Observational constraints on f(T) gravity from varying fundamental constants
Eur. Phys. J. C
Observational constraints on f (T ) gravity from varying fundamental constants
Rafael C. Nunes 2
Alexander Bonilla 2
Supriya Pan 1
Emmanuel N. Saridakis 0 3 4
0 Instituto de Fi?sica, Pontificia Universidad de Cato?lica de Valparai?so , Casilla, 4950 Valparai?so , Chile
1 Department of Physical Sciences, Indian Institute of Science Education and Research , Kolkata, Mohanpur, West Bengal 741246 , India
2 Departamento de Fi?sica, Universidade Federal de Juiz de Fora , 36036330 Juiz de Fora, MG , Brazil
3 CASPER, Physics Department, Baylor University , Waco, TX 767987310 , USA
4 Physics Division, National Technical University of Athens , Zografou Campus, 15780 Athens , Greece
We use observations related to the variation of fundamental constants, in order to impose constraints on the viable and most used f (T ) gravity models. In particular, for the finestructure constant we use direct measurements obtained by different spectrographic methods, while for the effective Newton constant we use a modeldependent reconstruction, using direct observational Hubble parameter data, in order to investigate its temporal evolution. We consider two f (T ) models and we quantify their deviation from CDM cosmology through a sole parameter. Our analysis reveals that this parameter can be slightly different from its CDM value, however, the bestfit value is very close to the CDM one. Hence, f (T ) gravity is consistent with observations, nevertheless, as every modified gravity, it may exhibit only small deviations from CDM cosmology, a feature that must be taken into account in any f (T ) modelbuilding.

Modified gravity [1] is one of the two main roads one can
follow in order to provide an explanation for the early and
latetime universe acceleration (the second one in the introduction
of the darkenergy concept [2, 3]). Furthermore, apart from
the cosmological motivation, modified gravity has a
theoretical motivation too, namely to improve the renormalizability
properties of standard general relativity [4].
In constructing a gravitational modification, one usually
starts from the Einstein?Hilbert action and extends it
accordingly. Thus, one can obtain f ( R) gravity [5], Gauss?Bonnet
a email:
b email:
c email:
d email:
and f (G) gravity [6, 7], gravity with higherorder
curvature invariants [8, 9], massive gravity [10] etc. Nevertheless,
one could start from the equivalent, torsional formulation
of gravity, namely from the Teleparallel Equivalent of
General Relativity (TEGR) [11?13], in which the gravitational
Lagrangian is the torsion scalar T , and construct various
modifications, such as f (T ) gravity [14?29] (see [30] for
a review), teleparallel Gauss?Bonnet gravity [31, 32],
gravity with higherorder torsion invariants [33], etc.
An important question in the above gravitational
modifications is what are the forms of the involved unknown functions,
and what are the allowed values of the various parameters.
Excluding forms and parameter regimes that lead to
obvious contradictions and problems, the main tool we have in
order to provide further constraints is to use observational
data. For the case of torsional gravity one can use solar
system data [34?37], or cosmological observations from
Supernovae type Ia, cosmic microwave background and baryonic
acoustic oscillations [38?41].
On the other hand, in some modified cosmological
scenarios one can obtain a variation of the fundamental constants,
such as the finestructure constant and the Newton constant.
Such a possibility has been investigated in the literature since
Dirac [42, 43] and Milne and Jordan [44?47] times. Later on,
Brans and Dicke proposed the time variation of the Newton
constant, driven by a dynamical scalar field coupled to
curvature [48, 49], while Gamow triggered subsequent
speculations on the possible variation of the finestructure constant
[50]. Similarly, in recent modified gravities, which involve
extra degrees of freedom compared with general relativity,
one may obtain such a variation of the fundamental constants
[51?63]. However, since experiments and observations give
strict bounds on these variations [64?76], one can use them
in order to constrain the theories at hand.
In the present work we are interested in investigating the
constraints on f (T ) gravity by observations related to the
variation of fundamental constants. In particular, since f (T )
gravity predicts a variation of the finestructure constant and
the Newton constant, we will use the recent observational
bounds of these variations in order to constrain the f (T )
forms as well as the range of the involved parameters. The
plan of the work is the following: In Sect. 2 we give a brief
review of f (T ) gravity and cosmology. In Sect. 3 we
investigate the constraints on specific f (T ) gravity models arising
from the observational bounds of the finestructure constant
variation, while in Sect. 4 we study the corresponding
constraints that arise from the observational bounds of the
Newton constant variation. Finally, in Sect. 5 we summarize our
results.
2 f (T ) gravity and cosmology
In this section we provide a short review of f (T ) gravity
?
and cosmology. We use the tetrad fields eA, which form an
orthonormal base at each point of the tangent space of the
underlying manifold (M, g?? ), where g?? = ?AB e?Ae?B is
the metric tensor defined on this manifold (we use Greek
indices for the coordinate space and Latin indices for the
tangent one). Furthermore, instead of the torsionless
LeviCivita connection which is used in the Einstein?Hilbert
action, we use the curvatureless Weitzenb?ck connection
w?
?? ? e?A ??e?A [13]. Hence, the gravitational field in such a
formalism is described by the following torsion tensor:
? A
T??? ? eA ??e? ? ?? e?A .
Subsequently, the Lagrangian of the teleparallel equivalent of
general relativity, namely the torsion scalar T , is constructed
by contractions of the torsion tensor as [13]
T ? 41 T ??? T??? + 21 T ??? T??? ? T??? T ??? .
One may consider generalized theories in which the
Lagrangian T is extended to an arbitrary function f (T ),
similarly to the f (R) extension of curvaturebased gravity. In
particular, such a gravitational action will read
where e = det(e?A) = ??g, and GN is the Newton constant.
Additionally, along the gravitational action (3) we consider
the matter sector, and hence the total action writes as
where Lm (e?A, M ) is the total matter Lagrangian including
the electromagnetic field. Finally, variation in terms of the
tetrad fields give rise to the field equations as
e?1??(ee?A S? ?? ) fT ? fT e?A T ? ?? S? ?? + 41 e?A f (T )
? ? (m)? ? ,
+eA S? ?? ??(T ) fT T = 4? GNeAT
where fT = ? f /? T , fT T = ?2 f /? T 2, and with T (m)? ? the
total matter energymomentum tensor. In the above
equation we have inserted for convenience the ?superpotential?
tensor S??? = 21 K??? + ???T??? ? ??? T??? , defined in terms
of the cotorsion tensor K??? = ? 21 T??? ? T??? ? T??? .
Applying f (T ) gravity in a cosmological framework we
consider a spatially flat FLRW universe with line element
ds2 = ?dt 2 +a2(t )[dr 2 +r 2 d? 2 +sin2 ? d?2], which arises
from the diagonal tetrad e?A = diag(1, a(t ), a(t ), a(t )), with
a(t ) the scale factor. In this case, the field equations (5)
become
H 2 =
where ? and p are, respectively, the total matter energy
density and pressure, and where ?T , pT are the effective
darkenergy density and the pressure of gravitational origin, given
by
4H? (2T fT T + fT ? 1) ? ?T .
In the above expressions we have used
T = ?6H 2,
which arises straightforwardly from (2) in the FLRW
universe. Finally, from Eqs. (8), (9) we can define the effective
darkenergy equation of state (EoS) as
2H? 2
w = ?1 ? 3H 2 = ?1 + 3 H (1 + z)
where as usual we use the redshift z = aa0 ? 1, as the
independent variable, and for simplicity we set a0 = 1. Clearly,
w has a dynamical nature.
In the following we focus on two wellstudied, viable
f (T ) models, which correspond to a small deviation from
CDM cosmology, and which according to [38?41] are the
ones that fit the observational data very efficiently.
where ? , b are the two free model parameters, out of
which only one is independent. Inserting this f (T ) form
into the first Friedmann equation (6) at present time, i.e.
at redshift z = 0, one may derive that
where m0 = 8?3GH?02m0 is the corresponding density
parameter at present. Hence, and using additionally that
?m = ?m0(1 + z)3, Eq. (6) for this model can be written
as
Lastly, note that this model reduces to CDM cosmology
for p ? +?. Hence, in the following, for this model
it will be convenient to set b ? 1/ p, and hence CDM
cosmology is obtained for b ? 0+.
Finally, the first Friedmann equation (6) for this model
can be written as
e? pHH0(z) ?1
? The first scenario is the powerlaw model (hereafter
f1CDM) introduced in [14], with
3 Observational constraints from finestructure constant variation
Lastly, we mention that the above model for b = 0
reduces to CDM cosmology, while for b = 1/2 it gives
rise to the Dvali?Gabadadze?Porrati (DGP) model [77].
? The second scenario is the squarerootexponential
(hereafter f2CDM) of [15], with
f (T ) = T + ? T0(1 ? e? p?T/T0 ),
in which ? and p the two free model parameters out of
which only one is independent. Inserting this f (T ) form
into (6) at present time, one obtains
In this section we will use observational data of the variation
of the finestructure constant ?, in order to constrain f (T )
gravity. Let us first quantify the ?variation in the
framework of f (T ) cosmology. In general, in a given theory the
finestructure constant is obtained using the coefficient of the
electromagnetic Lagrangian. In the case of modified
gravities, this coefficient generally depends on the new degrees
of freedom of the theory [56?63,78]. Even if one starts from
the Jordanframe formulation of a theory, with an
uncoupled electromagnetic Lagrangian, and although the
electromagnetic Lagrangian is conformally invariant, and it is not
affected by conformal transformations between the Jordan
and Einstein frames, thus it will acquire a dependence on the
extra degree(s) of freedom due to quantum effects [79,80].
In particular, if ? is the extra degree of freedom that arises
from the conformal transformation g??? = 2g?? from the
Jordan to the Einstein frame, then quantum effects such as
the presence of heavy fermions (note that this does not
necessarily require new physics till the Planck scale) will induce
a coupling of ? to photons, namely [79,80]
d4x ??g BF (?)F?? F ?? ,
where F?? is the electromagnetic tensor, gbare the bare
coupling constant, and
with Mpl = 1/(8? GN) the Planck mass and ?? = O(1)
a constant (we have assumed that ?? ? Mpl). Hence,
the scalar coupling to the electromagnetic field will imply
a dependence of the finestructure constant of the form [78?
80]
where the subscripts denote the Einstein and Jordan frames,
respectively, or equivalently
The above factor is in general time (i.e. redshift) dependent.
Therefore, it proves convenient to normalize it in order to
have ? = 0 at present (z = 0), which in the case where
BF (z = 0) ? BF0 = 1 is obtained through a rescaling
F?? ? ?BF0 F?? and BF ? BF /BF0. Thus, we result to
Although the above procedure is straightforward in cases
where a conformal transformation from the Jordan to the
Einstein frame exists, it becomes more complicated for
theories where such a transformation is not known. In case of
f (T ) gravity, it is well known that a conformal
transformation does not exist in general, since transforming the metric as
g??? = 2g?? , with 2 = fT a smooth nonvanishing
function of spacetime coordinates, one obtains Einstein gravity
plus a scalar field Lagrangian, plus the transformed matter
Lagrangian, plus a nonvanishing term 2 ?6?? ? 2T? ??? [81].
This additional term forbids the complete transformation to
the Einstein frame, and hence in every application one has
indeed to perform calculations in the more complicated
Jordan one.
In order to avoid performing calculations in the
Jordan frame we will make the reasonable assumption that
f (T ) = T + const. + corrections, which has been shown to
be the case according to observations [34?41], and holds for
the two forms considered in this work, namely (12) and (15),
too. Hence, 2 = 1 + corrections, and then ?? ? 2 is
negligible, which implies that the above extra term can be neglected.
In the end of our investigation, we will verify the validity of
the above assumption. Thus, we can indeed obtain an
approximate transformation to the Einstein frame, and in particular
the introduced degree of freedom reads ? = ??3/ fT [81].
Hence, inserting this into (19) we acquire
?
3??
BF (?) = 1 ? Mpl fT + ? ? ? ,
and thus inserting into Eq. (22), we can easily extract the
variation of the finestructure constant as
Mpl fT 0 ?
Mpl fT (z) ?
? 1,
where fT 0 = fT (z = 0). Lastly, since ?? = O(1), the above
relation becomes
Hence, for a general f (T ), the ratio ?/? indeed depends
on z, through the fT (z) function (we recall that according
to (10), T (z) = ?6H 2(z)), while in the case of standard
CDM cosmology, where f (T ) = T + , ?/? becomes
zero.
In the following, we confront Eq. (25) with observations
of the finestructure constant variation, in order to impose
constraints on f (T ) gravity (it proves that the neglected
term between (24) and (25) imposes an error of the order
of 10?9 and hence our approximation is justified). We use
direct measurements of the finestructure constant that are
obtained by different spectrographic methods, summarized
in Table 1. Additionally, along with these data sets, and in
order to diminish the degeneracy between the free
parameTable 1 Compilation of recent measurements of the finestructure
constant obtained by different spectrographic methods. For details in each
case, see the corresponding references
ters of the models, we use 580 Supernovae data (SNIa) from
Union 2.1 compilation [88], as well as data from BAO
observations, adopting the three measurements of A(z) obtained
in [89], and using the covariance among these data given in
[90].
In the following two subsections, we analyze two viable
models, namely f1CDM of (12) and f2CDM of (15),
separately.
3.1 Model f1CDM: f (T ) = T + ? (?T )b
For the powerlaw f1CDM model of (12), we easily acquire
fT (z) = 1 ? b
(b?1)
where we have used also (10). Inserting (26) into (25) we can
derive the evolution of ?/? as
1 ? b
1 ? b
(b?1)
? 1,
where the ratio H 2(z)/H02 is given by (14).
We mention that while analyzing the model for the data
set of ?/? of Table 1, we have marginalized over m0, and
thus the statistical information focuses only on the parameter
b. For the fittings ?/? + SNIa and ?/? + SNIa + BAO,
we have considered m0 as a free parameter, and we have
found that m0 = 0.23 ? 0.13 (for ?/? + SNIa) and
m0 = 0.293 ? 0.023 (for ?/? + SNIa + BAO) at 1?
confidence level.
Fig. 1 1? and 2? confidence regions for the f1CDM powerlaw model
of (12), obtained from the joint analysis ?/? + SNIa + BAO. The
cross marks the bestfit value
Table 2 Summary of the bestfit values of the parameter b of the
f1CDM powerlaw model of (12), for three different observational data
sets with reduced ? 2: ? m2in/d.o.f. (d.o.f. stands for the ?degrees of
freedom?)
Finally, in Fig. 1 we present the 68.27 and 95.45%
confidence regions in the plane m0 ? b, considering the
observational data ?/? + SNIa + BAO. Note that these results are
in qualitative agreement with those of different observational
fittings [38?40], and show that CDM cosmology (which is
obtained for b = 0) is inside the obtained region. In fact,
one may notice from Table 2 that the reduced ? 2 for ?/?
+ SNIa and ?/? + SNIa + BAO data are very close to 1,
while for single data from ?/? its value slightly exceeds 1
although not significantly.
Additionally, in order to examine the latetime asymptotic
behavior of the scenario at hand, in Fig. 2 we depict the
evolution of the equationofstate parameter given in (11),
applying a reconstruction at 1? confidence level via error
propagation using the joint analysis ?/? + SNIa + BAO.
As we can see, w at late times acquires values very close
to ??1?, as expected. For a more detailed investigation of
the latetime asymptotics up to the far future one must apply
the method of dynamical system analysis as it was done in
Fig. 2 The evolution of the equationofstate parameter given in (11),
for the f1CDM powerlaw model of (12), applying a reconstruction at
1? confidence level via error propagation using the joint analysis ?/?
+ SNIa + BAO
[91?93], where it was thoroughly shown that the universe
will end in a de Sitter phase.
In summary, it is clear that f1CDM model, under all the
above three different combinations of statistical data sets,
remains close to CDM cosmology as expected. Lastly, note
that this is a selfconsistent verification for the validity of our
assumption that f (T ) = T + const. + corrections, which
allowed us to work in the Einstein frame.
3.2 Model f2CDM: f (T ) = T + ? T0(1 ? e? p?T /T0 )
For the squarerootexponential f2CDM model of (15), we
easily obtain
? 1,
where the ratio H 2(z)/ H02 is given by (17).
We mention that while analyzing the model for the data
set of ?/? of Table 1, we have marginalized over m0, and
thus the statistical information focuses only on the parameter
Fig. 3 1? and 2? confidence regions for the f2CDM
squarerootexponential model of (15), obtained from the joint analysis ?/? +
SNIa + BAO. The cross marks the bestfit value
b. For the fittings ?/?+ SNIa and ?/? +SNIa +BAO we
have taken m as a free parameter, and we note that m0 =
0.277 ? 0.019 (for ?/? + SNIa) and m0 = 0.283 ? 0.016
(for ?/? + SNIa + BAO) at 1? confidence level.
Finally, in Fig. 3 we present the 68.27% and 95.45%
confidence regions in the plane m0 ? b, considering the
observational data ?/? + SNIa + BAO (we have taken b 0.001
in order to avoid divergences in the function H (z) at high
redshifts). Note that these results are in qualitative agreement
with those of different observational fittings [38?40], and
show that CDM cosmology (which is obtained for b ? 0+)
is inside the obtained region. Furthermore, and similarly to
the f1CDM model, from Table 3 we deduce that although
the data from ?/? alone show a slightly deviating nature
(reduced ? 2 = 1.1) from CDM scenario, but for ?/?
+ SNIa and ?/? + SNIa + BAO data it is implied that
the model is very close to CDM cosmology. This is also a
selfconsistent verification for the validity of our assumption
that f (T ) = T + const. + corrections, which allowed us to
work in the Einstein frame.
)
z
(
w1.003
Fig. 4 The evolution of the equationofstate parameter given in (11),
for the f2CDM squarerootexponential model of (15), applying a
reconstruction at 1? confidence level via error propagation using the
joint analysis ?/? + SNIa + BAO
Lastly, in order to examine the latetime asymptotic
behavior of f2CDM model, in Fig. 4 we depict the evolution of the
equationofstate parameter given in (11), applying a
reconstruction at 1? confidence level via error propagation using
the joint analysis ?/? + SNIa + BAO, where one can
see that w at late times acquires values very close to ??1?, as
expected. Similarly to the previous model, for a more detailed
investigation of the latetime asymptotics one must apply the
method of dynamical system analysis [91?93], where it can
thoroughly be shown that the universe will end in a de Sitter
phase.
4 Observational constraints from the Newton constant
variation
In this section we will directly use the observational
constraints imposed on f (T ) models in order to examine the
variation of the gravitational constant GN. Let us first quantify
the GNvariation in the framework of f (T ) cosmology. As is
well known, a varying effective gravitational constant is one
of the common features in many modified gravity theories
[1]. In case of f (T ) gravity, the effective Newton constant
Geff can be straightforwardly extracted as [40, 94]
Hence, for the f1CDM powerlaw model of (12), and using
(26), we obtain
Geff =
Geff (z) =
1 ? b
(b?1) ,
where the ratio H 2(z)/H02 is given by (14) (clearly, for b = 0
we have Geff (z) = GN = const.). Similarly, for the f2CDM
squarerootexponential model of (15), and using (28), we
acquire
Geff (z) =
where the ratio H 2(z)/H02 is given by (17) (clearly, for b =
1/ p ? 0+ we have Geff (z) = GN = const.).
Let us now use the above expressions for Geff (z) and
confront them with the observational bounds of the Newton
constant variation. We use observational Hubble parameter data
in order to investigate the temporal evolution of the function
Geff (z), since such a compilation is usually used to constrain
cosmological parameters, due to the fact that it is obtained
from modelindependent direct observations. We adopt 37
observational Hubble parameter data in the redshift range
0 < z ? 2.36, compiled in [95], out of which 27 data points
are deduced from the differential age method, whereas 10
correspond to measures obtained from the radial baryonic
acoustic oscillation method.
We apply the following methodology: Firstly, we estimate
the error in the measurements associated with the function
Geff /GN, for both models of (31) and (32), via the standard
method of error propagation theory, namely
? m
and we fix the free parameters of the two models within the
values obtained in the joint analysis of [41] and of Sect. 3
of the current work. Then the measurements of Geff /GN are
calculated directly for each redshift defined in the adopted
compilation.
In the left graph of Fig. 5 we depict the 1? confidencelevel
estimation of the function Geff (z)/GN from 37 Hubble data
points, in the case of the f1CDM powerlaw model of (12).
Additionally, in the right graph of Fig. 5 we present the
corresponding 1? confidencelevel reconstruction of Geff (z)/GN
for b ? [?0.01, 0.01] from the observational Hubble
parameter data. When we perform the analysis within the known
range of the parameter b for this model (from [41] as well
as from Fig. 1 above), we find that Geff /GN ? 1.
Nevertheless, a minor deviation is observed for the fixed value of
b = 0.01 (see the left graph of Fig. 5). For instance, note
that Geff (z = 0.07)/GN = 0.992 ? 0.004 and Geff (z =
2.36)/GN = 0.99923 ? 0.00005, for the first and the last
data points of the redshift interval [0, 2.36], respectively.
In Fig. 6 we present the corresponding graphs for the
f2CDM squarerootexponential model of (15). When we
perform the analysis within the known range of the
parameter b ? 1/ p for this model (from [41] as well as from Fig.
3 above), we find that Geff /GN ? 1, similarly to the case of
f1CDM model.
In summary, from the analysis of this section, we verify the
results of the previous section, namely that the parameter b,
Fig. 6 Results for the f2CDM squarerootexponential model of (15). Left graph estimation of Geff /GN as a function of the redshift, from 37
Hubble data points. Right graph reconstruction of Geff /GN as a function of the redshift, from the observational Hubble parameter data, for b ? 1/ p
? [0, 0.5]
which quantifies the deviation of both f1CDM and f2CDM
models from CDM cosmology, is very close to zero. These
results are in qualitative agreement with previous
observational constraints on f (T ) gravity, according to which only
small deviations are allowed, with CDM paradigm being
inside the allowed region [34?41].
5 Conclusions
In the present work we have used observations related to the
variation of fundamental constants, in order to impose
constraints on the viable and most used f (T ) gravity models. In
particular, since f (T ) gravity predicts a variation of the
finestructure constant, we used the recent observational bounds
of this variation, from direct measurements obtained by
different spectrographic methods, along with standard probes
such as Supernovae type Ia and baryonic acoustic
oscillations, in order to constrain the involved model parameters of
two viable and wellused f (T ) models.
For both the f1CDM powerlaw model and the f2CDM
squarerootexponential model, we found that the parameter
that quantifies the deviation from CDM cosmology can
be slightly different from its CDM value, nevertheless the
bestfit value is very close to the CDM one. Additionally,
since f (T ) gravity predicts a varying effective gravitational
constant, we quantified its temporal evolution with the use
of the previously constrained model parameters. For both the
f1CDM and the f2CDM models, we found that the deviation
from CDM cosmology is very close to zero.
These results are in qualitative agreement with previous
observational constraints on f (T ) gravity [38?41], however,
they have been obtained through completely independent
analysis. In summary, f (T ) gravity is consistent with
observations, and thus it can serve as a candidate for modified
gravity, although, as every modified gravity, it may have only a
small deviation from CDM cosmology, a feature that must
be taken into account in any f (T ) modelbuilding.
Acknowledgements The authors would like to thank P. Brax for useful
discussions. Additionally, they thank an anonymous referee for
clarifying comments. The work of SP is supported by the National
PostDoctoral Fellowship (File No: PDF/2015/000640) under the Science
and Engineering Research Board (SERB), Govt. of India. This article
is based upon work from COST Action ?Cosmology and Astrophysics
Network for Theoretical Advances and Training Actions?, supported
by COST (European Cooperation in Science and Technology).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. S. Capozziello , M. De Laurentis , Phys. Rep . 509 , 167 ( 2011 )
2. E.J. Copeland , M. Sami , S. Tsujikawa , Int. J. Mod. Phys. D 15 , 1753 ( 2006 )
3. Y.F. Cai , E.N. Saridakis , M.R. Setare , J.Q. Xia , Phys. Rep. 493 , 1 ( 2010 )
4. K.S. Stelle , Phys. Rev. D 16 , 953 ( 1977 )
5. S. Nojiri , S.D. Odintsov , Phys. Rep. 505 , 59 ( 2011 )
6. S. Nojiri , S.D. Odintsov , Phys. Lett . B 631 , 1 ( 2005 )
7. A. De Felice , S. Tsujikawa , Phys. Lett . B 675 , 1 ( 2009 )
8. A. Naruko , D. Yoshida , S. Mukohyama , Class. Quant. Grav. 33 ( 9 ), 09LT01 ( 2016 )
9. E.N. Saridakis , M. Tsoukalas , Phys. Rev. D 93(12) , 124032 ( 2016 )
10. C. de Rham, Living Rev . Relativ. 17 , 7 ( 2014 )
11. A. Einstein , Sitz. Preuss. Akad. Wiss., p. 217 ; ibid p. 224 ( 1928 )
12. K. Hayashi , T. Shirafuji , Phys. Rev. D 19 , 3524 ( 1979 ) [Addendumibid. 24 , 3312 ( 1982 )]
13. R. Aldrovandi , J.G. Pereira , Teleparallel Gravity : An Introduction (Springer, Dordrecht, 2013 )
14. G.R. Bengochea , R. Ferraro , Phys. Rev. D 79 , 124019 ( 2009 )
15. E.V. Linder , Phys. Rev. D 81 , 127301 ( 2010 )
16. S.H. Chen , J.B. Dent , S. Dutta , E.N. Saridakis , Phys. Rev. D 83 , 023508 ( 2011 )
17. R.J. Yang, Eur. Phys. J. C 71 , 1797 ( 2011 )
18. M. Li , R.X. Miao , Y.G. Miao , JHEP 1107 , 108 ( 2011 )
19. M.H. Daouda , M.E. Rodrigues , M.J.S. Houndjo , Eur. Phys. J. C 72 , 1890 ( 2012 )
20. Y.P. Wu , C.Q. Geng , Phys. Rev. D 86 , 104058 ( 2012 )
21. K. Bamba , R. Myrzakulov , S. Nojiri , S.D. Odintsov , Phys. Rev. D 85 , 104036 ( 2012 )
22. K. Karami , A. Abdolmaleki , JCAP 1204 , 007 ( 2012 )
23. V.F. Cardone , N. Radicella , S. Camera , Phys. Rev. D 85 , 124007 ( 2012 )
24. G. Otalora , JCAP 1307 , 044 ( 2013 )
25. J. Amoros , J. de Haro , S.D. Odintsov , Phys. Rev. D 87 , 104037 ( 2013 )
26. K. Bamba , S. Capozziello , M. De Laurentis , S'i Nojiri , D. S?ezG?mez, Phys. Lett . B 727 , 194 ( 2013 )
27. S. Bahamonde , C.G. B?hmer , M. Wright , Phys. Rev. D 92 , 104042 ( 2015 )
28. J. de Haro , J. Amoros , Phys. Rev. Lett . 110 ( 7 ), 071104 ( 2013 )
29. A. Paliathanasis , J.D. Barrow , P.G.L. Leach , Phys. Rev . D 94 ( 2 ), 023525 ( 2016 )
30. Y.F. Cai , S. Capozziello , M. De Laurentis , E.N. Saridakis , arXiv:1511.07586 [grqc]
31. G. Kofinas , E.N. Saridakis , Phys. Rev. D 90 , 084044 ( 2014 )
32. G. Kofinas , E.N. Saridakis , Phys. Rev. D 90 , 084045 ( 2014 )
33. G. Otalora , E.N. Saridakis , arXiv:1605.04599 [grqc]
34. L. Iorio , E.N. Saridakis , Mon. Notices R. Astron . Soc. 427 , 1555 ( 2012 )
35. M.L. Ruggiero , N. Radicella , Phys. Rev. D 91 , 104014 ( 2015 )
36. L. Iorio , N. Radicella , M.L. Ruggiero , JCAP 1508 ( 08 ), 021 ( 2015 )
37. G. Farrugia , J.L. Said , M.L. Ruggiero , Phys. Rev. D 93(10) , 104034 ( 2016 )
38. P. Wu , H.W. Yu , Phys. Lett . B 693 , 415 ( 2010 )
39. S. Capozziello , O. Luongo , E.N. Saridakis , Phys. Rev. D 91(12) , 124037 ( 2015 )
40. S. Nesseris , S. Basilakos , E.N. Saridakis , L. Perivolaropoulos , Phys. Rev. D 88 , 103010 ( 2013 )
41. R.C. Nunes , S. Pan , E.N. Saridakis , JCAP. arXiv:1606 .04359 [grqc] (to appear)
42. P.A.M. Dirac , Nature 139 , 323 ( 1937 )
43. P.A.M. Dirac , Proc. R. Soc . Lond . A 165 , 198 ( 1938 )
44. E.A. Milne , Relativity, Gravitation and World Structure (Clarendon Press, Oxford, 1935 )
45. E.A. Milne , Proc. R. Soc. A3 , 242 ( 1937 )
46. P. Jordan , Naturwiss. 25 , 513 ( 1937 )
47. P. Jordan , Z. Phys . 113 , 660 ( 1939 )
48. P. Jordan , Nature 164 , 637 ( 1949 )
49. C. Brans , R.H. Dicke , Phys. Rev. D 124 , 925 ( 1961 )
50. G. Gamow , Phys. Rev. Lett . 19 , 759 ( 1967 )
51. G.R. Dvali , M. Zaldarriaga , Phys. Rev. Lett . 88 , 091303 ( 2002 )
52. T. Chiba , K. Kohri , Prog. Theor. Phys. 107 , 631 ( 2002 )
53. L. Anchordoqui , H. Goldberg , Phys. Rev. D 68 , 083513 ( 2003 )
54. C. Wetterich , Phys. Lett . B 561 , 10 ( 2003 )
55. E.J. Copeland , N.J. Nunes , M. Pospelov , Phys. Rev. D 69 , 023501 ( 2004 )
56. M.d.C. Bento , O. Bertolami , N.M.C. Santos , Phys. Rev. D 70 , 107304 ( 2004 )
57. V. Marra , F. Rosati , JCAP 0505 , 011 ( 2005 )
58. P.P. Avelino , Phys. Rev. D 78 , 043516 ( 2008 )
59. H.B. Sandvik , J.D. Barrow , J. Magueijo , Phys. Rev. Lett . 88 , 031302 ( 2002 )
60. D.F. Mota , J.D. Barrow , Mon. Notices R. Astron . Soc. 349 , 291 ( 2004 )
61. J.D. Barrow , D. Kimberly , J. Magueijo , Class. Quant. Grav. 21 , 4289 ( 2004 )
62. H. Wei , Phys. Lett . B 682 , 98 ( 2009 )
63. H. Wei , X.P. Ma , H.Y. Qi , Phys. Lett . B 703 , 74 ( 2011 )
64. J. Magueijo , Rep. Prog. Phys. 66 , 2025 ( 2003 )
65. J.P. Uzan, Living Rev . Relativ. 14 , 2 ( 2011 )
66. P.P. Avelino , C.J.A.P. Martins , N.J. Nunes , K.A. Olive , Phys. Rev. D 74 , 083508 ( 2006 )
67. E. GarciaBerro , J. Isern , Y.A. Kubyshin , Astron. Astrophys. Rev . 14 , 113 ( 2007 )
68. J.D. Barrow, Ann. Phys. 19 , 202 ( 2010 )
69. T. Chiba , Prog. Theor. Phys. 126 , 993 ( 2011 )
70. J.K. Webb , J.A. King , M.T. Murphy , V.V. Flambaum , R.F. Carswell , M.B. Bainbridge , Phys. Rev. Lett . 107 , 191101 ( 2011 )
71. J.A. King , J.K. Webb , M.T. Murphy , V.V. Flambaum , R.F. Carswell , M.B. Bainbridge , M.R. Wilczynska , F.E. Koch , Mon. Notices R. Astron . Soc. 422 , 3370 ( 2012 )
72. J. Sol? , Mod. Phys. Lett. A 30(22) , 1502004 ( 2015 )
73. A.M.M. Pinho , C.J.A.P. Martins, Phys. Lett . B 756 , 121 ( 2016 )
74. H. Fritzsch , R.C. Nunes , J. Sola , arXiv:1605.06104 [hepph]
75. Y.V. Stadnik , V.V. Flambaum , Phys. Rev. Lett . 114 , 161301 ( 2015 )
76. W.W. Zhu et al., Astrophys. J . 809 ( 1 ), 41 ( 2015 )
77. G.R. Dvali , G. Gabadadze , M. Porrati , Phys. Lett . B 485 , 208 ( 2000 )
78. K.A. Olive , M. Pospelov , Phys. Rev. D 65 , 085044 ( 2002 )
79. P. Brax , A.C. Davis , B. Li , H.A. Winther , Phys. Rev. D 86 , 044015 ( 2012 )
80. P. Brax , C. Burrage , A.C. Davis , D. Seery , A. Weltman , Phys. Lett . B 699 , 5 ( 2011 )
81. R.J. Yang, Europhys. Lett. 93 , 60001 ( 2011 )
82. A. Songaila , L.L. Cowie , Astrophys . J. 793 , 103 ( 2014 )
83. T.M. Evans et al., Mon . Notices R. Astron . Soc. 445 ( 1 ), 128 ( 2014 )
84. P. Molaro , D. Reimers , I.I. Agafonova , S.A. Levshakov , Eur. Phys. J. ST 163 , 173 ( 2008 )
85. H. Chand , R. Srianand , P. Petitjean , B. Aracil , R. Quast , D. Reimers , Astron. Astrophys. 451 , 45 ( 2006 )
86. P. Molaro et al., Astron. Astrophys . 555 , A68 ( 2013 )
87. I.I. Agafonova , P. Molaro , S.A. Levshakov , J.L. Hou , Astron. Astrophys. 529 , A28 ( 2011 )
88. N. Suzuki et al., Astrophys. J . 746 , 85 ( 2012 )
89. C. Blake et al., Mon . Notices R. Astron . Soc. 418 , 1707 ( 2011 )
90. K. Shi , Y. Huang , T. Lu , Mon. Notices R. Astron . Soc. 426 , 2452 ( 2012 )
91. P. Wu , H.W. Yu , Phys. Lett . B 692 , 176 ( 2010 )
92. C. Xu , E.N. Saridakis , G. Leon , JCAP 1207 , 005 ( 2012 )
93. G. Kofinas , G. Leon , E.N. Saridakis , Class. Quant. Grav. 31 , 175011 ( 2014 )
94. R. Zheng , Q.G. Huang , JCAP 1103 , 002 ( 2011 )
95. X.L. Meng , X. Wang , S.Y. Li , T.J. Zhang , arXiv:1507.02517 [astroph.CO]