# An overview of f(R) theories

**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. Jor\'as in this volume.

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**ABSTRACT:**The field equations in FRW background for the so called C-theories are presented and investigated. In these theories the usual Ricci scalar is substituted with $f(\mathcal{R})$ where $\mathcal{R}$ is a Ricci scalar related to a conformally scaled metric $\hat{g}_{\mu\nu} = \mathcal{C}(\mathcal{R})g_{\mu\nu}$, where the conformal factor itself depends on $\mathcal{R}$. It is shown that homogeneous perturbations of this Ricci scalar around general relativity FRW background of a large class of these theories are either inconsistent or unstable.05/2013; - SourceAvailable from: Tomi S. Koivisto[Show abstract] [Hide abstract]

**ABSTRACT:**In the first order formalism of gravitational theories, the spacetime connection is considered as an independent variable to vary together with the metric. However, the metric still generates its Levi-Civita connection that turns out to determine the geodesics of matter. Recently, "hybrid" gravity theories have been introduced by constructing actions involving both the independent Palatini connection and the metric Levi-Civita connection. In this study a method is developed to analyse the field content of such theories, in particular to determine whether the propagating degrees of freedom are ghosts or tachyons. New types of second, fourth and sixth order derivative gravity theories are investigated and the so called f(X) theories are singled out as a viable class of "hybrid" extensions of General Relativity.Physical review D: Particles and fields 04/2013; 87(10).

Page 1

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.

Page 2

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].

Page 3

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.

Page 4

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].

Page 5

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.

Page 6

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)

Page 7

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).

Page 8

ACKNOWLEDGMENTS

The author acknowledges support from CNPQ, FAPERJ, UERJ, and ICRANet-Pescara.

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