Strong Constraints on Cosmological Gravity from GW170817 and GRB 170817A
T. Baker,
1
E. Bellini,
1
P. G. Ferreira,
1
M. Lagos,
2
J. Noller,
3
and I. Sawicki
4
1
University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom
2
Kavli Institue for Cosmological Physics, The Univer sity of Chicago, Chicago, Illinois 60637, USA
3
Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland
4
CEICO, Fyzikální ústav Akademie věd ČR, Na Slovance 2, 182 21 Praha 8, Czech Republic
(Received 16 October 2017; published 18 December 2017)
The detection of an electromagnetic counterpart (GRB 170817A) to the gravitational-wave signal
(GW170817) from the merger of two neutron stars opens a completely new arena for testing theories of
gravity. We show that this measurement allows us to place stringent constraints on general scalar-tensor and
vector-tensor theories, while allowing us to place an independent bound on the graviton mass in bimetric
theories of gravity. These constraints severely reduce the viable range of cosmological models that have
been proposed as alternatives to general relativistic cosmology.
DOI: 10.1103/PhysRevLett.119.251301
Introduction.—The advanced Laser Interferometer
Gravitational Observatory (aLIGO) and the VIRGO inter-
ferometer, have recently announced the detection of gravi-
tational waves (GW170817) from the merger of a neutron
star (NS) binary located near NGC 4993 [1].Aγ-ray burst
(GRB 170817A), occurring within 1.7 sec, and in the
vicinity, of GW170817, was observed by the Fermi
Gamma-ray Burst Monitor, and the Anti-Coincidence
Shield for the Spectrometer for the International
Gamma-Ray Astrophysics Laboratory [2,3]. There is
strong evidence that this event is an electromagnetic
counterpart to the NS-NS merger [4,5]. Comparing the
travel time of light and gravitational waves (GW), we can
place stringent constraints on cosmological gravity, and
cosmology more generally [6–14].
We will assume that constraints on Lorentz violation in
the electromagnetic sector are sufficiently strong that the
speed of light is c ¼ 1. In vacuum, Lorentz symmetry
implies that all massless waves propagate at the speed of
light. However, when a medium is present, Lorentz
symmetry is spontaneously violated and propagation
speeds can differ. Alternative theories of gravity, directly
coupling extra degrees of freedom (d.o.f.) to curvature,
provide such a medium when the new d.o.f. takes a
configuration that defines a preferred direction (such as
the time direction in cosmology). The action for linearized
gravitational waves in such a medium takes the form
S
h
¼
1
2
Z
d
3
xdtM
2
½
_
h
2
A
− c
2
T
ð∇h
A
Þ
2
: ð1Þ
We have decomposed the metric as g
αβ
¼ η
αβ
þ h
αβ
—with
η
αβ
the Minkowski metric—by choosing locally inertial
coordinates with time chosen to be the direction defined by
the medium. We have expanded h
αβ
in polarization states
ε
A
, with amplitudes h
A
, where A ¼ ×; þ. M
is the
effective Planck mass, which in media provided by alter-
native gravity theories can differ from the standard M
P
. c
T
is the speed of gravitational waves; we will find it
convenient to parametrize this as [15],
c
2
T
¼ 1 þ α
T
: ð2Þ
In principle, α
T
could adopt either positive or negative
values. However, negative values (c
T
<c) are constrained
to α
T
> −10
−15
by a lack of observed gravi-Čerenkov
radiation from cosmic rays [16]. Up to now, the only upper
bound on the propagation of GWs comes from measuring
the travel time between the two detectors of aLIGO, and is
α
T
< 0.42 [17,18].
In the regime we are considering (a gravitational wave
propagating in effectively empty space, other than the
medium provided by the new d.o.f.) the linearized action
(1) is sufficient. It is conceivable (but unlikely) that there
may be some exotic behavior close to the GW sources, in
regions of strong gravity (for example, as occurs with the
screening of scalar forces) that leads to nonlinear correc-
tions. Such effects could alter GW production, but will have
no bearing on the gravitational-wave propagation during
the bulk of its travel time. Also, though Eq. (1) is valid for a
wide range of gravitational theories, it does not encompass
bimetric theories.
Constraint on tensor speed excess.—Let us illustrate
how aLIGO and the Fermi monitor have obtained the
constraint in Ref. [1]. We consider the geometric optics
limit of Eq. (1) so that c
T
is indeed the speed of
gravitational waves. Let t
s
be the time of emission for
both the gravitational waves and photons; there can be a
delay of up to 1000 sec which will not change our
conclusions. Let t
T
be the merger time identified in the
gravitational-wave train, and t
c
be the measured peak
brightness time in the optical signal. The transit distance
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of the GW and photon signals are c
T
ðt
T
− t
s
Þ¼d
s
and
ðt
c
− t
s
Þ¼d
s
, where d
s
≃ 40 Mpc is the distance to the
source. Defining Δt ≡ t
c
− t
T
,wehaveΔt=d
s
¼ 1 − 1=c
T
.
Taylor expanding this gives α
T
≃ 2Δt=d
s
; an arrival delay
of Δt ≃ 1.7 sec implies that
jα
T
j ≲ 1 × 10
−15
: ð3Þ
Comparing this to current cosmological constraints
(where σ
α
T
∼ 1 [19]) or forecast cosmological constraints
(where σ
α
T
∼ 0.1 [20]), this constraint is remarkable. For all
intents and purposes, we will hereafter consider α
T
≃ 0 and
attempt to understand its consequences for cosmology.
Implications for scalar-tensor theories.—We begin by
considering scalar-tensor theories. The Horndeski action is
the most general scalar-tensor theory with second-order
equations of motion [21,22], and is given by S ¼
R
d
4
x
ffiffiffiffiffiffi
−g
p
f
P
5
i¼2
L
i
½ϕ;g
μν
þL
M
½g
μν
; …g, where L
M
is
the minimally coupled matter action. The scalar field
Lagrangian is built of four terms: two minimally coupled
to gravity, L
2
¼ K and L
3
¼ −G
3
□ϕ, and two terms
explicitly involving the Ricci curvature R and the
Einstein tensor G
μν
:
L
4
¼ G
4
R þ G
4;X
fð□ϕÞ
2
− ∇
μ
∇
ν
ϕ∇
μ
∇
ν
ϕg;
L
5
¼ G
5
G
μν
∇
μ
∇
ν
ϕ −
1
6
G
5;X
fð∇ϕÞ
3
− 3∇
μ
∇
ν
ϕ∇
μ
∇
ν
ϕ□ϕ
þ 2∇
ν
∇
μ
ϕ∇
α
∇
ν
ϕ∇
μ
∇
α
ϕg: ð4Þ
Here K and G
i
are functions only of ϕ and
X ≡ −∇
ν
ϕ∇
ν
ϕ=2, and subscript commas denote deriva-
tives. On a cosmological background, Horndeski models
give [23,24]
M
2
α
T
≡ 2 X½2G
4;X
− 2G
5;ϕ
− ð
ϕ −
_
ϕHÞG
5;X
; ð5Þ
where M
2
≡ 2 ðG
4
− 2XG
4;X
þ XG
5;ϕ
−
_
ϕHXG
5;X
Þ.
One way of satisfying α
T
∼ 0 is through a delicate
cancellation between G
4;X
, G
5;ϕ
and G
5;X
.IfG
5;X
¼ 0,
then G
5;ϕ
can be integrated by parts in the action to a form
equivalent to G
4
[25,26]. This cancellation is then just the
statement about G
4;X
¼ 0. A nontrivial cancellation would
not only have to be time dependent, but also sensitive to the
matter content of the Universe due to the dependence on H
and
ϕ. Thus, even a small change in, e.g., the dark matter
density, or deviations from isotropy and homogeneity,
would severely violate it. Furthermore, any such a can-
cellation would be accidental, with no symmetry to protect
it. Some shift symmetric Horndeski actions (i.e., not
dependent on ϕ) are, to some degree, stable to radiative
corrections. In flat spacetime, for K; G
i
linear in X
(Galileons [27]), there exists an exact quantum nonrenorm-
alization theorem [28–30]—there are no corrections to
these operators. The corrections remain under control when
the Galilean symmetry is weakly broken [31], as it must be
in curved spacetime. In this case, the Horndeski inter-
actions are suppressed by a scale Λ
3
, whereas quantum
corrections enter suppressed by the parametrically larger
scale Λ
2
≫ Λ
3
, which satisfies Λ
4
2
¼ M
Pl
Λ
3
3
[31]. A typical
value is Λ
3
∼ 10
−13
eV, leading to Λ
3
=Λ
2
∼ 10
−10
. With
relatively mild assumptions on the G
i
functions, this can be
shown to lead to order ðΛ
3
=Λ
2
Þ
4
∼ 10
−40
corrections on the
G
i
[31,32]. Thus, however difficult a classical cancellation
in α
T
, Eq. (5), at the required level of 10
−15
would be,
arranging for it to remain under radiative control is feasible.
Nonetheless, a more natural interpretation of the con-
straint (3) is that each of the terms (G
4;X
, G
5;ϕ
, G
5;X
)
contributing to α
T
is zero, i.e., that L
5
∝ G
μν
∇
μ
∇
ν
ϕ,
vanishing identically as a result of the Bianchi identity,
while L
4
¼ fðϕÞR, i.e., the coupling to gravity can at most
be of the Jordan-Brans-Dicke (JBD) type. Setting G
5;X
¼ 0
means that we avoid dependence on a fine-tuned back-
ground through H as well as on the (quantum) corrections it
receives. Radiative corrections to α
T
, even in the presence
of a G
3
term, are still 10
−40
.
Such a restriction reduces the viable Horndeski models to
two classes: in class (i), the scalar does not evolve signifi-
cantly on cosmological time scales. This is the generalized
JBD class, including models such as fðRÞ gravity. Such
models require chameleonic screening to evade Solar
System tests of gravity, and, therefore, cannot have a
background evolution significantly different from that of
concordance cosmology; they do not self-accelerate cos-
mological expansion [33,34]. The sound speed of the scalar
fluctuations is equal to that of light. On the other hand, the
strength of the fifth force, f
;ϕ
, is allowed to be similar to
gravity.
In class (ii), the scalar evolves quickly, X ∼ H
2
M
2
, and
noncanonical kinetic terms in K and G
3
play a significant
role: they can give rise to acceleration without a cosmo-
logical constant, significantly changing the equation of
state and the sound speed (see also Ref. [35]). If they are to
be the mechanism for acceleration, constraints on the
evolution of the Planck mass [36,37] restrict the strength
of coupling to gravity f
;ϕ
to be small, since the scalar runs
during the entire history of the universe in these models. We
reiterate that perturbative control of quantum corrections in
the fast-moving models depends on shift symmetry, which
would disallow any dependence on ϕ in the action,
specifically the conformal coupling fðϕÞ.
Horndeski theory is not the most general scalar-tensor
theory propagating one single extra d.o.f. New terms can be
added to construct the “beyond” Horndeski Lagrangian
[38–40] at the price of third derivatives in equations of
motion and new constraints to remove any extra d.o.f.
naively implied by them. This extension is described by
two new free functions,
~
G
4
ðϕ;XÞ and
~
G
5
ðϕ;XÞ correcting
L
4
and L
5
(see Ref. [39] for the complete expressions) and
modifying Eq. (5) to
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α
T
M
2
¼ 4XðG
4;X
−
~
G
4;X
− G
5;ϕ
Þ − 2
ϕXG
5;X
þ 2
_
ϕHXðG
5;X
−
~
G
5;X
Þ; ð6Þ
where M
2
¼ 2G
4
− 4XðG
4;X
−
~
G
4;X
Þþ2XG
5;ϕ
− 2
_
ϕHX×
ðG
5;X
−
~
G
5;X
Þ.
It is clear from Eq. (6) that one option is to set all the terms
contributing to α
T
to zero, as in the Horndeski case. An
intriguing alternative is to choose G
;5X
¼
~
G
5;X
¼ 0 and
~
G
4;X
¼ G
4X
− G
5;ϕ
, which indeed leads to α
T
¼ 0 but also
allows for M
≠ M
P
and α
H
≠ 0, where α
H
is the additional
beyond-Horndeski parameter introduced in Ref. [39].
Although it is beyond the scope of this work to discuss
the properties of this particular model, we should emphasize
that this is the only algebraic choice for the G
i
functions that
ensures α
T
¼ 0, regardless of the underlying cosmology.
In our discussion of scalar-tensor theories, we should
briefly mention degenerate higher-order scalar-tensor
(DHOST) theories [41,42]. DHOST theories are con-
structed to be a further generalization of Horndeski, but
have to include new constraints to avoid Ostrogradsky
instabilities. The result is a long list of classes of theories
(≃30) having disjoint parameter spaces, but which on a
cosmological background reduce to just two types [43].
One is unstable and thus irrelevant here. The other can be
transformed to beyond Horndeski with a conformal trans-
formation of the form
~
g
μν
¼ CðXÞg
μν
. Conformal trans-
formations leave null geodesics null. Thus, if a DHOST
model describes gravity in cosmology, then the require-
ments for α
T
¼ 0 listed above apply to the beyond-
Horndeski counterpart of the DHOST theory.
To conclude, if we assume that it is not possible to
enforce precise cancellations for the reasons discussed
above, the constraint on α
T
excludes such models as the
quartic and quintic Galileon or a generic beyond
Horndeski, leaving only models which are conformally
coupled to gravity. On the other hand, models where
gravity remains minimally coupled remain unconstrained:
fast-moving models such as kinetic gravity braiding [44]
can give rise to self-acceleration and admit an interpretation
as the dynamics of a superfluid [45], rather than as a
modification of gravity. Finally, quintessence models
remain unconstrained.
Implications for vector-tensor theories.—We now turn to
vector tensor theories of gravity, i.e., theories where the
additional gravitational d.o.f. is given by a 4-vector, A
μ
. First,
we consider generalized Einstein-Aether gravity, where A
μ
is
timelike and the action is S ¼
R
d
4
x
ffiffiffiffiffiffi
−g
p
½ðM
2
P
=2ÞRþ
F ðKÞþλðA
μ
A
μ
þ 1Þ,whereλ is a Lagrange multiplier,
K ¼ c
1
∇
μ
A
ν
∇
μ
A
ν
þ c
2
ð∇
μ
A
μ
Þ
2
þ c
3
∇
μ
A
ν
∇
ν
A
μ
(with c
i
constants) and F ðxÞ is an arbitrary function (we have not
included the “c
4
” term as it does not affect tensor modes)
[46,47].Inthismodelα
T
¼ −ðc
1
þ c
3
ÞF
;K
=½1 þðc
1
þ
c
3
ÞF
;K
, so the constraint on α
T
implies c
1
¼ −c
3
.On
perturbed Minkowski space, this reduces the theory to the
Maxwell action (with a timelike constraint) supplemented
by c
2
ð∇
μ
A
μ
Þ
2
. On a cosmological background, we have
3M
2
P
H
2
¼ðρ − F =2Þð1 − 3c
2
F
;K
Þ, whereas the effective
Planck mass in Eq. (1), which is generally given by
M
2
¼ M
2
P
½1 − ðc
1
þ c
3
ÞF
;K
, will reduce to the GR value.
A second class of vector-tensor theories of interest are ge-
neralized Proca theories [48,49], whose 4D action is, much
like Horndeski theory, given by S ¼
R
d
4
x
ffiffiffiffiffiffi
−g
p
ðL þ L
M
Þ;
L ¼
P
6
i¼2
L
i
, where the vector field Lagrangian is built so
that precisely one extra (longitudinal) scalar mode prop-
agates in addition to the two usual Maxwell-like transverse
polarizations. The individual L
i
are given by two minimally
coupled terms L
2
¼ G
2
ðX; F; YÞ and L
3
¼ G
3
ðXÞ∇
μ
A
μ
,
and two nontrivial terms given by
L
4
¼ G
4
ðXÞR þ G
4;X
ðXÞ½ð∇
μ
A
μ
Þ
2
þ c
2
∇
ρ
A
σ
∇
ρ
A
σ
− ð1 þ c
2
Þ∇
ρ
A
σ
∇
σ
A
ρ
;
L
5
¼ G
5
ðXÞG
μν
∇
μ
A
ν
−
1
6
G
5;X
ðXÞ½ð∇
μ
A
μ
Þ
3
− 3d
2
∇
μ
A
μ
∇
ρ
A
σ
∇
ρ
A
σ
− 3ð1 − d
2
Þ∇
μ
A
μ
∇
ρ
A
σ
∇
σ
A
ρ
þð2 − 3d
2
Þ∇
ρ
A
σ
∇
γ
A
ρ
∇
σ
A
γ
þ 3d
2
∇
ρ
A
σ
∇
γ
A
ρ
∇
γ
A
σ
:
ð7Þ
As usual, F
μν
¼ ∇
μ
A
ν
− ∇
ν
A
μ
, c
2
and d
2
are constants,
G
3;4;5
are arbitrary functions of X ¼ −
1
2
A
μ
A
μ
and G
2
is a
function of X; F ¼ −
1
4
F
μν
F
μν
;Y ¼ A
μ
A
ν
F
α
μ
F
να
. There is
also an additional term L
6
with a free function G
6
ðXÞ and
L
5
can be extended by a term controlled by a free function
g
5
ðXÞ [50,51], but here we will not give these terms
explicitly, since they do not affect (linearized) tensors.
On a cosmological background A
μ
¼ðA;
0Þ and α
T
is
given by
α
T
¼ A
2
½2G
4;X
− ðHA −
_
AÞG
5;X
=q
T
; ð8Þ
where q
T
¼ 2G
4
− 2A
2
G
4;X
þ HA
3
G
5;X
. Analogously to
the scalar-tensor case considered above, if α
T
¼ 0 we either
then have to carefully tune the functional dependence of G
4
and G
5
to satisfy this criterion (all the considerations about
radiative stability, time dependence, and background sym-
metry we discussed for Horndeski theories hold), or
consider a theory with minimal higher-order interactions
by requiring G
4;X
¼ G
5;X
¼ 0 leading to L
4
∝ R and
L
5
∝ G
μν
∇
μ
A
ν
. In the latter case, ghost freedom for tensor
perturbations then enforces G
4
> 0, while ghost and
gradient instabilities for vector modes are automatically
satisfied.
In generalized Proca theories the equation of motion for
A
μ
separates the evolution into two branches, one with a
nondynamical scalar d.o.f. and a second one with full
dynamics for all three d.o.f., which we will focus on here.
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Requiring G
4;X
¼ G
5;X
¼ 0 (and hence α
T
¼ 0) as above,
the modified Friedman equation then becomes
3H
2
¼ðρ − G
2
Þ=ð2G
4
Þ, and thus 2G
4
describes a rescaled
constant Planck mass. We note that on the de Sitter fixed
point of this model [52], in the limit ρ ¼ 0, consistency will
enforce G
2
< 0, due to the ghost-freedom condition for
tensor perturbations G
4
> 0.
One can go a step beyond generalized Proca theories and
consider the beyond generalized Proca model of Ref. [53].
Here four new free functions enter at the level of the action,
denoted f
4
, f
5
,
~
f
5
;
~
f
6
. Of the new functions only f
4
and f
5
affect the background evolution and that of linear tensor
perturbations, whereas the remaining functions only affect
linear vector and scalar perturbations. The α
T
¼ 0 con-
straint now implies G
5;X
ðHA −
_
AÞ − 2G
4;X
¼ 2f
4
A
2
þ
6f
5
HA
3
, which depends on the new functions f
4
, f
5
.If
we choose to set all participating functions to zero to ensure
α
T
¼ 0, this means both the background and tensor
perturbations will behave exactly as in the generalized
Proca case considered above.
Implications for bigravity theories.—We now consider
models with two coupled metrics. The only nonlinear Lorentz
invariant ghost-free possible interactions are given by the
deRham-Gabadadze-Tolley (dRGT) potential [54–56].
The action is given by S¼ðM
2
g
=2Þ
R
d
4
x
ffiffiffiffiffiffi
−g
p
R
g
þðM
2
f
=2Þ×
R
d
4
x
ffiffiffiffiffiffi
−f
p
R
f
−m
2
M
2
g
R
d
4
x
ffiffiffiffiffiffi
−g
p
P
4
n¼0
β
n
e
n
ð
ffiffiffiffiffiffiffiffiffiffi
g
−1
f
p
Þ, where
we have two dynamical metrics g
μν
and f
μν
with their
associated Ricci scalars R
g
and R
f
, and constant mass scales
M
g
and M
f
, respectively. Here, β
n
are free dimensionless
coefficients, while m is an arbitrary constant mass scale. The
dRGT potential is defined in terms of the functions e
n
ðXÞ,
which correspond to the elementary symmetric polynomials
of the matrix X ¼
ffiffiffiffiffiffiffiffiffiffi
g
−1
f
p
. For simplicity, let us assume that
matter fields are coupled minimally to the metric g
μν
, and all
the parameter βs are of order 1.
The bigravity action generally propagates one massive
and one massless graviton; and the field g
μν
will be a
combination of both modes. The massless mode has a
dispersion relation given by E
2
0
¼ k
2
, while the massive
mode has E
2
m
¼ k
2
þ m
2
(with omitted factors of β sof
order 1) on Minkowski space (and a de Sitter phase, i.e.,
late times). Let us first discuss the restricted case of massive
gravity, when M
f
=M
g
→ ∞, and only the massive graviton
propagates (while the metric f
μν
is frozen). In this case, the
dispersion relation of gravitational waves is E
2
¼ k
2
þ m
2
.
As a result, the speed of GW will be frequency dependent
leading to a phase difference in the waveforms. Bounds
from GW150914 led to m ≤ 1.2 × 10
−22
eV [57]. With an
EM counterpart to the GWs, the bound of 1.7 sec on the
time delay also leads to m ≲ 10
−22
eV (note that we have
considered a frequency region of interest of 10–100 Hz and
ignored the frequency dependency of the velocity, which is
small) which is uncompetitive with Solar System fifth-force
constraints of order m ≲ 10
−30
eV (see Ref. [58] for quartic
Galileons). In the case of massive bigravity, assuming
similar amplitudes for both modes and M
g
¼ M
f
¼ M
P
,
one has a fast oscillation with a slowly modulated ampli-
tude. The frequency of the modulated wave is proportional
to m and hence negligible compared to the time scale of the
NS merger. The dispersion relation of the fast mode is
effectively that of a massive graviton E
2
¼ k
2
þ m
2
(omit-
ting again factors or order 1), and thus one obtains the same
constraint as for massive gravity. Note that we assumed the
mass to be smaller than LIGO’s relevant frequencies, so this
result holds for bigravity models that can play a cosmo-
logical role but do not describe dark matter [59–61].
Unlike for scalar-tensor and vector-tensor theories, in
massive gravity, local constraints from GW propagation
have no bearing on cosmology. In particular, the existence
of scalar and tensor instabilities [62,63], in particular,
branches of the background cosmology will be uncon-
strained by the measurements discussed in this Letter. Note
that we have constrained these bimetric models only using
information on the propagation speed of GW, although
further constraints can be found from the entire GW
waveform. Further discussion on the waveforms can be
found in Refs. [64–66]. Constraints in the case where both
metrics are coupled to matter are discussed in Ref. [67].
Caveats.—We now address possible caveats. For a start,
the source lies at a very low redshift (z
s
¼ 0.01); thus our
constraint is on the speed of GWs today. It would of course
be a great coincidence if α
T
were to vanish now with such
precision, but not at other times. However, this is, in
principle, a possibility.
Another uncertainty is the extent to which the effective
metric relevant for the propagation of perturbations with
wavelengths similar to the size of the universe, as studied in
cosmology, is the same one that is experienced by the GW
with the wavelength of 3000 km (to which aLIGO/VIRGO
are sensitive). For cosmological modes with wavelengths of
10–100 Mpc, taking the background—the medium in
which fluctuations propagate—to be isotropic and homo-
geneous is a good approximation. Wavelengths probed by
aLIGO/VIRGO are much shorter than the typical size of
structures in the universe, so the GW should be sensitive to
the inhomogeneities. Indeed, one can argue that, apart from
the initial exit from the source galaxy and the final entrance
into the Milky Way, the GW was mostly propagating
through space with density of matter significantly below the
current cosmic average, when averaged over scales of the
order of the GW’s wavelength.
Some alternative theories of gravity depend crucially on
a highly nonlinear response to the matter density by the
extra d.o.f. (the need for screening on Solar System scales).
This may well mean that the GW speed predicted for an
averaged cosmology, and that for the matter density along
the particular trajectory this GW took could be different.
Thus, there would not be a simple connection between the
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22 DECEMBER 2017
251301-4
time delay observed and the properties of gravity on
cosmological scales. We would argue that, if such an effect
is relevant, then the GW would be propagating with a speed
which the cosmological modes will experience when the
universe has emptied out to the same extent as the averaged
density along the trajectory of the GW. If α
T
is evolving, we
may well have measured its asymptotic future value.
Conclusions.—The detection of GW170817, together
with its EM counterpart (GRB 170817A), bounds the speed
of gravitational waves to deviate from c by no more that
one part in 10
15
. This single fact has profound repercus-
sions for extended gravity models motivated by cosmic
acceleration. We stress that models of fifth forces acting
only on astrophysical scales remain viable. One way to see
this is to note the presence of G
4
in, for example, the
denominator of Eq. (5); one can have G
4
≃ M
2
P
≫
G
4;ϕ
;G
4;X
. Therefore, for theories with heavy d.o.f. acting
on subHubble scales, the denominator of Eq. (5) domi-
nates, and α
T
≃ 0. Likewise, a d.o.f. that does not evolve on
cosmological time scales (at the background level—such
that X ≃ 0), also satisfies α
T
≃ 0 .
We summarize here the key consequences explained in
this Letter: (i) Assuming no finely-tuned cancellations
between Lagrangian functions occur, the only viable
Horndeski scalar-tensor theories have a coupling to gravity
of the form ∝ fðϕÞR (plus nongravitational terms), i.e.,
conformally coupled theories. This eliminates, for example,
the quartic and quintic Galileons (and hence all Galileon
cosmologies, given ISW constraints on the cubic Galileon
[68]). (ii) In this remaining class, the only surviving self-
accelerating theories must be shift symmetric or very nearly
so, and thus can have at most a small conformal coupling to
gravity. Models in this category include kinetic gravity
braiding and k essence. (iii) The beyond Horndeski exten-
sion of scalar-tensor theories introduces only one further
surviving model, which is also conformally coupled to
gravity. (iv) For vector fields, assuming no finely-tuned
cancellations, (generalized) Einstein-Aether models are now
subject to the stringent relation c
1
¼ −c
3
. (v) Beyond and
standard generalized Proca models, assuming no finely-
tuned cancellations, behave identically at background level,
with vastly simplified higher order gravitational inter-
actions, such as a coupling to R, where the proportionality
constant acts as a rescaled Planck mass in the Friedmann
equations. (vi) In the bimetric theories the mass of the
graviton is constrained to be m ≲ 10
−22
eV (assuming equal
Planck masses for bigravity), which is weaker than current
Solar System bounds but entirely independent of them. This
constraint has no bearing on cosmology.
For the first time, powerful and general statements can be
made about the structure of (non-)viable gravitational
actions, and some current popular models are ruled out
(also see Refs. [32,69–71]). These decisive statements will
undoubtedly shape the direction of future research into
extensions of general relativity.
We acknowledge conversations with Rob Fender,
Constantinos Skordis, Filippo Vernizzi, and Miguel
Zumalacárregui, and the discussions made possible by
the DARKMOD workshop at IPhT Saclay. T. B. is sup-
ported by All Souls College, University of Oxford. E. B. is
supported by the ERC and BIPAC. P. G. F. acknowledges
support from STFC, BIPAC, the Higgs Centre at the
University of Edinburgh, and ERC. M. L. is supported at
the University of Chicago by the Kavli Institute for
Cosmological Physics through an endowment from the
Kavli Foundation and its founder Fred Kavli. J. N.
acknowledges support from Dr. Max Rössler, the Walter
Haefner Foundation, and the ETH Zurich Foundation. I. S.
is supported by ESIF and MEYS (Project No. CoGraDS—
CZ.02.1.01/0.0/0.0/15_003/0000437).
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