Mapping the Weak Field Limit of Scalar-Gauss-Bonnet Gravity Benjamin Elderand Jeremy Sakstein

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Mapping the Weak Field Limit of

Scalar-Gauss-Bonnet Gravity

Benjamin Elder∗and Jeremy Sakstein†

Department of Physics and Astronomy, University of Hawai’i,

2505 Correa Road, Honolulu, HI 96822, USA

Abstract

We derive the weak ﬁeld limit of scalar-Gauss-Bonnet theory and place novel bounds on the

parameter space using terrestrial and space-based experiments. In order to analyze the theory

in the context of a wide range of experiments, we compute the deviations from Einstein gravity

around source masses with planar, cylindrical, and spherical symmetry. We ﬁnd a correction to

the Newtonian potential around spherical and cylindrical sources that can be larger than PPN

corrections suﬃciently close to the source. We use this to improve on laboratory constraints on

the scalar-Gauss-Bonnet coupling parameter Λ by two orders of magnitude. Present laboratory

and Solar System bounds reported here are superseded by tests deriving from black holes.

1 Introduction

Despite the many successes of general relativity (GR), we still do not know whether it is the full

theory of gravity. Although this theory is compatible with all direct tests to date, the persistence of

several unexplained phenomena such as the cosmological constant problem [1–3] and the accelerated

cosmological expansion [4] has motivated the introduction of a wide range of alternative models

to, and extensions of, GR [5–7]. Likewise, ongoing and ever-more precise tests of gravity in the

laboratory, the Solar System, and in space are searching for hints of new physics beyond GR [8–16].

Any extension of GR will increase the number of degrees of freedom in the theory beyond the

nominal two [17,18]. From this perspective, a minimal modiﬁcation of gravity is to explicitly

add a single degree of freedom in the form of a new scalar ﬁeld φ. The scalar’s couplings to the

metric tensor, matter ﬁelds, and itself determine a wide range of possible phenomenologies [5–

7,13]. Infra-red (IR) modiﬁcations of gravity in which there is a direct coupling to the Ricci scalar

φR or, equivalently, a Yukawa coupling to matter ﬁelds φ¯

ψψ, result in ﬁfth force whose range

is set by the inverse mass of the scalar ﬁeld. This force modiﬁes the Newtonian 1/r2force law

and there are ongoing searches for this modiﬁcation from microscopic to cosmological scales [19–

21]. Alternatively, ultra-violet (UV) modiﬁcations of gravity where the scalar-graviton couplings

∗bcelder@hawaii.edu

†sakstein@hawaii.edu

arXiv:2210.10955v1 [gr-qc] 20 Oct 2022

are non-renormalizable, for example φR2, φRµν Rµν , give rise to deviations from GR in strongly

gravitating systems e.g. black holes, neutron stars, and binary pulsars [22].

In this work, we investigate the prospect for constraining UV modiﬁcations of GR focusing on

scalar-Gauss-Bonnet gravity (SGB) in which a scalar ﬁeld couples linearly to the Gauss-Bonnet

topological invariant1

G=R2−4Rµν Rµν +Rµναβ Rµναβ .(1.1)

Such theories are extremely interesting from a theoretical perspective for several reasons. First,

couplings of the form φGrepresent the leading-order scalar-graviton interaction in shift-symmetric

theories [23]. Second, the equations of motion resulting from such couplings are second-order, mean-

ing that the theory does not suﬀer from an Ostrogradski ghost instability. Third, more generalized

couplings of the form f(φ)Garise naturally in string theory [24,25], may explain the accelerated

expansion of the universe [26,27], and give rise to the novel phenomenomenon of spontaneous

black hole scalarization [25,28,29]. Studying the simplest coupling φGwill lay the foundation for

constraining these more complicated theories.

The eﬀects of SGB gravity are most pronounced in the strong-ﬁeld regime, and hence black

holes are powerful probes of this theory [30–33]. This is helped by the fact that although smooth

extended objects cannot obtain a scalar charge in the theory, black holes can [23,34]. Furthermore,

certain types of couplings can be restricted based on theoretical arguments alone [35]. It is natural

to wonder how laboratory and Solar System experiments compare against astrophysical tests of

this theory. There currently exists a wide range of tabletop experiments that employ radically

diﬀerent source mass geometries to test gravity in the weak ﬁeld regime. Experiments like torsion

balances [36], atom interferometers [37,38], and Casimir force sensors [39–41] have proven in recent

years to be extraordinarily useful thanks to their high accuracy and their ability to be tuned to

search for eﬀects in speciﬁc theories (see [42] for a review of laboratory tests of gravity). For

instance, large atom interferometers have performed some of the most precise measurements to

date on Newton’s constant G[38], while miniature ones have proven sensitive to screened modiﬁed

gravity theories that are otherwise very diﬃcult to constrain [12,43–46].

Some of the ﬁrst experimental constraints on SGB gravity were derived from Solar System tests

and focused on the speciﬁc case in which the scalar ﬁeld drives the accelerated expansion of the

universe [26]. The PPN expansion of the theory has been computed, and it was found that the

theory is indistinguishable from GR at second post-Newtonian order [47]. This does not mean that

the theory is impossible to constrain via local tests of gravity, only that the theory does not ﬁt

into the PPN framework. Deviations from GR were computed around point particles in [48], which

were then used to place bounds from Solar System and laboratory tests. In this work we relax the

point particle assumption, which enables us to use a larger range of experimental tests to constrain

the theory. Speciﬁcally, we compute the weak ﬁeld limit of SGB gravity and study the deviations

from GR around extended objects with planar, cylindrical, and spherical symmetry. We ﬁnd a

1We use units in which c=~= 1 and have deﬁned the reduced Planck mass as MPl ≡(8πG)−1/2.

1/r8force around spherical objects, a 1/r5force around cylindrical objects, and no modiﬁcation

beyond GR around planar objects. We use results from a recent atom interferometry experiment

to improve on the bound on the scalar-Gauss Bonnet coupling by two orders of magnitude relative

to previous studies on laboratory and Solar System tests. Our bounds are still weaker than those

deriving from black holes by 13 orders of magnitude. It is unlikely that even future experiments

will be able to reach the same sensitivity as strong-ﬁeld tests.

The rest of this paper is organized as follows. In Section 2we brieﬂy review the salient features

of SGB gravity. In Section 3we expand the theory in the weak ﬁeld limit, and compute its leading-

order deviations from GR around extended bodies with planar, cylindrical, and spherical symmetry.

We also provide a simpliﬁed proof showing that extended bodies do not acquire a scalar charge in

the theory, the details of which are in Appendix A. In Section 4we compare our bounds to those

coming from black holes. We discuss the implications of our results and conclude in Section 5.

2 Scalar-Gauss-Bonnet Gravity

The action for SGB gravity that we will study is

S=Zd4x√−gM2

2R−1

2(∂µφ)2+φ

ΛG+LSM[gµν ;ψ],(2.1)

where Λ is a new mass scale that parameterizes the SGB coupling, LSM represents the Standard

Model Lagrangian, and the Standard Model matter ﬁelds ψcouple minimally to the metric gµν .

Currently, the strongest bound on Λ is Λ >1.2×10−48 coming from black hole inspirals [32].

The action above describes the theory of a shift-symmetric, parity-even2scalar coupled to gravity.

The resulting equations of motion are second-order so there is no Ostragradski ghost instability

and the theory propagates precisely three degrees of freedom (the two helicity-2 modes of the

graviton and one helicity-0 scalar mode). Note that we have not included higher-dimensional scalar

self-interactions or other shift-symmetric scalar-graviton couplings since these lead to high-order

equations of motion and suﬀer from the Ostragradski instability. The status of SGB gravity as

an eﬀective ﬁeld theory (EFT) is not well-studied and we will not attempt to do so in this work3.

One can break the shift symmetry, which then allows for the addition of a scalar potential or a

generalized coupling of the form f(φ)G. We will not study these generalized couplings in this work

but we brieﬂy comment on them in Section 5.

2Taking the ﬁeld to instead be a pseudo-scalar results in a coupling between the scalar and the Pontryagin density

of the form φ˜

Rµναβ Rµναβ where ˜

Rµναβ is the dual Riemann tensor. We will not study such couplings — referred to

as Chern-Simons couplings [49] — in this work.

3Reference [23] have found that SGB equations of motion are well-posed when the theory is treated as an EFT

and expanded in the coupling parameter.

3 Weak ﬁeld limit

In this section we derive the non-relativistic, weak-ﬁeld limit of SGB theory in the weak-coupling

regime φ/Λ1. We then proceed to solve for the gravitational ﬁelds around highly symmetric

source objects. We introduce a bookkeeping parameter λinto the action (that we will later set to

unity) (2.1) so that our action reads

S=Zd4x√−gM2

2R−1

2(∂µφ)2+λ

ΛφG+LSM[gµν ;ψ].(3.1)

We expand this action in the weak-ﬁeld limit about ﬂat spacetime, choosing the Newtonian gauge:

ds2=−(1 + 2Φ)dt2+ (1 −2Ψ)d~x2,(3.2)

where |Φ|,|Ψ|  1 are small, time-independent perturbations. In this limit we ﬁnd that the

Einstein-Hilbert SEH and scalar Sφ(inluding the Gauss-Bonnet coupling) parts of the action are,

respectively,

SEH =Zd4xMPl2(~

∇Ψ)2−2MPl2~

∇Ψ·~

∇Φ,

Sφ=Zd4x−1

2(~

∇φ)2−1

2(Φ −Ψ)(~

∇φ)2+ 8λφ

Λ∇i∇jΨ∇i∇jΦ−~

∇2Ψ~

∇2Φ.(3.3)

The expansion of the matter Lagrangian about ﬂat space is:

SSM =Zd4xLSM[ψ]−√−g

2Tµν (gµν −ηµν ).(3.4)

If the matter is pressureless and non-relativistic then T00 =ρis the only non-zero component and

we have

SSM =Zd4x(LSM[ψ]−ρΦ) ,(3.5)

where the ﬁrst term is the ﬂat-space matter action and the second term its coupling to gravity.

From the above expression it immediately follows that the action for a non-relativistic point particle

ρ=mδ3(~x) is

Spp =Zdt 1

2m˙

~x2−mΦ.(3.6)

The equation of motion for the point particle is the familiar

¨xpp =−~

∇Φ.(3.7)

Varying Eq. (3.4) with respect to the ﬁelds φ, Φ, and Ψ gives the gravitational ﬁeld equations

∇2Ψ = 1

2MPl2ρ+1

2(~

∇φ)2−8λ

Λ∇i∇j(φ∇i∇jΨ) −~

∇2(φ~

∇2Ψ),

∇2Φ = ~

∇2Ψ−1

2MPl21

2(~

∇φ)2+ 8 λ

Λ∇i∇j(φ∇i∇jΦ) −~

∇2(φ~

∇2Φ),

∇2φ=~

∇ · (Ψ −Φ)~

∇φ−8λ

Λ∇i∇jΨ∇i∇jΦ−~

∇2Ψ~

∇2Φ.(3.8)

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MappingtheWeakFieldLimitofScalar-Gauss-BonnetGravityBenjaminElder*andJeremySaksteinDepartmentofPhysicsandAstronomy,UniversityofHawai'i,2505CorreaRoad,Honolulu,HI96822,USAAbstractWederivetheweakeldlimitofscalar-Gauss-Bonnettheoryandplacenovelboundsontheparameterspaceusingterrestrialandspace-basedex...

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Mapping the Weak Field Limit of Scalar-Gauss-Bonnet Gravity Benjamin Elderand Jeremy Sakstein

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