Screening fifth forces in generalized Proca theories

0.47 MB·

by  Antonio De Felice

YITP-16-42
Screening fifth forces in generalized Proca theories
Antonio De Felice,1 Lavinia Heisenberg,2 Ryotaro Kase,3 Shinji Tsujikawa,3 Ying-li Zhang,4,5 and Gong-Bo Zhao4,5
1Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan
2Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland
3Department of Physics, Faculty of Science, Tokyo University of Science,
1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
4National Astronomy Observatories, Chinese Academy of Science, Beijing 100012, People’s Republic of China
5Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
(Dated: May 12, 2016)
For a massive vector field with derivative self-interactions, the breaking of the gauge invariance
allows the propagation of a longitudinal mode in addition to the two transverse modes. We con-
6 sider generalized Proca theories with second-order equations of motion in a curved space-time and
1 study how the longitudinal scalar mode of the vector field gravitates on a spherically symmetric
0 background. We show explicitly that cubic-order self-interactions lead to the suppression of the
2
longitudinalmodethroughtheVainshteinmechanism. Provided thatthedimensionless coupling of

y theinteraction isnotnegligible, thisscreeningmechanism issufficientlyefficient togiverise totiny
a corrections to gravitational potentials consistent with solar-system tests of gravity. We also study
M thequartic interactions with the presence of non-minimal derivativecoupling with theRicci scalar
and find the existence of solutions where the longitudinal mode completely vanishes. Finally, we

1 discuss the case in which the effect of the quartic interactions dominates over the cubic one and
1 show that local gravity constraints can be satisfied under a mild bound on the parameters of the
theory.

]
c
q
I. INTRODUCTION
-
r
g
[ TheconstructionoftheoriesbeyondGeneralRelativity(GR)ismotivatednotonlybytheultravioletcompletionof

gravitybutalsobytheaccumulatingobservationalevidenceofthelate-timecosmicacceleration. Ifwemodifygravity
2
fromGR,however,additionaldegreesoffreedom(DOF)generallyarise[1–5]. Tokeepthe theorieshealthy,thesenew
v
DOF should give rise to neither ghosts nor instabilities. If the equations of motion are of second order, the lack of
1
7 higher-order derivatives forbids the propagationof further dangerous DOF associated with Ostrogradskiinstabilities
3 [6]. In the presence of one scalar degree of freedom, it is known that Horndeski theories [7] are the most general
0 scalar-tensor theories with second-order equations of motion in curved space-times. Independently of the original
0 work, the same action was rederived by extending the so-called Galileon action (“scalar Galileons”) [8, 9] to curved
.
2 space-time with the second-order property maintained [10–16].
0 In1976,Horndeskiderivedthe mostgeneralactionofanAbelianvectorfield witha non-minimalcouplingyielding
6 second-orderequationsofmotion,undertheassumptionthattheMaxwellequationsarerecoveredintheflatspace-time
1
[17]. The cosmologyand the stability of such Horndeskivector-tensortheories were recently studied in Refs. [18, 19].
:
v There have been attempts for constructing theories of Abelian vector fields analogous to scalar Galileons [20–22].
i If we try to preserve the U(1) gauge invariance for one vector field and stick to second-order equations of motion,
X
there exists a no-go theorem stating that the Maxwell kinetic term is the only allowed interaction [23, 24]. However,
r
droppingthe U(1)gaugeinvarianceallowsustogeneratenon-trivialtermsassociatedwith“vectorGalileons”[25,26]
a
(see also Refs. [27–32] for related works).
In relativistic field theory, it is well known that introduction of the mass term for a Maxwell vector field breaks
the U(1) gauge invariance. In this massive vector Proca theory, there is one propagating degree of freedom in the
longitudinal direction besides two DOF corresponding to the transverse polarizations. In the presence of derivative
interactionslikethose appearingforGalileons,itis naturalto askwhether they do notmodify the number ofDOFin
Procatheory. InRef.[25],oneoftheauthorsderivedageneralizedProcaactionforavectorfieldAµ withsecond-order
equations of motion on curved space-times. The analysis based on the Hessian matrix showed that only three DOF
propagateasintheoriginalProcatheory[25,31]. Theactionhasnon-minimalderivativecouplingstotheRicciscalar
R and the Einstein tensor G , whose structure is similar to that in scalar Horndeski theories. In fact, taking the
µν
limit Aµ µπ, the resulting action for the scalar field π reproduces that of scalar Galileons with suitable choices
→ ∇
of free functions [25, 26].
ItwasshowninRefs.[26,27]thatasub-classofthesegeneralizedProcatheoriescanleadtothe self-accelerationof
theUniverse. Ifweapplythesetheoriestothepresentcosmicacceleration,notonlyaviablecosmicexpansionhistory
could be realized but also the gravitational interaction similar to GR could be recovered inside the solar system. In
thispaper,theissueofhowthevectorfieldgravitatesinthepresenceofderivativeself-interactionsisaddressedonthe
spherically symmetric space-time with a matter source. We first show that the transverse components of the spatial
2
vector Ai vanish on the spherically symmetric background by imposing their regularities at the origin. Hence the
longitudinal scalar component is the only relevant contribution to Ai in addition to the time component of Aµ.
We study how the longitudinal propagation affects the behavior of gravitational potentials in the presence of the
vector Galileon interactions. We shall consider two cases: (i) the self-interacting Lagrangian = β X Aµ exists,
3 3 µ
L ∇
and (ii) the non-minimal derivative coupling β X2R is taken into account in the Lagrangian in addition to .
4 4 3
L L
We show that, due to derivative self-interactions, the screening mechanism of the longitudinal mode can be at work.
This leads to the suppression of the propagation of the fifth force in such a way that the theories are consistent with
local gravity constraints. This is analogous to the Vainshtein mechanism [33] for scalar Galileons [8, 34–37], but the
propertyof screenedsolutions exhibits some difference due to the non-trivialcoupling betweenthe longitudinalmode
and the time component of Aµ.
Thispaperisorganizedasfollows. InSec.II wepresentthe actionofthe generalizedProcatheoriesinthe presence
of a matter source and derive the equations of motion up to the Lagrangian on general curved backgrounds. In
4
L
Sec. III we obtain the equations of motion on the spherically symmetric background (with coefficients given in the
Appendix). In Sec. IV we derive the vector field profiles in the presence of the Lagrangian both analytically and
3
L
numerically and compute corrections to leading-order gravitational potentials induced by the longitudinal scalar. In
Sec. V we study the cases in which the contribution of the Lagrangian dominates over that of and also obtain
4 3
L L
analytic field profiles as well as gravitationalpotentials. Sec. VI is devoted to conclusions.
II. GENERALIZED PROCA THEORIES
We begin with the generalized Proca theories described by the four-dimensional action
5
S = d4x√ g( + ) , = + , (2.1)
m F i
− L L L L L
Z i=2
X
where g denotes the determinant of the metric tensor g , the matter Lagrangian, and = (1/4)F Fµν is
µν m F µν
L L −
the standardkinetic term of the vector field A with F = A A ( is the covariantderivative operator).
µ µν µ ν ν µ µ
∇ −∇ ∇
The Lagrangians encode the non-trivial derivative interactions [25]
i
L
= G (X), (2.2)
2 2
L
= G (X) Aµ, (2.3)
3 3 µ
L ∇
= G (X)R+G (X) ( Aµ)2+c A ρAσ (1+c ) A σAρ , (2.4)
4 4 4,X µ 2 ρ σ 2 ρ σ
L ∇ ∇ ∇ − ∇ ∇
1
= G (X)G µAν (cid:2)G (X)[( Aµ)3 3d Aµ A ρAσ 3(1(cid:3) d ) Aµ A σAρ
5 5 µν 5,X µ 2 µ ρ σ 2 µ ρ σ
L ∇ − 6 ∇ − ∇ ∇ ∇ − − ∇ ∇ ∇
+(2 3d ) A γAρ σA +3d A γAρ Aσ], (2.5)
2 ρ σ γ 2 ρ σ γ
− ∇ ∇ ∇ ∇ ∇ ∇
where R is the Ricci scalar, G is the Einstein tensor, G as well as c ,d are arbitrary functions of
µν 2,3,4,5 2 2
1
X A Aµ, (2.6)
µ
≡−2
∗
andG ∂G /∂X. NotethatwecouldhaveallowedanycontractionsofthevectorfieldA withF andF (with
F∗ beiin,Xgt≡he duialofF)inthe functionG ,forinstanceinthe formofA A FµρFν...etc, orµcontractµioνnsbetwµeνenthe
2 µ ν ρ
vectorfieldandtheEinsteintensorG AµAν,sincetheydonotcontainanytimederivativeapplyingonthetemporal
µν
componentofthevectorfield,butforthepurposeofourpresentanalysisofscreenedsolutionsweshallsimplyassume
G (X).
2
The Lagrangians given above keep the equations of motion up to second-order. They can be constructed
2,3,4,5
L
from the Lagrangian[25, 26]
1
L˜i+2 =−(4 i)!Gi+2(X)Eα1···αiγi+1···4Eβ1···βiγi+1···4∇β1Aα1···∇βiAαi, (2.7)
−
where i = 0,1,2,3, and is the anti-symmetric Levi-Civita tensor. For i = 0 and i = 1, we have that
Eµ1µ2µ3µ4
= ˜ and = ˜ , respectively. For i = 2,3, besides the terms ˜ and ˜ , there are other Lagrangians ¯ and
2 2 3 3 4 5 4
fLL¯o5r,ire=Lspe2ctainvdeLly,Fder(LiXve)d by exchanβg1βin2βg3γt4he inAdiαc1es inAEαq2. (2α.37A), e.gfo.,rL−i(=1/23),FLw4(hXer)eEαF1α(2Xγ3)γ4aEnβd1β2Fγ3(γX4∇)βa1rAeβ2a∇rLbαit1rAaαry2
− 5 Eα1α2α3γ4E ∇β1 ∇β2 ∇ β3 4 5
functions of X. Since = ˜ + ¯ and = ˜ + ¯ , the coefficients c and d appearing in Eqs. (2.4) and (2.5)
4 4 4 5 5 5 2 2
L L L L L L
3
correspond to c = F (X)/G (X) and d = F (X)/G (X), respectively. Throughout this paper, we assume that
2 4 4 2 5 5
c and d are constants. In Eqs. (2.4) and (2.5) the non-minimal coupling terms G (X)R and G (X)G µAν are
2 2 4 5 µν
∇
included to guarantee that the equations of motion are of second order [25].
The Proca Lagrangian corresponds to the functions G = m2X and G = 0, where m corresponds to the mass
2 3,4,5
of the vector field. The generalized Proca theories given by Eq. (2.1) generally break the U(1) gauge symmetry. It
is possible to restore the gauge symmetry by introducing a Stueckelberg field π [38], as A A +∂ π. To zero-th
µ µ µ
→
orderinA ,we canextractthe longitudinalmode ofthe vectorfield[25]. Forthe functionalchoicesG =X,G =X
µ 2 3
and G = X2,G = X2, this procedure gives rise to the scalar covariant Galileon Lagrangian originally derived in
4 5
Refs.[8,9]byimposingtheGalileansymmetry∂ π ∂ π+b inflatspace-time. Thedependenceontheparameters
µ µ µ
→
c and d present in Eqs. (2.4) and (2.5) disappears for the Stueckelberg field π. In fact, the terms multiplied by the
2 2
coefficients c and d are proportional to G F Fµν and G [( Aλ)F Fµν/2+ A νA Fρµ], respectively,
2 2 4,X µν 5,X λ µν µ ν ρ
∇ ∇ ∇
which are both expressed in terms of F [25, 31].
µν
In the following we focus on theories given by the action (2.1) up to the Lagrangian . We do not consider the
4
L
Lagrangian duetoitscomplexity,butweleavesuchananalysisforafuturework. Wedefinetheenergy-momentum
5
L
tensor of the matter Lagrangian , as
m
L
2 δ(√ g )
T(m) = − Lm . (2.8)
µν −√ g δgµν
−
Assuming that matter is minimally coupled to gravity, there is the continuity equation
µT(m) =0. (2.9)
∇ µν
Variation of the action (2.1) with respect to gµν and A leads to
ν
1
δS = d4x√ g T(m) δgµν + νδA , (2.10)
− Gµν − 2 µν A ν
Z (cid:20)(cid:18) (cid:19) (cid:21)
where
δ 1 δ
L g , ν L . (2.11)
Gµν ≡ δgµν − 2 µνL A ≡ δA
ν
The equation of motion of the gravity sector on general curved space-times is given by
1
= T(m), (2.12)
Gµν 2 µν
with
4
= (F)+ (i). (2.13)
Gµν Gµν Gµν
i=2
X
Here each term comes from the standard kinetic term and the Lagrangians (2.2)-(2.4), as
1 1
(F) = g ( A ρAσ A σAρ) A ρA + A Aρ 2 A Aρ , (2.14)
Gµν 4 µν ∇ρ σ∇ −∇ρ σ∇ − 2 ∇ρ µ∇ ν ∇µ ρ∇ν − ∇ρ (ν∇µ)
1 1 (cid:2) (cid:3)
(2) = g G G A A , (2.15)
Gµν −2 µν 2− 2 2,X µ ν
1
(3) = G A A Aρ+g AλA Aρ 2A A Aρ , (2.16)
Gµν −2 3,X µ ν∇ρ µν ρ∇λ − ρ (µ∇ν)
(cid:2)1 (cid:3)
(4) = G G G A A R
Gµν 4 µν − 2 4,X µ ν
1
+ G g [( Aρ)2 (2+c ) A ρAσ +(1+c ) A σAρ 2A (cid:3)Aρ+2Aρ Aσ]
4,X µν ρ 2 ρ σ 2 ρ σ ρ ρ σ
2 ∇ − ∇ ∇ ∇ ∇ − ∇ ∇
+G [(1+c ) A Aρ Aρ A (1+2c ) A Aρ+(1+c ) A ρA
4,X 2 µ ρ ν ρ (µ ν) 2 ρ (ν µ) 2 ρ µ ν
∇ ∇ −∇ ∇ − ∇ ∇ ∇ ∇
+A Aρ Aρ A +A (cid:3)A 2A Aσ +A Aρ]
ρ (µ ν) ρ (µ ν) (ν µ) (ν µ) σ (µ ρ ν)
∇ ∇ − ∇ ∇ − ∇ ∇ ∇ ∇
1
G A A [( Aρ)2+c A ρAσ (1+c ) A σAρ]+2A A Aρ Aσ
4,XX µ ν ρ 2 ρ σ 2 ρ σ ρ σ µ ν
−2 { ∇ ∇ ∇ − ∇ ∇ ∇ ∇
2A Aα[Aρ A A Aρ A ρA 2g A[ρ σ]A ] 4A ( Aσ)A Aα , (2.17)
α ρ (µ ν) (ν µ) (ν µ) µν σ α σ (ν µ)
− ∇ ∇ − ∇ − ∇ − ∇ − ∇ ∇ }
4
where A ( A + A )/2andA[ρ σ]A (Aρ σA Aσ ρA )/2. Theequationofmotionforthevector
(µ ν) µ ν ν µ σ σ σ
field A∇corresp≡on∇ds to ν∇=0, i.e., ∇ ≡ ∇ − ∇
ν
A
Fµν G Aν +2G A[µ ν]A RG Aν 2G [ ν Aµ+c (cid:3)Aν (1+c ) µ νA ]
µ 2,X 3,X µ 4,X 4,X µ 2 2 µ
∇ − ∇ − − ∇ ∇ − ∇ ∇
G [Aν ( Aµ)2+c A ρAσ (1+c ) A σAρ
4,XX µ 2 ρ σ 2 ρ σ
− ∇ ∇ ∇ − ∇ ∇
2A νAρ Aµ 2c A µAρ Aν +2(1+c )A µAρ νA ]=0. (2.18)
− ρ∇ ∇(cid:8)µ − 2 ρ∇ ∇µ 2 ρ∇ ∇ (cid:9)µ
In GR we have G = M2/2, where M is the reduced Planck mass, so (4) simply reduces to (M2/2)G .
4 pl pl Gµν pl µν
Existence of the vector field with derivative self-couplings induces additional gravitational interactions with matter
throughEq.(2.12). We shallstudy whethersuchafifth forcecanbe suppressedinlocalregionswithamattersource.
III. EQUATIONS OF MOTION ON THE SPHERICALLY SYMMETRIC BACKGROUND
Wederivetheequationsofmotiononthesphericallysymmetricandstaticbackgrounddescribedbythelineelement
ds2 = e2Ψ(r)dt2+e2Φ(r)dr2+r2 dθ2+sin2θdϕ2 , (3.1)
−
whereΨ(r)andΦ(r)arethegravitationalpotentialsthatdependo(cid:0)nradiusrfromth(cid:1)ecenterofsphere. Forthematter
Lagrangian ,weconsiderthe perfectfluidwiththe energy-momentumtensorTµ =diag( ρ ,P ,P ,P ), where
Lm ν − m m m m
ρ is the energy density and P is the pressure. Then, the matter continuity equation (2.9) reads
m m
′ ′
P +Ψ(ρ +P )=0, (3.2)
m m m
where a prime represents a derivative with respect to r.
We write the vector field Aµ in the form
Aµ = φ,Ai , (3.3)
where i = 1,2,3. From Helmholtz’s theorem, we can dec(cid:0)ompos(cid:1)e the spatial components Ai into the transverse and
longitudinal modes, as
A =A(T)+ χ, (3.4)
i i ∇i
where A(T) obeys the traceless condition iA(T) =0 and χ is the longitudinal scalar. On the spherically symmetric
i ∇ i
configuration, it is required that the θ and ϕ components of A(T) (i.e., A(T) and A(T)) vanish. Then, the traceless
i 2 3
condition gives the following relation
′ 2
A(T) + A(T) Φ′A(T) =0, (3.5)
1 r 1 − 1
whose solution is given by
eΦ
A(T) =C , (3.6)
1 r2
where C is anintegrationconstant. For the regularityof A(T) atr =0, we requirethat C =0. This discussionshows
1
(T)
that the transverse vector A vanishes, so we only need to focus on the propagation of the longitudinal mode, i.e.,
i
A = χ. Then, the components of Aµ on the spherical coordinate (t,r,θ,ϕ) are given by
i i
∇
Aµ = φ(r),e−2Φχ′(r),0,0 . (3.7)
(cid:0) (cid:1)
5
The (0,0), (1,1) and (2,2) components of Eq. (2.12) reduce, respectively, to1
Ψ′2+ + C3 Ψ′+ + C5 Φ′+ + C7 + C8 = e2Φρ , (3.8)
C1 C2 r C4 r C6 r r2 − m
(cid:18) (cid:19) (cid:18) (cid:19)
Ψ′2+ + C11 Ψ′+ + C13 + C14 =e2ΦP , (3.9)
C9 C10 r C12 r r2 m
(cid:18) (cid:19)
/4+
Ψ′′+ Ψ′2+ Ψ′Φ′+ + C3 C15 Ψ′+ C13 + C19 Φ′+ + C21
15 16 17 18 20
C C C C r − 2 r C r
(cid:18) (cid:19) (cid:18) (cid:19)
=e2ΦP , (3.10)
m
where the coefficients (i = 1,2, ,21) are given in the Appendix. The mass term (2.6) can be decomposed as
i
C ···
X =X +X , where
φ χ
1 1
X e2Ψφ2, X e−2Φχ′2. (3.11)
φ χ
≡ 2 ≡−2
The ν =0 and ν =1 components of Eq. (2.18) reduce, respectively, to
(Ψ′′+Ψ′2)+ Ψ′Φ′+ + D4 Ψ′+ + D6 Φ′+ + D8 + D9 =0, (3.12)
D1 D2 D3 r D5 r D7 r r2
(cid:18) (cid:19) (cid:18) (cid:19)
Ψ′2+ + D12 Ψ′+ + D14 + D15 =0, (3.13)
D10 D11 r D13 r r2
(cid:18) (cid:19)
where we introduced the short-cut notations for convenience
=2φ(2c G 1), =2φ[1 2c (G +2X G )],
1 2 4,X 2 2 4,X χ 4,XX
D − D −
=φχ′G φ′[3 2c (3G +2X G )] 4c e−2Φφχ′χ′′G ,
3 3,X 2 4,X φ 4,XX 2 4,XX
D − − −
′ ′
=4φ(2c G 2X G 1), = φχG +φ[1 2c (G +2X G )],
4 2 4,X χ 4,XX 5 3,X 2 4,X χ 4,XX
D − − D − −
=4φ(G +2X G ),
6 4,X χ 4,XX
D
=e2ΦφG +φχ′′G φ′′(1 2c G )+c (e2Ψφφ′2 2e−2Φφ′χ′χ′′)G ,
7 2,X 3,X 2 4,X 2 4,XX
D − − −
=2φχ′G 2φ′(1 2c G )+4e−2Φφχ′χ′′G , = 2φ[(1 e2Φ)G +2X G ],
8 3,X 2 4,X 4,XX 9 4,X χ 4,XX
D − − D − −
=8c e−2Φχ′X G , =2(X X )G +4c e2Ψ−2Φφφ′χ′G ,
10 2 φ 4,XX 11 χ φ 3,X 2 4,XX
D D −
=4e−2Φχ′[G +2(X X )G ], = χ′G e2Ψφφ′G +c e2Ψ−2Φφ′2χ′G ,
12 4,X χ φ 4,XX 13 2,X 3,X 2 4,XX
D − D − −
=4X G 4e2Ψ−2Φφφ′χ′G , = 2χ′[(1 e−2Φ)G 2e−2ΦX G ]. (3.14)
14 χ 3,X 4,XX 15 4,X χ 4,XX
D − D − − −
Among the six equations of motion (3.2), (3.8)-(3.10), and (3.12)-(3.13), five of them are independent. For a given
densityprofileρ ofmatter,solvingfiveindependentequationsofmotionleadstothesolutionstoP ,Ψ,Φ,φ,χwith
m m
appropriate boundary conditions.
For the consistency with local gravity experiments within the solar system, we require that the gravitational
potentials Ψ and Φ need to be close to those in GR. In GR without the vector field Aµ, we have G = G = 0,
2 3
G =M2/2 and φ=0=χ′, so Eqs. (3.8) and (3.9) read
4 pl
2M2 M2
plΦ′ pl 1 e2ΦGR =e2ΦGRρ , (3.15)
r GR− r2 − m
2M2 M2 (cid:0) (cid:1)
plΨ′ + pl 1 e2ΦGR =e2ΦGRP . (3.16)
r GR r2 − m
(cid:0) (cid:1)
Since Φ and Ψ would be the leading-order contributions to gravitational potentials under the operation of the
GR GR
screening mechanism, we first derive their solutions inside and outside a compact body. We assume that the change
ofρm occursrapidly atthe distance r∗, sothatthe matter density canbe approximatedasρm(r) ρ0 for r <r∗ and
≃
ρm(r) 0forr >r∗. Thisconfigurationisequivalenttothatofthe Schwarzschildinteriorandexteriorsolutions. For
≃
1 Wenotethatthe(0,1)componentofEq.(2.12)reducestothesameformasEq.(3.13).
6
r <r∗, integrationof Eq. (3.2) leads to Pm = ρm+ e−Ψ(r), where is an integrationconstant knownby imposing
− C C
the condition Pm(r∗)=0.
Matching the interior and exterior solutions of Ψ and Φ at r =r∗ with appropriate boundary conditions (at r =0
and r ), the gravitational potentials inside and outside the body are given by
→∞
−1/2
eΨGR = 3 1 ρ0r∗2 1 1 ρ0r2 , eΦGR = 1 ρ0r2 , (3.17)
2s − 3Mp2l − 2s − 3Mp2l − 3Mp2l!
for r <r∗, and
1/2 −1/2
eΨGR = 1 ρ0r∗3 , eΦGR = 1 ρ0r∗3 , (3.18)
− 3Mp2lr! − 3Mp2lr!
for r > r∗. In the following, we employ the weak gravity approximation under which Ψ and Φ are much smaller
| | | |
than 1, i.e.,
ρ r2
0 ∗
Φ∗ 1. (3.19)
≡ M2 ≪
pl
This condition means that the Schwarzschild radius of the source rg ≈ρ0r∗3/Mp2l is much smaller than r∗. Then, the
solutions (3.17) and (3.18) reduce, respectively, to
ρ ρ r2
Ψ 0 r2 3r2 , Φ 0 , (3.20)
GR ≃ 12M2 − ∗ GR ≃ 6M2
pl pl
(cid:0) (cid:1)
for r <r∗, and
ρ r3 ρ r3
0 ∗ 0 ∗
Ψ , Φ , (3.21)
GR ≃−6M2r GR ≃ 6M2r
pl pl
for r>r∗. For the theories with the action (2.1), the vector field interacts with gravity through the derivative terms
′′ ′ ′
Ψ ,Ψ,Φ in Eqs. (3.12) and (3.13). The leading-order contributions of such gravitational interactions follow from
′′ ′ ′
the derivatives Ψ ,Ψ ,Φ of the GR solutions (3.17)-(3.18). Then, we can integrate Eqs. (3.12) and (3.13) to
GR GR GR
′
obtain the solutions to φ and χ. The next-to-leading order corrections to Ψ and Φ can be derived by substituting
′
the solutions of φ and χ into Eqs. (3.8) and (3.9). In Secs. IV and V we apply this procedure to concrete theories.
IV. THEORIES WITH THE CUBIC LAGRANGIAN
Let us first consider theories in which the function G corresponds only to the Einstein-Hilbert term, i.e.,
4
M2
pl
G = , (4.1)
4
2
whereM isthe reducedPlanckmass. Inthis casetheG terminthe Lagrangian vanishes,butthe Lagrangian
pl 4,X 4
L
givesrisetoanon-trivialgravitationalinteractionwiththevectorfield. Theequationsofmotion(3.12)and(3.13)
3
L
reduce, respectively to
1 d 1 d
(r2φ′) e2ΦG φ G φ (r2χ′)
r2dr − 2,X − 3,X r2dr
4φ
+2φ Ψ′′+Ψ′2 Ψ′Φ′ φχ′G 3φ′ Ψ′+(φχ′G φ′)Φ′ =0, (4.2)
3,X 3,X
− − − − r −
(cid:18) (cid:19)
(cid:0) 2(cid:1)
χ′G + e2Ψφφ′+ e−2Φχ′2 G + e2Ψφ2+e−2Φχ′2 G Ψ′ =0. (4.3)
2,X 3,X 3,X
r
(cid:18) (cid:19)
(cid:0) (cid:1)
For concreteness, we shall focus on the theories given by the functions
G (X)=m2X, G (X)=β X, (4.4)
2 3 3
where m is the mass of the vector field, and β is a dimensionless constant. The choice of G (X) given above is
3 3
related with that of scalar Galileons. In what follows, we obtain analytic solutions to Eqs. (4.2) and (4.3) under the
approximation of weak gravity.
7
A. Analytic vector field profiles
1. Solutions for r<r∗
For the distance r smaller than r∗, we substitute the derivatives of Eq. (3.20) into Eqs. (4.2) and (4.3) to derive
leading-order solutions to φ and χ′. The terms containing e2Ψ and e−2Φ provide the contributions linear in Ψ and
′ ′
Φ [say, ΨφφG in Eq. (4.3)]. After deriving analytic solutions to φ and χ, however, we can show that such
3,X
terms give rise to contributions much smaller than the leading-order solutions. Hence it is consistent to employ the
approximations e2Ψ 1 and e−2Φ 1 in Eqs. (4.2) and (4.3), such that
≃ ≃
d d ρ
(r2φ′) m2r2φ β φ (r2χ′)+ 0 [6φ+r(φ′+β χ′φ)]r2 0, (4.5)
dr − − 3 dr 6M2 3 ≃
pl
2 ρ φ2
m2χ′+β φφ′+ χ′2+ 0 r 0. (4.6)
3 r 6Mp2l !≃
From Eq. (4.6) it follows that
m2r 8β2 ρ φ2
χ′ = 1+ 1 3 φφ′+ 0 r . (4.7)
4β3 − vu − m4r 6Mp2l !
u
 t 
The sign of (4.7) has been chosen in such a way that χ′ vanishes for β /m2 0, which can be regarded as the
3
→
GR limit. Since we are interested in how the screening mechanism is at work in the presence of the Lagrangian
3
L
for a very light field (e.g., the vector field associated with the late-time cosmic acceleration), we take another limit
β /m2 in the discussion below. In other words, we focus on the case m 0 with a non-zero dimensionless
3
→ ∞ →
coupling β . For β >0, Eq. (4.7) reduces to
3 3
r ρ φ2
χ′ = φφ′+ 0 r . (4.8)
vu−2 6Mp2l !
u
t
′
For the consistency of Eq. (4.8) we require the condition φφ <0.
We search for solutions where the scalar potential φ does not vary much with respect to r, i.e.,
φ(r)=φ +f(r), f(r) φ , (4.9)
0 0
| |≪| |
where φ is a constantand f(r) is a function of r. We also focus onthe case where φ(r) decreaseswith the growthof
0
′ ′ ′
r, such that φ(r) <0 with φ >0. In Eq. (4.5) we also neglect the terms r(φ +β χφ) relative to 6φ. The validity
0 3
′
of this approximation can be checked after deriving the solutions to φ and χ. Substituting Eq. (4.8) into Eq. (4.5)
with Eq. (4.9), we obtain the integrated solution
r ρ φ r ρ φ
r2f′ β φ3/2r2 f′+ 0 0 + 0 0r3 =C, (4.10)
− 3 0 vu−2 6Mp2l ! 3Mp2l
u
t
′
where C is a constant. Under the boundary condition φ(0)=0, we can fix C =0 and hence
r ρ φ r ρ φ
f′ β φ3/2 f′+ 0 0 = 0 0r. (4.11)
− 3 0 vu−2 6Mp2l ! −3Mp2l
u
t
Clearly, there is a solution of the form f′(r) r. Substituting the solution f(r) = Br2 into Eq. (4.11), we find
∝ − −
that the positive constant B, which remains finite in the limit β , is given by
3
→∞
ρ φ
0 0
B = (s ), (4.12)
6M2 F β3
pl
8
where
3(β φ M )2
3 0 pl
s , (4.13)
β3 ≡ 4ρ
0
s
(s ) (1+s ) 1 β3 . (4.14)
F β3 ≡ β3 − 1+s
(cid:18) r β3(cid:19)
Then, we obtain the following analytic field profiles
ρ
φ(r) = φ 1 (s ) 0 r2 , (4.15)
0" −F β3 6Mp2l #
ρ φ2 1
χ′(r) = 0 0 (s ) r. (4.16)
s6Mp2l (cid:20)F β3 − 2(cid:21)
As s increases from 0 to , the function (s ) decreases from 1 to 1/2. This means that the terms inside the
squareβr3oot of Eq. (4.8) rema∞ins positive. SinFce β3(s )ρ r2/(6M2) 1 from the condition (3.19) of weak gravity,
F β3 0 pl ≪
the solution (4.15) is consistent with the assumption (4.9). In the limit that s 1, the field profiles (4.15) and
β3 ≪
(4.16) reduce, respectively, to
ρ ρ φ2
φ(r) φ 1 0 r2 , χ′(r) 0 0 r, (4.17)
≃ 0 − 6Mp2l ! ≃s12Mp2l
whereas, for s 1, it follows that
β3 ≫
ρ ρ
φ(r) φ 1 0 r2 , χ′(r) 0 r. (4.18)
≃ 0 − 12Mp2l ! ≃ 6β3Mp2l
The amplitude of χ′(r) in Eq. (4.18) is about s−1/2 times smaller than that in Eq. (4.17). For a larger coupling β ,
β3 | 3|
thescreeningeffectisefficienttosuppressthepropagationofthelongitudinalmode. Onusingthesolutions(4.15)and
′ ′
(4.16), wecanconfirmthatthe terms r(φ +β χφ) inEq.(4.5)is muchsmallerthan6φandthatthe approximations
3
e2Ψ 1 and e−2Φ 1 employed in Eq. (4.6) are also justified.
≃ ≃
2. Solutions for r>r∗
Employing the GR solution (3.21) of gravitational potentials in the regime r > r∗ and substituting them into
Eqs. (4.2) and (4.3), it follows that
ddr(r2φ′)−m2r2φ−β3φddr(r2χ′)+ 9Mρ0r4∗3r2 ρ0r∗3φ+3Mp2lr2(2φ′−β3χ′φ) ≃0, (4.19)
pl
(cid:2) (cid:3)
2 ρ φ2r3
m2χ′+β φφ′+ χ′2+ 0 ∗ 0. (4.20)
3 r 6Mp2lr2!≃
′
Taking the m 0 limit and considering the branch χ >0, Eq. (4.20) gives the following relation
→
r ρ φ2r3
χ′ = φφ′+ 0 ∗ . (4.21)
vu−2 6Mp2lr2!
u
t
The term (ρ0r∗3)2φ/(9Mp4lr2) in Eq. (4.19) is at most Φ∗ times as small as the term ρ0φ/Mp2l in Eq. (4.5). Moreover,
after deriving the solutions to φ and χ′, we can confirm that the contributions 3M2r2(2φ′ β χ′φ) in Eq. (4.19) is
pl − 3
at most of the order of ρ r3φ. Hence it is a good approximation to neglect the terms inside the square bracket of
0 ∗
Eq.(4.19). Substituting Eq.(4.21)intoEq.(4.19)withthe approximation(4.9)andmatchingthe integratedsolution
at r =r∗ on account of Eq. (4.11), we obtain
r ρ φ r3 ρ φ r3
r2φ′ β φ3/2r2 φ′+ 0 0 ∗ 0 0 ∗ . (4.22)
− 3 0 vu−2 6Mp2lr2!≃− 3Mp2l
u
t
9
′
More explicitly, the field derivative φ can be expressed as
ρ φ r3 r3
′ 0 0 ∗
φ(r)= (ξ), ξ s . (4.23)
−3Mp2lr2F ≡ β3r∗3
From Eq. (4.21) the longitudinal mode is given by
ρ r3φ2 1
χ′(r)= 0 ∗ 0 (ξ) . (4.24)
s6Mp2lr (cid:20)F − 2(cid:21)
If sβ3 ≫1, then ξ ≫1 for r >r∗. In this case it follows that
ρ φ r3 ρ r3
′ 0 0 ∗ ′ 0 ∗
φ(r) , χ(r) . (4.25)
≃−6M2r2 ≃ 6β M2r2
pl 3 pl
Ifs <1,thereisthe transitionradiusr atwhichther dependence ofthe longitudinalmodechanges. Theradius
β V
r can∼be identified by the condition ξ =1, i.e.,
V
r∗
r = . (4.26)
V s1/3
β3
For the distance r∗ <r rV we have 1, so the solutions reduce to
≪ F ≃
ρ φ r3 ρ r3φ2
φ′(r) 0 0 ∗ , χ′(r) 0 ∗ 0 . (4.27)
≃−3Mp2lr2 ≃s12Mp2lr
For r r we have ξ 1 and hence the resulting solutions are given by Eq. (4.25). In this regime, the longitudinal
V
≫′ ≫
modeχ(r)decreasesfasterthanthatforr∗ <r rV withasuppressedamplitude. ThedistancerV canberegarded
′ ≪ ′
as the Vainshtein radius above which χ(r) starts to decay quickly. If β obeys the condition s 1, χ(r) is
| 3| β3 ≫
strongly suppressed both inside and outside the body due to the Vainshtein mechanism, see Eqs. (4.18) and (4.25).
Meanwhile, for s < 1, the screening of the longitudinal mode manifests for the distance r >r . The fact that the
β3 V
suppressionofthelo∼ngitudinalmodeoccursoutsidetheradiusr forsmall β isauniquefeatureofvectorGalileons.
V 3
| |
B. Numerical solutions for the vector field
To confirm the validity of the analytic solutions derived above, we shall numerically solve Eqs. (4.2) and (4.3)
coupled with the gravitationalEqs. (3.8)-(3.10). For concreteness we consider the density distribution given by
ρ (r)=ρ e−ar2/r∗2, (4.28)
m 0
where a is a positive constant of the order of 1. With this profile, the matter density starts to decrease significantly
for r >r∗. For the numerical purpose, it is convenient to introduce the following dimensionless quantities
∼
′
r φ χ
x= , y = , z = , (4.29)
r∗ φ0 φ0
where φ is the value of φ at r =0. In the massless limit with G =β X, we can express Eqs. (4.2) and (4.3) in the
0 3 3
forms
d2y 2dy dz 2 d2Ψ dΨ 2 dΨdΦ
dx2 + xdx −β3r∗φ0y dx + xz +2y dx2 + dx − dx dx
(cid:18) (cid:19) " (cid:18) (cid:19) #
dy 4 dΨ dy dΦ
β3r∗φ0yz 3 y + β3r∗φ0yz =0, (4.30)
− − dx − x dx − dx dx
(cid:18) (cid:19) (cid:18) (cid:19)
−1
dy dΨ dΨ
z =eΨ+Φ xy +y 2+x , (4.31)
s− dx dx dx
(cid:18) (cid:19)(cid:18) (cid:19)
10
FIG. 1: The numerical solutions to y = φ/φ0, −dy/dx, and z = χ′/φ0 as a function of x = r/r∗ for the matter profile
cρomnd=itρio0nes−4orf2/Ψr∗2,wΦi,thy,Φa∗n=d1d0y−/4d.xEaarcehcphaonseenlctoorrebsepoconndssisttoensβt3w=ith10E−4qs(.le(f3t.)1a7n)dansβd3(=4.115()riagthtx),=res1p0e−c3ti.veTlyh.eTvheertbicoaulndlinareys
represent thescales r=r∗ and rV =20r∗ (left panel) and the scale r=r∗ (right panel).
where the quantity β3r∗φ0 is related with sβ3 as β3r∗φ0 = 4sβ3Φ∗/3. We take the x derivative of Eq. (4.31) and
then eliminate the term dz/dx by combining it with Eq. (4.30) to obtain the second-order equation for y(x). To
p
derive the leading-ordergravitationalpotentials Φ and Ψ , we also solve Eqs. (3.15) and (3.16) with a vanishing
GR GR
pressureP . Thisproceduregivesrisetothesolutionsderivedundertheweakgravityapproximation,e.g.,Eq.(3.17).
m
Numerically, we confirmed that the approximation substituting Φ and Ψ into Eqs. (4.30) and (4.31) provides
GR GR
practically identical results to those obtained by solving full Eqs. (3.8)-(3.10).
InFig.1weplotthefieldprofileforρm =ρ0e−4r2/r∗2 andΦ∗ =10−4 withtwodifferentvaluesofsβ3. Theboundary
conditions of y and dy/dx around the center of body are chosen to match with Eq. (4.15). As we see in Fig. 1,
′ ′
both φ(r) and χ(r) linearly grow in r for the distance smaller than r∗. The left panel of Fig. 1 corresponds to
sβ3 =−10−4, in which case the solutions to φ(r) and χ′(′r) are well described by Eq. (4.17) in the regime r < r∗. For
s larger than the order of 1, the longitudinal mode χ(r) tends to be suppressed according to Eq. (4.18). The right
β3 ′
panel of Fig. 1, which corresponds to s = 1, is the case in which the suppression of χ(r) occurs in a mild way for
β3
r <r∗.
theFodristthaencdeisrtVanisceofrtlhaergoerrdethraonf1r0∗,r∗b.oHthen−cφe′t(hr)easonldutχio′(nrs)tsotaφr′t(rt)oadnedcrχe′a(sre)wariethgitvheengbryowEtqh.(o4f.2r7.)Wfohrern∗s<β3r=<1100−r4∗,
and by Eq. (4.25) for r > 10r∗. In the left panel of Fig. 1, we can confirm that the qualitative behavior o∼f χ′(r)
changes around r 10r∗∼(i.e., from χ′(r) r−1/2 to χ′(r) r−2). Note that φ′(r) decreases as φ′(r) r−2 for
≈ ∝ ∝ | | | | ∝
r >r∗.
Whens =1,asseenintherightpanelofFig.1,wefindthatthereisalmostnointermediateregimecorresponding
β3
to the solution χ′(r) r−1/2 and that the longitudinal mode decreases as χ′(r) r−2 for r > r∗. This reflects the
∝ ∝
fact that, even when s =O(1), the quantity ξ in Eq.(4.23) quickly becomes muchlarger than 1 with the growthof
β3
rβ(>, trh∗e).suTphperne,ssfioornsfβo3r>∼th1e,atmheplsitouludteioonfsχi′n(rt)hteenredgsimtoebre>mro∗reasriegnwiefilclaanptporuoxtsimidaettehdebbyoEdyq.. (4.25). For increasing
3
| |
In Fig. 1 we also find that φ(r) stays nearly constant in the whole regime of interest. This is associated with the
fact that the r-dependent correction to φ(r) is at most of the order of φ0Φ∗, i.e., much smaller than φ0 under the
′
weak gravity approximation. The numerical solutions to φ(r) and χ(r) are fully consistent with the analytic field
profiles derived under the assumption (4.9), so we resort to the analytic solutions for discussing corrections to the
leading-order gravitationalpotentials in Sec. IVC.