PreprintPDF Available

On the equivalence between f (R) theories and Einstein gravity

Preprints and early-stage research may not have been peer reviewed yet.


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.
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.
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
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
16 πZd4xR+R2+R3+OR4+Lm,
where Rnare general curvature invariants of dimension
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-
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
16πZd4xg[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.
The famous 4-dimensional Einstein-Hilbert action in
the presence of matter is
SEH =1
16πZd4xgR +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,
16πZd4xg[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].
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]
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
P l
where the first term is the Einstein-Hilbert action and
the higher dimensional operators are
µανβ +cGB R2
M2d1RR+d2Rµν Rµν +d3R3
µν +d5RR2
µναβ +d6R3
µν +d7Rµν RαβRµναβ
d8Rµν Rµαβγ Rαβγ
µν Rγσ
αβ Rµν
+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
P l 2cW2Rµν +cR2+1
3cW2gµν R
P l
M2d2Rµν d4RRµν d6Rα
µRναβγ +gµµ d1+d2
2R+ (d3
αβ +d5+d8
which leads to the Lagrangian (6) becoming as follows
µν Rγσ
αβ Rµν
+d10Rα β
µ ν Rγ σ
α β Rµ ν
γ σ i+O1
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.
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
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].
The conformal metric transformation
¯gαβ = 2gαβ ,(12)
induces the following Ricci scalar conformal transforma-
tion in 4-dimensional space-time (see e.g. [33]):
R= 2R6 12,(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=gαβαβ
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= 4gwe
present (14) in the following form:
Let us observe first that for f(R) = 0 the expression (5)
reduces to the expression (4), which implies that = 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,
= 1 + X
where Pk[R] = Pk(R, 2R, ...) are sought polynomials
composed of the Ricci scalar and its covariant derivatives.
Then, the conformal factor 2can be computed from (16)
for a given λ-order nof f(R) as follows:
2= 1 +
λk 2Pk+
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:
(6Pnk2PkR PnkPk)2R Pn+cnRn+1 ,(18)
where n1. Each λ-order expression should vanish
independently modulo the boundary term. As a result,
we can find the following nonlinear recurrence relation of
order n2 for Pn:
k=1 3R1Pnk2Pk1
As an example, for f(R) of λ-order 2 terms are
2R , P2=c2
They define the following conformal factor:
2= 1 + λ1c1R+λ2c2R2+3
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
The Brans-Dicke theory with ω0= 0 parameter with-
out matter fields reads (see, e.g. [14, 18, 35])
16πZd4xg[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
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
where λ1 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+ 1λkRk+1 .(26)
Thus we see that taking a0= 1 and ak=ck(k+ 1) for
k1 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).
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
Then, we apply the conformal transformations (12), (13)
with the conformal factor
and present the kinetic term modulo the boundary term
as follows: 6ϕ1/22ϕ1/26(ϕ1/2)(ϕ1/2). This
brings us to the action
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
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).
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
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.
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
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.
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
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
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 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= 4g, (A2)
Substituting the above in Eq. (A1) leads to the following
g[R+f(R)] g2R6 = 0.(A4)
Canceling gon both sides one obtains
[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-
energy length scale, Phys. Rev. Lett. 98, 021101 (2007),
[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. 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),
[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-
[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,
[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),
[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-
[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
[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 λ2R1, and have R(1 + λ2R+λ3R2+
· · · ). One can establish a transformation between these
[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-
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
The ever-increasing number of detections of gravitational waves (GWs) from compact binaries by the Advanced LIGO and Advanced Virgo detectors allows us to perform ever-more sensitive tests of general relativity (GR) in the dynamical and strong-field regime of gravity. We perform a suite of tests of GR using the compact binary signals observed during the second half of the third observing run of those detectors. We restrict our analysis to the 15 confident signals that have false alarm rates ≤10⁻³yr⁻¹. In addition to signals consistent with binary black hole (BH) mergers, the new events include GW200115_042309, a signal consistent with a neutron star--BH merger. We find the residual power, after subtracting the best fit waveform from the data for each event, to be consistent with the detector noise. Additionally, we find all the post-Newtonian deformation coefficients to be consistent with the predictions from GR, with an improvement by a factor of ~2 in the -1PN parameter. We also find that the spin-induced quadrupole moments of the binary BH constituents are consistent with those of Kerr BHs in GR. We find no evidence for dispersion of GWs, non-GR modes of polarization, or post-merger echoes in the events that were analyzed. We update the bound on the mass of the graviton, at 90% credibility, to m_g ≤ 1.27×10⁻²³eV/c². The final mass and final spin as inferred from the pre-merger and post-merger parts of the waveform are consistent with each other. The studies of the properties of the remnant BHs, including deviations of the quasi-normal mode frequencies and damping times, show consistency with the predictions of GR. In addition to considering signals individually, we also combine results from the catalog of GW signals to calculate more precise population constraints. We find no evidence in support of physics beyond GR.
Full-text available
The detections of gravitational-wave (GW) signals from compact binary coalescence by ground-based detectors have opened up the era of GW astronomy. These observations provide opportunities to test Einstein’s general theory of relativity at the strong-field regime. Here we give a brief overview of the various GW-based tests of General Relativity (GR) performed by the LIGO-Virgo collaboration on the detected GW events to date. After providing details for the tests performed in four categories, we discuss the prospects for each test in the context of future GW detectors. The four categories of tests include the consistency tests, parametrized tests for GW generation and propagation, tests for the merger remnant properties, and GW polarization tests.
Full-text available
We investigate the propagation of gravitational waves on a black hole background within the low-energy effective field theory of gravity, where effects from heavy fields are captured by higher-dimensional curvature operators. Depending on the spin of the particles integrated out, the speed of gravitational waves at low energy can be either superluminal or subluminal as compared to the causal structure observed by other species. Interestingly, however, gravitational waves are always exactly luminal at the black hole horizon, implying that the horizon is identically defined for all species. We further compute the corrections on quasinormal frequencies caused by the higher-dimensional curvature operators and highlight the corrections arising from the low-energy effective field.
Full-text available
The understanding of stellar structure represents the crossroads of our theories of the nuclear force and the gravitational interaction under the most extreme conditions observably accessible. It provides a powerful probe of the strong field regime of General Relativity, and opens fruitful avenues for the exploration of new gravitational physics. The latter can be captured via modified theories of gravity, which modify the Einstein–Hilbert action of General Relativity and/or some of its principles. These theories typically change the Tolman–Oppenheimer–Volkoff equations of stellar’s hydrostatic equilibrium, thus having a large impact on the astrophysical properties of the corresponding stars and opening a new window to constrain these theories with present and future observations of different types of stars. For relativistic stars, such as neutron stars, the uncertainty on the equation of state of matter at supranuclear densities intertwines with the new parameters coming from the modified gravity side, providing a whole new phenomenology for the typical predictions of stellar structure models, such as mass–radius relations, maximum masses, or moment of inertia. For non-relativistic stars, such as white, brown and red dwarfs, the weakening/strengthening of the gravitational force inside astrophysical bodies via the modified Newtonian (Poisson) equation may induce changes on the star’s mass, radius, central density or luminosity, having an impact, for instance, in the Chandrasekhar’s limit for white dwarfs, or in the minimum mass for stable hydrogen burning in high-mass brown dwarfs. This work aims to provide a broad overview of the main such results achieved in the recent literature for many such modified theories of gravity, by combining the results and constraints obtained from the analysis of relativistic and non-relativistic stars in different scenarios. Moreover, we will build a bridge between the efforts of the community working on different theories, formulations, types of stars, theoretical modellings, and observational aspects, highlighting some of the most promising opportunities in the field.
Full-text available
We construct the general effective field theory of gravity coupled to the Standard Model of particle physics, which we name GRSMEFT. Our method allows the systematic derivation of a non-redundant set of operators of arbitrary dimension with generic field content and gravity. We explicitly determine the pure gravity EFT up to dimension ten, the EFT of a shift-symmetric scalar coupled to gravity up to dimension eight, and the operator basis for the GRSMEFT up to dimension eight. Extensions to all orders are straightforward.
Full-text available
We consider Einstein gravity with the addition of R2 and RμνRμν interactions in the context of effective field theory, and the corresponding scattering amplitudes of gravitons and minimally coupled heavy scalars. First, we recover the known fact that graviton amplitudes are the same as in Einstein gravity. Then we show that all amplitudes with two heavy scalars and an arbitrary number of gravitons are also not affected by these interactions. We prove this by direct computations, using field redefinitions known from earlier applications in string theory, and with a combination of factorization and power-counting arguments. Combined with unitarity, these results imply that, in an effective field theory approach, the Newtonian potential receives neither classical nor quantum corrections from terms quadratic in the curvature.
Full-text available
QuasiNormal modes or the mathematical description of gravitational waves emitted during the ring-down of a perturbed black hole, provide critical information about the structure of these compact objects. Since regions around Black Holes have some of the strongest gravitational fields in the Universe, hence, Quasi-Normal Modes can be a tool for strong field tests of General Relativity and possible deviations from it. In the case of General Relativity, it is known for a long time that a relation between two types of Black Hole perturbations: even parity (Zerilli) and odd parity (Regge-Wheeler), leads to an equality of reflection coefficients for both parities. With the direct detection of Gravitational waves, it is now natural to ask: whether the same relation (between even and odd parity perturbations) holds for modified gravity theories? If not, whether one can use this as a way to probe deviations from General Relativity. As a first step, this thesis shows explicitly that the above relation between Regge-Wheeler and Zerilli potentials break down for modifications to gravity. Hence, the two perturbations do not share the equality of reflection coefficients. This thesis also discusses the implication of this inequality on the gravitational wave observations.
Full-text available
This work explores an alternative solution to the problem of renormalizability in Einstein gravity. In the proposed approach, Einstein gravity is transformed into the renormalizable theory of four-derivative gravity by applying a local field redefinition containing an infinite number of higher derivatives. It is also shown that the current–current amplitude is invariant with the field redefinition, and thus the unitarity of Einstein gravity is preserved.
We analyze possible violations of the equivalence principle in scalar-tensor gravity at finite temperature T. We first present an approach where the equivalence principle violation is achieved within the framework of quantum field theory. Then, we rely on an alternative approach first proposed by Gasperini, which leads to the same outcome obtained in the framework of quantum field theory (one-loop corrections) at finite T. Finally, we exhibit the application of the above formalism both to a generic diagonal metric, cast in spherical coordinates, and to the Brans-Dicke theory. In the last case, we show that it is possible to put a significant constraint on the free parameter of the theory by means of experimental bounds on the equivalence principle.
The LIGO detection of GW150914 provides an unprecedented opportunity to study the two-body motion of a compact-object binary in the large-velocity, highly nonlinear regime, and to witness the final merger of the binary and the excitation of uniquely relativistic modes of the gravitational field. We carry out several investigations to determine whether GW150914 is consistent with a binary black-hole merger in general relativity. We find that the final remnant's mass and spin, as determined from the low-frequency (inspiral) and high-frequency (postinspiral) phases of the signal, are mutually consistent with the binary black-hole solution in general relativity. Furthermore, the data following the peak of GW150914 are consistent with the least-damped quasinormal mode inferred from the mass and spin of the remnant black hole. By using waveform models that allow for parametrized general-relativity violations during the inspiral and merger phases, we perform quantitative tests on the gravitational-wave phase in the dynamical regime and we determine the first empirical bounds on several high-order post-Newtonian coefficients. We constrain the graviton Compton wavelength, assuming that gravitons are dispersed in vacuum in the same way as particles with mass, obtaining a 90%-confidence lower bound of 10^{13} km. In conclusion, within our statistical uncertainties, we find no evidence for violations of general relativity in the genuinely strong-field regime of gravity.