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On the equivalence between f(R)theories and Einstein gravity
Soham Bhattacharyya1, ∗
1Department of Physics, IIT Madras, Chennai, India
In this brief note we present a somewhat trivial result. Namely, we show that perturbative off-shell
f(R)-theory is equivalent to Einstein gravity, as well as to the Brans-Dicke theory and the Einstein
scalar field model. We also discuss possible generalisation of this result to higher-order gravitational
field models.
I. INTRODUCTION
Significant strides were made in the early parts of
the twentieth century in the formulation of gravity as
a classical field theory. Out of that effort came forth
General Relativity (GR), which passed almost all the
tests thrown at it, starting from the short, terrestrial (or
weak gravity region) length scales [1], to solar system
tests [2], to binary pulsar (strong gravity region) tests
[2], and finally constraints from Gravitational Wave
(GW) (extreme gravity region) observations in [3].
GR is a theory with two massless degrees of freedom.
The degrees of freedom manifest at asymptotic infinity as
the so called plus and cross polarizations in gravitational
wave detectors. Efforts have been made to ascertain
whether there are more than two degrees of freedom in
the observed gravitational wave signal, as in [4–6], and
others. However, so far there is no conclusive proof of
the presence of more than two degrees of freedom. On
the other hand, there are more than enough theoretical
and observational considerations that GR cannot be
a complete theory of gravity. In the theoretical side,
prediction of singularities in extreme situations like
black holes or big bang render the theory useless in
such scenarios. Similarly, in the observational side, the
presence of flat velocity profile of galaxies instead of
decaying ones, or the observational challenges faced by
the ΛCDM model provide much needed motivation to
come up with modified theories of gravity.
However, modifying gravity has its costs. Most
modifications to gravity leads to certain observational
features that have so far eluded experiments. Generally
speaking, modifications to GR leads to either increase
in degrees of freedom or violations of one or more of the
postulates of GR. On the increase in degrees of freedom
aspect, it is possible for gravity to have six degrees of
freedom, given that we live in a four dimensional space-
time. In the most naive sense, one can make coefficients
of the modification to GR go to very small values in
the action level and claim that the modifications are
un-excitable. However, that theory loses any predictive
power. However, there are other ways in which one can
∗xeonese@gmail.com
show modifications to gravity can be “pushed” to sub-
dominant regimes. For example, in some special cases it
can be shown that the extra degrees of freedom turn out
to be non-propagating (hanging around perturbed black
holes as some massive scalar cloud), like for the case of
Ricci flat solutions of f(R) gravity (see Chapter 6 of the
thesis [7]), or it is shielded by means of the so called
Chameleon mechanism [8]. Such a mechanisms dictate
that extra degrees of freedom become highly massive
(that is they cannot be excited by standard solar system
excitations) in dense environments and light in very low
density scenarios, leading to expansion of the universe
as a whole. Similarly, in the violations of postulates
aspect, for example, in Chern-Simons theory of gravity,
a pseudo-scalar fields breaks the parity invariance of GR
and leads to the birefringence of vacuum, as in [9].
There exists another little known method to “push”
leading order modifications to GR to sub-leading orders.
This is the case of the field redefinition. In such cases,
the metric is redefined in a particular manner such that
leading order effects (such as quadratic modifications
to GR) can be pushed via conformal transformations
to cubic orders or higher. For example, in [10], it was
shown that under field redefinition of the metric tensor,
certain combinations of the curvature scalars appearing
as modifications to the Einstein-Hilbert action can be
pushed to higher orders. That is, any observer whose
effective metric is the redefined metric will no longer
see any quadratic modification, as well as some cubic
modifications. Since the energy scales at which quadratic
modifications appear are lower than the energy scales
of the cubic modifications, it becomes harder for any
astrophysical source to perturb or excite the additional
degrees of freedom in the redefined metric scenario. In
the language of [10], dimension four terms have been
pushed to dimension eight under the field redefini-
tion. In this article, we generalize the quadratic order
calculations of [10] to arbitrary powers of the Ricci scalar.
At the low energy limit, string theoretic/loop quantum
gravity or in other words, effective field theory actions
can lead to a Lagrangian of the following form
S=1
16 πZd4xR+R2+R3+OR4+Lm,
(1)
where Rnare general curvature invariants of dimension
2
2nconsisting of the Ricci scalar, Ricci tensor, and the
Riemann tensor. Lmis the matter Lagrangian with a
minimal coupling to the curvature terms, and is also a
functional of the metric tensor. If in Eq. (1), one only
considers till curvature squared terms of the most general
form like that of the Gauss-Bonnet type,
R2=a R2+b Rµν Rµν +c Rµρνσ Rµρν σ,(2)
then it was shown by [11] that such a model is renor-
malizable.
Authors like [12] decided to approach the inverse prob-
lem while trying to perturbatively renormalize gravity.
They started with GR, and under field redefinition was
able to obtain the most general quadratic theory, that is
Eq. (1) truncated till R2.
In this article we do not pursue all possible type of
independent scalar invariants at each dimension, but re-
strict ourselves to only vacuum polynomial f(R) theories
of the form
S=1
16πZd4x√−g[R+f(R)] ,(3)
where Ris the Ricci scalar and the form of f(R) is
given in Eq. (11). Whereas GR has only two massless
spin-2 (or purely tensorial) degrees of freedom (dof),
f(R) theories of gravity on the other hand contain, in
addition to the massless dof of GR, an extra massive
scalar degree of freedom. This is best understood by
going to the so called ‘Einstein frame’ by means of
a conformal transformation of the original frame (or
‘Jordan frame’) metric of the f(R) theory which frames
the theory effectively as GR plus scalar field with a
potential (the form of which is fixed by the particular
form of the f(R) theory at hand). Examples of such
can be found in [13–18].
We however find that in perturbative f(R) theories
of gravity, with f(R) of the form of Eq. (11), under a
suitable field redefinition, can be mapped back to GR ad
infinitum. We find that the field redefinition responsible
for mapping the theory back to GR is a conformal
transformation on the metric.
Our paper is organized as follows. In Sec. II we
briefly review the Einstein and f(R)-gravity theories.
In Sec. III we review the field redefinition method
to excise quadratic (and some cubic) modifications
to GR. In Sec. IV we prove the perturbative off-shell
equivalence of these theories. Section V contains a
brief review of Brans-Dicke theory and describe its
off-shell equivalence with f(R)-theory. In Sec. VI we
review off-shell equivalence between f(R)-gravity and
the Einstein-scalar field model. In Sec. VII we establish
perturbative off-shell equivalence between f(R)-gravity,
Brans-Dicke theory, and the Einstein-scalar field model
and illustrate it via equivalence maps. Finally, in
Sec. VIII we conjecture on the potential effects of the
conformal transformation when matter fields are present.
In this paper we shall use the geometrized system of
units, setting c=G= 1. We use space-time signature
+2 and conventions adopted in the book [19], unless ex-
plicitly stated otherwise.
II. EINSTEIN AND f(R)-GRAVITY MODELS
The famous 4-dimensional Einstein-Hilbert action in
the presence of matter is
SEH =1
16πZd4x√−gR +Sm(gαβ , ψ),(4)
where gis determinant of space-time metric gαβ in the
chosen coordinates xα,Ris the Ricci scalar, and the in-
tegral is taken over the space-time manifold. The matter
action Smcontains matter fields ψ. One of the simplest
generalization of the Einstein-Hilbert action is the so-
called f(R) model,
S=1
16πZd4x√−g[R+f(R) + Lm],(5)
where f(R) is an arbitrary function of the Ricci scalar.
f(R) theories shot to prominence with the quadratic
version of Eq. (5), or the ‘Starobinsky model’ given
by R+α R2([20]), which was shown to demonstrate
inflation, and remains a viable alternative to scalar field
inflationary models.
Equations of motion or field equations which have
higher than second order derivatives suffer from an
instability where their total energy or the Hamiltonian
is unbounded from below. Implying that in general
these theories do not have a stable vacuum state.
This is referred to as the Ostrogradski instability in
literature. f(R) theories of gravity however do not
suffer from the Ostrogradski instability, as was found
in [21]. This is in contrast to more generalized theories
of gravity containing scalar invariants like Ricci tensor
and Riemann contractions whose field equations are
also fourth order in nature, and they suffer from the so
called ‘ghost’ degrees of freedom [11] whose energy is
unbounded from below.
There are two different formalisms of f(R) theories
of gravity, the metric and the Palatini (or metric-affine)
formalism. While the metric formalism is the standard
that assumes the metric to be the dynamical field
in the space-time, the Palatini formalism takes the
connection to be an independent variable as well in
addition to the metric tensor (a review in [22]), lead-
ing to some interesting features, like the suppression of
the massive scalar degree of freedom as was found in [23].
3
It has been noticed by more and more accurate ex-
periments that there exist in nature, an equivalence (or
a strict equality) between the gravitational ‘charge’ (or
gravitational mass) and the inertial mass. GR as a theory
satisfies the equivalence principle. The fate of the equiva-
lence principle in f(R) theories of gravity is still debated
in the gravity community. On one hand, various authors
argue that the principle is violated, for example in the
Palatini formalism in [24]. Also, since f(R) theories can
be mapped to scalar-tensor theories of a particular form
(as in [25]), it was shown in [26] that equivalence prin-
ciple is violated in such theories. On the other hand,
it was shown in [27] that for transition from dense re-
gions of space-time to vacuum, the exterior solution can
be represented by a Schwarzschild-like metric, with the
extra Yukawa type field (arising due to f(R) modifica-
tion) only appearing as a very thin shell around the dense
object. In such a case, the weak equivalence principle is
shown to hold to a very high accuracy, like for solar sys-
tem tests of the equivalence principle. Constraints on
fifth force parameters can also be found in [27]
III. THE FIELD REDEFINITION METHOD
Following the notations of [10] only for this section, we
briefly review the field redefinition method here. For a
Lagrangian of the following form [28] in vacuum
L=√−gM2
P l
2R+LD4+LD6+ORiemann4
M4,
(6)
where the first term is the Einstein-Hilbert action and
the higher dimensional operators are
LD4=√−gcR2R2+cW2W2
µανβ +cGB R2
GB,(7)
LD6=√−g
M2d1R□R+d2Rµν □Rµν +d3R3
d4RR2
µν +d5RR2
µναβ +d6R3
µν +d7Rµν RαβRµναβ
d8Rµν Rµαβγ Rαβγ
ν+d9Rαβ
µν Rγσ
αβ Rµν
γσ
+d10Rα β
µ ν Rγ σ
α β Rµ ν
γ σ i,(8)
where Wµανβ is the Weyl tensor and R2
GB is the Gauss-
Bonnet topological term which can be ignored since in
this article we restrict ourselves to four dimensions. One
can perform a perturbative field redefinition of the metric
as follows
gµν →gµν −2
M2
P l −2cW2Rµν +cR2+1
3cW2gµν R
−2
M2
P l
1
M2−d2□Rµν −d4RRµν −d6Rα
µRνα
−d8Rαβγ
µRναβγ +gµµ d1+d2
2□R+ (d3
+d4
2R2+d6+d7
2R2
αβ +d5+d8
2R2
αβγσ,
(9)
which leads to the Lagrangian (6) becoming as follows
L=√−g"M2
pl
2R+1
M2d9Rαβ
µν Rγσ
αβ Rµν
γσ
+d10Rα β
µ ν Rγ σ
α β Rµ ν
γ σ i+O1
M4.(10)
One can see that due to the field redefinition entirety
of the quadratic and some of the cubic terms, except
pure Weyl cube terms which cannot be field redefined
away, have been pushed to higher orders. In the follow-
ing section, ignoring Ricci tensor and Riemann tensor
terms in the Lagrangian, we generalize the field redefini-
tion method to higher powers of the Ricci scalar.
IV. EQUIVALENCE WITH EINSTEIN
GRAVITY
Here we shall consider perturbative approach to f(R)
gravity. Namely, we introduce a parameter λwhich char-
acterizes strength of perturbation, such that λR ≪1.
This parameter can be considered as squared character-
istic length of the model (measured in units of mP). In
the string effective action it corresponds to the string
slope parameter α′, which in natural units is equal to the
squared string length. Hence, the dimension of λis length
squared. The perturbative f(R) takes the following form
(see, e.g. [10, 29]):
f(R) = X
k≥1
ckλkRk+1 ,(11)
where ck’s are dimensionless expansion coefficients [30].
Our goal is to establish perturbative equivalence between
the f(R) theory (5) with f(R) given by (11) and the Ein-
stein gravity (4). Such an equivalence was established be-
tween Lagrangian density with quadratic and cubic terms
in R,Rαβ, and the Riemann tensor components in ear-
lier works by means of field redefinition (see, for example,
[17, 28, 29, 31]). In the case of Rterms only, the field
redefinition is equivalent to conformal transformation of
space-time metric. Here we shall use the same approach,
which can also be suggested by the Scherk-Schwarz for-
malism for performing dimensional reduction, which we
do not apply here [32].
4
The conformal metric transformation
¯gαβ = Ω2gαβ ,(12)
induces the following Ricci scalar conformal transforma-
tion in 4-dimensional space-time (see e.g. [33]):
¯
R= Ω−2R−6 Ω−1∇2Ω,(13)
where ∇is the covariant derivative operator associated
with the space-time metric gαβ,∇αgβγ = 0. Here and in
what follows, we shall use the notations ∇2=gαβ∇α∇β
and (∇Ω)2=gαβ(∇αΩ)(∇βΩ). It is to be noted that
under the conformal transformation (12), the coupling of
matter with the space-time curvature changes. This is
because of the fact that the matter Lagrangian density
Lmexplicitly contains the metric, and any redefinition
of the metric will correspondingly feature in Lmas well.
The point has been illustrated in [34]. However, for sim-
plicity, and to illustrate our point, we will only consider
the vacuum case.
Now we require that for a certain conformal factor the
f(R) theory (5) is equivalent to the Einstein gravity (4),
that is
√−g[R+f(R)] ⊜√−¯g¯
R , (14)
where ¯gis determinant of ¯gαβ and the symbol ⊜stands
for equality modulo total derivatives, which give vanish-
ing within our theory boundary terms. Let us remark
that expression with ⊜is always understood under the
integral sign, i.e. we can multiply the expression like (14)
by a constant, but not by a scalar function, except for
a function of the metric gαβ and its determinant. Using
the expression (13) and the relation √−¯g= Ω4√−gwe
present (14) in the following form:
6Ω∇2Ω−(Ω2−1)R+f(R)⊜0.(15)
Let us observe first that for f(R) = 0 the expression (5)
reduces to the expression (4), which implies that Ω = 1,
which is a trivial solution to (15) with f(R) = 0. Be-
cause f(R) is given in terms of series (11), we shall look
for a solution to (15) in the perturbative form of the cor-
responding to f(R) order,
Ω = 1 + X
k≥1
λkPk[R],(16)
where Pk[R] = Pk(R, ∇2R, ...) are sought polynomials
composed of the Ricci scalar and its covariant derivatives.
Then, the conformal factor Ω2can be computed from (16)
for a given λ-order nof f(R) as follows:
Ω2= 1 +
n
X
k=1
λk 2Pk+
k−1
X
l=1
Pk−lPl!,(17)
where for brevity we dropped the [R] notation in the
Pk’s terms. Substituting (16) into (15) and using (11)
we derive the following λnorder term:
n−1
X
k=1
(6Pn−k∇2Pk−R Pn−kPk)−2R Pn+cnRn+1 ,(18)
where n≥1. Each λ-order expression should vanish
independently modulo the boundary term. As a result,
we can find the following nonlinear recurrence relation of
order n−2 for Pn:
Pn=cn
2Rn+
n−1
X
k=1 3R−1Pn−k∇2Pk−1
2PkPn−k.
(19)
As an example, for f(R) of λ-order 2 terms are
P1=c1
2R , P2=c2
2R2+c2
1
43∇2R−1
2R2.(20)
They define the following conformal factor:
Ω2= 1 + λ1c1R+λ2c2R2+3
2c2
1∇2R.(21)
The recurrence relation (19) together with (17) solves
equation (15) and proves by induction that the perturba-
tive f(R) theory (11) is equivalent to the Einstein gravity
(4).
V. EQUIVALENCE WITH BRANS-DICKE
THEORY WITH ω0= 0
The Brans-Dicke theory with ω0= 0 parameter with-
out matter fields reads (see, e.g. [14, 18, 35])
S=1
16πZd4x√−g[ϕ R −V(ϕ)] ,(22)
where ϕis the Brans-Dicke scalar field and V(ϕ) is its
potential. The corresponding field equations are
Gαβ =1
ϕ(∇α∇βϕ−gαβ∇2ϕ)−gαβ
2ϕV(ϕ),(23)
R=V′(ϕ),(24)
where the prime stands for the derivative with respect to
ϕ. Assume now that the scalar field depends perturba-
tively on Rand consider the following expansion:
ϕ(R) = X
k≥0
akλkRk,(25)
where λ≪1 as before and ak’s are constant expansion
coefficients. Then, integrating the expression (24) and
imposing the condition V(0) = 0 we derive
V(R) = X
k≥0
akk
k+ 1λkRk+1 .(26)
Thus we see that taking a0= 1 and ak=ck(k+ 1) for
k≥1 the Brans-Dicke action (22) is equivalent to the
f(R) theory (5) and, as it follows from Sec. II, can be
transformed to the Einstein gravity (4).
5
VI. EQUIVALENCE WITH EINSTEIN-SCALAR
FIELD MODEL
The Brans-Dicke model (22) can be viewed as Einstein-
scalar field model in the Jordan frame. Here we present
it in the Einstein frame. Let us first rewrite the action
(22) in the “barred” form
S=1
16πZd4x√−¯gϕ¯
R−V(ϕ).(27)
Then, we apply the conformal transformations (12), (13)
with the conformal factor
Ω2=ϕ−1.(28)
and present the kinetic term modulo the boundary term
as follows: 6ϕ1/2∇2ϕ−1/2⊜−6(∇ϕ1/2)(∇ϕ−1/2). This
brings us to the action
S=1
16πZd4x√−gR−3
2ϕ−2(∇ϕ)2−ϕ−2V(ϕ).
(29)
The final step is to write the kinetic term in the standard
form by defining new scalar field φ, such that
ϕ= exp φ/√3(30)
and the related potential
U(φ) = exp −2φ/√3V(φ).(31)
This results in the Einstein-scalar field model
S=1
16πZd4x√−gR−1
2(∇φ)2−U(φ).(32)
This procedure is well known and presented in many
works (see, e.g. [14, 18]). Our goal is to establish its re-
lation with the perturbative f(R) model (5), (11), and,
as a result, with the Einstein gravity (4). We observe
that relations inverse to (28), (30), and (31) transform
the Einstein-scalar field model (32) to the Brans-Dicke
model (22). Then, as it was illustrated in the previous
section, the Brans-Dicke model can also be transformed
perturbatively via the expressions (25) and (26) to the
Einstein gravity (4).
VII. EQUIVALENCE MAPS
In the previous sections we established off-shell equiv-
alence between perturbative f(R) model, the Einstein
gravity, the Brans-Dicke with ω0= 0 and the Einstein-
scalar field models. Here we explore this equivalence in
detail.
It is well known that f(R) gravity is equivalent to
Brans-Dicke theories (see, e.g. [14, 18] and the equiva-
lence map presented therein.) This equivalence exists for
any f(R) function of class C2and is established by means
of the Legendre and conformal transformations. Equiva-
lence between the Einstein gravity and f(R) model can
be established by means of the conformal transforma-
tion (12) where the conformal factor solves equation (15).
This equation is a nonlinear PDE of the first (or ignoring
boundary term second) order. We do not know whether
a general solution to this equation for arbitrary f(R) of
class C2exists and if so, how to find it. It could be that
there is some class of transformations or generating tech-
niques allowing to find for a given f(R) corresponding
solutions to this equation. An analysis of this problem
goes beyond the scope of this work.
We were able to find a perturbative solution to this
equation for the corresponding form of f(R) function
(11). This equivalence implies that the perturbative f(R)
and the related to it Brans-Dicke and Einstein scalar field
models are conformally equivalent to the Einstein gravity.
In other words, all these models lie within the Einstein
gravity and can be “perturbatively revealed” by a suit-
able conformal factor. One can say accordingly, that the
Einstein theory is rich enough to accommodate pertur-
batively via conformal transformation f(R) and related
gravity models.
VIII. SCENARIO IN THE PRESENCE OF
MATTER AND APPLICATIONS TO
COSMOLOGY
As the current work deals with vacuum scenarios, it
is important to note that the scenario maybe completely
different in the presence of matter. While it is clear from
this work that in vacuum the extra degree of freedom is
suppressed infinitely for f(R) theories (and related theo-
ries that f(R) can be mapped to), the presence of matter
will lead to complications. To be precise, the Lagrangian
for the matter fields contain the metric; any changes to
the metric, like a conformal transformation, will lead to
a change in the matter Lagrangian. In effect, the confor-
mal transformation in the current work and other such
transformations will lead to the non-minimal coupling of
the matter field with gravity, resulting in an actual de-
viation from GR. As an example, consider the conformal
transformation on the metric which takes an f(R) theory
in Jordan frame to its equivalent Einstein+scalar model
in the Einstein frame. The conformal transformation is
given as follows
˜gµν =f′(R)gµν ,(33)
where f′(R) = df(R)
dR . In the Einstein frame, due to
this particular conformal transformation, the matter La-
grangian picks up the conformal factor and hence be-
comes non-minimally coupled to gravity. This particu-
lar non-minimal coupling leads to a variable mass of the
scalar field which depends on the ambient matter density
[36]. To be precise, the mass of the scalar field becomes
6
heavy in regions of high mass density (leading to a sharp
fall off of the scalar field profile outside a mass distribu-
tion) like the surface of planets, stars, etc. On the other
hand when the mass density is low, the mass of the scalar
field becomes light, which can support cosmic accelera-
tion. This effect is known as the Chameleon mechanism.
Now consider the leading order f(R) theory of the form
f(R)≡R+α R2.(34)
The conformal transformation required to take this the-
ory into the Einstein frame is then given by
f′(R) = 1 + 2 α R. (35)
Compare this with the leading order conformal trans-
formation for taking the (R) theory to Einstein gravity
in Eq. (21) which goes as 1 + λ c1R. This is not a
coincidence. While the former serves to convert higher
powers of the Ricci scalar (and in essence higher powers
of derivatives of the metric) into kinetic and potential
terms of the scalar field, the latter ‘pushes’ the quadratic
powers of the Ricci scalar into cubic and higher powers.
Both conformal transformations have the same effect of
getting rid of the quadratic Ricci term. Both conformal
transformations will lead to a non-minimal coupling of
the matter fields with gravity, and in return will gener-
ate Chameleon screening of the scalar field, which then
becomes important for cosmological scenarios.
IX. DISCUSSIONS AND FUTURE WORK
In this article we were able to show the pertubative
equivalence between polynomial f(R) theories of gravity
and GR. At the same time equivalences between Brans-
Dicke theories, scalar-tensor theories, and polynomial
f(R) were also reviewed. This exercise shows that it
is possible to generate theories of higher complexity and
field content from GR itself through proper field redefini-
tions. Specifically, the graviton, with it’s three degrees of
freedom in f(R) theories of gravity is seen to apparently
shed one massive scalar degree of freedom through the
particular field redefinition in Eqs. (12) and (21), and
have only two massless tensor degrees of freedom of GR,
to which it was mapped to. The equivalences shown im-
ply that if a solution to the vacuum field equations of GR
is known, one can generate the corresponding solutions
of the vacuum field equations of the modified theories of
gravity that were discussed in this article. However, this
will require further study and will be published elsewhere.
The equivalences raise the question about the ‘physi-
cality’ of the conformal ‘frames’. That is, whether a black
hole solution in GR in the conformally transformed GR
‘frame’ is the physical solution, or whether the GR frame
is simply a convenient frame to generate solutions, and
the physical object in question (the black hole solution)
is described by the solution of the vacuum field equations
of modified gravity frame. However, this question is be-
yond the scope of this article and will be pursued in a
following publication.
In this article, we have only pursued vacuum La-
grangian densities. However, field redefinitions of the
form (12) induce non-minimal coupling of the curvature
with the matter Lagrangian density since it is an explicit
function of the metric tensor as well. This will lead to
the comformally transformed Lagrangian having a more
complicated form in the GR ‘frame’ and require further
investigation.
For future work we will also generalize from f(R) the-
ories to the fully general Lagrangian of Eq. (1), and will
aim to come up with a field redefinition that maps it to
GR.
ACKNOWLEDGMENTS
The author would like to acknowledge Andrey Shoom
for giving significant help and directions throughout the
duration of the study. The author would also like to
thank Alok Laddha and Ashoke Sen for giving helpful
comments. The author wishes to express his gratitude to
the MPI f¨ur Gravitationsphysik (Hannover) and ICTS-
TIFR (Bangalore) for hospitality. The author would also
like the thank the anonymous reviewers for their ques-
tions and suggestions.
Appendix A: Derivation of the master equation
From Eq. (14), one has the following
√−g[R+f(R)] −√−¯g¯
R= 0.(A1)
The conformally transformed square root of metric de-
terminant and the Ricci scalar transform as
√−¯g= Ω4√−g, (A2)
¯
R=R
Ω2−6□Ω
Ω3.(A3)
Substituting the above in Eq. (A1) leads to the following
√−g[R+f(R)] −√−gΩ2R−6 Ω □Ω= 0.(A4)
Canceling √−gon both sides one obtains
6Ω□Ω−(Ω2−1)R+f(R)=0.(A5)
[1] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gund-
lach, B. R. Heckel, C. D. Hoyle, and H. E. Swanson, Tests
of the gravitational inverse-square law below the dark-
7
energy length scale, Phys. Rev. Lett. 98, 021101 (2007),
arXiv:hep-ph/0611184.
[2] C. M. Will, The Confrontation between General Rela-
tivity and Experiment, Living Rev. Rel. 17, 4 (2014),
arXiv:1403.7377 [gr-qc].
[3] A. Ghosh, N. K. Johnson-Mcdaniel, A. Ghosh, C. K.
Mishra, P. Ajith, W. Del Pozzo, C. P. L. Berry, A. B.
Nielsen, and L. London, Testing general relativity using
gravitational wave signals from the inspiral, merger and
ringdown of binary black holes, Class. Quant. Grav. 35,
014002 (2018), arXiv:1704.06784 [gr-qc].
[4] B. P. Abbott et al. (LIGO Scientific, Virgo), Tests of
general relativity with GW150914, Phys. Rev. Lett. 116,
221101 (2016), [Erratum: Phys.Rev.Lett. 121, 129902
(2018)], arXiv:1602.03841 [gr-qc].
[5] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA),
Tests of General Relativity with GWTC-3, (2021),
arXiv:2112.06861 [gr-qc].
[6] N. V. Krishnendu and F. Ohme, Testing General Rela-
tivity with Gravitational Waves: An Overview, Universe
7, 497 (2021), arXiv:2201.05418 [gr-qc].
[7] S. Bhattacharyya, Distinguishing general relativity and
modified theories of gravity using quasinormal modes,
Ph.D. thesis, IISER, Trivandrum (2019).
[8] A. Hees and A. F¨uzfa, Can the Chameleon Mechanism
Explain Cosmic Acceleration While Satisfying Solar Sys-
tem Constraints?, in 13th Marcel Grossmann Meeting
on Recent Developments in Theoretical and Experimental
General Relativity, Astrophysics, and Relativistic Field
Theories (2015) pp. 1140–1142, arXiv:1302.6527 [gr-qc].
[9] R. Jackiw and S. Y. Pi, Chern-Simons modification
of general relativity, Phys. Rev. D 68, 104012 (2003),
arXiv:gr-qc/0308071.
[10] C. de Rham, J. Francfort, and J. Zhang, Black Hole Grav-
itational Waves in the Effective Field Theory of Gravity,
Phys. Rev. D 102, 024079 (2020), arXiv:2005.13923 [hep-
th].
[11] K. S. Stelle, Renormalization of Higher Derivative Quan-
tum Gravity, Phys. Rev. D 16, 953 (1977).
[12] B. Slovick, Renormalization of Einstein gravity through a
derivative-dependent field redefinition, Mod. Phys. Lett.
A33, 1850016 (2017), arXiv:1309.5945 [hep-th].
[13] Y. Shtanov, On the Conformal Frames in f(R) Gravity,
Universe 8, 69 (2022), arXiv:2202.00818 [gr-qc].
[14] T. P. Sotiriou and V. Faraoni, f(r) theories of gravity,
Rev. Mod. Phys. 82, 451 (2010).
[15] A. De Felice and S. Tsujikawa, f(R) theories, Living Rev.
Rel. 13, 3 (2010), arXiv:1002.4928 [gr-qc].
[16] B. Whitt, Fourth-order gravity as general relativity plus
matter, Physics Letters B 145, 176 (1984).
[17] J. D. Barrow and S. Cotsakis, Inflation and the conformal
structure of higher-order gravity theories, Physics Letters
B214, 515 (1988).
[18] T. P. Sotiriou, f(r) gravity and scalar–tensor theory, Clas-
sical and Quantum Gravity 23, 5117 (2006).
[19] Gravitation (W. H. Freeman and Co., San Francisco,
1973).
[20] A. A. Starobinsky, A New Type of Isotropic Cosmological
Models Without Singularity, Phys. Lett. B 91, 99 (1980).
[21] R. P. Woodard, Avoiding dark energy with 1/r modi-
fications of gravity, Lect. Notes Phys. 720, 403 (2007),
arXiv:astro-ph/0601672.
[22] G. J. Olmo, Palatini Approach to Modified Gravity: f(R)
Theories and Beyond, Int. J. Mod. Phys. D 20, 413
(2011), arXiv:1101.3864 [gr-qc].
[23] G. J. Olmo, D. Rubiera-Garcia, and A. Wojnar, Stel-
lar structure models in modified theories of gravity:
Lessons and challenges, Phys. Rept. 876, 1 (2020),
arXiv:1912.05202 [gr-qc].
[24] G. J. Olmo, Violation of the Equivalence Principle in
Modified Theories of Gravity, Phys. Rev. Lett. 98,
061101 (2007), arXiv:gr-qc/0612002.
[25] T. P. Sotiriou, f(R) gravity and scalar-tensor the-
ory, Class. Quant. Grav. 23, 5117 (2006), arXiv:gr-
qc/0604028.
[26] M. Blasone, S. Capozziello, G. Lambiase, and
L. Petruzziello, Equivalence principle violation at finite
temperature in scalar-tensor gravity, Eur. Phys. J. Plus
134, 169 (2019), arXiv:1812.08029 [gr-qc].
[27] J. Khoury and A. Weltman, Chameleon cosmology, Phys.
Rev. D 69, 044026 (2004), arXiv:astro-ph/0309411.
[28] A. Tseytlin, Ambiguity in the effective action in string
theories, Physics Letters B 176, 92 (1986).
[29] R. Metsaev and A. Tseytlin, Curvature cubed terms in
string theory effective actions, Physics Letters B 185, 52
(1987).
[30] For the given choice of the perturbation parameter λthe
integrand in (5) reads R(1 + λR +λ2R2+· · · ). However,
one could also consider λas the characteristic length,
such that λ2R≪1, and have R(1 + λ2R+λ3R2+
· · · ). One can establish a transformation between these
choices.
[31] M. Accettulli Huber, A. Brandhuber, S. De Angelis, and
G. Travaglini, Note on the absence of R2corrections to
Newton’s potential, Phys. Rev. D 101, 046011 (2020),
arXiv:1911.10108 [hep-th].
[32] J. Scherk and J. H. Schwarz, How to Get Masses from
Extra Dimensions, Nucl. Phys. B 153, 61 (1979).
[33] R. M. Wald, General Relativity (The University of
Chicago Press, Chicago and London, 1984).
[34] M. Ruhdorfer, J. Serra, and A. Weiler, Effective Field
Theory of Gravity to All Orders, JHEP 05, 083,
arXiv:1908.08050 [hep-ph].
[35] J. O’Hanlon, Intermediate-range gravity: A generally co-
variant model, Phys. Rev. Lett. 29, 137 (1972).
[36] C. Burrage and J. Sakstein, Tests of Chameleon Gravity,
Living Rev. Rel. 21, 1 (2018), arXiv:1709.09071 [astro-
ph.CO].
