Instabilities of Spherical Solutions with Multiple Galileons and SO(N) Symmetry

Abstract

The 4-dimensional effective theory arising from an induced gravity action for
a co-dimension greater than one brane consists of multiple galileon fields
pi^I, I=1...N, invariant under separate Galilean transformations for each
scalar, and under an internal SO(N) symmetry. We study the viability of such
models by examining spherically symmetric solutions. We find that for general,
non-derivative couplings to matter invariant under the internal symmetry, such
solutions exist and exhibit a Vainshtein screening effect. By studying
perturbations about such solutions, we find both an inevitable gradient
instability and fluctuations propagating at superluminal speeds. These findings
suggest that more general, derivative couplings to matter are required for the
viability of SO(N) galileon theories.

Full text

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Instabilities of Spherical Solutions with Multiple Galileons and SO(N) Symmetry
Melinda Andrews,∗Kurt Hinterbichler,†Justin Khoury,‡and Mark Trodden§
Center for Particle Cosmology, Department of Physics and Astronomy,
University of Pennsylvania, 209 S 33rd Street, Philadelphia PA 19104-6396 USA
(Dated: March 1, 2011)
The 4-dimensional effective theory arising from an induced gravity action for a co-dimension greater
than one brane consists of multiple galileon fields πI, I = 1,...,N, invariant under separate Galilean
transformations for each scalar, and under an internal SO(N) symmetry. We study the viability of
such models by examining spherically symmetric solutions. We find that for general, non-derivative
couplings to matter invariant under the internal symmetry, such solutions exist and exhibit a Vain-
shtein screening effect. By studying perturbations about such solutions, we find both an inevitable
gradient instability and fluctuations propagating at superluminal speeds. These findings suggest
that more general, derivative couplings to matter are required for the viability of SO(N) galileon
theories.
I.INTRODUCTION
There has been much recent interest in theories of
gravity arising from scenarios with extra spatial dimen-
sions. Many examples of these are based on the Dvali-
Gabadadze-Porrati (DGP) model [1, 2] — a 4+1 dimen-
sional theory with action consisting simply of separate
Einstein-Hilbert terms in the bulk and on a codimension-
1 brane, to which standard model particles are also con-
fined.The model results in a 4D gravitational force
law at sufficiently small scales, which transitions to a
5D gravitational force law at a crossover length scale
rc ∼ M2
tional couplings M5 and MPl respectively. To yield in-
teresting cosmological dynamics, this crossover scale is
usually chosen to be of order the horizon size.
Much of the phenomenology of the DGP model is cap-
tured by its decoupling limit MPl, M5 → ∞ with the
strong-coupling scale Λ5∼ M2
this limit, the difference between DGP gravity and gen-
eral relativity is encoded in the behavior of a scalar degree
of freedom, π. The dynamics of this scalar are invariant
under internal galilean transformations π → π+c+bµxµ,
with c a constant and bµa constant vector. This sym-
metry proves to be extremely restrictive, with a leading
order self-interaction term which is a higher-derivative
coupling cubic in π, and yet which yields second or-
der equations of motion. Higher order couplings with
these properties were derived independently of the DGP
model [5–8] and dubbed “galileons.” See [9–17] for cos-
mological studies of galileon theories.
It is natural to explore induced gravity models in co-
dimension greater than one [18–26], and recently multi-
galileon actions arising in the relevant 4-dimensional de-
coupling limit have been derived [27–30]. The theories
Pl/M3
5, determined by the 5D and 4D gravita-
5/MPlkept fixed [3, 4]. In
∗mgildner@sas.upenn.edu
†kurthi@physics.upenn.edu
‡jkhoury@sas.upenn.edu
§trodden@physics.upenn.edu
studied in [30] are invariant under individual galilean
transformations of the π fields, and also under an inter-
nal SO(N) symmetry rotating the fields into one another,
thus forbidding the existence of terms containing an odd
number of π fields, in contrast to the co-dimension one
DGP case.
In this paper we explore the nature of spherically sym-
metric solutions in theories with an SO(N) internal sym-
metry among the galileon fields, and couplings to mat-
ter that respect this symmetry. Spherical solutions for
a more general bi-galileon action were discussed in [31],
for the specific case of a linear coupling ∼ πT to matter,
where T is the trace of the matter energy momentum ten-
sor. This form of coupling arises from decoupling limits of
DGP-like theories, because π arises through a conformal
mixing with the graviton. However, while this coupling
is therefore the natural form to consider in the case of a
single galileon field, it breaks the new internal symmetry
satisfied by multiple galileons (and breaks the galilean
symmetry if the matter is dynamical). We instead study
general non-derivative couplings to matter fields which
respect the SO(N) internal symmetry.
At the background level, our solution can always be
rotated to lie along a single field direction, say π1, while
the other field variables remain trivial, thus exhibiting
spontaneous symmetry breaking. The solution exhibits
Vainshtein screening [32, 33], characteristic of galileon
theories: we find π1∼ r sufficiently close to the source,
whereas π1∼ 1/r far away, with the crossover scale de-
termined by a combination of the galileon self-interaction
scale and the coupling to the source.
we turn to the stability of spherically symmetric solu-
tions under small perturbations, we find that, sufficiently
close to the source, perturbations in π1suffer from gra-
dient instabilities along the angular directions. Morever,
they propagate superluminally both along the radial and
angular directions (in the regime that angular perturba-
tions are stable). Perturbations in the remaining N − 1
galileon fields are stable but propagate superluminally in
the radial direction.
The gradient instability and superluminal propaga-
tion found here for the π1 field are multi-field general-
However, when
arXiv:1008.4128v2 [hep-th] 26 Feb 2011

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izations of single galileon instabilities [5]. Our findings
thus present significant hurdles for SO(N) galileon mod-
els with non-derivative matter coupling. One of the main
lessons to be drawn is that more general matter cou-
plings, including derivative interactions, are necessary for
the phenomenological viability of SO(N) multi-galileon
theories. For instance, the coupling ∼ ∂µπI∂νπITµνnat-
urally arises from brane-world constructions [30, 34] and
maintains both the galilean and the internal rotation
symmetries.
II.THE MODEL
In co-dimension N, the 4-dimensional effective theory
contains N fields πI, I = 1···N, representing the N
brane-bending modes of the full 4 + N-dimensional the-
ory. The extended symmetry of the vacuum lagrangian
is
δπI= ωI
µxµ+ ?I+ ωI
JπJ,(1)
where ωIµ, ?Iand ωI
rameters. (See [30] for the geometric setup and origin
of this symmetry).This transformation consists of a
galilean invariance acting on each of the πIfields, and an
SO(N) rotation symmetry under which πItransforms as
a vector. The unique four dimensional lagrangian density
respecting this is [28, 30]
Jare constant transformation pa-
Lπ=
−1
−λ?∂µπI∂νπJ?∂λ∂µπJ∂λ∂νπI− ∂µ∂νπI?πJ
where λ is a coupling with dimension [mass]−6, contain-
ing the strong interaction mass scale. The I,J indices
are raised and lowered with δIJ.
It remains to couple this theory to matter. The nat-
ural coupling we might consider, the lowest dimension
coupling that preserves the galilean and internal rotation
symmetries, is ∼ ∂µπI∂νπITµν. This is the coupling that
naturally arises from brane matter in the construction
of [30, 34]. However, for static non-relativistic sources
Tµν∼ ρδµ
there are no nontrivial spherically symmetric solutions
with this coupling.
Linear couplings Llinear ∼ πT arise naturally from
DGP-like setups, since the π’s conformally mix with the
graviton. These lead to spherical solutions [31], but break
the SO(N) internal symmetry.
We therefore do not consider these couplings fur-
ther, and instead concentrate on the most general non-
derivative coupling that preserves the SO(N) symmetry.
2∂µπI∂µπI
??,
(2)
0δν
0, and since ∂0π = 0 for static solutions
Lcoupling=T
2P(π2) , (3)
where P is an arbitrary function of the invariant π2≡
πIπI.
III. SPHERICALLY SYMMETRIC SOLUTIONS
Our focus is on the existence and viability of spher-
ically symmetric solutions sourced by a delta function
mass distribution1
T = −Mδ3(r) . (4)
The equations of motion, including the coupling (3), are
?πI− λ??πI?∂µ∂νπJ∂µ∂νπJ− ?πJ?πJ
= MP??π2(0)?πI(0)δ3(? r) ,
where P?(X) ≡ dP/dX. Restricting to spherically sym-
metric configurations πI(r), this reduces to
?
+ 2∂µ∂νπI?∂µ∂νπJ?πJ− ∂µ∂λπJ∂ν∂λπJ??
(5)
1
r2
d
dr
?r3?yI+ 2λyIy2??= MP??π2(0)?πI(0)δ3(? r) ,
(6)
where
yI≡1
r
dπI
dr
, (7)
and y2≡ yIyI. Note that, due to the shift symmetry of
the lagrangian, the equations of motion of galileon fields
always take the form of a total derivative. Thus we can
integrate once to obtain the equations of motion
yI+ 2λyIy2=
M
4πr3P??π2(0)?πI(0) . (8)
Dividing these equations by each other, we obtain the
relations
dπI/dr
dπJ/dr=πI(0)
πJ(0), (9)
which, when integrated from the origin, gives
πI(r)
πJ(r)=πI(0)
πJ(0).(10)
The various components of the solution are therefore
always proportional to each other. Thus, by a global
SO(N) rotation, we can rotate the solution into one di-
rection in field space, say the I = 1 direction, so that the
solution takes the form π1≡ π and πI= 0 for I ?= 1. This
model therefore exhibits a kind of spontaneous symme-
try breaking of the internal SO(N) symmetry, since any
non-trivial solution must pick a direction in field space.
Equation (8) now takes the form
y + 2λy3=
M
4πr3P??π2(0)?π(0) .(11)
1Note that stable, non-trivial solutions without a source do not
exist [35].

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As r ranges from zero to infinity, the left hand side is
monotonic, and is positive or negative depending on the
sign of P??π2(0)?π(0). For there to be a continuous so-
invertible when it is positive (negative). For a solution
to exist, this requires (for non-trivial λ)
lution for y as a function of r, the left hand side must be
λ > 0 .(12)
Thus y is also positive (negative), is monotonic with r,
and ranges from zero to (negative) infinity as r ranges
from infinity to zero. This in turn implies that dπ/dr
does not cross zero, and hence π is monotonic.
Equation (11) yields a solution for y, and hence dπ/dr,
as a function of r and the parameters of the theory. In-
tegrated from r = 0 to infinity, this will give a relation
between π(0) and the asymptotic value of the field π(∞).
The asymptotic field value is essentially a modulus of
the theory — it will be set by whatever cosmological ex-
pectation value is present. It is a physically meaningful
parameter as it affects the coupling to the source by de-
termining π(0).
Near the source, where the non-linear term dominates,
the solution is linear in r,
πr?r∗(r) ∼ π(0) +
?M
8πλP??π2(0)?π(0)
?1/3
r ,(13)
whereas far from the source, where the linear term dom-
inates, the solution goes like 1/r,
πr?r∗(r) ∼ π(∞) −M
4πP??π2(0)?π(0)1
r,
(14)
where the transition between these regimes occurs at the
radius
?
Note that this crossover radius, and hence the distance
at which non-linearities become important, depends on
the modulus π(0). The equation of motion for π(r) is
readily solved numerically, and the solution obtained is
plotted schematically in Fig. 1.
r∗∼
λM2?P??π2(0)?π(0)?2?1/6
. (15)
IV.PERTURBATIONS: STABILITY AND
SUBLUMINALITY
While the existence of static, spherically-symmetric
configurations is encouraging, there are, of course, other
important checks that our solution must pass to be phys-
ically viable. Specifically, following [5], we must study
the stability of these spherically symmetric solutions and
to determine the speed at which fluctuations propagate,
since superluminal propagation can be an obstacle to
finding an ultraviolet completion of the effective the-
ory [36].
r � r∗
r � r∗
π(0)
π ∼ r
π ∼ −1
r
π(∞)
π
r ∼ r∗
FIG. 1: Schematic sketch of the solution for π(r).
We expand the field in perturbations around the back-
ground solution πI
0,
πI= πI
0+ δπI.(16)
Away from the source, the linearized equations of motion
for the perturbations are of the form
−Kt
I(r)∂2
tδπI+1
r2∂r
?r2Kr
I(r)∂rδπI?+KΩ
I(r), Kr
I(r)∂2
ΩδπI= 0 ,
(17)
where the coefficients Kt
on r through the background field πI(r). We find
I(r) and KΩ
I(r) depend
Kt
1=
1
3r2
d
dr
?r3?1 + 18λy2??
?r2?1 + 6λy2??
dr
;
Kr
1= 1 + 6λy2;
1
2r
1
3r2
I?=1= 1 + 2λy2;
1
2r
KΩ
1
=
d
dr
;
Kt
I?=1=
d
?r3?1 + 6λy2??
?r2?1 + 2λy2??
;
Kr
KΩ
I?=1=
d
dr
.(18)
Applying the implicit function theorem to the function
F(y,r) = y + 2λy3−
M
4πr3P??π2(0)?π(0) = 0, we have
y + 2λy3
1 + 6λy2.
dy
dr= −∂rF
∂yF= −3
r
(19)

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This allows us to eliminate dy/dr from (18):
Kt
1=
?1 − 6λy2?2
1 + 6λy2
;
Kr
1= 1 + 6λy2;
1 − 6λy2
1 + 6λy2;
1 + 12λ2y4
1 + 6λy2
I?=1= 1 + 2λy2;
1 + 2λy2
1 + 6λy2.
KΩ
1
=
Kt
I?=1=
;
Kr
KΩ
I?=1=
(20)
Stability of the spherically symmetric background so-
lutions against small perturbations requires K > 0 for
all K’s. The I ?= 1 directions in field space are stable,
but the π1direction exhibits a gradient instability suffi-
ciently close to the source along the angular directions.
In other words, KΩ
1< 0 near the source. Therefore, lo-
calized perturbations can be found near the source that
lower the energy of the solution through their gradients.
This instability plagues very short-wavelength fluctua-
tions, right down to the UV cutoff, so decay rates are
dominated by the shortest distances in the theory and
cannot be reliably computed within the effective theory.
Equations (20) also allow us to compute the speeds
of propagation of our small perturbations, in both the
radial and angular directions. These are given by
(c2)r
1=
Kr
Kt
KΩ
Kt
Kr
Kt
1
1
=
?1 + 6λy2
1
1 − 6λy2;
?1 + 2λy2??1 + 6λy2?
1 + 2λy2
1 + 12λ2y4.
1 − 6λy2
?2
;
(c2)Ω
1 =
1
1
=
(c2)r
I?=1=
I?=1
I?=1
=
1 + 12λ2y4
;
(c2)Ω
I?=1=
KΩ
Kt
I?=1
I?=1
= (21)
Note that (c2)r
ways propagate superluminally.
(c2)Ω
1, in regions where these perturbations are stable.
The speed (c2)r
always subluminal. Whether superluminal propagation
of signals is problematic for a low-energy effective theory
is still an arguable issue, but it seems that at the least it
may preclude the possibility of embedding the theory in
a local, Lorentz-invariant UV completion [36].
1> 1, and hence these perturbations al-
The same is true of
I?=1is always superluminal, and (c2)Ω
I?=1is
A.Other constraints
It is interesting to note in passing that if a mechanism
exists to tame the instabilities we have identified, then
precision tests of gravity within the solar system already
place useful constraints on multi-galileon theories. The
galileon is screened at radii below the Vainshtein radius
r∗, given by equation (15), restoring the behavior of gen-
eral relativity. Requiring the solar system to be screened
to r ∼ 1016m thus yields a constraint on λ and π(0).
However, lunar laser ranging data constrain the depar-
ture from the gravitational potential predicted by GR to
satisfyδΦ
and we may translate this into a constraint on a different
combination of λ and π(0)
For example, consider the choice of P(X)MPl ∼
?
are saturated, and detection of an effect is therefore im-
minent, the relevant constraint simply becomes
Φ< 2.4 × 10−11(at radius r = 3.84 × 1010cm),
πIπI, giving a linear coupling between the radial π field
and matter. In the interesting case when the constraints
1
λ1/6? 10−9eV .(22)
Note that this is an extremely low cutoff for the effective
theory, as is also found in the DGP model.
V.DISCUSSION
We have derived spherically-symmetric solutions in an
SO(N) multi-galileon theory with general, non-derivative
couplings to matter. These solutions exhibit a Vain-
shtein screening effect, characteristic of galileon models.
However, a study of the behavior of fluctuations around
these solutions shows that one of the fields has imagi-
nary sound speed along the angular directions, signaling
an instability to anisotropic modes of arbitrarily short
wavelength. Moreover fluctuations inevitably propagate
superluminally.
These results raise serious concerns about the phe-
nomenological viability of SO(N) multi-galileon theories.
(Of course, this does not preclude their effectiveness in
early universe physics [15, 37], for instance during infla-
tion, as long as they become massive or decouple before
the present epoch.) A key input in our analysis is the re-
striction to non-derivative coupling to matter. The main
lesson to be drawn is that more general, derivative cou-
plings are necessary. For instance, the lowest-dimensional
coupling invariant under the galilean and internal rota-
tion symmetries is ∼ ∂µπI∂νπITµν.
fact naturally arises in the higher-codimension brane pic-
ture [34]. As mentioned earlier, the galileon fields are
oblivious to static, spherically-symmetric sources in this
case; thus exhibiting a screening mechanism. However,
they will be excited by orbital motion, and we leave a
study of the phenomenological implication of this cou-
pling to future work.
Our analysis also highlights a distinct advantage to ex-
plicitly breaking the symmetry (1), for example through
the introduction of a sequence of regulating branes
of different co-dimensions, as in the cascading gravity
case [21, 22, 38]. The explicit breaking of SO(N) sym-
This coupling in

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metry allows for more general terms in the action, which
can lead to a healthier phenomenology [31].
Finally, should a creative cure for our instabilities be
found, then we have demonstrated that precision so-
lar system tests of gravity set interesting constraints on
multi-galileon theories.
Acknowledgments
This work is supported in part by NASA ATP grant
NNX08AH27G, NSF grant PHY-0930521, and by De-
partment of Energy grant DE-FG05-95ER40893-A020.
M.T. is also supported by the Fay R. and Eugene L.
Langberg chair. The work of K.H. and J.K. is supported
in part by funds from the University of Pennsylvania.
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