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 –