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arXiv:1107.5183v1 [gr-qc] 26 Jul 2011
An overview of f(R) theories
Santiago Esteban Perez Bergliaffa
Departamento de Física Teórica, Instituto de Física, Universidade do Estado do Rio de Janeiro,
Brazil
Abstract. A brief introduction to theories of the gravitational field with a Lagrangian that is a
function of the scalar curvature is given. The emphasis will be placed in formal developments,
while comparison to observation will be discussed in the chapter by S. Jorás in this volume.
Keywords: modified theories of gravity, cosmology, dark energy
INTRODUCTION
The predictions of General Relativity (GR) are confirmed to an impressive degree by
observations in number of situations [1]. In spite of this fact, theories that differ from
GR either in the limit of low or high curvature have been intensively studied lately,
and have a long tradition, starting with a paper by Weyl in 1918 [2]. Although Weyl’s
motivationwas related to the unification of GR and Electrodynamics, the current revival
of these theories is twofold. In the case of low curvature, the aim is to describe the
accelerated expansion of the universe that follows from several observations [3] (when
interpreted in the standard cosmological model [4])1. Regarding the high-curvature
regime, it is important to note that there is no observational evidence of the behaviour
of the gravitational field for very large values of the curvature. This makes objects such
as black holes and neutron stars the ideal places to look for deviations from General
Relativity in the strong regime. In fact, the Kerr solution is not unique in f(R) theories
[33]. Consequently, any deviation from Kerr’s spacetimee in compact objects will be
unequivocally signaling the need of changes in our description of strong gravity. The
task of understanding what kind of deviations can be expected, and their relation to
observable quantities is of relevance in view of several developments that offer the
prospect of observing properties of black holes in the vicinity of the horizon [11].
In this short review we will be concerned with gravitational theories described by the
action
S =
?
d4x√−gf(R),
(1)
where g is the determinant of the metric gµνand f is an arbitrary function of the the
curvature scalar R2. The function f must satisfy certain constraints, some of which
1The possibility of describing the current accelerated expansion of the universe using f(R) theories was
first discussed in [5].
2This choice if favoured by a theorem by Ostrogradski [12] over Lagrangians built with invariants
obtained from the the Ricci and Riemann tensors.
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are necessary for the theory to be well-defined ab initio, and others to account for
observational facts. Those in the first class will be discussed in this review, while those
in the second class are presented in the chapter by S. Jorás in this volume3. We shall
begin by reviewing in the next section some general features of this type of theories.
THE THREE VERSIONS OF f(R) THEORIES
We shall see in this section that f(R) theories can be classified in three different types,
according to the role attibuted to the connection. In all of the versions, the equation for
the energy-momentum conservation is valid, since the total (gravitational plus matter)
action is diffeomorphism-invariant and gravity and matter are minimaly coupled by
hypothesis (see for instance [10], [21]).
Metric version
In the metric version of f(R) theories, the action
S =
1
2κ
?
d4x√−gf(R)+SM(gµν,ψ),
(2)
is varied with respect to gµν. Here, SMis the matter action, which is independent of the
connection. The resultant equations of motion are of fourth order in the derivatives of
the metric tensor:
d f(R)
dR
Rµν−1
2f(R)gµν−?∇µ∇ν−gµν??d f(R)
dR
= κTµν,
(3)
where Tµνis the energy-momentum of he matter fields, defined by
Tµν= −
2
√−g
δSM
δgµν,
and the covariant derivative is defined using the usual Levi-Civita connection. Taking
the trace, we obtain
d f(R)
dR
which is to be compared to R = −κT, the result in GR.
R−2f(R)+3?d f(R)
dR
= κT,
3There are several reviews that deal with different aspects of f(R) theories, see [6],[7],[8], and also the
recent book [9].
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Equivalence with Brans-Dicke theory
As shown for instance in [13], the gravitational part of the action given in Eqn.(2) is
equivalent to the following action:
S =
?
d4x√−g
?φR
2κ−U(φ)
?
,
(4)
where
U(φ) =φχ(φ)− f(χ(φ))
2κ
,
(5)
φ = f,χ(χ), and χ = R, corresponding to a Brans-Dicke theory with ω = 0 [15]. Note
that the absence of a kinetic term for the scalar field does not mean that it is non-
dynamical: its evolution, due to the non-minimal coupling with the gravitational field, is
given by the variation of the action wrt φ:
3?φ +2U(φ)−φdU
dφ= κT.
(6)
Through a conformal transformation of the metric and a redefinition of the scalar field,
the action given in Eqn.(4) can be taken to that of a scalar field minimally coupled
with gravity, and with nonzero kinetic term and potential. These representations of f(R)
theories show that there is a massive scalar degree of freedom, which manifests as a
longitudinal mode in gravitational radiation (see for instance [23] for the cosmological
case).
We close this section by stating that the equivalent representations are convenient
sincetheassociatedequationsofmotionareofordertwo,butaword ofcautionisneeded
because sometimes the potential in Eqn.(5) is typically multivalued (see for instance
[16][17]). It may be better to work directly in the original representation, as for instance
in Ref.[14] in the case of compact stars.
Palatini version
In this second type of f(R) theories, the metric and the connection are taken as
independent fields, and the matter action SMis independent of the connection. So the
starting point is the action
S =
1
2κ
?
d4x√−g f(R)+SM(gµν,ψ),
(7)
where R = gµνRµν, and the corresponding Riemann tensor is constructed with a
connection Γ a priori independent of the metric.
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From the variation of the action wrt the metric and Γ we get4(see for instance [8])
f′(R)R(µν)−1
¯∇λ(√−g f′(R)gµν) = 0,
2f(R)gµν= κTµν,
(8)
(9)
where the prime denotes derivative wrt R, and the barred covariant derivative is built
with the connection Γ. GR is recovered by setting f(R) = R in these equations. Taking
the trace of Eqn.(8) we obtain
f′(R)R−2f(R) = κT,
(10)
which shows that in this case the relation between R and T is algebraic, hence no scalar
mode is present.
From Eqn.(9), it follows that [8]
Γλ
µν=
1
f′(R)gλσ?∂µ(f′(R)gνσ)+∂ν(f′(R)gµσ)−∂σ(f′(R)gµν)?.
Since this expression relates Γ to R and the metric, and R and T are in principle
interchangeable through Eqn.(10), the connection can be in principle expressed in terms
of the matter fields and the metric. In other words, it is an auxiliary field. In fact, Eqn.(8)
can be rewritten as
(11)
Gµν
=
κ
f′Tµν−1
−3
2
f′2
2gµν
(∇µf′)(∇νf′)−1
?
R −f
f′
?
+1
f′(∇µ∇ν−gµν?)f′
?
(12)
1
?
2gµν(∇f′)2
where the Einstein tensor and the covariant derivatives are built with the Levi-Civita
connection, and R is expressed in terms of T using Eqn.(10). It follows that this version
of f(R) theories can be interpreted as GR with a modified source. Perhaps the most
important modification is that third order derivatives of the matter fields appear on the
rhs of Eqn.(13). As reported in [24], this feature may cause serious problems in static
spherically symmetric solutions with a polytropic fluid with index 3/2 < γ < 2 as a
source. Note however that this result result was challenged in the review [25]5.
4In the case of GR, this method furnishes the same result as the metric case, but this is not the case in
more general theories as discussed for example in [19].
5It has also been claimed that the Cauchy problem is not well-posed for the Palatin version of f(R)
theories [8]. For an updated discussion see [26].
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Metric-affine theories
In thiscase, the matteraction depends of theconnection, which is a prioriindependent
of the metric. The action is given by
S =
1
2κ
?
d4x√−gf(R)+SM(gµν,Γλ
µν,ψ).
(13)
Depending on the matter fields, the theory may display non-propagating torsion and
non-metricity (see [8] and [35] for details).
NONMINIMAL COUPLING
Metric f(R) theories havebeen generalized by allowinga nonminimalcoupling between
the curvature and the matter Lagrangian, with action given by
S =
? ?1
2f1(R)+[1+λ f2(R)]Lm
?√−gd4x,
(14)
where f1and f2are arbitrary functions of R, and λ is a constant. A particular case of
this action was considered in [27] in the context of the accelerated expansion of the
universe. Later, it was shown in [28] that this type of theory leads to a modification of
the conservation law of the matter energy-momentum tensor, namely
∇µT(m)
µν =
λ
1+λ f2f′
2
?
gµνLm−T(m)
µν
?
∇µR.
(15)
The presence of a nonzero rhs leads to non-geodesic motion, and it was suggested in
[28] that this may be related to MOND.
A more general type of theories was propposed in [29], with action given by
S =
?
f(R,Lm)√−gd4x,
(16)
where f is an arbitrary function of R and of the matter Lagrangian. As in the previous
case, an extra force, perpendicular to the 4-velocity, accelerates the particles.
ASSORTED APPLICATIONS
f(R)theories havebeen used to describe different aspects ofrelativisticastrophysicsand
cosmology. Since the low curvature limit, which has been studied primarily to explain
the accelerated expansion of the universe, is discussed in the chapter by S. Joras in
this volume, only one example will be given here in this regime. Afterwards, some
applications in the strong-curvature regime will be discussed.
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Low curvature
In the case of the k = 0 Friedmann-Lemâitre-Robertson-Walker metric, the EOM (3)
can be written as
ρ = −f′Rtt−f
p = −f′
Let us remark that it is safe to asssume that most of the current matter content of the
universe (assumed here to be normal matter, as opposed to dark energy) is pressureless.
This matter must satisfy the conditions ρ0≥ 0 and p0= 0, where the subindex 0 means
that the quantity is evaluated today. Using Eqns.(17) and (18), we shall rewrite these
conditions in terms of following kinematical parameters: the Hubble and deceleration
parameters, the jerk, and the snap, respectively given by [41]
2+3f′′˙ a˙R
¨R−2˙ a˙R
a,
(17)
3(Rtt+R)+f
2− f′′
?
a
?
− f′′′˙R2.
(18)
H =˙ a
a,
q = −1
H2
¨ a
a,
j =
1
H3
...a
a,
s =
1
H4
....
a
a.
While the current value of the first two parameters is relatively well-established today,
the value of j0is not known with high precision, and no acceptable value of s0has been
reported yet [42]. By writing ρ0≥ 0 in terms of the kinematical parameters we get
0−f0
3q0H2
0f′
2−18H4
0f′′
0(j0−q0−2) ≥ 0.
(19)
This inequality gives a relation that the parameters and the derivatives of a given f(R)
must satisfy today and, as shown in [43], it limits the possible values of the parameters
of a given theory. Notice that Eqn.(18) involves the snap (through ¨R). If we had a
measurement of s0, we could use the equation p0= 0 to obtain another constraint on
f(R). Since this is not the case, we shall express p0= 0 in such a way that it gives a
forecast fo the possible current values of the snap for a given f(R):
s0=
f′
0
0f′′
6H2
0
(q0−2)+6H2
0
f′′′
0
f′′
0
(−q0+ j0−2)2−[q0(q0+6)+2(1+ j0)]−
f0
12H4f′′
0
.
(20)
Strong curvature
• The possibility of nonsingular cosmological solutions in f(R) theories has been
considered in [30] and [31]. In the latter article, a necessary condition for a bounce
to occur in a Friedmann-Lemâtre-Robertson-Walker setting was obtained, and it is
given by
¨ a0
a0
= −ρ0
f′
0
+
fb
2f′
0
,
(21)
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with
R0= 6
?¨ a0
a0+K
a2
0
?
,
(22)
and the subindex b means that the quantity is evaluated at the bounce. Contrary to
the case of GR, a bounce may occur for any value of K.
• It was shown in [32] that the theory given by
f(R) = R+
R2
(6M)2
(23)
has an inflationary solution given by
H ≈ Hi−M2
Hi(t−ti)−M2
6(t−ti),
a ≈ aiexp
?
12(t−ti)2
?
,
where timarks the beginning of the inflationary epoch. Several features of this
model have been studied in detail (see references in [7]). The results of WMAP
constraint M ≈ 1013GeV, and the spectral index for this model is nR≈ 0.964,
which is in the range allowed by WMAP 5-year constraint. The tensor to scalar
ratio r also satisfies the current observational bound, but is different from that of
the chaotic inflation model. Hence, future observations such as the Planck satellite
may be able to discriminate between these two models.
• Compact stars have been repeatedly studied for a number of f(R) theories, either
in the conformal representation [36], or directly in the fourth-order version (see for
instance [14]).
• Regarding black holes in f(R) theories, it was shown in [33] that an observational
verification of the Kerr solution for an astrophysical object cannot be used in
distinguishingbetween GR and f(R) theories. Hence, the observation of deviations
from the Kerr spacetime may point to changes in our understanding of gravitation.
Other features of black holes in f(R) theories have been analyzed in [34].
CONCLUSIONS
Inthisshortreview,Iintendedtoshowthatseveralaspectsof f(R)theories(initsvarious
representations) have been extensively studied in the literature. There are many other
aspects that I had to left aside such as “good propagation” (i.e. absence of shocks) [37],
the loop representation [38], and the Hamiltonianrepresentation [39]. Although the low-
curvature regime and its consequences has attracted a lot of attention due to its possible
relevance in Cosmology, the high-curvature regime is also of interest independently
of the low-curvature regime, and the consequences of a modification in such regime
may be testable in the near future (as in the inflationary model in [32], and through the
observation of electromagnetic [11] and gravitational waves [40] in the case of compact
objects).
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ACKNOWLEDGMENTS
The author acknowledges support from CNPQ, FAPERJ, UERJ, and ICRANet-Pescara.
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