# Dynamic Dark Energy from the Local Limit of Nonlocal Gravity

Javad Tabatabaei1<EMAIL_ADDRESS>Abdolali Banihashemi1
<EMAIL_ADDRESS>Shant Baghram1,a<EMAIL_ADDRESS>Bahram
Mashhoon2,3<EMAIL_ADDRESS>1Department of Physics, Sharif University
of Technology, Tehran 11155-9161, Iran
2School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran 19395-5531, Iran
3Department of Physics and Astronomy, University of Missouri, Columbia,
Missouri 65211, USA
aCorresponding Author

(
” Essay written for the Gravity Research Foundation 2023 Awards for Essays on
Gravitation)

###### Abstract

Nonlocal gravity (NLG), a classical extension of Einstein’s theory of
gravitation, has been studied mainly in linearized form. In particular,
nonlinearities have thus far prevented the treatment of cosmological models in
NLG. In this essay, we discuss the local limit of NLG and apply this limit to
the expanding homogenous and isotropic universe. The theory only allows
spatially flat cosmological models; furthermore, de Sitter spacetime is
forbidden. The components of the model will have different dynamics with
respect to cosmic time as compared to the standard $\Lambda$CDM model;
specifically, instead of the cosmological constant, the modified flat model of
cosmology involves a dynamic dark energy component in order to account for the
accelerated phase of the expansion of the universe.

Gravitation

###### pacs:

04.20.Cv

The cosmological observations of cosmic microwave background (CMB) radiation
Planck:2018vyg  as well as large scale structure (LSS) surveys BOSS:2016wmc
indicate that in terms of energy, we live in a dark universe with about $70\%$
dark energy, about $25\%$ dark matter, and only about $5\%$ visible matter.
The source of the two main dark sectors is still a mystery to us since we only
know about their gravitational effects. These two components of the universe
contribute to the standard model of cosmology only through gravitation. The
standard model of cosmology known as $\Lambda$CDM is based on the cosmological
principle and Einstein’s theory of general relativity (GR). This model assumes
that the cosmological constant $\Lambda$ is the cause of the accelerated
universe Weinberg:2013agg  and dark matter is due to a cold component (CDM).
This theory is truly ignorant about its two important players and yet gives us
acceptable results for describing the universe on large scales as a homogenous
coarse-grained fluid. The standard model also gives a reasonable connection
between the universe in its early times and the one we observe at low
redshifts through the distribution of structures in the universe Dodelson2020
.

The continuing failure of experiments to find the particles of dark matter
naturally leads to the possibility that what appears as dark matter in
astrophysics and cosmology is indeed an aspect of the gravitational
interaction. While the endeavor to unmask the true identity of the dark
sectors continues, there is a persistent notion that perhaps we do not know
the gravitational interaction well enough even at the classical level.
Einstein’s theory of gravitation has been well tested and quite successful on
the scales of the solar system and isolated binary star systems Will ;
moreover, the detection of gravitational radiation is further evidence in
support of GR in connection with binary systems LIGOScientific:2016aoc ;
Akhshi:2021nsy . However, “dark” gravitational effects of unknown origin seem
to appear on the scales of galaxies and beyond. A starting point to understand
general relativity more deeply would be to investigate the physical basis of
the assumptions that underlie Einstein’s elegant theory in order to align it
with reality.

As a field theory of gravitational interaction, general relativity has been
patterned after Maxwell’s electrodynamics. Maxwell originally formulated the
basic equations of electromagnetism in terms of the electromagnetic fields
$(\mathbf{E},\mathbf{B})$ and their excitations in a material medium
$(\mathbf{D},\mathbf{H})$. The latter contains the response of the medium in
terms of its polarizability and magnetizability, respectively; therefore,
there are constitutive relations that connect these fields. In their simplest
forms, we have $\mathbf{D}=\epsilon\,\mathbf{E}$, where $\epsilon$ is the
electric permittivity of the medium and $\mathbf{B}=\mu\,\mathbf{H}$, where
$\mu$ is the corresponding magnetic permeability. Even in Maxwell’s time,
observations pointed to the necessity of nonlocal and possibly nonlinear
constitutive relations Hop ; Poi . Indeed, history dependence must be taken
into account in the description of the properties of material media; for
instance, magnetic materials generally exhibit hysteresis. In the process of
averaging the response of the atomic medium, the memory of past events must be
taken into account. The resulting nonlocal constitutive relations contain
kernels that incorporate the atomic and molecular physics of the background
medium Jackson ; L+L ; HeOb . Similarly, it appears rather natural that in
describing the large scale structure and evolution of the universe, averaging
procedures may be necessary for cosmology. This leads to the problem of
averaging spacetime; in this connection, see VanDenHoogen:2017nyy  and the
references cited therein. On the other hand, nonlocality could be an intrinsic
feature of the universal gravitational interaction BMB .

Physics is local within the frameworks of special and general theories of
relativity Einstein . Here, locality means that the observer can make physical
predictions on the basis of information contained in the local spacetime patch
around itself; indeed, there is no need to keep track of the memory of past
events. In extending Lorentz invariance to accelerated systems in Minkowski
spacetime, Lorentz transformations are applied point by point along
accelerated world lines. Acceleration, ignored locally, is taken into account
only through the variation in the velocity of motion that appears in the
pointwise Lorentz transformations. This locality principle is physically
justified if the velocity of the accelerated system is essentially constant
during an elementary act of measurement. For pointlike coincidences of
classical point particles and rays of radiation, locality holds; however,
according to the Huygens principle waves are intrinsically nonlocal. In
connection with classical field measurements, Bohr and Rosenfeld have shown
that spacetime averaging is necessary BoRo . Furthermore, acceleration can
deflect the observer’s path from a geometrically defined geodesic, and it may
rotate the observer’s frame, diverting the frame from how it should naturally
evolve along the path via parallel propagation; hence, accelerated motion
involves local invariant scales of length and time. The challenge here mainly
arises when we ask how an accelerated observer can measure wave phenomena and
also how quantum effects are manifested in accelerated frames. It, therefore,
appears that the past history of the accelerated motion should in general be
taken into account through a nonlocal kernel that depends upon acceleration.
The nonlocality of accelerated systems eventually leads to nonlocal special
relativity theory Mashhoon:2008vr .

The cornerstone of general relativity, namely, Einstein’s principle of
equivalence, is based on the local connection between gravitation and inertia,
the same inertia that governs the dynamics of accelerated systems. This
circumstance suggests that gravity should be nonlocal as well. The most
natural approach to a classical nonlocal generalization of GR would be to
introduce history dependence into the theory through a nonlocal constitutive
relation in close analogy with the nonlocal electrodynamics of media
Hehl:2008eu ; Hehl:2009es . To this end, we first need to express GR in a form
that resembles electrodynamics. The key idea here is the introduction of an
orthonormal tetrad frame field adapted to preferred observers in spacetime and
the utilization of an extended geometric framework that employs both the
Riemann curvature of spacetime and the Weitzenböck torsion of the preferred
frame field. That is, in the extended framework we have one spacetime metric
tensor and two metric-compatible connections: the standard torsion-free
symmetric Levi-Civita connection and the curvature-free Weitzenböck connection
defined via the preferred frame field. The Weitzenböck connection We  renders
the spacetime manifold parallelizable. In the framework of teleparallelism
Itin:2018dru , two distant vectors are considered parallel if they have the
same components with respect to their local preferred frame fields. It turns
out that in this framework, Einstein’s theory becomes the well-known
teleparallel equivalent of general relativity (TEGR), which is the gauge
theory of the four-parameter Abelian group of spacetime translations Cho .
Therefore, TEGR, though nonlinear, is formally analogous to electrodynamics
and then we can use our intuition about electrodynamics and modify TEGR.

To introduce history dependence within the extended GR framework, let us
consider an admissible system of spacetime coordinates $x^{\mu}$ in spacetime
with metric

$ds^{2}=g_{\mu\nu}(x)\,dx^{\mu}\,dx^{\nu}\,.$ (1)

In this gravitational field, free test particles and null rays follow the
geodesics of the spacetime manifold. In our convention, 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 $c=1$. We assume the existence of a
preferred set of observers with adapted tetrads $e^{\mu}{}_{\hat{\alpha}}(x)$
that are orthonormal, namely,

$g_{\mu\nu}(x)\,e^{\mu}{}_{\hat{\alpha}}(x)\,e^{\nu}{}_{\hat{\beta}}(x)=\eta_{\hat{\alpha}\hat{\beta}}\,.$
(2)

Here, indices without hats are normal spacetime indices, hatted indices refer
to the tetrad axes in the local tangent space, and $\eta_{\alpha\beta}={\rm
diag}(-1,1,1,1)$ is the Minkowski metric tensor.

The curvature-free Weitzenböck connection is defined in terms of the tetrad
frame field as

$\Gamma^{\mu}_{\alpha\beta}=e^{\mu}{}_{\hat{\rho}}~{}\partial_{\alpha}\,e_{\beta}{}^{\hat{\rho}}\,.$
(3)

It simply follows from this definition that
$\nabla_{\nu}\,e_{\mu}{}^{\hat{\alpha}}=0$, where $\nabla$ denotes covariant
differentiation via the Weitzenböck connection. In this way, the preferred
tetrad frames are parallel throughout spacetime and provide a natural
scaffolding. The _torsion_ tensor corresponding to the Weitzenböck connection
is the gravitational field strength within the framework of teleparallelism
and is given by

$C_{\mu\nu}{}^{\alpha}=\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\nu\mu}=e^{\alpha}{}_{\hat{\beta}}\Big{(}\partial_{\mu}e_{\nu}{}^{\hat{\beta}}-\partial_{\nu}e_{\mu}{}^{\hat{\beta}}\Big{)}\,.$
(4)

A remark is in order here regarding the analogy between the gravitational
field strength $C_{\mu\nu}{}^{\hat{\alpha}}$ and the electromagnetic field
strength $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$; that is,
for each ${\hat{\alpha}}={\hat{0}},{\hat{1}},{\hat{2}},{\hat{3}}$ in the
torsion tensor

$C_{\mu\nu}{}^{\hat{\alpha}}=e_{\rho}{}^{\hat{\alpha}}C_{\mu\nu}{}^{\rho}=\partial_{\mu}e_{\nu}{}^{\hat{\alpha}}-\partial_{\nu}e_{\mu}{}^{\hat{\alpha}}\,,$
(5)

we have an analogue of the electromagnetic field tensor defined in terms of
the vector potential $e_{\mu}{}^{\hat{\alpha}}$. The extended GR framework
also contains the _contorsion_ tensor

$K_{\mu\nu}{}^{\alpha}=\,^{0}\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\mu\nu}\,,$
(6)

where ${}^{0}\Gamma^{\alpha}_{\mu\nu}$ denotes the symmetric Levi-Civita
connection. It follows from the metric compatibility of the Weitzenböck
connection that the contorsion tensor is related to the torsion tensor,
namely,

$K_{\mu\nu\rho}=\frac{1}{2}\,(C_{\mu\rho\nu}+C_{\nu\rho\mu}-C_{\mu\nu\rho})\,.$
(7)

Therefore, the Levi-Civita connection given by the Christoffel symbol is the
sum of the Weitzenböck connection and the contorsion tensor. The Einstein
tensor and the gravitational field equations, via the Levi-Civita connection,
can then be expressed in terms of the teleparallelism framework resulting in
the teleparallel equivalent of GR, namely, TEGR BMB . Einstein’s field
equations expressed in terms of torsion thus become the TEGR field equations

$\frac{\partial}{\partial
x^{\nu}}\,\mathfrak{H}^{\mu\nu}{}_{\hat{\alpha}}+\frac{\sqrt{-g}}{\kappa}\,\Lambda\,e^{\mu}{}_{\hat{\alpha}}=\sqrt{-g}\,(T_{\hat{\alpha}}{}^{\mu}+\mathbb{T}_{\hat{\alpha}}{}^{\mu})\,,$
(8)

where $\Lambda$ is the cosmological constant, $\kappa:=8\pi G$, and the
auxiliary torsion field $\mathfrak{H}_{\mu\nu\rho}$ is defined via the
auxiliary torsion tensor $\mathfrak{C}_{\alpha\beta\gamma}$ as

$\mathfrak{H}_{\mu\nu\rho}:=\frac{\sqrt{-g}}{\kappa}\,\mathfrak{C}_{\mu\nu\rho}\,,\qquad\mathfrak{C}_{\alpha\beta\gamma}:=C_{\alpha}\,g_{\beta\gamma}-C_{\beta}\,g_{\alpha\gamma}+K_{\gamma\alpha\beta}\,.$
(9)

Here, $C_{\mu}:=C^{\alpha}{}_{\mu\alpha}=-C_{\mu}{}^{\alpha}{}_{\alpha}$ is
the torsion vector, $T_{\mu\nu}$ is the symmetric energy-momentum tensor of
matter, and $\mathbb{T}_{\mu\nu}$ is the traceless energy-momentum tensor of
the gravitational field

$\mathbb{T}_{\mu\nu}:=(\sqrt{-g})^{-1}\,(C_{\mu\rho\sigma}\,\mathfrak{H}_{\nu}{}^{\rho\sigma}-\tfrac{1}{4}g_{\mu\nu}\,C_{\rho\sigma\delta}\,\mathfrak{H}^{\rho\sigma\delta})\,.$
(10)

Inspired by the electrodynamics of nonlocal media, it is possible to introduce
history dependence and equip TEGR with the memory of the past. In the
resulting nonlocal theory of gravitation, the gravitational field would be
locally defined but would satisfy partial integro-differential field
equations. That is, we expect that the modified gravitational field equations
would contain a certain average of the gravitational field over past events.
Such a term should involve a weight or “kernel” function that in the case of
electrodynamics is determined by the nature of the atoms and molecules of the
medium. The main difference with the electrodynamics of nonlocal media has to
do with the absence of the background atomic medium. Therefore, the nonlocal
kernel of the theory in the case of gravitation must ultimately be determined
by observation.

We can view the relationship between $\mathfrak{H}_{\mu\nu\rho}$ and the
torsion tensor in Equation (9) as the local constitutive relation of TEGR. In
the nonlocal electrodynamics of media, the field equations of electrodynamics
are not changed, only the constitutive relation is made nonlocal. In a similar
fashion, in nonlocal gravity (NLG), only the constitutive relation of TEGR is
modified. This means that we replace $\mathfrak{H}$ in Equations (8) and (10)
by $\mathcal{H}$ given by

$\mathcal{H}_{\mu\nu\rho}=\frac{\sqrt{-g}}{\kappa}(\mathfrak{C}_{\mu\nu\rho}+N_{\mu\nu\rho})\,,$
(11)

where $N_{\mu\nu\rho}=-N_{\nu\mu\rho}$ is a tensor that is nonlocally related
to the torsion tensor. That is, the components of $N_{\mu\nu\rho}$ measured by
the preferred observers of the theory with adapted tetrads
$e^{\mu}{}_{\hat{\alpha}}$ are associated with the corresponding measured
components of $X_{\mu\nu\rho}$ that is directly connected to the torsion
tensor, namely,

$N_{\hat{\mu}\hat{\nu}\hat{\rho}}(x)=\int\mathcal{K}(x,x^{\prime})\,X_{\hat{\mu}\hat{\nu}\hat{\rho}}(x^{\prime})\sqrt{-g(x^{\prime})}\,d^{4}x^{\prime}\,,$
(12)

where $\mathcal{K}(x,x^{\prime})$ is the basic causal kernel of NLG and BMB ;
Puetzfeld:2019wwo ; Mashhoon:2022ynk

$X_{\hat{\mu}\hat{\nu}\hat{\rho}}=\mathfrak{C}_{\hat{\mu}\hat{\nu}\hat{\rho}}+\check{p}\,(\check{C}_{\hat{\mu}}\,\eta_{\hat{\nu}\hat{\rho}}-\check{C}_{\hat{\nu}}\,\eta_{\hat{\mu}\hat{\rho}})\,.$
(13)

Here, $\check{C}^{\mu}$ is the torsion pseudovector defined via the Levi-
Civita tensor $E_{\alpha\beta\gamma\delta}$ by

$\check{C}_{\mu}:=\frac{1}{3!}C^{\alpha\beta\gamma}\,E_{\alpha\beta\gamma\mu}\,$
(14)

and $\check{p}\neq 0$ is a constant dimensionless parameter. The resulting
theory of nonlocal gravity (NLG) is rather intricate and no exact solution is
known except for the trivial result that the spacetime is Minkowskian in the
absence of gravity. Furthermore, de Sitter spacetime is not an exact solution
of NLG Mashhoon:2022ynk . On the other hand, linearized NLG has been
extensively studied BMB ; in fact, in the Newtonian regime of NLG, it has been
possible to account for the rotation curves of nearby spiral galaxies as well
as for the solar system data Rahvar:2014yta ; Chicone:2015coa ; Roshan:2021ljs
; Roshan:2022zov . Thus far, the nonlinearity of NLG has made it impossible to
find exact solutions for the strong-field regimes such as those involving
black holes or cosmological models. However, it is possible that features of
NLG survive in the local limit of the theory, just as in electrodynamics the
local permittivity $\epsilon(x)$ and permeability $\mu(x)$ functions can
preserve features of nonlocal electrodynamics of media.

The local limit of Equation (12) can be obtained by assuming that the kernel
is proportional to the 4D Dirac delta function, namely,

$\mathcal{K}(x,x^{\prime}):=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x^{\prime})\,,$
(15)

where $S(x)$ is a dimensionless scalar function that must be determined based
on observational data Tabatabaei:2022tbq . In this case, the nonlocal
constitutive relation (11) reduces to $N_{\mu\nu\rho}(x)=S(x)X_{\mu\nu\rho}$;
that is,

$N_{\mu\nu\rho}(x)=S(x)\,[\mathfrak{C}_{\mu\nu\rho}(x)+\check{p}\,(\check{C}_{\mu}\,g_{\nu\rho}-\check{C}_{\nu}\,g_{\mu\rho})]\,$
(16)

and the constitutive relation takes the form

$\mathcal{H}_{\mu\nu\rho}=\frac{\sqrt{-g}}{\kappa}[(1+S)\,\mathfrak{C}_{\mu\nu\rho}+S\,\check{p}\,(\check{C}_{\mu}\,g_{\nu\rho}-\check{C}_{\nu}\,g_{\mu\rho})]\,.$
(17)

Here, the susceptibility function $S(x)$ is a characteristic of the background
spacetime just as $\epsilon(x)$ and $\mu(x)$ are features of the medium in
electrodynamics. For $S(x)=0$, we recover TEGR; otherwise, we have a
generalization of GR that we expect may retain features of NLG through $S(x)$.
Indeed, Equation (17) implies that to have GR as a limit, we must impose the
requirement that $1+S>0$. In this local limit of nonlocal gravity, explicit
deviations from the locality have vanished; however, nontrivial aspects of NLG
may have survived through $S(x)$ that would be interesting to study. We,
therefore, explore the cosmological implications of this local limit of NLG.

The very first application of the local limit of NLG to the expanding universe
has a dramatic result and implies spatial flatness. We begin with the FLRW
metric for a dynamic (expanding or contracting) universe that is homogeneous
and isotropic and assume that $S$ is only a function of _cosmic time_ $t$ with
$dS/dt\neq 0$. That the susceptibility $S$ should be time dependent is
consistent with the assumption of a dynamic universe model. The field
equations of the local limit of NLG imply that the only model consistent with
these assumptions is the spatially _flat_ model. Therefore, we assume a metric
of the form

$ds^{2}=-dt^{2}+a^{2}(t)\,\delta_{ij}\,dx^{i}\,dx^{j}\,,$ (18)

where $a(t)$ is the scale factor. The gravitational field equations imply that
our modified flat model is governed by

$3(1+S)\left(\frac{\dot{a}}{a}\right)^{2}=\Lambda+8\pi G\rho\,$ (19)

and

$2(1+S)\frac{\ddot{a}}{a}+(1+S)\left(\frac{\dot{a}}{a}\right)^{2}=\Lambda-8\pi
GP-2\frac{dS}{dt}\frac{\dot{a}}{a}\,,$ (20)

where $\rho(t)$ and $P(t)$ are the energy density and pressure of the
background cosmic medium, respectively. Moreover, we recall that $1+S>0$;
however, to ensure that this relation holds as $S$ varies with cosmic time, we
further assume that $dS/dt>0$. Next, differentiating Equation (19) with
respect to cosmic time $t$ and using Equation (20), we find

$\frac{d\rho}{dt}=-3(\rho+P)\frac{\dot{a}}{a}-\frac{3}{8\pi
G}\frac{dS}{dt}\left(\frac{\dot{a}}{a}\right)^{2}\,,$ (21)

which implies, for $\rho+P\geq 0$ and $dS/dt>0$, that $\rho$ monotonically
decreases as the universe expands.

Equations (19) and (20) govern the dynamics of the modified flat model of
cosmology. In Equation (19), $1+S$, $\Lambda$, and $\rho$ are all positive;
hence, the universe expands forever and as $t\to\infty$, $a(t)$ approaches
infinity, while $\rho$ and $P$ approach zero. It appears that the universe is
then dominated by the cosmological constant $\Lambda$; however, the variation
of $S$ with cosmic time leads to inconsistency.

Let us assume that our model is dominated by the cosmological constant as
$t\to\infty$ and $\rho=0$. Therefore, Equation (19) implies

$\frac{\dot{a}}{a}=(\Lambda/3)^{1/2}(1+S)^{-1/2}\,.$ (22)

Taking the derivative of this relation with respect to time, we find

$\frac{\ddot{a}}{a}-\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{1}{2}(\Lambda/3)^{1/2}(1+S)^{-3/2}\,\frac{dS}{dt}\,.$
(23)

Let us multiply this relation by $2(1+S)>0$ and subtract the result from
Equation (20) to get

$3(1+S)\left(\frac{\dot{a}}{a}\right)^{2}=\Lambda-2\frac{dS}{dt}\frac{\dot{a}}{a}+(\Lambda/3)^{1/2}(1+S)^{-1/2}\,\frac{dS}{dt}\,.$
(24)

Plugging Equation (19) in Equation (24) results in

$2\frac{\dot{a}}{a}\frac{dS}{dt}=(\Lambda/3)^{1/2}(1+S)^{-1/2}\,\frac{dS}{dt}\,,$
(25)

which contradicts Equation (22), since $dS/dt$ does not vanish. Therefore, de
Sitter spacetime is not a solution of our modified flat model. This
circumstance is in conformity with the fact that de Sitter spacetime is not a
solution of nonlocal gravity (NLG) Mashhoon:2022ynk . The modified flat
cosmological model under consideration here will never asymptotically approach
a de Sitter phase.

To conclude our discussion of the local limit of nonlocal gravity (NLG), we
have shown that the first approximation of the NLG, which is its local limit,
has an important effect on the evolution of the densities of constituents of
the universe with cosmic time. All of the components of the universe evolve
differently from their standard $\Lambda$CDM prediction. Accordingly, the
modified flat model replaces the cosmological constant with a dark energy
component with an evolving equation of state. An important part of future
studies in cosmology is related to the idea to see if the cause of the
accelerated expansion of the universe is indeed due to a cosmological constant
or not. The fundamental nonlocal feature of gravitational interaction predicts
that we will find dynamic dark energy in upcoming decades instead of a
cosmological constant Annis:2022xgg ; Ferraro:2022cmj .

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