Preprint typeset in JHEP style - HYPER VERSION

The Speed of Galileon Gravity

Philippe Brax

Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA,CNRS,

F-91191Gif sur Yvette, France

E-mail: philippe.brax@cea.fr

Clare Burrage

School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD,

United Kingdom

E-mail: Clare.Burrage@nottingham.ac.uk

Anne-Christine Davis

DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA, UK

E-mail: A.C.Davis@damtp.cam.ac.uk

Abstract: We analyse the speed of gravitational waves in coupled Galileon models with

an equation of state ω

φ

= −1 now and a ghost-free Minkowski limit. We ﬁnd that the

gravitational waves propagate much faster than the speed of light unless these models are

small perturbations of cubic Galileons and the Galileon energy density is sub-dominant to

a dominant cosmological constant. In this case, the binary pulsar bounds on the speed

of gravitational waves can be satisﬁed and the equation of state can be close to -1 when

the coupling to matter and the coeﬃcient of the cubic term of the Galileon Lagrangian

are related. This severely restricts the allowed cosmological behaviour of Galileon models

and we are forced to conclude that Galileons with a stable Minkowski limit cannot account

for the observed acceleration of the expansion of the universe on their own. Moreover any

sub-dominant Galileon component of our universe must be dominated by the cubic term.

For such models with gravitons propagating faster than the speed of light, the gravitons

become potentially unstable and could decay into photon pairs. They could also emit

photons by Cerenkov radiation. We show that the decay rate of such speedy gravitons into

photons and the Cerenkov radiation are in fact negligible. Moreover the time delay between

the gravitational signal and light emitted by explosive astrophysical events could serve as

a conﬁrmation that a modiﬁcation of gravity acts on the largest scales of the Universe.

arXiv:1510.03701v1 [gr-qc] 9 Oct 2015

Contents

1. Introduction 1

2. Galileons 3

2.1 The Models 3

2.2 Cosmological Galileons 4

2.3 The Speed of Gravitons 5

3. The Speed of Gravitons and Screening 6

3.1 Screening Eﬀects 6

3.2 Cubic Galileons 9

4. Graviton Instability 10

4.1 Graviton Decay 10

4.2 Cerenkov Radiation 13

5. Time Delay 14

6. Conclusion 15

1. Introduction

Gravitational waves have now been predicted for nearly a century and despite decades of

experimental eﬀorts, their existence is only conﬁrmed by indirect evidence coming from

the time drift of the period of binary pulsars. New experiments such as the advanced

Laser Interferometry Gravitational-Wave Observatory (a-LIGO) [1], the advanced VIRGO

interferometer [2], the Kamioka Wave Detector (KAGRA) [3], the space based mission

DECIGO [4] or eLISA [5] will be able to test directly the existence of gravitational waves

to improved levels. Gravity waves are also important probes for theories going beyond

Einstein’s General Relativity (GR) [6]. These theories are motivated by the discovery of

the recent acceleration of the expansion of the Universe [7] whose origin is still unknown.

Models such as the quartic Galileons [8] where a coupling between a scalar ﬁeld and gravity

is present predict a background dependent speed of gravitational waves.

In this work we focus on Galileon models [8]. These are a subset of the Horndeski

action [9,10] describing the most general scalar tensor model with second order equations

of motion. The Galileon terms on ﬂat space are protected by a symmetry, the so called

Galileon symmetry, which is softly broken on a curved spacetime background [11]. In these

models the cosmic acceleration is due to the presence of higher order terms in the derivatives

– 1 –

compared to quintessence models where a non-linear potential, typically containing a term

equivalent to a cosmological constant, provides the required amount of vacuum energy.

In vacuum the scalar mediates a ﬁfth force of at least gravitational strength. Locally

close to massive sources the scalar ﬁeld is strongly inﬂuenced by matter and within the

Vainshtein radius GR is restored. On cosmological time scales, the scalar ﬁeld evolves.

This cosmological time drift is screened from matter ﬁelds whilst the average density of the

universe is suﬃciently high but has consequences for the dynamics of gravity locally [12].

In particular the speed of gravitational waves in a massive environment is not protected

from the evolution of the background cosmology by the Vainshtein mechanism [8], meaning

that it can diﬀer from the speed of light in a signiﬁcant manner [13]. We will review this

calculation in Section 3.

If we impose that the equation of state of the scalar ﬁeld should be close to -1 now

and the existence of a stable Minkowski limit of the theory in the absence of matter, both

necessary conditions for a viable cosmology dominated by Galileons at late times and a

meaningful embedding of the model in higher dimensions

1

[14], we ﬁnd that the speed

of gravitational waves would be much greater than one. This would increase the rate of

emission of gravitational waves from binary pulsars. As a result, the speed of gravity in such

a Galileon model is not compatible with the bound that positive deviations of the speed of

gravity from the speed of light cannot be more than one percent [13,15]. We then conclude

that these Galileon models cannot lead to the acceleration of the Universe on their own

and a certain amount of dark energy must be coming from a pure cosmological constant.

This forces the quartic Galileon terms to be subdominant to the cubic terms in order that

the binary pulsar bound can be satisﬁed. When this is the case, the time delay between

gravity and light or even neutrinos can be as large as a few thousand years for events

like the SN1987A supernova explosion. This would essentially decouple any observation of

supernovae gravitational waves from the corresponding photon or neutrino signal coming

from such explosive astrophysical events. On the other hand, a time diﬀerence as low as

the uncertainty on the diﬀerence in emission time signal between neutrinos and gravity,

e.g. up to 10

−3

s for supernovae [16], would allow one to bound deviations of the quartic

Galileon model from its cubic counterpart at the 10

−14

level.

One possible caveat to these results would be if the superluminal gravitational waves

do not reach our detectors because they either decay into two photons or lose all their

energy through Cerenkov radiation [17]. We will show that superluminal gravitational

waves with a speed as large as one percent higher than the speed of light are not excluded

by particle physics processes. A related possibility is at the origin of the stringent bounds

on subluminal gravitational waves which could be Cerenkov radiated by high energy cosmic

rays. As these high energy rays are observed the speed of gravitons cannot be signiﬁcantly

smaller than that of the particle sourcing the cosmic ray [18, 19]. We analyse the decay

and the Cerenkov eﬀect for superluminal gravitational waves and we ﬁnd that their eﬀects

are negligible.

1

We require this embedding in higher dimensional brane models with positive tension branes as a pre-

requisite ﬁrst step towards a possible extension to fundamental theories such as string theory.

– 2 –

Galileons have been widely studied both on purely theoretical grounds, with results

showing that this kind of models arise also in the context of massive gravity [20] and

braneworld models [21]. Constraints on the allowed cosmology of Galileon theories can

be obtained from a wide variety of observations, unveiling a very rich phenomenology

[12,22–36]. Here we consider for the ﬁrst time the constraints that current and near future

observations of gravitational waves can place on these theories.

In section 2, we recall details about Galileon models and show that quartic models

with an equation of state close to -1 lead to very fast gravitons. In section 3, we consider

the inﬂuence of the Vainshtein mechanism on the propagation of gravity and we check

that the screening mechanism does not protect the speed of gravity from large deviations

compared to the speed of light. We also introduce models of subdominant Galileons whose

gravitational waves have a speed which satisﬁes the binary pulsar bounds. In section 4

we consider the decay rate of gravitons into two photons, and the Cerenkov radiation.

We show that these processes are negligible for allowed diﬀerences between the speed of

gravitons and photons. Finally In Section 5 we discuss the time delay in the arrival time

of gravitons and photons from explosive astrophysical sources. We conclude in section 6.

2. Galileons

2.1 The Models

In this paper, we are interested in models of modiﬁed gravity with a Galilean symmetry.

They are potential candidates to explain the late time acceleration of the expansion of

the Universe. They also lead to a modiﬁcation of gravity on large scales. Such Galileons

are scalar ﬁeld theories which have equations of motion that are at most second order in

the derivatives. Moreover they are interesting dark energy candidates where an explicit

cosmological constant is not compulsory. Their Lagrangian reads in the Jordan frame

deﬁned by the metric g

µν

L =

1 + 2

c

0

φ

m

Pl

R

16πG

N

−

c

2

2

(∂φ)

2

−

c

3

Λ

3

φ(∂φ)

2

−

c

4

Λ

6

L

4

−

c

5

Λ

9

L

5

. (2.1)

The common scale

Λ

3

= H

2

0

m

Pl

(2.2)

is chosen to be of cosmological interest as we focus on cosmological Galileon models which

can lead to dark energy in the late time Universe. We also require that c

2

> 0 to avoid

the presence of ghosts in a Minkowski background. This theory could be rewritten in the

Einstein frame where the conformal coupling of the scalar ﬁeld to matter would be given

by

A(φ) = 1 +

c

0

φ

m

Pl

(2.3)

– 3 –

where c

0

is a constant. The complete Galileon Lagrangian depends on operators with

higher order terms in the derivatives which are given by

L

4

=(∂φ)

2

2(φ)

2

− 2D

µ

D

ν

φD

ν

D

µ

φ − R

(∂φ)

2

2

L

5

=(∂φ)

2

(φ)

3

− 3(φ)D

µ

D

ν

φD

ν

D

µ

φ + 2D

µ

D

ν

φD

ν

D

ρ

φD

ρ

D

µ

φ (2.4)

−6D

µ

φD

µ

D

ν

φD

ρ

φG

νρ

] .

These terms play an important role cosmologically. In the following and in the study of

the cosmological evolution, we focus on the coupling of the Galileon to Cold Dark Matter

(CDM) as the coupling to baryons is more severely constrained by the time variation

of Newton’s constant in the solar system, at the one percent level, and does not play a

signiﬁcant role for the background cosmology [38].

This model is a subset of terms in the Horndeski action describing the most general

scalar tensor theory with second order equations of motion

L = K(φ, X) − G

3

(X, φ)φ + G

4

(X, φ)R + G

4,X

(φ)

2

− (D

µ

D

ν

φ)

2

+

G

5

(X, φ)G

µν

D

µ

D

ν

φ −

1

6

G

5,X

h

(φ)

3

− 3φ(D

µ

D

ν

φ)

2

+ 2(D

µ

D

α

φ)(D

α

D

β

φ)(D

β

D

µ

φ)

i

with the particular functions

K = c

2

X, G

3

(X) = −2

c

3

Λ

3

X, G

4

(X, φ) =

A

2

(φ)

16πG

N

+ 2

c

4

Λ

6

X

2

, G

5

(X) = −6

c

5

Λ

9

X

2

(2.5)

where X = −

(∂φ)

2

2

is the kinetic energy of the ﬁeld. In the following we shall focus on

quartic Galileons with c

5

= 0 as this leads to both interesting cosmology and a non-trivial

speed for gravitational waves.

2.2 Cosmological Galileons

We focus on the behaviour of Galileon models on cosmological scales in a Friedmann-

Robertson-Walker background

ds

2

= a

2

(−dη

2

+ dx

2

) (2.6)

where η is conformal time and we have set the speed of light c = 1. The equations of

motion of the Galileons can be simpliﬁed using the variable x = φ

0

/m

Pl

where a prime

denotes

0

= d/d ln a = −d/d ln(1 + z), a is the scale factor and z the redshift. We deﬁne

the scaled ﬁeld ¯y =

φ

m

Pl

x

0

, the rescaled variables ¯x = x/x

0

and

¯

H = H/H

0

where H is the

Hubble rate, and the rescaled couplings [36] ¯c

i

= c

i

x

i

0

, i = 2 . . . 5, ¯c

0

= c

0

x

0

, ¯c

G

= c

G

x

2

0

where x

0

is the value of x now. Notice that x

0

is not determined by the dynamics and is

a free parameter of the model. The cosmological evolution of the Galileon satisﬁes [37]

¯x

0

= −¯x +

αλ − σγ

σβ − αω

¯y

0

= ¯x

¯

H

0

= −

λ

σ

+

ω

σ

σγ − αλ

σβ − αω

– 4 –