Vainshtein screening in a cosmological background in the most general second-order scalar-tensor theory

0.29 MB·English

by  Rampei Kimura

Vainshtein screening in a cosmological background in the most general second-order
scalar-tensor theory
Rampei Kimura,1,∗ Tsutomu Kobayashi,2,3,† and Kazuhiro Yamamoto1,‡
1Department of Physical Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
2Hakubi Center, Kyoto University, Kyoto 606-8302, Japan
3Department of Physics, Kyoto University, Kyoto 606-8502, Japan
A generic second-order scalar-tensor theory contains a nonlinear derivative self-interaction of
the scalar degree of freedom φ `a la Galileon models, which allows for the Vainshtein screening
mechanism. Weinvestigate thiseffect onsubhorizon scales in acosmological background,based on
2 themost generalsecond-orderscalar-tensor theory. Ouranalysistakesintoaccount all therelevant
1 nonlinear terms and the effect of metric perturbations consistently. We derive an explicit form of
0 Newton’sconstant, which in general istime-dependentand henceisconstrained from observations,
2 as suggested earlier. It is argued that in the most general case the inverse-square law cannot be
reproduced on thesmallest scales. Some applications of ourresults are also presented.
n
a
PACSnumbers: 04.50.Kd,95.36.+x
J

9
1
I. INTRODUCTION mological observations, we therefore need to clarify the

behavior of gravity around and below the scale at which
]
O the relevant nonlinearities set in. In this paper, we ex-
Since the discoveryofthe acceleratedexpansionofthe
plorethe consequencesofthe lattermechanismindetail,
C Universe [1], numerous attempts have been proposed to
taking into account the nonlinear effect.
. explainthe originof this biggestmystery in modern cos-
h
mology. A vast class of models for cosmic acceleration We study gravity sourced by a density perturbation
p
- invokes a scalar degree of freedom, φ, which may cou- of nonrelativistic matter, on subhorizon scales in a cos-
o ple minimally or nonminimally to gravity and ordinary mological background, using the (quasi)static approx-
r matter. In the case of nonminimal coupling, the mod- imation. To provide generic results, we work in the
t
s elsarecommonlycalledmodifiedgravity(or“darkgrav- mostgeneralscalar-tensortheorywithsecond-orderfield
a
[ ity”) rather than dark energy, as φ participates in the equations [7], which can be derived by generalizing the
long-range gravitational interactions and thereby accel- Galileontheory [8, 9] andtherefore is expected to be en-
2
erates the cosmic expansion. Modified gravity models dowedwiththe Vainshteinmechanism. Inthe contextof
v
must be designedwith care,because otherwise the effect theGalileon,previousworksfocusonlyonthescalar-field
9
ofmodificationcouldpersistdownto smallscales,which equation of motion to see the profile of (the gradientof)
4
7 could easily be inconsistent with stringent tests in the φ, ignoring gravity backreaction [5, 10].1 Since our ap-
6 solar system and laboratories. For this reason, screening proach follows the cosmological perturbation theory on
. mechanisms for scalar-mediated force are crucial. subhorizon scales [12], the effect of metric perturbations
1
1 There are mainly two approaches for screening the cannaturallybetakenintoaccountconsistently. There-
1 scalardegreeoffreedominmodifiedgravitymodels. The sults in this paper can be applied to various aspects of
1 first one is the Chameleon mechanism [2], by which the cosmology and astrophysics.
: scalaracquireslargemassinahighdensityenvironment.
v This paper is organized as follows. We define the the-
i This is employed in viable f(R) gravity [3], which is oryweconsiderandthenpresenttheequationsgoverning
X
equivalent to a scalar-tensor theory with an appropri-
the background cosmological dynamics in the next sec-
r ate potential. The second one is the Vainshtein mech-
a tion. In Sec. III we derive the perturbation equations
anism [4]. In this case φ’s kinetic term becomes effec-
with relevant nonlinear contributions using the subhori-
tively large in the vicinity of matter due to some nonlin-
zonapproximation. Wethenexploresphericallysymmet-
ear derivative interaction, suppressing the effect of non-
ric solutions of the perturbation equations in Sec. IV. In
minimal coupling. The Vainshtein screening is typical
Sec.Vwepresentsomesimpleapplicationsofourresults
in Galileon-like models [5] and nonlinear massive grav-
and finally we conclude in Sec. VI. In Appendix A, we
ity (e.g., [6]). In the Vainshtein case, nonlinearities play
summarizethe definitions ofcoefficients inthe equations
an important role in possible recovery of usual gravity
in the main text. In Appendix B, we discuss a possible
on small scales even in a weak gravity regime. To test
models of modified gravity against experiments and cos-
1 In the course of the preparation of this manuscript, we became
aware of the very recent paper by De Felice, Kase, and Tsu-
∗Email: rampei”at”theo.phys.sci.hiroshima-u.ac.jp jikawa[11],inwhichthemetricundertheinfluenceoftheVain-
†Email: tsutomu”at”tap.scphys.kyoto-u.ac.jp shtein mechanism is obtained for a static and spherically sym-
‡Email: kazuhiro”at”hiroshima-u.ac.jp metricconfigurationinasubclassofthemostgeneraltheory.
2
varietyofsolutionsofthekeyequation(50). InAppendix where
C, we present the Fourier transform of the perturbation
equations. := 2XKX K+6Xφ˙HG3X 2XG3φ
E − −
6H2G +24H2X(G +XG )
4 4X 4XX
−
12HXφ˙G 6Hφ˙G
4φX 4φ
− −
+2H3Xφ˙(5G +2XG )
5X 5XX
6H2X(3G +2XG ), (5)
II. COSMOLOGY IN THE MOST GENERAL − 5φ 5φX
SCALAR-TENSOR THEORY := K 2X G +φ¨G +2 3H2+2H˙ G
3φ 3X 4
P −
12H2X(cid:16)G 4HX˙G(cid:17) (cid:16) (cid:17)
4X 4X
We consider a theory whose action is given by − −
8H˙XG 8HXX˙G +2 φ¨+2Hφ˙ G
4X 4XX 4φ
− −
(cid:16) (cid:17)
+4XG +4X φ¨ 2Hφ˙ G
S = d4x√ g( + ), (1) 4φφ − 4φX
GG m
− L L (cid:16) (cid:17)
Z 2X 2H3φ˙ +2HH˙φ˙ +3H2φ¨ G
5X
−
(cid:16) (cid:17)
where 4H2X2φ¨G5XX +4HX X˙ HX G5φX
− −
+2 2(HX)˙+3H2X G(cid:16)+4HXφ˙(cid:17)G , (6)
5φ 5φφ
= K(φ,X) G (φ,X)✷φ
LGG − 3 and ρm is(cid:2)the (nonrelativist(cid:3)ic) matter energy density,
+G4(φ,X)R+G4X (✷φ)2 ( µ νφ)2 while the scalar-fieldequation of motion is
− ∇ ∇
1
+G5(φ,X)Gµν∇µ∇ν(cid:2)φ− 6G5X (✷φ)3 (cid:3) S :=J˙+3HJ −Pφ =0, (7)
3✷φ( µ νφ)2+2( (cid:2)µ νφ)3 ,(2) where
− ∇ ∇ ∇ ∇
(cid:3) J := φ˙K +6HXG 2φ˙G
X 3X 3φ
−
with four arbitrary functions, K,G ,G , and G , of φ
3 4 5 +6H2φ˙(G +2XG ) 12HXG
and X := (∂φ)2/2. Here GiX stands for ∂Gi/∂X, and 4X 4XX − 4φX
hereafter w−e will use such a notation without stating so. +2H3X(3G5X +2XG5XX)
aTnhde sLcaaglarra-nfigeiladnpLorGtGioniss,aasmtihxetuRreicocif stchaelagrraRviatantdiotnhael −6H2φ˙(G5φ+XG5φX), (8)
EinsteintensorGµν areincluded. Noteinparticularthat Pφ := Kφ−2X G3φφ+φ¨G3φX
ifaconstantpieceispresentinG4thenitgivesrisetothe +6 2H2+(cid:16) H˙ G +6H(cid:17)X˙ +2HX G
Einstein-Hilbertterm. Weassumethatmatter,described 4φ 4φX
by Lm, is minimally coupled to gravity. 6H(cid:16)2XG5φφ+(cid:17) 2H3Xφ˙G5(cid:16)φX. (cid:17) (9)
−
The Lagrangian (2) gives the most general scalar-
An overdot denotes differentiation with respect to t and
tensor theory with second-order field equations in four
H =a˙/a.
dimensions. Themostgeneraltheorywasconstructedfor
thefirsttimebyHorndeski[7]inadifferentformthan(2),
and later it was rediscovered by Deffayet et al. [9] as a
III. PERTURBATIONS WITH RELEVANT
generalizationoftheGalileon. Theequivalenceofthetwo
NONLINEARITIES
expressions is shown by the authors of Ref. [13]. In this
paper, we employ the Galileon-like expression (2) since
We work in the Newtonian gauge, in which the per-
it is probably more useful than its original form when
turbed metric is written as
discussing the Vainshtein mechanism. The gravitational
and scalar-field equations can be found in the Appendix
ds2 = (1+2Φ)dt2+a2(1 2Ψ)dx2, (10)
of Ref. [13]. − −
We now replicate the cosmological background equa- withtheperturbedscalarfieldandmatterenergydensity,
tions in the theory (1) [13, 14]. For φ = φ(t) and the
φ φ(t)+δφ(t,x), (11)
background metric ds2 = dt2+a2(t)dx2, the gravita- →
tional field equations are − ρm ρm(t)[1+δ(t,x)]. (12)
→
It will be convenient to use
= ρm, (3) δφ
E − Q:=H , (13)
= 0, (4) φ˙
P
3
which is dimensionless. The(00)componentofthegravitationalfieldequations
Wewishtoknowthebehaviorofthegravitationaland reads
scalar fields on subhorizon scales sourced by a nonrel- a2 B
ativistic matter overdensity δ. To do so, we may ig- GT∇2Ψ = 2 ρmδ−A2∇2Q− 2a2H2 2Q(2)
nore time derivatives in the field equations, while keep-
B
ing spatial derivatives. We assume that Φ, Ψ, and Q 3 2Ψ 2Q ∂ ∂ Ψ∂ ∂ Q
−a2H2 ∇ ∇ − i j i j
are small, but nevertheless we do not neglect terms that
are schematically written as (∂2ǫ)2 and (∂2ǫ)3, where ∂ C1 (cid:0) (3), (cid:1) (23)
represents a spatial derivative and ǫ is any of Φ,Ψ, and −3a4H4Q
Q. This is because L2(t)∂2ǫ could be larger than (1) where
O
below certain scales, where L(t) is a typical length scale (3) := 2Q 3 3 2Q(∂ ∂ Q)2+2(∂ ∂ Q)3. (24)
associated with Gi which may be as large as the present Q ∇ − ∇ i j i j
Hubble radius. [As we will see, ((∂2ǫ)4) terms do not Finally, t(cid:0)he equ(cid:1)ation of motion for φ reduces to
O
appear.]
B
The traceless part of the gravitational field equations A0∇2Q−A1∇2Ψ−A2∇2Φ+ a2H02Q(2)
is given by
B
1 2Ψ 2Q ∂ ∂ Ψ∂i∂jQ
j( Ψ Φ A Q) −a2H2 ∇ ∇ − i j
Di 2BFT −GT2B− 1 2B B2 (cid:0) 2Φ 2Q ∂ ∂ Φ∂i∂jQ(cid:1)
= 1 Z j + 3 Y j + 3 Z˜j, (14) −a2H2 ∇ ∇ − i j
a2H2 i a2H2 i a4H4 i
B3 (cid:0) 2Φ 2Ψ ∂ ∂ Φ∂i∂jΨ(cid:1)
where we defined a derivative operator −a2H2 ∇ ∇ − i j
C0 (cid:0)(3) C1 (3) =0, (cid:1) (25)
1
j :=∂ ∂j δj 2, 2 =∂ ∂i, (15) −a4H4Q − a4H4U
Di i − 3 i ∇ ∇ i
where
and (3) := (2) 2Φ 2 2Q∂ ∂ Q∂i∂jΦ
i j
U Q ∇ − ∇
1 +2∂ ∂ Q∂j∂kQ∂ ∂iΦ. (26)
Z j := 2Q jQ ∂ ∂ Q∂j∂kQ+ δj(∂ ∂Q)2, i j k
i ∇ Di − i k 3 i k l
Equations (22), (23), and (25), supplemented with the
(16)
matter equations of motion (see Sec. VA) govern the
Y j := 2Φ jQ+ 2Q jΦ ∂ ∂ Q∂j∂kΦ (quasi)staticbehaviorofthe gravitationalpotentialsand
i ∇ Di ∇ Di − i k
2 the scalar field on subhorizon scales.
−∂i∂kΦ∂j∂kQ+ 3δij∂k∂lΦ∂k∂lQ, (17) Note that in deriving Eqs. (23) and (25) we have ne-
glected the “mass terms” (∂ /∂φ)δφ and (∂ /∂φ)δφ.
Z˜ij := −Q(2)DijQ+2∇2Q∂i∂kQ∂j∂kQ These contributions could be lEargerthan the hiSgher spa-
2∂i∂kQ∂j∂lQ∂k∂lQ tial derivative terms, and in that case the fluctuation δφ
−
2 willnotbeexcited. Wedonotconsiderthisrathertrivial
+ δj (∂ ∂Q)3 2Q(∂ ∂ Q)2 . (18)
3 i k l −∇ k l situationandfocus onthe casewhere∂ /∂φand∂ /∂φ
E S
(cid:2) (cid:3) can safely be ignored. Though restricted to the linear
The coefficients such as T, A1, B1, ... that appear in analysis, these terms have been considered in Ref. [14].
F
the field equations here and hereafter are defined in Ap- Let us end this section with a short remark. One may
pendix A. notice that all the terms (except δ) in Eqs. (22), (23),
Applyingtheoperator∂j∂i totheabovequantities,we and (25) can be written as total divergences, as
find
2Φ 2Q ∂ ∂ Φ∂i∂jQ=∂ ∂iΦ 2Q ∂ Φ∂i∂jQ ,
i j i j
1 ∇ ∇ − ∇ −
∂j∂iZij = 6∇2Q(2), (19) U(3) =∂i ∂iΦQ(2)−2∂jΦ∇2Q(cid:0)∂i∂jQ (cid:1)
1 (cid:16)
∂ ∂iY j = 2 2Φ 2Q ∂ ∂ Φ∂i∂jQ , (20) +2∂jΦ∂ ∂ Q∂k∂iQ .
j i 3∇ ∇ ∇ − i j k j
∂ ∂iZ˜j = 0, (cid:0) (cid:1) (21) Therefore, those equations can be integrated over(cid:17)a spa-
j i
tial domain , and then one is left with the boundary
where (2) := 2Q 2 (∂ ∂ Q)2. Thus,fromthetrace- terms and thVe enclosed mass,
i j
Q ∇ −
less equation (14) we obtain
(cid:0) (cid:1) δM =ρ (t) δ(t,x′)d3x′, (27)
m
2( Ψ Φ A Q)= B1 (2) ZV
∇ FT −GT − 1 2a2H2Q as a consequence of neglecting the “mass terms” men-
B tioned above. This fact will be used explicitly in the
+a2H32 ∇2Φ∇2Q−∂i∂jΦ∂i∂jQ .(22) next section.
(cid:0) (cid:1)
4
IV. SPHERICALLY SYMMETRIC The spherical symmetry allows us to write
CONFIGURATIONS
a−2 2Q = r−2(r2Q′)′,
Wenowwanttoconsiderasphericallysymmetricover- ∇
density on a cosmological background. For this purpose a−4 2Φ 2Q ∂i∂jΦ∂i∂jQ = 2r−2(rΦ′Q′)′,
∇ ∇ −
it is convenient to use the coordinate r = a(t) δijxixj. (cid:0) a−6 (3(cid:1)) = 2r−2 Φ′(Q′)2 ′,
We are primarily interested in scales much smaller than U
p
the horizon radius, rH 1. Under this circumstance (cid:2) (cid:3)
the background metric m≪ay be written as ds2 dt2+ where a prime denotes differentiation with respect to r.
dr2 +r2dΩ2, where dΩ2 is the line element of≃th−e unit The gravitational field equations and scalar-field equa-
two-sphere. tion of motion can then be integrated once, leading to
Ψ′ Φ′ Q′ β Q′ 2 β Φ′Q′
c2 α = 1 +2 3 , (28)
h r − r − 1 r H2 r H2 r r
(cid:18) (cid:19)
Ψ′ Q′ 1 δM(t,r) β Q′ 2 β Ψ′Q′ 2 γ Q′ 3
2 3 1
+α = 2 , (29)
r 2 r 8π r3 − H2 r − H2 r r − 3H4 r
GT (cid:18) (cid:19) (cid:18) (cid:19)
Q′ Ψ′ Φ′ β Q′ 2 β Ψ′Q′ β Φ′Q′ β Φ′Ψ′ γ Q′ 3 γ Φ′ Q′ 2
0 1 2 3 0 1
α α α = 2 + + + + + ,
0 r − 1 r − 2 r "−H2 (cid:18) r (cid:19) H2 r r H2 r r H2 r r H4 (cid:18) r (cid:19) H4 r (cid:18) r (cid:19) #
(30)
where we defined A. G4X =0, G5 =0
r
δM(t,r)=4πρm(t) δ(t,r)r′2dr′, (31) A simple example for which nonlinear terms can op-
Z erate is the model with G = 0 = G and G = 0,
4X 5 3X
6
c2 := / , and dimensionless coefficients i.e.,
h FT GT
α (t):= Ai, β (t):= Bi, γ (t):= Ci. (32) L=G4(φ)R+K(φ,X)−G3(φ,X)✷φ. (37)
i i i
GT GT GT In this case, we have β1 = β2 = β3 = γ0 = γ1 = 0 and
Note that ch is the propagation speed of gravitational c2h =1. We also have the relation
waveswhichmaybedifferentfrom1ingeneral[13]. Note α
1
also that in deriving Eqs. (28)–(30) we have set the in- β0 = +α2 (=0). (38)
2 6
tegration constants to be zero, requiring that Φ′ =Ψ′ =
Q′ =0 is a solution if δM =0. The Lagrangian (37) corresponds to a nonminimally
Forsufficientlylarger,wemayneglectallthenonlinear coupled version of kinetic gravity braiding [15], and has
terms in the above equations. The solution to the linear been studied extensively in the context of inflation [16]
equations is given by anddark energy/modifiedgravity[17–19]. Of course the
nonminimalcouplingcanbeundonebyperformingacon-
Φ′ = c2hα0−α21 µ, (33) formal transformation, but in the present analysis we
α +(2α +c2α )α r2 have no particular reason to do so. Note that even in
0 1 h 2 2
Ψ′ = α0+α1α2 µ, (34) tvhaetucraeseatofthGe4l=evceolnsotftthheesficaelladreφquisatcioounp,lewdhticohthsiegncuarls-
α +(2α +c2α )α r2
0 1 h 2 2 “braiding.”
α +c2α µ
Q′ = 1 h 2 . (35) Using Eqs. (28) and (29), Eq. (30) reduces to a
α0+(2α1+c2hα2)α2r2 quadratic equation
where we defined µ := δM/8π T. In this regime, the Q′ 2 2Q′ µ
parametrized post-Newtonian paGrameter γ is given by B + =2 , (39)
H2 r r Cr3
(cid:18) (cid:19)
α +α α
γ = 0 1 2, (36) where
c2α α2
h 0− 1 4β α +α
0 1 2
:= , := . (40)
which in general differs from unity. B α +2α α +α2 C α +2α α +α2
0 1 2 2 0 1 2 2
5
Equation (39) can easily be solved to give With G = 0, the problem reduces to solving a cu-
5X
bic equation one can handle. Indeed, using Eqs. (28)
Q′ H2 2 µ and (29) to remove Φ′ and Ψ′ from Eq. (30), we obtain
r = B r1+ HB2Cr3 −1!. (41) the cubic equation for Q′:
µ
At short distances, r3 r3 := µ/H2, one finds (Q′)3+ H2r(Q′)2+ C1H4r2 H2 Q′
≪ ∗ BC C2 2 − Cβr
(cid:18) (cid:19)
H 2 µ H4 µ
Q′ BC . (42) Cα =0, (50)
≃ r − 2
Br
In order for this solution to be real, we require where r-independent coefficients are defined as
>0 G (XG +G )>0. (43) α +c2α
BC ⇔ 3X 3X 4φ := 1 h 2 ,
Cα 2β β +c2β2 γ
In this regime, the metric potentials are given by 1 2 h 2 − 0
β +c2β
:= 1 h 2 ,
G δM H(α +α ) 2 G δM Cβ 2β β +c2β2 γ
Φ′ N 1 2 BC N , (44) 1 2 h 2 − 0
≃ r2 − B r r := α0+(2α1+c2hα2)α2,
G δM Hα 2 G δM C1 2β β +c2β2 γ
Ψ′ N 2 BC N , (45) 1 2 h 2 − 0
≃ r2 − B r r := 2β0+3 α1β2+α2β1+c2hα2β2 . (51)
where C2 2(2(cid:0)β1β2+c2hβ22−γ0) (cid:1)
1 We note the expressions for Ψ′ and Φ′ in terms of Q′:
8πG := . (46)
N
2G
4
µ Q′2
If (1), the typical length scale r∗ (µ/H2)1/3 Φ′ = c2hr2 −(α1+c2hα2)Q′−(β1+c2hβ2)H2r,
BC ∼ O ∼
can be estimated using Eq.(46) as µ Q′2
Ψ′ = α Q′ β . (52)
µ 1/3 H 2/3 δM 1/3 r2 − 2 − 2H2r
0
120 pc, (47)
H2 ≃ H M LinearizingEq.(50)atr3 ( / )µ/H2,oneobtains
(cid:16) (cid:17) (cid:18) (cid:19) (cid:18) ⊙(cid:19) the solution (35) as expecte≫d. CTβhiCs1will be matched to
where H0 =70km/s/Mpc. one of the following three solutions at short distances:
Note that in this case the Friedmann equation can be
written as µ µ H2r
Q′ +H , H , Cα . (53)
β β
≃ C r − C r − 2
3H2 =8πGcos(ρm+ρφ), (48) r r Cβ
If simply = (1),2 the two regimes
w6Xheφ˙rHe GGcos =2X1/G16πG64H=φ˙GGN .aTndhuρsφ,t=he2eXffeKctXiv−egKrav+- are connecCtαed∼atCβar∼ouCn1d∼r ∼C2r∗, wOhere
3X 3φ 4φ
− −
itationalcouplinggoverningshort-distancegravityis the 2 µ 1/3 µ
same as the one in the Friedmann equation. r := Cα for Q′ H ,
∗ 2 H2 ≃± Cβr
The Vainshtein mechanism successfully screens the ef- (cid:18)C1Cβ (cid:19) r
fect of the fluctuation δφ, so that the two metric po- µ 1/3 H2r
r := Cβ for Q′ Cα . (54)
tentials coincide and exhibit the Newtonian behavior at ∗ − H2 ≃− 2
leadingorder. However,G istime-dependent sinceitis (cid:18) C1 (cid:19) Cβ
N
a function ofthe time-dependent field φ(t), whichmeans If > 0 and > 0, the solution with the bound-
β 1
C C
thattheVainshteinmechanismcannotsuppressthetime ary condition (35) at large r can be matched either to
variationof GN in a cosmologicalbackground. This fact Q′ +H βµ/r or to Q′ H βµ/r. We call
was first noticed in Ref. [20]. We will discuss this point this≃situationCCase I. This is p≃oss−ible foCr ( , ) in the
2 α
further in the next subsection. shadedregiponinFig.1. For >0(repspectiCvelCy <0),
α α
C C
the short-distance solution is given by +H µ/r (re-
β
C
spectively H µ/r). Outside this region one cannot
B. G5X =0 − Cβ p
p
Let us consider a class of models with G = 0. In
5X
thiscase,weseethatβ3 =γ1 =0. Fortheothernonzero 2 Thisisprobablythe mostnatural caseifoneconsiders amodel
coefficients we have the following relations: thataccounts forthepresentcosmicaccelerationandthecoeffi-
cients are evaluated at present time, because in that case there
c2h =1+2β1(6=1), β1+β2+2γ0 =0. (49) isonlyonetypicallengthscaleL=H0−1.
6
FIG. 1: Relation between the coefficients and the short- FIG.3: Coefficientsforwhicharealsolutionexists(CaseII).
distance solution for Case I. In the shaded region one gets
a real solution whose behaviorat short distances is noted.
1.0´10-10
r2Y'
7.0´10-11
r2F'
5.0´10-11
3.0´10-11 r2Q'
2.0´10-11
1.5´10-11 Γ=1 Γ¹1HEq.36L
1.0´10-11
10-5 10-4 0.001 0.01
Hr FIG.4: Coefficientsforwhicharealsolutionexists(CaseII).
FIG. 2: r2Φ′, r2Ψ′, and r2Q′ as a function of r. H−1 = 1. where
c2h =1.2, α1 =α2 =1, α0 =2, β2 =2, µ=10−10. Inside the
Vainshteinradiusγ =1isreproduced,buttheconcretevalue C = −β12−c2hγ0 , (56)
of the radius depends on the model under consideration (see Φ 2β β +c2β2 γ
1 2 h 2 − 0
Eqs. (47) and (54)).
β β γ
1 2 0
C = − . (57)
Ψ 2β β +c2β2 γ
1 2 h 2 − 0
Although the coefficients look apparently different, now
find a solution that is real for r (0, ). A typical
we use the relations (49) for the first time to show that
∈ ∞
behavior of the Case I solution is plotted in Fig. 2. If
QCβ′C1 <(0,/the)nHt2hre/2s.oluWtieoncacllanthbisesimtuaattcihoendCoansley ItIo. CΦ =CΨ = C2β, (58)
α β
≃ − C C
Thisispossiblefor( , )intheshadedregioninFigs.3
C2 Cα i.e., thetwometricpotentialsactuallycoincide. Wethus
( > 0, < 0) and 4 ( < 0, > 0). Outside this
Cβ C1 Cβ C1 obtain the Newtonian behavior
region no real solutions can be found. If < 0 and
β
C1 <0 then no real solutions can be found eCither. Φ′ ≃Ψ′ ≃ GNr2δM, GN := 8CπΦ (>0). (59)
Let us then evaluate the metric perturbationsfor each GT
solution Q′. We begin with the Case I, Q′ µ/r. It is interesting to note that the above conclusion holds
β
≃± C
The metric potentials at short distances are given by even for the generic propagation speed of gravitational
p waves, c2 =1. Explicitly, one finds
h 6
C δM C δM 1
Φ′ ≃ 8πGΦT r2 , Ψ′ ≃ 8πGΨT r2 , (55) 8πGN = 2(G4−4XG4X −4X2G4XX +3XG5φ).(60)
7
Asinthecaseoftheprevioussubsection,G isingen- For the solution Q′ ( / )H2r/2, we find
N α β
≃− C C
eraltime-dependent,asitisafunctionoftime-dependent
φ(t) and X =φ˙2(t)/2. We thus illustrate how the Vain- Φ′ c2h δM + (r), Ψ′ 1 δM + (r),(67)
shtein mechanism fails to suppress the time variation of ≃ 8π r2 O ≃ 8π r2 O
T T
G G
G in a cosmological background within the context of
N
implying that the parametrized post-Newtonian param-
some generic scalar-tensortheories minimally coupled to
eter γ is given by γ =1/c2. Therefore, c2 is tightly con-
matter. The claim was originally suggested using the h h
strained from solar-system tests in this case: 1 γ <
Einstein frame action in Ref. [20]. Here we explicitly 2.3 10−5 [23]. | − |
give the concrete formula with which one can evaluate ×
When the coefficients , , , and have hi-
the time variation of GN for a given model. Cα Cβ C1 C2
erarchies in their values, we find a variety of solutions
If G happens to vary very slowly, we can say that
N
to Eq. (50) on an intermediate scale between the linear
the usual Newtonian gravity is reproduced in the vicin-
regime at large r and the small r limit of Case I or Case
ity of the source. However, in general, one expects that
II. The details are summarized in Appendix B, which
G varies on cosmological time scales. The time varia-
N
tion G˙/G isconstrainedfromlunarlaserrangingexper- could be potentially confronted with observations.
| |
iments to be G˙ /G <0.02H [21].
N N 0
| |
Atthisstageitisinterestingtolookatthebackground
C. G5X 6=0
evolution for the models with G = 0. The (modified)
5X
Friedmann equation (3) can be written as
Let us finally discuss the most general case where all
3H2 =8πGcos(ρm+ρφ), (61) the coefficients in Eqs. (28)–(30) are nonzero. Although
onecanstilleliminateΦ′andΨ′togetanequationsolely
where the gravitationalcoupling in the Friedmannequa-
in terms of Q′, the resulting equation will be a sextic
tion read off from the above exactly coincides with the
equation. This hinders us from analyzing a variety of
expression for G ,
N possible solutions in detail. However, for β = 0 and
3
6
G =G , (62) γ = 0 one can show that there is no solution such that
cos N Φ1′ 6 Ψ′ 1/r2 on sufficiently small scales. To show
and this≃, one s∼ubstitutes Φ′ Ψ′ 1/r2 to Eq. (28). The
≃ ∼
ρ := 2XK K 2XG second term in the right-hand side can be compensated
φ X 3φ
− − by the other provided that Q′ r or Q′ 1/r2. If Q′
+6H Xφ˙G3X 2Xφ˙G4φX φ˙G4φ . r, one cannot find a term tha∼t compen∼sates the fort∼h
− −
(cid:16) (cid:17) term in the right-handside of Eq. (30) which behaves as
The situation here is the same as what we have seen
Φ′Ψ′/r2 1/r6. If Q′ 1/r2, then one cannot find a
in the previous subsection. We refer to a constraint in ∼ ∼
term that compensates the last term in the right-hand
Ref. [22], obtained by translating the big bang nucle-
side of Eq. (29). Thus, there is no consistent solution
osynthesis (BBN) bound on extra relativistic degrees of
with Φ′ Ψ′ 1/r2 on sufficiently small scales. This
freedom, as ≃ ∼
implies that the typical length scale associated with B
3
1 GN|BBN .0.1, (63) and C1 must be as small as O(100 µm) [24], though it
− G is uncertain whether or not we can have the Newtonian
(cid:12)(cid:12) N|now (cid:12)(cid:12) behavior of the potentials on intermediate scales.
whereG (r(cid:12)espectively,G(cid:12) )isevaluatedatthe
N|BBN (cid:12) (cid:12)N|now
time of BBN (respectively, today).
Having thus seen that the Newtonian behavior is re- V. APPLICATIONS
produced with time-dependent G , let us then evaluate
N
leading order corrections to the potentials. In this case
A. Evolution of density perturbations
we need to keep the subleading term in Q′:
µ H2( 2 )r Since it is assumed that matter is minimally coupled
Q′ H + Cα− CβC2 . (64)
β toφ,nomodificationismadefortheenergyconservation
≃± C r 4
r Cβ equation and the Euler equation for matter. Therefore,
From this we obtain the corrections ∆Φ′ = Φ′ the nonlinear evolution equation for δ is given by
G δM/r2 and ∆Ψ′ =Ψ′ G δM/r2 as −
N N
− 4 δ˙2 2
β 2G δM δ¨+2Hδ˙ =(1+δ)∇ Φ. (68)
∆Φ′ = H α + 2 ( 2 ) N ,(65) − 31+δ a2
2 α β 2
∓ 2 C − C C r
(cid:20) Cβ (cid:21)r However, as we have seen in the previous section, the
β +c2β
∆Ψ′ = H α +c2α + 1 h 2 ( 2 ) relation between Φ and δ is modified. One may tackle
∓ 1 h 2 2 Cα− CβC2
(cid:20) Cβ (cid:21) the nonlinear equations using the perturbative approach
2G δM (e.g., [25]). We derive the Fourier transform of the non-
N
.(66)
linear equations in Appendix C.
× r
r
8
For simplicity, we here consider the cases G = 0 =
4X
HaL G andG =0asdescribedinIVAandIVB.Although
2.0 5 5X
the mass is proportional to r, one can check that three
HbL solutionsatshortdistancesremainthe sameasEq.(53).
1.5 To see the effects of modification of gravity, we consider
2N the velocity V(r) of a test particle in a circular motion
V
2(cid:144) withradiusr,V2(r)=rΦ′,whichreducesto2σ2 inNew-
V
1.0 tonian gravity. In the generalized model, V2(r) depends
ontheradiusr,whoseasymptoticbehaviorcanbefound,
0.5 HcL G
V2 effV2 (73)
≃ G N
0.001 0.01 0.1 1 10 100 N
r Hh-1MpcL for large r, and
V2 V2 (74)
FIG. 5: The ratio of thecircular speed V2 of a test particle ≃ N
in the generalized model to that in Newtonian gravity, as a for small r, where V2 = 2σ2. A typical behavior of
N
functionofradiusr(h−1Mpc). WeusedtheHubbleparameter the circular speed divided by the one in general rela-
at present H = 100 hkm/s/Mpc where h = 0.7, and the tivity, V2/V2, is demonstrated in Fig. 5. The line (a)
N
velocity dispersion σ = 100 km/s. Each line corresponds to represents the minimally coupled model, corresponds to
thefollowingcases: (a)c2h =1,α0 =−2.0,α2 =1.0,β0 =1.0, G = M2/2 and G = 0. The lines (b) and (c) show
α1 =β1 =β2 =γ0 =0.0, (b) c2h =0.9, α0 =−2.0, α1 =0.1, th4e modePllwith G 5= 0 and the parameters in (b) and
5X
α2 = 0.9, β0 = 0.9, β1 = −0.05, β2 = 0.1, γ0 = −0.025, (c) (c) are chosen so that the solutions at short distances
c2h =0.9, α0 =2.0, α1 =1.0, α2 =0.1, β0 =1.0, β1 =−0.05, become Q′ ( / )H2r/2 and Q′ +H µ/r,
β2 =0.1, γ0 =−0.025. Dotted and dashed lines are given by ≃ − Cα Cβ ≃ Cβ
respectively. As one can see in Fig. 5, the Vainshtein
Eq. (73) and (74), respectively.
p
radius r (1) Mpc for σ =100km/s.
∗
∼O
For spherical perturbations, 2Φ/a2 can be expressed
in terms of δ using the results i∇n the previous section. It VI. CONCLUSION
follows that
Based on the most general scalar-tensor theory with
2
∇ Φ 4πG ρ δ (69) second-orderfieldequations,wehavestudiedmetricper-
a2 → eff m
turbations on a cosmological background under the in-
where the effective gravitational coupling for large-scale fluence of the Vainshtein screening mechanism. We have
perturbations (but well inside the Hubble horizon) is derivedthe perturbation equations with relevantnonlin-
earitiesbytakingintoaccounttheeffectsofcosmological
G := 1 c2hα0−α21 , (70) background. WehaveclarifiedhowtheVainshteinmech-
eff 8π α +(2α +c2α )α anism operates in the two subclasses: (i) G =G =0
GT 0 1 h 2 2 4X 5
and (ii) G = 0. The situation in the first case
5X
and
G = G = 0 is very similar to the Vainshtein mech-
4X 5
2 anism in the Galileon theory, which contains only X(cid:3)φ
∇a2 Φ→4πGNρmδ (71) in the Lagrangian. However,the second case G5X =0 is
consideredforthefirsttimeinacosmologicalbackground
for small-scale ones. in this paper. We have explicitly shown that below
the Vainshtein scale r there are three possible solutions
∗
for Q′: Q′ H µ/r and Q′ ( / )H2r/2.
β α β
B. Halo Model ≃ ± C ≃ − C C
We have found that two metric perturbations coincide
p
well inside the Vainshtein radius r in the first case,
∗
We considerasimple halomodelto investigateachar- while the parametrized post-Newtonian parameter γ is
acteristic feature of general second-order scalar-tensor related to the propagation speed of gravitational waves,
theories [26]. For simplicity, let us assume the density c2,wellinsider inthe secondcase. Inbothcases,New-
h ∗
of matter follows ton’s constant G , its time variation G˙ /G , and the
N N N
| |
σ2 parametrized post-Newtonian parameter γ can be con-
δρ(r)= , (72) strained from BBN and the experiments such as lunar
2πG r2
N laser ranging. These could provide powerful constraints
where σ is the parameter of the velocity dispersion. In on the most general scalar-tensor theories. In the case
this model we have δM(r) = 2σ2r/G . This model is G = 0, we have demonstrated that the inverse-square
N 5X
6
the singular isothermal sphere in Newtonian gravity. law cannot be recoveredat sufficiently small r.
9
Acknowledgments No. 21244033 and JSPS Core-to-Core Program “Inter-
national Research Network for Dark Energy”. R.K. ac-
ThisworkwassupportedinpartbyJSPSGrant-in-Aid knowledges support by a research assistant program of
for Research Activity Start-up No. 22840011(T.K.), the Hiroshima University.
Grant-in-Aid for Scientific Research No. 21540270 and
Appendix A: Coefficients in the field equations
Here we summarize the definitions of the coefficients in the field equations:
:= 2 G X φ¨G +G , (A1)
T 4 5X 5φ
F −
h (cid:16) (cid:17)i
:= 2 G 2XG X Hφ˙G G , (A2)
T 4 4X 5X 5φ
G − − −
Θ := hφ˙XG +2HG (cid:16)8HXG 8HX(cid:17)i2G +φ˙G +2Xφ˙G
3X 4 4X 4XX 4φ 4φX
− − −
H2φ˙ 5XG +2X2G +2HX(3G +2XG ), (A3)
5X 5XX 5φ 5φX
−
Θ˙ (cid:0)Θ ˙T(cid:1) +
A := + + 2 2G E P, (A4)
0 H2 H FT − GT − H − 2H2
1 d
T
A := G + , (A5)
1 T T
H dt G −F
Θ
A := , (A6)
2 T
G − H
X
B := φ˙G +3 X˙ +2HX G +2XX˙G 3φ˙G +2φ˙XG
0 3X 4XX 4XXX 4φX 4φXX
H −
(cid:26)
(cid:16) (cid:17)
+ H˙ +H2 φ˙G +φ˙ 2HX˙ + H˙ +H2 X G +Hφ˙XX˙G 2 X˙ +2HX G
5X 5XX 5XXX 5φX
−
(cid:16) (cid:17) h (cid:16) (cid:17) i (cid:16) (cid:17)
φ˙XG X X˙ 2HX G , (A7)
5φφX 5φXX
− − −
(cid:16) (cid:17) (cid:27)
B := 2X G +φ¨(G +XG ) G +XG , (A8)
1 4X 5X 5XX 5φ 5φX
−
h i
B := 2X G +2XG +Hφ˙G +Hφ˙XG G XG , (A9)
2 4X 4XX 5X 5XX 5φ 5φX
− − −
B := Hφ˙X(cid:16)G , (cid:17) (A10)
3 5X
2X2
C := 2X2G + 2φ¨G +φ¨XG 2G +XG , (A11)
0 4XX 5XX 5XXX 5φX 5φXX
3 −
C := Hφ˙X(G +XG(cid:16) ). (cid:17) (A12)
1 5X 5XX
Appendix B: Intermediate regime for G5X =0
Here we would like to point out that there could be an interesting intermediate regime where the quadratic term
in Eq. (50) comes into play so that we have the solution
µ
Q′ H Cα . (B1)
≃± 2 r
r C2
This regime can be found in the range
1/3 1/3
µ µ
β α 2
C r C C (Case I),
2 H2 ≪ ≪ 2 H2
(cid:18)C2 (cid:19) (cid:18) C1 (cid:19)
2 µ 1/3 µ 1/3
Cβ r CαC2 (Case II). (B2)
CαC2H2! ≪ ≪(cid:18) C12 H2(cid:19)
10
This intermediate regime can be seen if or ( or ) is sufficiently large (small) compared with others. It is
α 2 β 1
C C C C
also interesting to see the behavior of the metric perturbations in the intermediate regime (B2). In this regime, the
metric potentials can be obtained by substituting Eq. (B1) into the relations (52),
µ
Φ′ c2 Cα (β +c2β ) , (B3)
≃ h− 2 1 h 2 r2
(cid:20) C2 (cid:21)
µ
Ψ′ 1 β Cα . (B4)
≃ − 22 r2
(cid:18) C2(cid:19)
The metric potentials in this regime differ from Eqs. (59) and (67), and the parametrizedpost-Newtonian parameter
γ,
2 β
2 β 2 α β
γ = C C − C C , (B5)
2c2 c2β β
hC2Cβ − h 2C2Cα− 1C2Cα
is not equal to unity.
We also notice another intermediate regime if is sufficiently large compared with the other coefficients. In this
α
regime Q′ becomes constant, C
H4 µ 1/3
Q′ Cα . (B6)
≃ 2
(cid:18) (cid:19)
This solution can be seen between Eq. (B1) and Eq. (53),
3 µ 1/3 µ 1/3
Cβ r Cα . (B7)
Cα2 H2! ≪ ≪(cid:18)C23 H2(cid:19)
The metric potentials in this regime are given by
µ µ
Φ′ c2 , Ψ′ . (B8)
≃ hr2 ≃ r2
Thus, the parametrizedpost-Newtonianparameteris γ =1/c2 in this regime. Ifthese regimes(B2)and (B7) include
h
oursolar-systemscales,itispossibletoconstraintheparametrizedpost-Newtonianparameterγ asintheformercase.
Appendix C: Equations in the Fourier space
In this Appendix we summarize the coupled equations for the evolution of the matter density perturbations in
Fourier space, which will be useful in the perturbative approach[25]. The matter density perturbations follow
∂δ 1
+ [(1+δ)v]=0, (C1)
∂t a∇·
∂v 1 1
+Hv+ (v )v= Φ, (C2)
∂t a ·∇ −a∇
where v is the velocity field. Assuming the irrotational fluid, we introduce the velocity divergence θ = v/(aH).
∇·
Then, due to the Fourier transform, the above equations can be rephrased as
1 ∂δ(p) 1 k k
+θ(p)= dk dk δ(3)(k +k p) 1+ 1· 2 θ(k )δ(k ), (C3)
H ∂t −(2π)3 1 2 1 2− k2 2 1
Z (cid:18) 2 (cid:19)
1 ∂θ(p) H˙ p2
+ 2+ θ(p) Φ(p)
H ∂t H2! − a2H2
1 1 (k k )k +k 2
= dk dk δ(3)(k +k p) 1· 2 | 1 2| θ(k )θ(k ). (C4)
−2(2π)3 1 2 1 2− k2k2 1 2
Z (cid:18) 1 2 (cid:19)
Similarly, Eqs. (22), (23), and (25) give
B 1
p2( Ψ(p) Φ(p) A Q(p))= 1 dk dk δ(3)(k +k p) k2k2 (k k )2 Q(k )Q(k )
− FT −GT − 1 2a2H2(2π)3 1 2 1 2− 1 2− 1· 2 1 2
Z
B 1 (cid:0) (cid:1)
+ 3 dk dk δ(3)(k +k p) k2k2 (k k )2 Q(k )Φ(k ), (C5)
a2H2(2π)3 1 2 1 2− 1 2 − 1· 2 1 2
Z
(cid:0) (cid:1)