Palatini formulation of f(R, T) gravity theory, and its cosmological implications
Eur. Phys. J. C
Palatini formulation of f ( R, T ) gravity theory, and its cosmological implications
Jimin Wu 2
Guangjie Li 2
Tiberiu Harko 0 1 2
ShiDong Liang 2 3
0 Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
1 Department of Physics, BabesBolyai University , Kogalniceanu Street, 400084 ClujNapoca , Romania
2 School of Physics, Sun YatSen University , Guangzhou 510275 , People's Republic of China
3 State Key Laboratory of Optoelectronic Material and Technology, Guangdong Province Key Laboratory of Display Material and Technology , Guangzhou , People's Republic of China
We consider the Palatini formulation of f ( R, T ) gravity theory, in which a nonminimal coupling between the Ricci scalar and the trace of the energymomentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the LeviCivita connection of an auxiliary, energymomentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the nonconservation of the energymomentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f ( R, T ) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type f = R − α2/ R + g(T ) and f = R + α2 R2 + g(T ) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Palatini formulation of f ( R, T ) gravity . . . . . . . .
Appendix A: f (R, T ) field equations in the metric
formulation . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B: Divergence of the matter energymomentum
tensor in the metric formalism . . . . . . . . . . . . .
Appendix C: The geometric quantities in the FRW geometry
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
The observational discovery of the recent acceleration of
the Universe [
1–5
] has raised the fundamental theoretical
problem if general relativity, in its standard formulation, can
fully account for all the observed phenomena at both
galactic and extragalactic scales. The simplest theoretical
explanation for the observed cosmological dynamics consists in
slightly modifying the Einstein field equations, by adding to
it a cosmological constant [
6
]. Together with the
assumption of the existence of another mysterious component of the
Universe, called dark matter [
7,8
], assumed to be cold and
pressureless, the Einstein gravitational field equations with
the cosmological constant included, can give an excellent fit
to all observed data, thus leading to the formulation of the
standard cosmological paradigm of our present days, called
the Cold Dark Matter ( CDM) model. However, despite
its apparent simplicity and naturalness, the introduction of
the cosmological constant raises a number of important
theoretical and observational question for which no convincing
answers have been provided so far. The CDM model can fit
the observational data at a high level of precision, it is a very
simple theoretical approach, it is easy to use in practice, but
up to now no fundamental theory can explain it. Why is the
cosmological constant so small? Why is it so finetuned? And
why did the Universe begin to accelerate only recently? And,
after all, would a cosmological constant really be necessary
to explain all observations?
From a theoretical point of view two possible answers to
the questions raised by the observation of the recent
acceleration of the Universe can be formulated. The first, called the
dark energy approach, assumes that the Universe is filled by
a mysterious and unknown component, called dark energy
[
9–12
], which is fully responsible for the acceleration of the
Universe, as well as for its mass–energy balance. The
cosmological constant corresponds to a particular phase of
the dynamical dark energy (the ground state of a potential,
say), and the recent de Sitter phase may prove to be just an
attractor of the dynamical system describing the
cosmological evolution. A second approach, the dark gravity approach,
assumes the alternative possibility that at large scales the
gravitational force may have a very different behavior as
compared to the one suggested by standard general relativity.
In the general relativistic description of gravity, the starting
point is the Hilbert–Einstein action, which can be written
down as S = R/2κ2 + Lm √−g d4x , where R is the
Ricci scalar, κ is the gravitational coupling constant, and Lm
is the matter Lagrangian, respectively. Hence in dark
gravity type theories for a full understanding of the gravitational
interaction a generalization of the Hilbert–Einstein action is
necessary.
There are (at least) two possibilities to construct dark
gravity theories. The first is based on the modification of
the geometric part of the Hilbert–Einstein Lagrangian only.
An example of such an approach is the f (R) gravity
theory, introduced in [
13,14
], and in which the geometric part
of the action is generalized so that it becomes an
arbitrary function f (R) of the Ricci scalar. Hence in f (R)
gravity the total Hilbert–Einstein action can be written as
S = f (R)/2κ2 + Lm √−g d4x . The recent
cosmological observations can be satisfactorily explained in the f (R)
theory, and a solution of the dark matter problem, interpreted
as a geometric effect in the framework of the theory, can also
be obtained [15]. For reviews and in depth discussions of
f (R) and other modified gravity theories see [
16–24
].
A second avenue for the construction of the dark gravity
theories consists in looking for maximal extensions of the
Hilbert–Einstein action, in which the matter Lagrangian Lm
plays an equally important role as the Ricci scalar. Hence in
this more general approach one modifies both the
geometric and the matter terms in the Hilbert–Einstein action, thus
allowing a coupling between matter and geometry [
25,39
].
The first possibility for such a coupling is to replace the
gravitational action by an arbitrary function of the Ricci scalar
and the matter Lagrangian Lm, thus obtaining the socalled
f (R, Lm) class of modified gravity theories [26]. This class
of theories has the potential of explaining the recent
acceleration of the Universe without the need of the cosmological
constant, and can give some new insights into the dark
matter problem, and on the nature of the gravitational motion.
The cosmological and physical implications of this theory
have been intensively investigated [
27–37
]. For a review of
f (R, Lm) type theories see [
38
].
A second extension of the Hilbert–Einstein action can
be obtained by assuming that the gravitational field
couples to the trace T of the energymomentum tensor of the
matter. This assumption leads to the f (R, T ) class of
gravitational theories [
39
]. f (R, T ) theory may give some hints
for the existence of an effective classical description of the
quantum properties of gravity. As pointed out in [
40
], by
using a nonperturbative approach for the quantization of
the metric, proposed [
41–43
], as a consequence of the
quantum fluctuations of the metric, a particular type of f (R, T )
gravity naturally emerges, with the Lagrangian given by
L = (1 − α)R/2κ2 + (Lm − αT /2) √−g, where α is a
constant. This interesting theoretical result suggests that a
deep connection may exist between the quantum field
theoretical description of gravity, which naturally involves
particle production in the gravitational field, and the
corresponding effective classical description of the f (R, T ) gravity
theory [
44
]. The astrophysical and cosmological implications of
f (R, T ) gravity theory were investigated in [
45–66
].
Einstein’s theory of general relativity can be obtained by
starting from two different theoretical approaches, called the
metric and the Palatini formalism, respectively, the latter
being introduced by Albert Einstein [
67–69
]. In the Palatini
variational approach one takes as independent field variables
not only the ten components gμν of the metric tensor, but also
the components of the affine connection αβμ, without
assuming, a priori, the form of the dependence of the connection on
the metric tensor, and its derivatives [
70,71
]. When applied to
the Hilbert–Einstein action, these two approaches lead to the
same gravitational field equations. Moreover, the Palatini
formalism also provides the explicit form of the symmetric
connection as determined by the derivatives of the metric tensor.
However, in f (R) modified gravity, as well as in other
modified theories of gravity, this does not happen anymore. In fact
it turns out that the gravitational field equations obtained by
using the metric approach are generally different from those
obtained by using the Palatini variation [
70,71
]. An important
difference is related to the order of the field equations. The
metric formulation usually leads to higherorder derivative
field equations, while in the Palatini formalism the obtained
gravitational field equations are always secondorder partial
differential equations. A number of new algebraic relations
also do appear in the Palatini variational formulation, which
describe the subtle relation between the matter fields and the
affine connection, which can be determined from a set of
equations that couples it not only to the metric, but also to
the matter fields. The astrophysical and cosmological
implications of the Palatini formulation of f (R) gravity have also
been intensively investigated [
72–76
].
Based on a hybrid combination of the metric and Palatini
mathematical formalisms, an extension of the f (R) gravity
theory was proposed in [
77
] and was used to construct a new
type of gravitational Lagrangian [
77,78
]. A simple example
of such an hybrid metricPalatini theory can be constructed
by adopting for the gravitational Lagrangian the expression
R + f R(g, ˜ ) , where R(g, ˜ ) is the Palatini scalar curva
ture. A similar formalism that interpolates between the metric
and Palatini regimes was proposed in [
79,80
] for the study
of f (R) type theories. This approach is called Ctheory. A
generalization of the hybrid metricPalatini gravity was
introduced in [81].
Despite the intensive investigations of the theoretical and
observational aspects of the modified gravity theories with
geometry–matter coupling, their Palatini formulation and
properties have attracted considerably less attention. The
Palatini formulation of the linear f (R, Lm) gravity was
introduced in [
82
], where the field equations and the
equations of motion for massive test particles were derived.
The independent connection can be expressed as the
LeviCivita connection of an auxiliary, matter Lagrangian
dependent metric, which is related to the physical metric by
means of a conformal transformation. Similar to the
metric case, the field equations impose the nonconservation of
the energymomentum tensor. The study of Palatini
formulation of f (R, T ) gravity was initiated in [
83
]. Analogously
to its metric counterpart, the field equations impose of the
f (R, T ) gravity in the Palatini formulation implies the
nonconservation of the energymomentum tensor, which leads
to nongeodesic motion, and to the appearance of an extra
force.
It is the purpose of the present paper to derive the
gravitational field equations of the generalized f (R, T ) type
gravity models, with nonminimal coupling between matter,
described by the trace of the energymomentum tensor, and
geometry, characterized by the Ricci scalar By taking
separately two independent variations of the gravitational action
with respect to the metric and the connection, respectively,
we obtain the field equations and the connection associated
to the Ricci tensor, which, due to the coupling between the
trace of the energymomentum tensor and the geometry, is
also a function of T . The metric that defines the new
independent connection is conformally related to the initial
spacetime metric, with the conformal factor given by a function of
the trace of the energymomentum tensor, and of the Ricci
scalar. After the conformal factor is obtained, the
gravitational field equations can be written down easily in both
metrics. Similar to the case of the metric f (R, T ) gravity,
after taking the divergence of the gravitational field equations
we obtain the important result that the energymomentum
tensor of the matter is not conserved. Similar to the metric
case, the motion of the particles is not geodesic, and due to
the matter–geometry coupling, an extra force arises.
However, this force has the same expression as in the metric case,
and therefore no new physics is expected to arise during the
motion of massive test particles in the Palatini formulation
of f (R, T ) gravity. As the next step in our analysis we
investigate in detail the cosmological implications of the Palatini
formulation of the f (R, T ) gravity theory. We obtain the
generalized Friedmann equations, which explicitly contain
the extra terms generated by the coupling between the trace
of the energymomentum tensor and geometry. The general
properties of the cosmological evolution are obtained,
including the behavior of the deceleration parameter, of the
effective energy density and pressure, and of the parameter of the
equation of state of the dark energy. Cosmological models
with Lagrangians of the type L = R − α2/R + g(T ) and
L = R + α2 R2 + g(T ) are considered in detail, and it is
shown that these models lead to evolution equations whose
solutions tend to a de Sitter type Universe at late times.
The present paper is organized as follows. After a brief
review of the metric formalism, the field equations of
f (R, T ) gravity theory are obtained by using the Palatini
formalism of gravitational theories in Sect. 1. The energy
and momentum balance equations are obtained, after taking
the divergence of the energymomentum tensor, in Sect. 3.
The thermodynamical interpretation of the theory is also
briefly discussed. The cosmological implications of the
Palatini f (R, T ) theory is investigated in Sect. 4. We discuss and
conclude our results in Sect. 5. The details of the
derivation of the field equations in the metric formalism are given
in Appendix A, while the divergence of the matter
energymomentum tensor is derived in Appendix B. The explicit
computations of the various geometric quantities for the
Friedmann–Robertson–Walker geometry are presented in
Appendix C.
2 Palatini formulation of f ( R, T ) gravity
In the present section, after a brief review of the metric
formulation of f (R, T ) gravity theory, we derive the field equations
of the theory by using the Palatini formalism.
2.1 The metric formalism
The f (R, T ) gravity theory is described by the action [39]1
S =
√16−πg f (R, T ) + √−g Lm
d4x ,
where g ≡ det gμν , f is an arbitrary function of the Ricci
scalar R = R(g) and of the trace T = gμν Tμν of the matter
energymomentum tensor Tμν ; the matter Lagrangian Lm
is assumed to be independent of ∂λgμν . Tμν is generally
obtained as [
89
]
Tμν ≡ √
−2
−g
∂ √−g Lm
∂gμν
+ ∂λ
∂ √−g Lm
∂ (∂λgμν )
−2 ∂ √−g Lm
= √−g ∂gμν
∂ Lm
= −2 ∂gμν + gμν Lm.
To describe the variation of the energymomentum tensor
with respect to the metric, we also introduce the tensor μν ,
defined as
μν ≡ gαβ δδgTμαβν .
(1)
(2)
(3)
1 Throughout this article we use the natural system of units with
c = G = 1. For the metric tensor we adopt the signature convention
(−, +, +, +).
fT
∇μT νμ = 8π − fT
1
− 2 ∇ν T
≡ Qν .
For a perfect fluid characterized by its energy density ρ
and isotropic pressure P only, the energymomentum tensor
is given by
where the fourvelocity uμ satisfies the normalization
condition uμuμ = − 1. In the comoving frame its components are
uμ = (− 1, 0, 0, 0), and in this frame the components of the
μ
energymomentum tensor become T ν = (− ρ , P, P, P).
The components of the affine connection are defined to be
ρμν (g) =
gρσ
2
∂ν gσ μ + ∂μgσ ν − ∂σ gμν ,
which are currently regarded as functions of the metric. In
the following we assume that the connection is symmetric,
that is, ρμν = ρνμ. Varying Eq. (1) with respect to gμν , we
obtain
δ S =
f R δ R(g) + fT δT − g2μν f δgμν
− 8π Tμν δgμν d4x ,
where f R ≡ ∂∂ Rf , fT ≡ ∂∂Tf and
δ R(g) = Rμν (g)δgμν + gμν δ Rμν (g).
From the condition δ S = 0 we obtain the field equations of
f (R(g), T ) gravity theory as (for the computational details
see Appendix A),
Rμν (g) + gμν
gμν f = 8π Tμν .
− 2
− ∇μ∇ν f R + Tμν +
μν fT
The field equations (8) can be rewritten with the help of
the Einstein tensor Gμν (g) = Rμν (g) − gμν R(g)/2 as
1
Gμν (g) = f R
8π T νμ − T μ
ν +
μν fT
μ
+ δ2ν [ f − R(g) f R ] + ∇μ∇ν − δμν
From Eq. (9) it follows that the matter energymomentum
tensor is not conserved, and its divergence is given by [
52
]
(for the computational details see Appendix B),
(4)
(5)
(6)
(7)
(8)
f R . (9)
(10)
2.2 Palatini formulation of f (R, T ) gravity
2.2.1 Field equations from metric variation
An alternative formulation of gravitational theories can be
obtained within the Palatini formalism, which consists in
taking separately in the gravitational action two
independent variations, with respect to the metric and the
connection, respectively. The action is formally identical to the
metric one, but the Riemann tensor and the Ricci tensor are
constructed with the independent symmetric connection ˜ .
Hence in the Palatini formulation the gravitational action of
f (R, T ) gravity is given by
S =
√−g f R g, ˜ , T
16π
+
√−g Lm (g, ψ ) d4x .
In Eq. (11) the Ricci scalar is defined as
the Palatini connection ˜ , with the connection coefficients
˜ μλν determined selfconsistently through the independent
variation of the gravitational field action Eq. (11), and not
constructed directly from the metric by using the usual
LeviCivita definition. In the following we define R˜μν ˜ with
the help of the yet undetermined Palatini connection as
R˜μν ˜
= ∂λ ˜ μν − ∂ν ˜ μλ + ˜ μλν ˜ λα − ˜ μαλ ˜ νλα.
λ λ α
(13)
The matter Lagrangian Lm (g, ψ ) is assumed to be a function
of the metric tensor g and of the physical fields ψ only.
We vary now the gravitational action (11) with respect to
the metric tensor gμν , under the assumption δ R˜μν ˜ = 0,
that is, by keeping the connection constant. As a result we
immediately obtain the field equations
R˜μν ˜ f R = 8π Tμν − Tμν +
μν fT + g2μν f.
By contracting the above equation with gμν we obtain for
the Ricci scalar (12) the expression
R g, ˜ f R = 8π T − (T +
) fT + 2 f,
where
=
μμ. In terms of the Einstein tensor
the Palatini field equations can be written as
We vary now the gravitational action (11) with respect to the
connection ˜ , by keeping the metric constant, so that
δ R g, ˜
= gμν δ R˜μν ˜ .
According to the Palatini identity [
82
], we have
δ R˜μν ˜
= ∇˜ λ δ ˜ μλν − ∇˜ ν δ ˜ μλλ ,
where ∇˜ λ is the covariant derivative associated with ˜ . Hence
the variation of the action (11) with respect to ˜ leads to
∇˜ λ δ ˜ μλν − ∇˜ ν δ ˜ μλλ
d4x , (20)
(18)
(19)
(21)
(22)
(23)
(24)
(25)
(16)
δ S =
√−g Aμν
16π
where we have denoted
Aμν = f R gμν .
We integrate now by parts to obtain
16π δ S =
∇˜ λ
√−g Aμν δ ˜ μλν − Aμλδ ˜ μαα
d4x
−
∇˜ λ
√−g Aμν δαλ − Aμλδν
α
δ ˜ μαν d4x .
The first term in Eq. (22) is a total derivative, and thus after
transforming it into a surface integral it vanishes. Therefore
the variation of the action with respect to the connection ˜
becomes
∇˜ λ
√−g Aμν δαλ − Aμλδν
α
Equation (23) can be significantly simplified by taking into
account that for α = λ the equation is identically zero. Hence
for the case λ = α, we find
with the conformal factor F defined as
F ≡ f R R g, ˜ , T .
Moreover, we have
−g˜ g˜μν =
√−g f R gμν =
−F 4g F −1gμν ,
where g˜ ≡ det g˜μν . Thus Eq. (24) gives the geometric
interpretation of the Palatini connection ˜ as the LeviCivita
connection corresponding to the metric g˜μν , conformally
related to gμν ,
λ 1 gλρ ∂ν g˜ρμ + ∂μg˜ρν − ∂ρ g˜μν .
˜ μν = 2 ˜
2.2.3 Conformal geometry and g frame field equations
Since the metrics gμν and g˜μν are conformally related, the
connection ˜ can be expressed in terms of the LeviCivita
connection , whose components are given in Eq. (5), as
λ
˜ μν =
λ 1
μν + 2F
δλμ∂ν + δλν ∂μ − gμν ∂λ F.
In terms of the tensor Rμν (g), constructed from the metric
by using the LeviCivita connection (5), the Ricci tensor R˜μν
in the conformally transformed metric is given by [
84,85
]
1
R˜μν = Rμν (g) + F
3 gμν
2F ∇μ F ∇ν F − ∇μ∇ν + 2
Since R˜ ≡ g˜μν R˜μν = F −1 R g, ˜ and G˜ μν ≡ R˜μν −
gμν R˜ /2 = Gμν g, ˜ , the Ricci scalar (12) and the
Ein˜
stein tensor in Eq. (17) can be obtained in the conformally
related frames [
84,85
]:
3
= F R˜ = R(g) + F
1
2F
(∇ F )2 −
F
1
= G˜ μν = Gμν (g) + F
gμν
− ∇μ∇ν F
,
respectively, with all the covariant derivatives and algebraic
operations performed with the help of the metric gμν .
Tμν → T˜μν =
and
T νμ → T˜ νμ = −2g˜μλ ∂∂gL˜λmν + δμν Lm = T νμ.
That is, the mixed components T νμ of the energymomentum
tensor are conformally invariant. Additionally, we have
μ μ
ν → ˜ ν =
μν , T → T˜ = T ,
→ ˜ =
.
(38)
By using the expression of the Einstein tensor as given
by Eq. (32) in Eq. (17), we obtain finally the f (R, T ) field
equations of the Palatini formulation expressed solely in the
g frame:
gμν (∇ F )2
− 2
−
(35)
(36)
(37)
In the previous subsection we have obtained the gravitational
field equations in the Palatini formulation of f (R, T ) gravity
as expressed in terms of the metric tensor gμν . This approach
usually involves higherorder differential equations for the
physical and geometrical quantities. An alternative approach,
which keeps the order of the differential equations of the
model not higher than two would be to solve first the
gravitational field equations in the conformal metric g˜μν , and to
recover the metric gμν with the use of the conformal
transformation (25). To obtain the field equations in the conformal
frame we follow the procedure introduced in [
83
].
First we multiply the Palatini field equation (14) with
g˜λμ = f R −1gλμ, thus obtaining
Rλ
1
˜ ν = F 2
8π T νλ − T λ
ν +
λν fT + δ2λν f ,
where R˜ λν is now a function of the metric g˜ only. As for the
energymomentum tensor, we have, respectively,
−2 ∂
−g˜
−g˜ Lm
∂ g˜μν
∂ Lm
= −2 ∂ g˜μν + g˜μν Lm
With the use of the above equations, and preserving only the
firstorder terms, the field equation (14) becomes
− 21 ∂2γ˜μν − 2∂μν w − η2μν f = 8π Tμν − Tμν +
For perfect fluids, T μν = diag(−ρ , P, P, P) and μν =
δμν P − 2T νμ (see Eq. (61)); besides, in the Newtonian limit,
P → 0 and ∂0 → 0. Hence we can obtain immediately the
generalized Poisson equation in the Palatini formulation of
f (R, T ) gravity:
− 21 ∇2γ˜00 = (8π + fT ) ρ − 2f .
The same result can also be found in [
83
].
2.4 Violation of the equivalence principle
By contracting Eq. (35) by taking ν = λ we obtain the
expression of the Ricci scalar in the conformal frame, thus
then some algebra gives the expression of the Ricci
tensor (30) as
(39)
Therefore in the conformal frame the full set of the Einstein
equations can be written with f = f F R˜ = R g, ˜ , T
as
μ 1
δ ν R˜ = F 2
G˜ μν = R˜ μν − 2
8π
T μ
These equations determine the conformal metric g˜ as a
function of the thermodynamic parameters that enter in the
definition of the matter energymomentum tensor.
2.3 The Newtonian limit
To investigate the Palatini f (R, T ) gravity under the weak
field, slow motion and static approximation, namely the
Newtonian limit, we assume the metric to be a Minkowski metric
plus a perturbation, given by
gμν = ημν + γμν , γμν
1.
Hence gμν = ημν − γ μν and the relation gαλgλβ = δαβ
still holds. In this context, we also assume the conformal
metric (25) to be nearly flat, so that
g˜μν = Fgμν = ημν + γ˜μν ,
where γ˜μν 1 is of the same order as γμν . Hence F ≈ 1.
From Eqs. (41) and (42) we obtain
(F − 1)ημν = γ˜μν − F γμν .
If we take F = e2W and expand it to F = 1 + 2W , then the
above equation shows that W = (γ˜ − γ ) /2 (4 + γ ), where
γ˜ ≡ ημν γ˜μν and γ ≡ ημν γμν . Thus W ∼ O(γ ) ∼ O(γμν ).
Let us now consider the Palatini field equation (14) under
the Newtonian limit. First we obtain the gframe Ricci tensor:
1
Rμν (g) = 2
∂λ∂μγνλ + ∂λ∂ν γμλ − ∂2γμν − ∂μν γ .
Omitting all higherorder terms with respect to O(γμν ) and
taking into account the gauge
∂μ
1
γμν − 2 ημν γ
= 0,
(46)
μν fT .
(47)
(48)
(41)
(42)
(43)
(44)
(45)
An interesting feature of the modified gravities, including
their Palatini extensions, is the violation of the equivalence
principle. In Palatini f (R) gravity this problem was
discussed in [
86
]. In the following we will generalize some
results from the case of f (R) theory to the Palatini f (R, T )
gravity. In the conformal frame the field equations of the
Palatini f (R, T ) gravity are given by Eq. (40), where f =
f F R˜ , T = f R g, ˜ , T .
In the weak field limit we can represent the gravitational
Lagrangian as
The above equation tells us that, similar to the metric and
the Palatini formulation of f (R) gravity [
86
], in the Palatini
formulation of f (R, T ) gravity it is impossible to recover
the flat Minkowski metric even in local frames with external
gravitational fields screened. This result violates the basic
postulate of general relativity according to which in locally
free falling frames the nongravitational laws of physics are
those of special relativity [
86
]. Since this postulate assumes
that the Einstein equivalence principle holds, it follows that,
similar to Palatini f (R) theory, in the Palatini formulation
of f (R, T ) gravity the equivalence principle does not hold
exactly. The deviation of the current metric with respect to
the Minkowski metric is ημν − gμν ≈ (∂ K /∂ R) ημν .
In order to give a quantitative estimate for the deviation
of the f (R, T ) metric from the Minkowski metric we will
consider the cosmological case, to be discussed in detail in
Sect. 4. We assume that the gravitational action takes the form
f (R(g, ˜ ), T ) = R(g, ˜ ) + 16απ R2(g, ˜ ) + 8πβ T , where
α, β → 0, as an example. From Eq. (141), to be derived in
Sect. 4, it follows that F = 1+β0αρ = 1+β0αρ0a−β1 , where
ρ0 is the presentday matter density, a is the scale factor, β0 =
1−3w +β(3−5w), β1 = 3(1+β)(1+w)/[1+β(3−w)/2],
and w is the parameter of the matter equation of state. By
estimating all quantities at the present time t = t0, then when
α, β → 0, and w = 0, a (t0) = 1, the deviation (∂ K /∂ R)
from the Minkowki metric can be obtained:
(51)
1 − F −1
= 1 − [1 + β0t=t0 αρ0a0− β1t=t0 ]−1
where H0 is the presentday value of the Hubble function, and
we have assumed for the presentday density of the Universe
the critical value.
Recently the first results obtained by the MICROSCOPE
satellite, whose aims are to constrain the weak equivalence
principle in outer space by determining the Eötvös
parameter η, have been published [
87
]. The Eötvös parameter is
defined as the normalized difference of accelerations between
two bodies i and j , located in the same gravitational field.
The MICROSCOPE determinations give for η the value
η = (− 1 ± 27) × 10−15 at a 2σ confidence level [
87
]. These
results allow one to constrain possible sources of violation of
the weak equivalence principle, like, for example, the
existence of light or massive scalar fields with coupling to matter
weaker than the gravitational coupling [
88
]. For a massive
scalar field of mass smaller than 10−12 eV, the coupling is
constrained as αC  < 10−11, if the scalar field couples to
the baryon number, and to αC  < 10−12 if the scalar field
couples to the difference between the baryon and the lepton
numbers, respectively. We expect a similar order of
magnitude for the coupling between matter and geometry in both
the metric and the Palatini formulations of f (R, T ) gravity.
3 Energy and momentum balance equations
An interesting and important consequence of modified
gravity theories with geometry–matter coupling is the
nonconservation of the matter energymomentum tensor. This
property of the theory has a number of far reaching
physical implications, and it may represent the main link between
the interpretation of f (R, T ) theory as an effective classical
description of the quantum theory of gravity. In the present
section we obtain the general expression of the divergence
of the energymomentum tensor in f (R, T ) gravity theory,
and, by using it, we obtain the energymomentum balance
equations, which describe the energy transfer processes from
geometry to matter, and the deviations from the geodesic
motion, respectively.
3.1 The divergence of the matter energymomentum tensor
We begin our analysis by calculating first the divergence of
the Einstein tensor in the Palatini frame. Since ∇μGμν (g) =
0, Eq. (32) yields
∇μG˜ μν = ∇μ G˜ μν − Gμν (g)
= 2 (∇ν
−
+∇ν (∇w)2
= − 2 Rμν (g)∇μw + 2 w∇ν w + 4∇μw∇μ∇ν w,
(52)
where to simplify the calculation we have taken F = e2w,
and we have used the mathematical identities [
84
]
(∇ν
−
∇ν ) w + 2 w∇ν w + 2∇μw∇μ∇ν w
= gαβ Rμβαν ∇μφ
= − Rμν ∇μφ
and
∇ν (∇φ)2 = ∇ν ∇μφ∇μφ = ∇μφ∇ν ∇μφ
+∇μφ∇ν ∇μφ = 2∇μφ∇ν ∇μφ
= 2∇μφ∇ν ∇μφ,
respectively, where φ (x μ) is an arbitrary scalar. We have also
used the relation gαβ Rμβαν = −gαβ Rβμαν = −Rμν in the
last step of Eq. (53). Note that the above two identities are
valid in both the metric and the Palatini formulations [
84
].
Substituting Eq. (30) into Eq. (52), we find
∇μG˜ μν = −2 Rμν (g)∇μw + 2 w∇ν w + 4∇μw∇μ∇ν w
∇μ F
= −2 R˜μν ∇μw = − F R˜μν .
(55)
(53)
(54)
The covariant divergence of the field equation (17) yields
(note that Gμν g, ˜ = G˜ μν and gμν R g, ˜ = g˜μν R˜ )
∇μ F Gμν g, ˜
+ g2μν R g, ˜ ∇μ F
=
G˜ μν − R˜μν + g˜2μν R˜ ∇μ F = 0
where we have used Eq. (55).
One can check now by comparison with Eq. (10) that the
above expression gives the same result as in the metric
formulation, except for the functional form of the function f ,
Multiplying Eq. (57) by uν [
46
] we obtain the f (R, T )
perfectfluid energy balance equation,
ρ˙ + 3(ρ + P)H = 8π−+fTfT (ρ + P)uμ∇μ ln  fT 
+uμ∇μ
, f = f R g, ˜ , T ,
where we have denoted H = ∇μuμ /3, and ˙ = d/ds =
uμ∇μ.
Multiplying (57) by the projection operator hνλ, defined
as hνλ ≡ δνλ + uν uλ [
46
], with the properties uν hνλ = 0,
hνλ∇μuν = ∇μuλ, and hνλ∇ν = gνλ + uν uλ ∇ν = ∇λ +
uλuν ∇ν , respectively, we obtain the (nongeodesic) equation
of motion of massive test particles:
uν ∇ν uλ = dd2sx2λ +
μλν uμuν =
−hνλ∇ν P + hνλ Qν
ρ + P
,
(66)
(67)
where
fT /2
hνλ Qν = 8π + fT
hνλ∇ν (ρ − P).
3.2 Balance equations in the conformal frame
In the following for notational simplicity we define first a
mixedcomponent vector field,
V μ
ν ≡ 8π
−
where V = − [8π T − (T + ) fT + 2 f ] =
−F R g, ˜ according to Eq. (15), and we have used Eq.
(28). Taking into account that F is a scalar and ∇˜ μ F =
∇μ F = ∂μ F , then
V
∇μV νμ = 2 ∂ν ln F  ,
(70)
or equivalently
∇μ 8π T νμ − T μ
ν +
= −
R g, ˜ ∂ν F
2
,
∇ν
μν fT − 2
where we have used the relation 8π T − (T + ) fT + f =
F R g, ˜ − f on the left hand side. Hence one can easily see
that the equations above are exactly the same as the energy
balance equation (57).
3.3 Thermodynamic interpretation of f (R, T ) gravity
theories
For the sake of completeness we briefly present the
thermodynamic interpretation of f (R, T ) gravity theories, as
discussed in [
46
]. The nonconservation of the matter
energymomentum tensor strongly suggests that, due to the matter–
geometry coupling, a particle creation processes may take
place during the cosmological evolution. This phenomenon
is also specific to quantum field theories in curved
spacetimes, as pointed out in [
90–92
], and it is a consequence
of a time varying gravitational field. Hence, f (R, T )
theory, which also involves particle creation, may lead to the
possibility of a semiclassical description of quantum field
theoretical processes in gravitational fields.
3.3.1 Particle and entropy fluxes, and the creation pressure
Particle creation implies that the covariant divergence of
the basic equilibrium quantities, including the particle and
entropy fluxes, as well as of the energymomentum tensor,
are now different from zero. Consequently, all the balance
equilibrium equations must be modified to include particle
creation [
93–95
]. In the presence of gravitationally generated
matter, the balance equation for the particle flux N μ ≡ nuμ,
where n is the particle number density, becomes
∇μ N μ = n˙ + 3H n = n ,
where is the particle production rate, which can be
neglected in the case that H . The entropy flux vector
is defined to be Sμ ≡ suμ = nσ uμ, where s is the entropy
density, and σ is the entropy per particle. The divergence of
the entropy flux gives
(72)
(73)
∇μ Sμ = nσ˙ + nσ
≥ 0.
∇μ Sμ = nσ
= s
≥ 0,
If we consider a specific σ which is constant, then
ρ˙ + 3H (ρ + P + Pc) = 0,
(74)
where the creation pressure Pc is defined as
that is, the variation of the entropy is entirely due to (adiabatic
gravitational) particle creation processes. Since s > 0, from
the above equation it follows that the particle creation rate
must satisfy the condition ≥ 0, that is, gravitational fields
can generate particles, but the inverse process is prohibited.
The energymomentum tensor of a fluid in the presence of
particle creation must also be modified to take into account
the second law of thermodynamics, so that [
96
]
T μν = Teμqν +
T μν ,
where Teμqν denotes the equilibrium component (4), and
T μν is the correction due to particle creation. Due to the
isotropy and homogeneity of spacetime, the extra
contribution to the equilibrium energymomentum tensor must be
represented by a scalar process. Generally one can write
T 00 = 0,
T ij = − Pcδ ji ,
where Pc is the dynamic creation pressure that describes
phenomenologically the thermodynamic effect of particle
creation in a macroscopic system. In a covariant representation
we have [
96
]
T μν = − Pchμν = − Pc gμν + uμuν ,
which immediately gives uμ∇ν T μν = 3H Pc. Therefore
in the presence of particle creation the total energy balance
equation uμ∇ν T μν = 0, which follows from Eq. (75),
immediately gives
ρ˙ + 3H (ρ + P + Pc) = 0.
The thermodynamic quantities must also satisfy the Gibbs
law, which can be formulated as [
94
]
s
nT d n
= nT dσ = dρ −
ρ + p dn,
n
where T is the thermodynamic temperature of the system.
3.3.2 Thermodynamic quantities in f (R, T ) gravity
After some simple algebraic manipulations the energy
balance Eq. (93) can be reformulated as
(75)
(76)
(77)
(78)
(79)
(80)
form ρ = ρ(n, T ) and p = p(n, T ), respectively. Then we
obtain
,
(81)
ρ˙ =
∂ρ
∂n
T
n˙ +
∂ρ
∂T
n
T˙ .
∂ρ
∂n
T
By using the energy and particle balance equations we find
− 3H (ρ + P + Pc) =
n (
− 3H ) +
Pc =
− fT (1 + w) ρ
8π + fT
1 − w 1
ln  fT  + 2 (1 + w) − 3H
× ∇μ uμ ln  fT  + (1 + w) ρ ∇μ uμ (1−w) ρ
1
2
where we have denoted w = P/ρ. Then the generalized
balance equation (93) can be derived from the divergence of
the total energy momentum tensor T μν , defined as
T μν = (ρ + P + Pc) uμuν + ( P + Pc) gμν .
On the other hand under the assumption of adiabatic particle
production, with σ˙ = 0, the Gibbs law gives
n
ρ˙ = (ρ + P) ˙
n = (ρ + P) (
− 3H ) ,
which together with the energy balance equation gives
immediately the relation between the particle creation rate and the
creation pressure as
−3H Pc
= ρ(1 + w)
In the framework of f (R, T ) gravity theory we obtain for
the particle creation rate the general expression
fT
= 8π + fT
1 − w
3H ln  fT  + 2 (1 + w)
− ∇μ uμ ln  fT  +(1 + w) ρ ∇μ uμ (1 − w) ρ
1
2
Hence the condition ≥ 0 imposes a strong constraint on the
physical parameters of the theory. In the case of pressureless
dust, with P = 0, w = 0, under the assumption fT > 0,
we obtain the following general cosmological constraint that
must be satisfied by the function fT for all times:
1
3H ln  fT  + 2
1
≥ ∇μ uμ ln  fT  + 2ρ ∇μ ρuμ . (86)
The divergence of the entropy flux vector can be
reformulated in terms of the creation pressure as
T
˙
T
T
Equation (89) yields the temperature evolution of a
relativistic fluid in the presence of matter creation:
− 3nσ H Pc
∇μ Sμ = ρ(1 + w)
Finally, we consider the temperature evolution in a system
with particle creation. In order to fully determine the time
behavior of a relativistic fluid we must add two equations
of state for the density and pressure, which have the general
In the particular case (∂ P/∂ρ)n = w = constant, we obtain
for the temperature–particle number dependence the simple
expression T ∼ nw.
3.3.3 The case w = −1
Based on the homogeneous and isotropic Friedmann–
Robertson–Walker metric, and on the energy conservation
equation ρ˙ + 3H (1 + w)ρ = 0, in [
97
], general
cosmological thermodynamic properties with an arbitrary, varying
equationofstate parameter w(a), where a is the scale
factor, were discussed. The w = − 1crossing problem of w
was explicitly pointed out, and the behaviors of the
quantities (ρ(a), μ(a), T (a), etc.) at/near w = −1 were discussed.
As a result of this study it was concluded that all
cosmological quantities must be regular and well defined for all values
of w(a), and indeed they are [
97
]. In the present
thermodynamical approach we have assumed that matter is created in
an ordinary form, and therefore all our previous results are
valid for w ≥ 0. However, the thermodynamic approach and
the interpretation of f (R, T ) gravity can be extended to the
case w < 0. In the following we will consider this problem,
and we show that our results are valid, in the sense of
regularity and of being well defined, even in the case of w = −1.
In particular, we concentrate on the temperature evolution
equation,
˙
T
T
=
∂ P
∂ρ
n
˙ ,
n n
(92)
which still holds even if w = P/ρ = −1. The
demonstration is as follows. First we consider the perfectfluid
energymomentum balance equation,
− fT
ρ˙ + 3(ρ + P)H = 8π + fT
+uμ∇μ
where Pc is the matter creation pressure. Under the
assumption of adiabatic particle production, with σ˙ = 0, the Gibbs
law gives
n
ρ˙ = (ρ + P) ˙
n = 0,
where σ is the entropy per particle. That is, from the above
two equations, we obtain
ρ˙ = Pc = 0.
Since ρ = ρ (n, T ), we have
∂ρ
∂n
n˙ +
∂ρ
∂T
n
T˙ = 0.
ρ˙ =
Combining the above equation and the thermodynamic
identity [
96
]
,
it immediately follows that Eq. (92) still holds even for
negative values of w = − 1. If w = −1 = constant, we can find
from Eq. (92) that nT is a constant, or T ∼ 1/n. This
relation shows that for very low density “dark energy” particles
their thermodynamic temperature is extremely high, while
high particle number (density) systems have a very low
temperature. In the limit n → ∞, the temperature of the “dark
energy” made system tends to zero.
4 Cosmology of Palatini f ( R, T ) gravity
In the present section we investigate the cosmological
implications of the Palatini formulation of f (R, T ) gravity. We
assume that the Universe is flat, homogeneous and isotropic,
with the metric given in comoving coordinates by the
Friedmann–Robertson–Walker metric,
ds2 = gμν dx μdx ν = −dt 2 + a2(t ) d x 2 + d y2 + d z2 ,
where a(t ) is the scale factor. We also introduce the Hubble
function, defined as H = a˙ /a. We assume that the matter
content of the Universe consists of a perfect fluid that can be
characterized by two thermodynamic parameters, the energy
density ρ, and the pressure P, respectively. As for the
relations of the geometric quantities in the g and g˜ frames, their
detailed computation is presented in Appendix C.
4.1 Generalized Friedmann equations in Palatini f (R, T )
gravity
Substituting the expression of the perfectfluid
energymomentum tensor as well as Eqs. (61), (62) and (C11) into
the Palatini field Eq. (33), from the 00 component we obtain
the first modified Friedmann equation as
3H 2
H˙ = 2F
When f (R, T ) → f (R), the energy balance equation
reduces to the ordinary conservation equation,
Hence the first modified Palatini f (R, T ) Friedmann
equation (100) reduces to the ordinary second Friedmann equation
(109) when f (R, T ) → R.
4.2 The energy balance equation
With the help of Eq. (C2), we can directly work out the
covariant divergence of the energymomentum tensor (4) as
∇μTiμ = ∂μTiμ +
= ∂i Tii +
= 0, i = 1, 2, 3,
μμν Tiν −
μμi Tii −
νμi T νμ
00i T 00 − 3 111T 11
and
μ μ
∇μT 0 = ∂μT 0 +
μμν T 0ν −
νμ0T νμ
Substituting the above two equations and Eqs. (61) and (62)
into the already known Palatini f (R, T ) energy balance
equation (93),
An important cosmological parameter, indicating the
accelerating/decelerating nature of the cosmological dynamics is
the deceleration parameter q, defined to be
d
q = dt
1
H
H
˙
− 1 = − H 2 − 1.
Using Eqs. (100) and (104), we immediately obtain
q = 3
3F˙ 2
(8π + fT ) (ρ + P) − 2F + F¨ − H F˙
3F˙ 2
8π (ρ + 3 P) + (ρ + P) fT + f − 2F − 6H F˙
For a vacuum Universe with ρ = P = 0, the condition
for accelerated expansion, q < 0, reduces to
F¨ − 3F˙ 2/2F − H F˙
f − 3F˙ 2/2F − 6H F˙
< .
The deceleration parameter can also be defined, by
analogy with the standard general relativistic cosmology, in terms
of the effective parameter weff of the equation of state of the
Universe as
q =
1 + 3weff ,
2
(112)
(113)
(115)
(116)
− 1.
(117)
(118)
(119)
By using the above definition we obtain for the effective
parameter of the equation of state of the Universe the
expression
weff = 2
3F˙ 2
(8π + fT ) (ρ + P) − 2F + F¨ − H F˙
3F˙ 2
8π (ρ +3 P)+(ρ + P) fT + f − 2F − 6H F˙
4.4 The de Sitter solution
Next, we investigate the possibility of the existence of a de
Sitter type solution in the framework of the Palatini
formulation of f (R, T ) gravity. The de Sitter solution corresponds
to H = H0 = constant, and H˙ = 0, respectively. Assuming
that the Universe is filled with a pressureless dust, we have
P = 0, and T = −ρ. Moreover, we adopt for the function
f the functional form f (R, T ) = k(R) + g(T ), where for
simplicity we take g(T ) = 8πβ T , with β a constant. Then
the energy balance equation (114) takes the form
with the general solution given by
ρ(t ) = ρ0e−α˜ t ,
where ρ0 is an integration constant, and we have denoted
giving
weff =
− 1.
(121)
(122)
(123)
(124)
(125)
(126)
α˜ =
3H0 (1 + β)
1 + 3β/2
.
Equation (104) becomes
3F˙ 2
F¨ − 2F
− H0 F˙ + 8π (1 + β) ρ0e−α˜ t = 0.
In the limit of large times the last, exponential term in the
above equation becomes negligibly small, and hence we can
approximate the solution of Eq. (125) by
F (t ) ≈
F0
eH0t − 1 2 ,
where F0 is an arbitrary constant of integration, and without
any loss of generality we have taken the second integration
f¨R − H f˙R
f + 3 f¨R + 3H f˙R
< .
constant as zero. Then the first generalized Friedmann
equation (100) gives the Lagrangian function of the model as
3F˙ 2
f (t ) = 6H02 F + 2F
+ 6H0 F˙ + 8π (1 + β) T .
(127)
F R + 3
or
On the other hand the trace of Eq. (103) gives (note that in
the following R = R(g))
F 2
˙
F + 4H0 F˙
= 8π (1 + β) T + 2 f,
Equations (127) and (129) give a parametric
representation of f as a function of R, with t taken as a parameter. Once
the function t = t (R) is obtained from Eq. (129), by
substituting it in Eq. (127) we can find the explicit dependence of
f on R.
4.5 Comparison with the metric f (R, T ) cosmology
Using the Friedmann–Robertson–Walker metric, and the
field equation (9), we can similarly derive the two
Friedmann equations in the metric formulation. The cosmological
equations in metric f (R, T ) gravity are different from their
counterparts in the Palatini formalism, due to the presence of
some dynamical terms related to f R , and they are given by
3H 2
−1
Besides, the deceleration parameter q = −H˙ /H 2 − 1 can
be obtained:
q = 3
(8π + fT ) (ρ + P) + f¨R − H f˙R
−1.
8π (ρ + 3 P) + (ρ + P) fT + f + 3 f¨R + 3H f˙R
(132)
For ρ = P = 0, an accelerated Universe with q < 0
requires that
(128)
(129)
(130)
(131)
(133)
(134)
(135)
The condition for accelerated expansion in the metric
f (R, T ) gravity is very different from the similar condition
given by Eq. (118), in the Palatini formulation of the theory.
The presence of the extra term, 3F˙ 2/2F , in Eq. (118) may
have a significant effect on the transition from the
decelerating to the accelerating phase. In the Palatini formulation the
moment of the transition to the accelerated expansion with
q ≤ 0 is determined by the equation
f¨R − f˙R2/ f R + H f˙R − f /3 = 0,
with f R = f R R g, ˜ , T , while in the metric f (R, T )
gravity the same condition is given by the much simpler
expression
6H f˙R + f = 0,
with f = f (R(g), T ). It is interesting to note that the
conditions for the transition to an accelerated expansion in the
vacuum case are independent in both approaches on fT .
However, the Palatini formulation of f (R, T ) gravity allows for
a much richer cosmological dynamics, as compared to the
metric formulation.
As for the energymomentum balance equation and the
nongeodesic equation of motion of massive test particles:
since they are all derived from the divergence of the
energymomentum tensor, and since ∇μT μν is the same in the two
formalisms, independently of the functional form of f , the
energy balance equations and the equations of motion have
the same functional form in both approaches.
4.6 Specific cosmological models in the Palatini f (R, T )
gravity
In the present section we will investigate two specific
cosmological models in the framework of the Palatini formulation
of f (R, T ) gravity. We will assume that the action of the
gravitational field has the general form
f (R, T ) = R g, ˜
+ k R g, ˜
where β is a constant, and k R g, ˜ and g(T ) are
arbitrary functions of the Ricci scalar and the trace of the matter
energymomentum tensor. For the function g(T ) for
simplicity we will assume a simple linear dependence on T , so
that g(T ) = T . As for the function k R g, ˜ , we will
consider two cases, corresponding to the Starobinsky model
k(R) ∼ R2 g, ˜ [
98
], and the 1/R g, ˜ case,
respectively.
ρ˙ = −β1 Hρ ,
where we have denoted
β1 ≡
3(1 + β)(1 + w)
1 + β2 (3 − w)
4.6.1 f (R, T ) = R g, ˜
+ 16απ R2 g, ˜
We consider a Palatini f (R, T ) model, specified by a
functional form of f (R, T ) given as
α
f (R, T ) = R g, ˜ + 16π R2 g, ˜ +8πβg(T ),
(137)
where α, β are constants, g(T ) is a function that depends
on T solely, and for simplicity we set g(T ) = T . For
this Lagrangian we immediately find F = f R = 1 +
(α/8π ) R g, ˜ , fT = 8πβgT , gT ≡ ∂g(T )/∂ T = 8πβ.
Moreover, we assume that the cosmological fluid satisfies
a linear barotropic equation of state of the form P = wρ,
w = constant.
Consequently, from the trace of (15) of the Palatini field
equations we first obtain
α R
1 + 8π
thus
R = 8π T − 8πβ (T +
) + 2 f,
R = 8π [(1 − 3w) + β(3 − 5w)] ρ ≡ 8πβ0ρ .
When β → 0, β0 → 1 − 3w, and R g, ˜ → −8π T ,
respectively. Substituting Eq. (139) back into the expressions
of f (R, T ) and F , we obtain
f = 8π β0 + β202 αρ + β(3w − 1) ρ
and
F = 1 + β0αρ .
Substituting Eqs. (139)–(141) into the Friedmann Eq. (102),
we find
β0αρ˙
H + 2 (1 + β0αρ)
2
=
1 + β2 (3 − w) + β402 αρ 8πρ
(1 + β0αρ)
Substitution of the expression of fT into the balance equation
(114) gives
Taking into account the limit αρ → 0, we have the series
expansion
2
(1 + β0αρ) 1 + β2 (3 − w) + β40 αρ
where we have denoted β2 = 1 + β(3 − w)/2 and β3 =
β0 {β0/4 − [1 + β(3 − w)/2] (1 − β1)}. Then Eq. (145)
takes the form
H 2 = 8π3ρ
(β2 + β3αρ) .
Equation (143) can be immediately integrated to give
≈ β2 + β3αρ , (146)
When β → 0, then β1 = 3(1 + w). Substituting Eq. (143)
into Eq. (142), we find
In the limit of large times we obtain a(t ) ∼ t 2/β1 , and
H (t ) = (2/β1) (1/t ), respectively. The deceleration
parameter in this model is given by q = β1/2 − 1, and once the
coefficient β1 satisfies the condition β1/2 < 1, the Universe
will experience an accelerated evolution. For an arbitrary w,
the condition for a power law type accelerated expansion
is β < − (1 + 3w) /4w, a condition which shows that, for
w > 0, β must take negative values.
4.6.2 f (R, T ) = R g, ˜
α2
− 3R g, ˜
Now let us consider the following f (R, T ) gravity Palatini
type model:
ρ = ρ0a−β1 .
a˙ =
8π
3
ρ0a1−β1 β2aβ1 + β3αρ0,
with the general solution given by
a(t ) =
2π β3 αρ0
3 β12β2ρ0t 2 − β2
1
β1
.
Hence Eq. (147) becomes a firstorder differential equation,
(145)
(147)
(148)
(149)
(150)
The algebraic equation (152) has two distinct solutions.
However, only one of them can be adopted as the physical solution,
more exactly the one which in the limit f → R, would give
R = −8π T , which is the trace of the Einstein field equation.
Hence when ≤ 0, the physical solution of Eq. (152) is [99]
When f (R, T ) → f (R), β → 0, and the above field
equations reduce to the field equations considered within the
Palatini formulation of f (R) gravity, considered in [
99
],2:
2 Note that in our result there is a minus sign before α, when compared
with Eq. (24) in [
99
]. The reason is that a different solution has been
chosen in Eq. (17) of [
99
], while we have adopted the solution given by
our Eq. (154).
(151)
(152)
(153)
(154)
(155)
(156)
(157)
.
(158)
With the equation of state w = P/ρ, and since T =
−ρ + 3 P for a perfect fluid, we have = −8πβ0ρ; besides,
we already know from Eq. (143) that ρ˙ = −β1 Hρ. Thus,
similar to Eq. (145), we have
2 α
= 1 + (3 + 4β) w + 3 β0 πρ + 12 .
In the firstorder approximation in ρ/α we obtain
Hence in the present model a cosmological constant α/12
is automatically generated, due to the 1/R modification of
the gravitational Lagrangian. Hence for ρ → 0, the Universe
will end in a de Sitter phase, with H = H0 = √α/12 =
constant. For ρ = 0, we have ρ = ρ0a−β1 , and the evolution
of the scale factor is determined by the equation
a˙ = a
α β4ρ0
12 + aβ1 ,
where we have denoted β4 = β0(5 − β1)/6 + (3 + 4β)w +
1 π , with the general solution given by
1
a(t ) = 12 β1
β4ρ0 sinh sinh−1
α
+
√αβ1 (t − t0)
4√3
2
β1
,
where we have used the initial condition a (t0) = a0.
For the Hubble function we obtain
H (t ) = 2√√α3 coth √αβ41 √(t3− t0)
aβ1/2
+ sinh−1 0 ,
√12β4ρ0/α
while the deceleration parameter of this model is given by
1 √αβ1 (t − t0)
q(t ) = 2 β1sech2 4√3
aβ1/2
+ sinh−1 0 − 1,
√12β4ρ0/α
(165)
where sech t = 1/cosh t . In the limit t → ∞, q → −1,
and hence the Universe ends in a de Sitter type accelerating
phase, independent of the matter equation of state.
(166)
(167)
(168)
.
(169)
a. The matter dominated phase The matter dominated phase
corresponds to the choice w = 0 in the matter equation of
state, that is, to a Universe filled with pressureless baryonic
matter. In order to investigate the behavior of the
cosmological model during matter domination, we consider the series
expansion of the cosmological parameters. Thus we obtain
a(t ) ≈ a0 +
These equations describe the main cosmological
parameters during the matter dominated era. The expansion is
decelerating, and, depending on the model parameters, the
deceleration parameter can have a large range of positive values. The
transition to the accelerating phase occurs at a time interval
ttr , which in the firstorder approximation is found as
ttr ≈ t0 +
a− β21
0
αa0β1
3β4ρ0 + 4 6 (β1 − 2) β4ρ0 − αa0β1
αβ12√β4ρ0
Since ttr must be greater than t0, it follows that in order
for the model to admit a matter dominated era followed by
a transition to an accelerated phase, the model parameters
must satisfy the condition 6 (β1 − 2) β4ρ0 − αa0β1 > 0, or,
equivalently,
a condition that can easily be satisfied by appropriately
choosing the free parameters in the gravitational action.
5 Discussions and final remarks
In the present paper we have considered in the framework
of the Palatini formalism the gravitational field equations for
the modified gravity f (R, T ) theory, implying a geometry–
matter coupling, with the trace of the energymomentum
tensor included as a field variable in the gravitational action. We
have derived the field equations by independently varying
the metric and the connection in the f (R, T ) type
gravitational action, and we have formulated them in both the initial
metric frame and the conformal one, in which the
independent connection can be expressed as the LeviCivita
connection of an auxiliary, energymomentum trace dependent
metric, which is related to the physical metric by a conformal
transformation. Similar to the metric case [
46
] the
energymomentum tensor of the matter is not conserved, and the
energy and momentum balance equations take the same form
as in the metric theory. Generally, Palatini type theories have
a number of special properties that make them especially
attractive for analyzing strong gravity phenomena, like, for
example, the dynamics of the early Universe or stellar
collapse processes [
100–107
]. The coupling of the trace of the
energymomentum tensor with the curvature scalar generates
some extra terms in the gravitational field equations, which
strongly depend on the possible functional forms for the
geometry– energy momentum trace coupling. If, for
example, it would be possible to generate through the geometry–
energy momentum trace coupling some repulsive forces, then
one could obtain cosmological models that are nonsingular
at extremely high densities and high geometric curvatures, or
one could even construct models for nonsingular collapsing
stars as viable alternatives for the black hole paradigm.
To obtain such repulsive gravitational forces, in
modified gravity theories with geometry–matter coupling no new
degrees of freedom in the matter side (exotic sources) or in
the gravitational side are required in the total action. In these
models the extra force is simply induced by the coupling
between matter and geometry. Our present results show that
Palatini type theories might play an important role in the
phenomenology of gravity at both high densities (energies), as
well as in the very low density limit. On the other hand, in
the variational process the assumption of the independence
between metric and connection is essential to obtain
secondorder differential equations for the metric tensor. It is thus
possible to assume that at large/small scales the effective
descriptions of the gravitational forces, going beyond
standard general relativity, could come from the Palatini
formulation of gravity theories.
In the Palatini type formulation of f (R, T ) gravity, the
equation of motion of massive particles is nongeodesic, and
in three dimensions and in the Newtonian limit, Eq. (66) can
be formally represented as an ordinary vector equation in
three dimensions of the form a = aN + a f where a
represents the total acceleration of the particle, aN denotes the
Newtonian gravitational acceleration, while a f is the
acceleration due to the presence of the extra force induced by
the coupling between geometry and matter. This shows that
one observational or experimental possibility of testing the
effects of the coupling between geometry and the trace of the
energymomentum tensor could be in the physical domain
of extremely small particle accelerations, with values of the
order of 10−10 m/s2. Such an acceleration could explain the
observed behavior of test particles rotating around
galaxies, which is usually explained by postulating the existence
of dark matter. However, as a possible astrophysical
application of the gravitational field equations derived with the
Palatini formalism one may consider an alternative view to
the dark matter problem, in which the mass discrepancy in
galaxies and clusters of galaxies as well as the galactic
rotation curves are explained by the existence of a nonminimal
coupling between matter and geometry.
We have also briefly investigated the intriguing feature
of the nonconservation of the energymomentum tensor of
the matter in f (R, T ) gravity theory by interpreting it in
the framework of the thermodynamics of open systems. We
have interpreted this effect as describing
phenomenologically the particle production in the cosmological fluid
filling the Universe, with the extra terms induced by the
nonminimal coupling between R and T assumed to describe
particle creation processes, with the gravitational field
acting as a source for particles. We have explicitly obtained the
particle creation rates, the entropy flux, the creation pressure
and entropy generation rate in a covariant form, as
functions of the Lagrangian density f (R, T ) of the theory, and
of its derivatives, respectively. On the other hand it is
natural to assume that such particle production processes are of
the same nature as the similar processes that appear in the
framework of quantum field theory in curved spacetimes.
A static gravitational field does not produce particles. But
a time dependent gravitational field can generate new
particles. This interesting analogy between gravitational theories
with geometry–matter coupling and quantum field theory in
curved spacetimes may open the possibility of an effective
classical description of quantum gravity on small geometric
scales.
As a cosmological application of the Palatini formalism
of f (R, T ) theory we have briefly considered two classes
of cosmological models, corresponding to two choices of
the gravitational Lagrange density f (R, T ) = k(R) + g(T ),
with k(R) ∼ R2 and k(R) ∼ 1/R, respectively. In both
cases we have assumed for the function g(T ) the simple
form g(T ) ∼ T . We have explicitly shown that both models
can generate an accelerating expansion of the Universe, with
a power law and an exponential form of the scale factor,
respectively.
In both metric and Palatini formulation of f (R, T )
gravity, dark energy is interpreted as a material–geometrical fluid,
with a negative parameter of the equation of state, for which
the function f (R, T ) is not known a priori. Hence, similar to
the case of f (R) gravity, there is a need of a model
independent reconstruction of the Lagrangian of theory, which can
be done by using some relevant cosmographic techniques to
determine which f (R, T ) model is favored with respect to
others. In the case of f (R) gravity, a cosmographic approach
was introduced in [
16
], by assuming that the cosmological
principle is valid, and that dark energy can be described
as a geometric fluid. Then, after expanding the
cosmological observables (the Hubble parameter, the luminosity
distance, the apparent magnitude modulus, the effective
pressure etc.) into a Taylor series, and matching the derivatives of
the expansions with cosmological data, one can obtain some
model independent constraints on the gravitational theory.
The coefficients of the power series of the expansion of the
scale factor, calculated at present time (at redshift z = 0) are
known as the cosmographic series. The importance of the
cosmographic approach is that it does not need the
assumption of a specific cosmological model. If the scalar
curvature is negligible, the Taylor series of the scale factor around
t = t − t0 = 0 can be represented as [
108
]
1 − a(t )
H0
t + q20 H0 t + j60 H02 t 3 − 2s04 H03 t 4 + · · · ,
(171)
where the jerk parameter j is defined as j (t ) = 1/a H 3 d3
a/dt 3 , while the snap parameter s is given by s(t ) =
1/a H 4 d4a/dt 4 [
110
]. A strategy to infer the transition
redshift zda , which indicates the passage of the Universe from
a decelerating to an accelerating phase, was proposed, in the
framework f (R) gravity, in [
109
]. This goal can be achieved
by numerically reconstructing f (z), that is, the
corresponding gravitational Lagrangian f (R) reexpressed as a
function of the redshift z, and by matching f (z) with
cosmography. The highredshift f (R) cosmography was considered in
[
110
], by adopting the technique of polynomial
reconstruction. Instead of considering the Taylor expansions that proved
to be nonpredictive for redshifts z > 1, the Padé
rational approximations were considered, by performing series
expansions that converge in the domains of high redshifts. A
first step in this strategy is the reconstruction of the function
f (z), by assuming that the Ricci scalar can be inverted with
respect to the redshift z.
The cosmographic approach developed in [
108–110
] for
the case of f (R) gravity can also be extended to both
(A1)
(A2)
metric and Palatini f (R, T ) gravities. To be more
specific, such an approach requires one to rewrite f (R, T )
or f R g, ˜ , T into a function f (z). Similar to the
approach introduced in [
16
] for f (R) gravity in the metric
formulation, in order to handle highz data one can rewrite
the f (R, T ) function into an f (z) function generally through
the use of Padé polynomials. As a next step data fitting based
on some general f (z) models is required. Hence one can
generalize the approaches of [
108–110
] for the case of f (R, T )
theory, in both metric and Palatini formulations, and
numerically determine the coefficients of the series expansions for
R and T in the f (R, T ) models through cosmological data
fitting. The cosmographic approach could help distinguish
between the roles and weights of the functions R and T in
the gravitational action, and it could lead to a full comparison
of the theory with the cosmological observations.
The cosmology of Palatini f (R, T ) gravity represents a
promising way for the explanation of the accelerated phases
in the dynamics of the Universe, which is characterized by
its evolution in both very early and late stages. In the present
paper we have introduced some basic theoretical tools
necessary for the indepth investigation of the cosmological and
astrophysical aspects of the Palatini formulation of f (R, T )
gravity.
Acknowledgements We would like to thank the two anonymous
reviewers for comments and suggestions that helped us to significantly
improve our manuscript. ShiDong Liang acknowledges the support
of the Natural Science Foundation of Guangdong Province (Grant no.
2016A030313313). T. H. would like to thank the YatSen School of
the Sun YatSen University in Guangzhou, P. R. China, for the kind
hospitality offered during the preparation of this work.
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.
Appendix A: f ( R, T ) field equations in the metric
formulation
We first introduce two basic formulas we are going to use,
where in Eq. (A2) we have taken into account the relations
2δ√−g
√−g
δ ρμν = 21 gρλ · 2gσ λδ σμν
= gμν δgμν = −gμν δgμν ,
∇μδgνλ + ∇ν δgλμ − ∇λδgμν ,
δ ∇μgνλ = δ ∇ν gλμ = δ ∇λgμν = 0. A variation δgμν
of the metric tensor then leads to
δT = δ gαβ Tαβ =
= Tμν +
μν δgμν ,
δgαβ
Tαβ δgμν +
μν δgμν
μν ≡ gαβ δTαβ
δgμν
and
ρ ρ
δ Rμν = δ ∂ρ μν − ∂ν μρ +
ρ λ
ρλ μν −
ρ λ
νλ μρ
= ∇ρ δ ρμν − ∇ν δ ρμρ ,
respectively, where the above relation is called the Palatini
identity. Then for the variation of the metric we obtain
δ S = √16−πg f R Rμν δgμν + gμν δ Rμν + fT δT
− g2μν f δgμν − 8π Tμν δgμν d4x
√−g
= 16π f R Rμν +gμν −∇μ∇ν + fT Tμν + μν
gμν f − 8π Tμν δgμν d4x ,
− 2
≡ gαβ ∇α∇β .
Assuming that the variation of δgμν vanishes at infinity, then
gμν gαβ f R∇α∇β − f R∇μ∇ν δgμν d4x
−gμν gαβ ∇α f R∇β δgμν + ∇μ f R∇ν δgμν d4x
gμν gαβ δgμν ∇β ∇α f R − δgμν ∇ν ∇μ f R d4x
√−g δgμν gμν − ∇μ∇ν f R d4x
16π
Rμν + gμν
− ∇μ∇ν f R
gμν f (R, T )−8π Tμν δgμν d4x .
μν fT − 2
Since δ S = 0, from the above relation we obtain
immediately the field equation (8) of f (R, T ) gravity theory.
Appendix B: Divergence of the matter energymomentum
tensor in the metric formalism
By taking the covariant divergence of Eq. (9), with the use
of the mathematical identity ∇μGμν (g) = 0 we obtain
μ
δ ν R(g)∇μ f R
∇μ f R Gμν (g) − Rμν (g)∇μ f R + 2
=
μ
δ ν R
Gμν (g) − Rμν (g) + 2
∇μ f R
and
∇ν f (R, T ) = f R ∇ν R + fT ∇ν T ,
respectively.
Appendix C: The geometric quantities in the FRW
geometry
For the metric (99), we have
∂λgμν =
2aa˙ , λ = 0 and μ = ν = 0,
0, others.
Hence the only nonzero components of the connection,
ρμν (g) = (gρσ /2) ∂ν gσ μ + ∂μgσ ν − ∂σ gμν , are now
0
ii = aa˙ = a2 H and i0i =
ii0 = aa˙ = H, i = 1, 2, 3.
=
=
=
and
δ S =
and
Hence
In that case,
ρ ρ
R00(g) = ∂ρ 00 − ∂0 0ρ +
ρ σ
σρ 00 −
ρ σ
0σ 0ρ
= −3 H˙ + H 2 ,
Rii (g) = a2
H˙ + 3H 2 ,
R(g) = gμν Rμν (g) = g00 R00(g) + 3g11 R11(g)
= 6 H˙ + 2H 2 .
G00(g) = R00(g) − g200 R(g) = 3H 2,
Gii (g) = Rii (g) − g2ii R(g) = −a2 2H˙ + 3H 2 .
and
R˜ii ˜
.
F
gii
Similarly, by substituting Eq. (C5) into Eq. (31) we find
3
= R(g) + F
(∇ F )2
2F
3
= 6 H˙ + 2H 2 − F
− F
F 2
˙
2F − F¨ − 3H F˙ .
Some combinations of the above equations lead to
G00 g, ˜
= R˜00 ˜ − 2
and
Gii g, ˜
1
= Gii (g) + F
= R˜ii ˜ − g2ii R g, ˜
gii
− ∇i ∇ j F
3 gii
+ 2F ∇i F ∇i F − 2
(∇ F )2
= a
2
− 2H˙ + 3H 2
1
+ 2F
3F˙ 2
2F
− 2F¨ − 4H F˙
.
Note that for the above calculations we have used the relations
∇μ F = ∂μ F =
F˙ , 0, 0, 0, ,
∇0∇0 F = ∂00 −
λ00∂λ F = ∂00 F = F¨ ,
∇i ∇i F = ∂ii − λii ∂λ F = − 0ii ∂0 F = −a2 H F˙ , (C15)
Substituting Eqs. (C3) and (C4) into Eq. (30) we obtain
and
(C8)
F
(C9)
(C10)
(C11)
(C12)
(C13)
(C14)
F = gμν ∇μν F = gμν ∂μ∂ν −
λμν ∂λ F
=
−∂00 − 3g11 011∂0 F = −F¨ −
3a
˙ F˙
a
= −F¨ − 3H F˙ ,
(C16)
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