsion tensor and the auxiliary torsion tensor are Moreover, in this case, Čα = 0; therefore, The Einstein tensor for conformally flat spacetimes is given by [44] 0 where We assume that the energy-momentum tensor of matter is given by a perfect fluid of energy density ρ and pressure P such that where u µ is the 4-velocity vector of the perfect fluid.As in the standard cosmological models, we assume the fundamental particles are comoving with the preferred observers and are thus spatially at rest, namely, and ρ and P are functions of conformal time η.
Let us return to the field Eqs. ( 25) and ( 26) and note that the constitutive relation (31) in this case reduces to Therefore, Using these relations in Q µν and N µν , we find Q µν is symmetric and The second field Eq. ( 26) implies N [µν] = 0, which means Hence, dS ∧ dU = 0.This equation has the natural solution that dS is proportional to dU , namely, The symmetric field Eq. ( 25) is simply Einstein's field equation together with a new source Let us write Eq. ( 25) in the form The left side of this equation can be written as To proceed, we note that in the context of standard flat cosmology, the natural choice for U (x) would be to assume U = U (η).Therefore, S = S(η) as well, in agreement with the time dependent nature of the background.Moreover, we introduce the scale factor a, With these assumptions, Eq. ( 25) has nonzero contributions for indices (µ, ν) = (0, 0) and (µ, ν) = (i, j).We find 3 and respectively.When S = 0, we have the standard GR results, as expected.
It is useful to express these equations in the more traditional form of standard flat cosmology.

V. MODIFIED FLAT COSMOLOGY
In the traditional flat model, we assume a metric of the form in terms of cosmic time t.With dt = adη, the metric can then be written as ds 2 = a 2 η µν dx µ dx ν , just as in the previous section.

Let us write
The traditional forms of the dynamical equations of our model become respectively.Next, differentiating Eq. (89) with respect to cosmic time t and using Eq. ( 90), we find which implies, for ρ + P ≥ 0 and dS/dt > 0, that ρ monotonically decreases as the universe expands.Indeed, we expect that as t → ∞, the scale factor a(t) approaches infinity and ρ and P approach zero.Finally, it follows from Eqs. ( 89) and (91) that Is the modified TEGR framework consistent only with the flat cosmological model?
It may be possible to extend our results to the standard Friedmann-Lemaître-Robertson-Walker (FLRW) models.This is the subject of the next section.

VI. MODIFIED FLRW COSMOLOGY
In our modified TEGR framework, it may be possible to extend the flat model of the previous section to the standard cosmological models.We begin with where a(t) is the scale factor, and k = 1, −1, or 0, for the closed, open, or flat FLRW model, respectively.For metric (93), the nonzero components of the Einstein tensor are given by and The fundamental observers are at rest in space and carry adapted tetrads that point along the coordinate directions.That is, and After some work, details of which we relegate to Appendix B, we find that the constraint equations imply k Ṡ = 0. Therefore, the equations of modified TEGR are consistent with S = S(t) in this case provided k = 0 and we thus recover the flat model of the previous section.On the other hand, it is possible that k = 0, in which case S must be constant.In the latter case, the main equations of FLRW model for constant S become For S = 0, we get the standard equations of FLRW model.
In a spatially homogeneous and isotropic background that varies with time, we expect that the susceptibility S would be time dependent as well, in close analogy with the electrodynamics of media.That is, a constant S would naturally correspond to a background that is independent of time, while a time-varying S(t) would normally belong to a dynamic background.It is therefore thought-provoking that for Ṡ = 0 only the flat model of standard cosmology is allowed within the framework of the modified theory of gravitation under consideration here.
It is interesting to explore some of the general consequences of our modified flat model.This is the purpose of the next section.

VII. IMPLICATIONS OF THE MODIFIED FLAT MODEL
To interpret the main equations of the modified flat model physically, let us first recall the implications of the standard model with S = 0.In the standard benchmark model, the positive cosmological constant (Λ > 0) can be replaced with a perfect fluid source with T µν = P Λ g µν , where P Λ = −ρ Λ = −Λ/(8πG); that is, the cosmological constant represents dark energy with an equation of state parameter w Λ = −1.As the universe expands, the energy densities of the matter and radiation monotonically decrease such that the cosmological constant eventually becomes the dominant source.For t → ∞, Λ generates de Sitter's solution of GR.Indeed, in the standard model (S = 0), the universe asymptotically approaches a de Sitter phase of exponential expansion such that a(t) increases as exp ( Λ/3 t).On the other hand, it is simple to demonstrate that de Sitter spacetime is not a solution of our modified flat cosmological model so long as dS/dt = 0. We prove this assertion by contradiction.
Let us start with Eqs. ( 89) and (90) that govern the dynamics of the modified flat model of cosmology and set ρ = P = 0.The result is the system of ordinary differential equations For an expanding universe, Eq. ( 101) implies Differentiating this equation with respect to time, we get Multiplying this relation by 2(1 + S) > 0 and subtracting the resulting equation from Eq. ( 102), we find which, in the light of Eq. ( 101), implies If dS/dt = 0, we can divide both sides of Eq. ( 106) by dS/dt; then, we are left with a relation that directly contradicts Eq. ( 103).We conclude that de Sitter spacetime is not a solution of our modified flat model.This is consistent with the fact that de Sitter spacetime is not a solution of nonlocal gravity (NLG) [25].Consequently, the universe model under consideration here will never asymptotically approach a de Sitter phase.Henceforth, we assume Λ = 0. To incorporate dark energy in our model, we resort to a perfect fluid component with positive energy density ρ de > 0 and negative pressure P de = w de ρ de such that w de < 0.
It is interesting to note that in Eq. ( 90), the important additional term due to dS/dt acts like added pressure.For S = constant, the basic local thermodynamic relation for adiabatic processes, namely, dU = −P dV is satisfied.That is, imagine an amount of energy U of the background perfect fluid contained within a local sphere of radius that expands with the universe; then, with we have On the other hand, the variation in S is related to variation in entropy S.More generally, the basic thermodynamic relation for a nonadiabatic process is where T is the temperature and S is the entropy.Writing the change in heat as δQ := TdS, we find in this case Usually, heat is generated by friction.As the universe expands and S increases, the corresponding entropy decreases.For a discussion of entropy variation in cosmology, see [45].
Within the context of standard cosmology, we expect that entropy increases as the universe expands.However, we deal here with comoving space and it can be shown that the local comoving entropy of matter and radiation remains constant in the standard model of cosmology and one can write the local law of energy conservation (or energy continuity equation) in the absence of heat flow.On the other hand, we find in our modified cosmology that there is more deceleration accompanied with negative entropy production.Normally, this should mean more order and less chaos.
Let us write the main equations of the modified flat model in terms of the Hubble (H) and deceleration (q) parameters that are defined as Then, in the absence of the cosmological constant we have It is clear from Eq. ( 113) that the extra term involving dS/dt > 0 contributes to the deceleration of the universe in the modified flat model.
To incorporate the accelerated expansion of the universe in the framework of the modified flat model, the dark energy component should be such that in accordance with Eq. ( 113).It follows from Eq. ( 114) that 1 + 3 w de < 0 or w de < −1/3.
Dark energy in the modified theory is thus generally different from the way it is incorporated in the standard model.
In terms of matter, radiation and dark energy components, the main equations of the modified flat model, namely, Eqs. ( 89) and (90), become and respectively.Here, as in the standard flat cosmological model, we have expressed the energy density and pressure as the sum of the matter, electromagnetic radiation and dark energy content of the model, namely, As before, we can differentiate Eq. ( 115) with respect to cosmic time t and combine the result with Eq. (116) to derive the analogue of Eq. (92) in this case, namely, Let us recall that in the standard model (S = 0), we have a consistent system of equations for the scale factor and it is sufficient to solve Eq. ( 89) with S = 0 to find a(t); indeed, this comes about because the standard energy continuity equation is valid for each component.
The same kind of consistency can be achieved in the modified model if we assume that for each component the modified energy continuity equation holds, namely, As usual, for each component the pressure is assumed to be proportional to the corresponding energy density with proportionality constant w.Then, Eq. (119) has the solution which for S = 0 reduces to the result of the standard cosmological model.Here, t = t 0 represents the present epoch in cosmology and we assume a(t 0 ) = 1.We note that the restrictions on w de in the modified flat model are different from those of the standard cosmological model; in particular, w de = −1 is allowed, which implies The main equation of the modified flat model is thus obtained by substituting Eq. (120) in Eq. ( 115), namely, To cast this equation in a form that would be similar to the standard flat cosmological model, let us define Ωi to be the ratio of ρ i (t 0 ) to a certain constant energy density ρ c given by where H0 is the Hubble parameter of the modified model at the present epoch.Then, Eq. ( 122) at the present epoch can be written as and the main evolution equation of our modified flat model can be expressed as This is the result that we need in order to confront the modified flat cosmological model with observational data; that is, by a suitable comparison of the numerical solutions of this equation with current cosmological data, one should be able to choose an appropriate susceptibility function S(t) and determine the energy content of the universe.In this process, the density parameters evolve differently due to dS/dt > 0 and the Ωi will in general turn out to be different from the density parameters of the standard benchmark model.We emphasize that the density parameters Ωm , Ωr and Ωde in this equation refer to matter, radiation and dark energy, respectively; in particular, Ωm here refers to the total of the visible baryonic Ωb and effective cold dark matter Ωc contributions, i.e., Ωm = Ωb + Ωc , at the present epoch.
We can interpret Eq. (125) as the total energy equation for a one-dimensional mechanical system.The net energy is zero, which is the sum of a positive effective kinetic energy part (K ef f ) plus a negative effective potential energy part (V ef f ), namely, and respectively.The general behavior of the negative effective potential energy is clear: its shape is roughly similar to an inverted harmonic oscillator potential; moreover, near a = 0, it is dominated by −1/a 2 due to radiation, while near a = ∞, the dark energy component dominates.In this way, we can illustrate the general behavior of a(t): Near the t = 0 singularity where a = 0, the universe expands very rapidly until V ef f approaches its maximum value where the expansion slows and there is a loitering phase around the maximum of the effective potential after which the universe expands rapidly again and may go through an accelerating phase depending on the strength of the dark energy component that could render q negative.The important point here is that this general picture is independent of the form of S(t).However, the details do depend on S(t), especially where the expansion of the universe might actually accelerate whenever q < 0, namely, To illustrate the nature of our main result, let us assume our cosmology contains only one component with an equation of state parameter w > −1; that is, ρ w (t 0 ) = ρ c and Ωw = 1.
Then, Eq. ( 125) reduces to a simple ordinary differential equation that can be easily solved and the solution is The Hubble (H) and deceleration (q) parameters can be simply computed in this case and we find In the case of the standard model (S = 0) for matter (w = 0), ξ(t) = t and we recover the Einstein-de Sitter model with a = (t/t 0 ) 2/3 , H = 2/(3t) and q = 1/2; for dark energy, on the other hand, −1 < w de < −1/3 and q(t) should be negative during accelerated epochs of the expansion of the universe.For dark energy with w de = −1, we find just as in Eq. (121).Finally, the case of dark energy with w de < −1 is discussed in Appendix C.
To go forward, we must compare the predictions of the modified flat model with observational data.We make a beginning in this direction in the next section; however, a more complete analysis necessitates a separate detailed investigation that is under consideration [46].
VIII.MODIFIED FLAT MODEL: H MTEGR (z) AND H 0 TENSION Within the ΛCDM framework, the standard benchmark cosmological model with S = 0 has had significant success in explaining the vast majority of observations in connection with the cosmic microwave background (CMB) [47] and large scale structure formation [48].
However, in recent years some discrepancies have been observed [49].These discrepancies could be hints for physics beyond the standard benchmark model of cosmology.The most recent discrepancy that is now under frequent discussion is related to the measurement of H 0 , the relative expansion rate of the universe at the present time [50].There is a 4-5 σ discrepancy between the measurement of the Hubble constant using local studies of the nearby supernovas, for instance, and the measurement of the recession rate using the CMB on the basis of the ΛCDM model [51].This inconsistency has opened up a new arena in cosmological studies.Many alternative models have been suggested to reconcile this tension, from the early dark energy models to late-time modified gravity theories [51].
To examine in detail the observational consequences of the modified flat model, we must integrate the ordinary differential Eq. ( 125) to determine the temporal evolution of this universe model and then compare the results with cosmological data.For this purpose, we need a separate extensive investigation that is beyond the scope of the present work [46].
On the other hand, to indicate that this model has the potential to help in the resolution of the H 0 tension, we can regard Eq. ( 125) as an algebraic relation that expresses the Hubble parameter H as a function of the cosmological redshift z, where Let us note that for the standard benchmark model while the Hubble parameter H MTEGR (z) for our modified flat TEGR model is given by Eq. ( 125), namely, Here, we have introduced The function Γ(z) is such that Γ(0) = 1, so that for the nearby universe, 0 ≤ z 1, the modified flat model is not substantially different from the standard benchmark model.This implies that to solve the problem of H 0 tension, we must resort to cosmological data in the early universe.To proceed, we adopt the simple expression Thus S(z) is a monotonically decreasing function of z; in fact, S is zero at z = ∞ and increases monotonically to a positive constant α at the present epoch (z = 0); moreover, Γ(z) is an increasing function of z, since S(z) monotonically decreases with z.
We need to compare observational data near the decoupling epoch with the predictions of Eqs. ( 134)-(136).To this end, we choose a cosmological standard candle, namely, the dimensionless number θ(z), which is the angle of view of the sound horizon.For a data point at redshift z, this can be calculated as where r d is the sound horizon at the drag epoch at which the baryons were free from the Compton drag of photons [52,53] and D(z) is the distance to the observed structure The Hubble parameter H in the calculation of r d is mostly dominated by the contribution of radiation, while the H in D is mainly calculated in the matter dominated era.Here, the sound speed is given by Before calculating these quantities for our modified flat model, let us note that for the standard flat ΛCDM model according to [47], The corrections that the local limit of NLG imposes on the above equation for θ can now be simply derived by using Eqs.( 134)-(136).That is, and As described below, we calculate the theoretical value of θ from this formula and compare the results with our observed values of θ and conclude that the early value of H 0 could possibly be larger and could resolve the H 0 tension.In particular, we note that Ωb = Ω b and Ωγ = Ω γ , as these are fixed by big-bang nucleosynthesis and CMB observations, respectively [54,55].
Consequently, c s (z) remains the same as in the standard model.We consider Ωc a parameter and calculate Ωde = 1 − Ωc − Ωb − Ωγ from the flatness condition.Moreover, we select w de = −1; this choice, as discussed in connection with Eq. (121), is permitted in the modified flat model, and it also happens to be preferred when we treat w de as a free parameter in our numerical experiments.In this way, we are left with a 4-dimensional parameter space consisting of α, β, H0 and Ωc .Within this parameter space, we compute θ| MTEGR and compare our result with observational data.To analyze the data, we employ the Monte Carlo Markov Chain (MCMC) algorithm to look for the best value of χ 2 within the space of parameters.

A. Numerical Analysis
The main purpose of our numerical work is to use some selected cosmological observations to constrain H0 , Ωc and S(z).The dataset consists of 16 observations; of these, 15 are data points for the Baryonic Acoustic Oscillation (BAO) in large-scale structures, and the remaining datum is related to the location of the first peak in the angular power spectrum of CMB temperature anisotropies.We employ CMB first peak location data from Planck [47], BAO data from 6dF galaxy survey [56], dark energy survey (DES) [57], main galaxy sample (MGS) from SDSS DR7 [58], WiggleZ [59], luminous red galaxy (LRG) from SDSS DR14 [60], Lyα [61,62], BOSSDR12 [48] and SDSS DR14 quasar surveys [63].We name this dataset CMB+BAO in our plots and table.
The simple choice of S(z) = α(1 + z) β adds two new parameters α and β to the set of parameters to constrain using our data points.Accordingly, the parameter set is: α, β, H0 and Ωc for our modified flat TEGR model (MTEGR).Searching this 4-dimensional parameter space via the MCMC algorithm for the best value of χ 2 that matches the theory with data, gives us the posterior diagram in Figure 1.While we do not impose prior conditions on the values of α and β, our results are in alignment with α > 0 and β < 0.
In Figure 2, we plot the confidence levels regarding matter density and Hubble parameter.
The green band represents the direct measurement of H 0 from the low redshift observations by Riess et al. [64].Hereafter, we refer to the work of Riess et al. [64] as R19.The red contour plots show the confidence levels for Ω c and H 0 in the ΛCDM model; indeed, in the latter case, the red contour does not touch the green band, demonstrating H 0 tension.On the other hand, the blue contours touch the green band, which means that our MTEGR model can resolve the H 0 tension.Once it becomes clear that our model can alleviate this tension, we can include Hubble constant data by Riess et al. [64] and repeat the χ 2 analysis.
In Table I, we show the best results for ΛCDM and MTEGR models with two data sets.
In the left subtable, we have the CMB+BAO data.In the right subtable, we have the  the locally modified flat model.
The constitutive relation of NLG permits a different cosmic expansion history as compared to the prediction of the standard ΛCDM benchmark model [26,27].A similar outcome is expected for the local limit of nonlocal gravity.Consequently, the locally modified TEGR cosmological models can give rise to a rich phenomenology to be confronted with observational data.In connection with recent tensions in cosmology, we note that there is current interest in quantum-inspired nonlocal cosmological models [66,67] as well as in various modified teleparallel gravity theories [68,69].
It should be mentioned that besides the H 0 tension, there is a less significant discrepancy known as σ 8 tension.The problem is related to cosmological observations that suggest a smaller amplitude in the matter perturbation in the late-time universe as compared to the prediction of ΛCDM on the basis of CMB data.The late-time observations that show this tension are mainly in the domain of weak lensing results [70][71][72] and growth-rate measure-  ments [49].The fading memory effect of NLG for all components of the universe can be a hint that the modified TEGR model can address these tensions as well.In future work, we plan to investigate numerically the cosmological implications of this model at both the background and perturbation levels.
Let e 1 and e 2 , e 1 • e 2 = 0, be constant unit vectors that are orthogonal to the direction of propagation of the null ray n; moreover, let e be a linear combination of e 1 and e 2 .Then, the solution of Eq. (A5) can be written as It follows that the torsion vector can be expressed as Moreover, for the contorsion tensor we have K 0µν = 0 and the only nonzero components can be written as Similarly, for the auxiliary torsion tensor we have C ij0 = 0, while the only nonzero components are Finally, the torsion pseudovector vanishes due to the symmetries of the torsion tensor in this particular case.
With Č = 0, the constitutive relation (31) reduces to N µνρ = S C µνρ ; hence, N ij0 = 0 and the only nonzero components of N µνρ are given by and Next, we must employ Eqs. ( 20) and ( 24) to calculate Q µν and N µν , respectively.Then, the modified Einstein's field equations ( 23) can be expressed as where To compute Q µν in this case, we first note that We find and Similarly, for N µν , we have and Finally, we can compute the components of R µν .The results are R 0i = −(x i /K)k Ṡ, R i0 = 0 and Substituting these results in Eq. (B7), we find that k Ṡ = 0. Therefore, either k = 0, as in the flat model investigated in Section V, or S = constant.In the latter case, the field equations reduce to Eqs. (99) and (100) of Section VI.

CMB+BAO+R19 data. The χ 2
and Akaike Information Criterion (AIC) values show that our model is preferred for describing the CMB+BAO+R19 dataset.In connection with future work, one can constrain the parameters of the model with the full CMB data by employing perturbation theory.

FIG. 1 :FIG. 2 :
FIG.1:Triangle plot[65] showing the posterior results by means of the MCMC algorithm using CMB+BAO dataset for the MTEGR model with S(z) = α(1 + z) β ansatz.The blue shaded regions show the 1σ and 2σ confidence levels.

F kl 0 k 1 e l 1 + f 12 (e k 1 e l 2 + e k 2 e l 1 ) + f 22 e k 2 e l 2 ]
(t, x) = [f 11 e F(e • x) [1 + S(t)] −1/2 , (A6) where f 11 , f 12 and f 22 are constants and F is a smooth function that varies slowly in the plane transverse to the direction of motion of the ray.This result is consistent with the temporal decay of the wave amplitude along the ray when S monotonically increases with time.Appendix B: Modified Standard Cosmological Models Starting with the tetrads given in Eqs.(97) and (98), one can show that C µν0 = 0 and the only nonzero components of the torsion tensor are given by

TABLE I :
Results from the MCMC algorithm for both MTEGR and ΛCDM models.In the left subtable (a), we use the CMB+BAO dataset.In the right subtable (b), we use the CMB+BAO+R19 dataset.Note that h ≡ H 0 /(100 km/s/Mpc).