Stability of spin-0 graviton and strong coupling in Horava-Lifshitz theory of gravity

0.48 MB·English

by  Anzhong Wang

Stability of spin-0 graviton and strong coupling in Horava-Lifshitz theory of gravity
Anzhong Wang∗
GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA
Qiang Wu†
Department of Physics, Zhejiang University of Technology, Hangzhou 310032, China
(Dated: January 24, 2011)
In this paper, we consider two different issues, stability and strong coupling, raised lately in the
newly-proposed Horava-Lifshitz (HL) theory of quantum gravity with projectability condition. We
find that all the scalar modes are stable in the de Sitter background, due to two different kinds of
effects,onefromhigh-orderderivativesofthespacetimecurvature,andtheotherfromtheexponen-
tialexpansionofthedeSitterspace. Combiningtheseeffectsproperly,onecanmaketheinstability
1 found in theMinkowskibackground neverappear evenfor small-scale modes, provided thattheIR
1 limit is sufficiently closed to the relativistic fixed point. At the fixed point, all the modes become
0 stabilized. We also show that the instability of Minkowski spacetime can be cured by introducing
2 masstothespin-0graviton. Thestrongcouplingproblemisinvestigatedfollowingtheeffectivefield
n theory approach, and found that it cannot be cured by the Blas-Pujolas-Sibiryakov mechanism,
a initially designed for the case without projectability condition, but might be circumvented by the
J Vainshteinmechanism,duetothenon-lineareffects. Infact,weconstructaclassofexactsolutions,
1 and show explicitly that it reduces smoothly to thede Sitterspacetime in the relativistic limit.
2
PACSnumbers: 04.60.-m;98.80.Cq;98.80.-k;98.80.Bp

]
h
t I. INTRODUCTION made of a , where
- i
p
e ai =∂iln(N), (1.1)
h
[ Properly formulating the theory of quantum gravity can cure the instability of the Minkowski background,
has been one of the main driving forces in gravitational where N is the lapse function. Ofcourse,this is possible
3 physicsoverpastseveraldecades[1]. Althoughthere are onlyfortheversionoftheHLtheorywithoutprojectabil-
v
8 severalverypromisingcandidates,includingLoopQuan- ity condition. Otherwise, N depends only on time, and
6 tum Gravity [2] and string/M theory [3], it is fair to see ai vanishes identically. By properly choosing the cou-
2 that our understanding of it is still very limited. Horava pling constants,the strongcoupling problemcan be also
0 recently proposed another alternative [4], motivated by addressedinsuchasetup[31]. Themainideaistointro-
9. the Lifshitz theory in solid state physics [5], for which it duce two energy scales, the UV cutoff scale M∗ and the
0 is often referred to as the Horava-Lifshitz (HL) theory. strong coupling scale Λk. If M∗ is low enough so that
0 It has various remarkable features, including its power-
1 countingrenormalizability[6], thedivergenceofits effec- M∗ <Λk, (1.2)
: tive speed of light in the ultra-violet (UV), which could ∼
v
then the linear perturbations become invalid before Λ
i potentiallyresolvethehorizonproblemwithoutinvoking k
X inflation[7]. Scale-invariantsuper-horizoncurvatureper- is researched, so that the strong coupling problem does
not show up at all [cf. Fig.5]. Applications of the BPS
r turbationscanalsobeproducedwithoutinflation[8–13],
a model to cosmology were studied recently in [32–34],
and dark matter and dark energy can have their geo-
whilesphericallysymmetricspacetimeswereinvestigated
metric origins [14, 15]. Furthermore, bouncing universe
in [35]. However, a price to pay in such a setup is the
canbeeasilyconstructedduetothehigh-orderderivative
enormous number of independent coupling constants. It
terms of the spacetime curvature [16–18]. For detail, we
can be shown that only the sixth-order derivative terms
refer readers to [19] and references therein.
in the potential are more than 60 [27]. It should be also
Despite all of these attractive features, the theory noted that giving up the projectability condition often
plagues with two serious problems: the instability of the causesthetheorytosuffertheinconsistenceproblem[36].
Minkowskibackground[4,11,20–22],andthestrongcou- However, Kluson recently showed that the Hamiltonian
pling [23–27]. To solvethese problems,variousmodifica- formalism of the BPS model is very rich, and that the
tions wereproposed[28, 29]. In particular,Blas, Pujolas algebra of constraints is well-defined [37].
and Sibiryakov (BPS) [30] found that inclusion of terms On the other hand, Sotiriou, Visser and Weinfurtner
(SVW) generalized the original version of the HL the-
ory to the most general form by giving up the detailed
balanceconditionbut stillkeepingthe projectabilityone
∗Electronicaddress: anzhong˙[email protected] [21]. In the SVW setup, the inconsistence problem does
†Electronicaddress: [email protected] notexist,andthegravitationalsectorcontainstotallyten
2
coupling constants, G, Λ, ξ, g (i = 2,3,...,8), where G II. STABILITY OF DE SITTER SPACETIME
i
andΛdenote,respectively,the4-dimensionalNewtonian
andcosmologicalconstants,ξ andgn’sareothercoupling To start with, in this section we shall first give a brief
constants, due to the breaking of the Lorentz invariance introductionto the SVW setup, andthenconsiderlinear
of the theory. Although the Minkowski background is perturbations in the de Sitter background.
stillnotstableinsuchasetup[11,21],deSitterspacetime
is [19]. Therefore, in such a setup, one may consider the
latter as its legitimate background,similar to what hap- A. The SVW Setup
pened in the massive gravity [38]. Moreover, the SVW
setup also faces the strong coupling problem [25]. Re-
Sotiriou, Visser and Weinfurtner (SVW) formulated
cently, Mukohyama showed that this problem could be
themostgeneralHLtheorywithprojectabilitycondition
solved by the Vainshtein mechanism [39], at least as far
but without the detailed balance [21]. Writing the 4-
as the spherically symmetric, static, vacuum spacetimes
dimensional metric in the ADM form,
are concerned [40].
ds2 = N2c2dt2+g dxi+Nidt dxj +Njdt ,
ij
−
(i, j =1,2,3), (2.1)
(cid:0) (cid:1)(cid:0) (cid:1)
the projectability condition requires that
In this paper, our purposes are two-folds. We first N =N(t), Ni =Ni(t,x), g =g (t,x). (2.2)
ij ij
generalizeourstudiesof[19]toincludehigh-orderderiva-
tiveterms,wherebyweshowexplicitlythatallthescalar Notethatin[11,13,45], theconstantc,representingthe
modes, including the short-scale ones, are stable in the speedoflight,wasabsorbedintoN. TheADMform(2.1)
de Sitter spacetime, by properly combining two different is preserved by the types of coordinate transformations,
kinds of effects, one from high-order derivatives of the
t f(t), xi ζi(t,x). (2.3)
spacetime curvature,and the other fromthe exponential
→ →
expansion of the de Sitter space. This is done in Sec.
Duetotheserestricteddiffeomorphisms,onemoredegree
II. Second, we systematically study the strong coupling
of freedom appears in the gravitational sector - a spin-
problem by following the effective theory approach [41],
0 graviton. This is potentially dangerous, and needs to
andshowclearlythattheBPSmechanismforsolvingthe
decouple in the Infrared (IR), in order to be consistent
strong coupling problem originally invented in the case
with observations. Similar problems are also found in
withoutprojectabilityconditioncannotbeappliedtothe
other modifiedtheories,suchas the massivegravity[43].
SVW case with projectability condition. This is consis-
Then, it can be shown that the most general action,
tentwiththeresultsfoundbyBPSusingtheStu¨ckelberg
which preserves the parity, is given by [21],
trick[42,43]. ThisisdoneinSec. III.InSec. IV,wecon-
structa classofnon-perturbativecosmologicalsolutions,
S =ζ2 dtd3xN√g +ζ−2 , (2.4)
andshowthatitreducessmoothlytothedeSitterspace- LK −LV LM
time(withrotation)intherelativisticlimit. Thisimplies Z (cid:0) (cid:1)
that the spin-0gravitonindeed decouples inthe IR limit where g = detgij, M denotes the matter Lagrangian
L
anddoesnotcauseadditionalproblem,oncenonlinearef- density, and
fects are included, a similar situationalso happens other
= K Kij (1 ξ)K2,
theories, such as the DGP model [44] 1. This can be LK ij − −
1
considered as the generalization of Mukohyama’s analy- = 2Λ R+ g R2+g R Rij
sis of the spherical case to the cosmological one. In Sec. LV − ζ2 2 3 ij
V, we present our main conclusions, and shown that the + 1 g R3+(cid:0)g RR Rij +g R(cid:1)iRjRk
Minkowski spacetime can also become stable by intro- ζ4 4 5 ij 6 j k i
ducing mass to the spin-0 graviton. 1 (cid:16) (cid:17)
+ g R 2R+g ( R ) iRjk , (2.5)
ζ4 7 ∇ 8 ∇i jk ∇
(cid:2) (cid:0) (cid:1)(cid:3)
where ζ2 = 1/16πG, and the covariant derivatives and
1 Itshouldbenotedthat, althoughinboththeoriesitisthenon- Ricci and Riemann terms are all constructed from the
linear interaction that makes the theories consistent with ob- three-metric g , while K is the extrinsic curvature,
ij ij
servations, there isafundamental difference between them: the
DGP model represents the modification of general relativity in 1
the IR, while the HL theory modifies general relativity mainly Kij = ( g˙ij + iNj + jNi), (2.6)
2N − ∇ ∇
in the UV. So, the coupling of the scalar graviton in the DGP
model is of order one, and the non-linear effects help to screen where N = g Nj. In the IR limit, all the high order
i ij
itscoupling toexternal sources. In theHLtheory, onthe other
curvature terms (with coefficients g , i = 2,...,8) drop
hand,thescalargravitonisself-interacting,anditisthisinterac- i
out, and the total action reduces when ξ = 0 to the
tionthatleadstothestrongcouplingproblem. Seetheanalysis
carriedinSec. III. Einstein-Hilbert action.
3
B. Linear Perturbations in de Sitter Background which has the general solution,
x=Ae−Ft/2e−iωt, (2.16)
With the conformal time η, the de Sitter spacetime is
given by ds2 = a2(η) dη2+δ dxidxj , where a(η) =
− ij where A is a constant. When >0, from the above ex-
1/(Hη)=eHt, and t denotes the cosmic time. F
− (cid:0) (cid:1) pressionwecanseethatthefreemodesωisexponentially
Linear scalar perturbations of the metric are given by
damped.
In the Minkowski background, we have a= Constant.
δg = a2(η)( 2ψδ +2E ),
ij − ij ,ij Without loss of generality, we can set a = 1. Then, we
δNi = a2(η)B,i δN =a(η)φ(η). (2.7) find that =0, and
F
Choosing the quasi-longitudinal gauge [11], k2 k4
ω2 = c 2k2 1 , (a=1). (2.17)
k −| ψ| − M2 − M4
φ=0=E, (2.8) (cid:18) A B(cid:19)
Therefore, if the scale of a mode is large enough so that
we find that the two gauge-invariant quantities defined
ω2 becomes negative, this mode is unstable. In particu-
in [11] reduce to, k
lar,withoutthehigh-ordercorrections,allthe modesare
Φ= B+B′, Ψ=ψ B, (2.9) unstable [11]. This is quite similar to the Jeans insta-
H −H
bility [47], for which there exists a characteristic Jeans
where =a′/a= 1/η, and ψ and B are given by [19] length λ = 1/k , where when scales are smaller than
H − J J
the Jeans length, the modes are stable. When scales are
(2 3ξ)ψ ′ = ξk2B, (2.10)
− k − largerthantheJeanslength,theybecomeunstable. The
ψk′′+2Hψk′+ωk2ψk =0, (2.11) largest instability occurs at
in the momentum space, where M2
k2 = B , (2.18)
k2 k4 M √r4+3+r2
ω2 = c 2k2 1+ + , (2.12)
k | ψ| − M2a2 M4a4 for which we have
(cid:18) A B (cid:19)
with c2ψ ≡ξ/(2−3ξ) and ωk(kM) = i|cψ|3M/2B r4+r2 r4+3+2 1/2
M Mpl , iΓ,B (cid:16) p (cid:17) (2.19)
A ≡ [2(8g +3g )]1/2 ≡
2 3
where
M
pl
M . (2.13)
B ≡ [4(8g7 3g8)]1/4 r4+3+r2,
− B ≡
1/4
Clearly, to have MA and MB real, we must assume that pMB (8g2+3g3)2
r = . (2.20)
8g2+3g3 0, 8g7 3g8 0, (2.14) ≡ MA 8g7−3g8 !
≥ − ≥
conditionsweshalltakeforgrantedintherestofthispa- The instability will grow significantly during a time t
per. Note that in writing the above expressions, we had tΓ Γ−1, or in other words, for any given time t0 o≥f
≡
assumed that ξ 0. When ξ = 0 the corresponding so- interest, only when t0 <tΓ, the growthof the instability
lutionsarestable≤,asshownin[19],sointhefollowingwe during t0 can be neglected.
shall not consider this case any further, and concentrate However, it is well-known that Jeans instability can
ourselves only to the case ξ < 0. Then, from the above be removed by Hubble friction in an expanding universe
one can see that the studies of stability of the de Sitter [47]. In the following we shall show that this is also
spacetime reduces to the studies of the master equation true in the HL theory. In particular, in the de Sitter
(2.11). Once ψ is known, from Eq. (2.10) one can find background, two important modifications occur: (a) For
k
Bk. Then, the gauge-invariantquantities Φk andΨk can any given k, ωk2 is always positive at sufficiently early
be read off from Eq.(2.9). From the latter one can see time, due to high-order corrections, as one can see from
thatthepropertiesofΦ andΨ areuniquelydetermined Eq.(2.12). (b) The damping force [= 2/η] is strictly
by ψ . In particular, ifkψ is nokt singular, so are Φ and non-negative and independent of HF. W−hen η 0− it
k k k →
Ψ . Therefore,inthe followingweshallconcentrateour- becomes infinitely large. For short-scalewaves,although
k
selves only on ψ . the spacetime canbe consideredas locally flat, the high-
k
To study the perturbations further, we notice that order derivatives can kick in at a very early time, if the
Eq.(2.11) is quite similar to an oscillator with a dissi- UVcutoffscaleisverylow. Astimeincreases,thedamp-
pative force [46], ing force becomes more and more important, and will fi-
F nally become dominant. Therefore, if the UV cutoff is
x¨+ x˙ +ω2x=0, (2.15) sufficiently low, by combining these two kinds of effects,
F
4
one worksin the IR (η 0−) and the other worksin the where X =√b. This yields
m
≃
UV (η 1), one might be able to stabilize the modes
| | ≫
ofboththeshort-andlarge-scales. Toseethatthisisin- MB Λstable, (2.29)
≤
deed possible here in the HL theory, we first notice that,
as the universe expands, a becomes larger and larger, where
and there exists a moment, say, η , at which ω2(η )=0,
c k c 1/2
where H 2 3/2 9
Λ r4+3 +r2 r4+ .
√2MB stable ≡ |cψ|(r4+4"(cid:16) (cid:17) (cid:16) 2(cid:17)#)
η (k)= . (2.21) (2.30)
c −Hk r2+√r4+4 1/2 It is remarkable to note that the condition (2.29) does
not depend on k. As a result, it is valid for any scale of
From this moment on, th(cid:0)e instability s(cid:1)tarts to develop
modes. In particular, once it is satisfied, the short-scale
until η = 0−, at which we have ω2(0−) = c 2k2.
Note that for the modes with k >k k ,−w|eψ|have modes become stabilized, too. Thus, for any given MA
stable andM , if ξ is sufficiently closedto its fixedpoint ξ =0
ηc(H0)/η0 < 1, that is, the instabi∼lity of these modes (at whBich one has c = 0) 2, Λ becomes large, and
|has not oc|cu∼rred within the age of our universe, where ψ stable
the condition (2.29) can be easily satisfied. At the fixed
η H−1 denotes the current conformal time of our
0 ≃ − 0 point ξ = 0, we have Λstable = , that is, now for any
universe, and ∞
given g (or equivalently for any given M and M ), all
i A B
the modes, of large- and small-scales, are stable.
1/2
k r2+ r4+4 M . (2.22) The above can be further seen from the following lim-
stable B
≡
iting cases. First, when r 1, we have
(cid:16) p (cid:17) ≪
Therefore, the only possible unstable modes that occur
within the age of our universeare those with their wave- 27 1/4 H
Λ = , (r 1). (2.31)
liefntghtehUs Vλ >cutλosfftabslcea,lwehMereisλslotawbleen≡ou1g/hk,stsaobleth.aHtotwheeveexr-, stable (cid:18) 4 (cid:19) |cψ| ≪
∗
ponentially damping force kicks in before these modes Thus,evenH istakentobe thecurrentHubble constant
become unstable, that is, if H ,Λ canstillbelarge,aslongerasc issufficiently
0 stable ψ
closed to its fixed point c =0.
ψ
(η) 2 ωk(η) 0, (η >ηc), (2.23) When r 1, on the other hand, we have
F − | |≥ ≃
then these unstable modes will be stabilized, and never
27 H
show up, where Λ = , (r 1). (2.32)
stable
5 c ≃
r | ψ|
M =min M , M . (2.24)
∗ A B
{ } Onceagain,ifξ issufficientlyclosedtoξ =0,Λ will
stable
Setting belarge,andtheconditionMB Λstable canbesatisfied
≤
for a given non-zero H.
M4 When r 1, Eq.(2.29) reduces to
X H2η2+ B , (2.25) ≫
≡ 3k2M2
A 2H
M , (r 1). (2.33)
the condition (2.23) can be written as A ≤ c ≫
ψ
| |
D(X) X3 3bX+2d 0, (2.26) TakingH =H ,onecanseethattheconditions(2.31)-
≡ − ≥ 0
(2.33) can be written as
where
H
M4r4 M < (1) 0 , (2.34)
b ≡ 9Ak4 r4+3 , ∗ ∼O |cψ|
(cid:16) (cid:17)
M6 9 27H2 which is equivalent to the condition that the instabil-
d B r6+ r2+ . (2.27)
≡ 27k6 2 2|cψ|2MB2! iwtyithfoinuntdheinagtheeoMf oiunrkouwnsivkeirbsaec[k2g5r,o3u1n]d. does not happen
Fig. 1 schematically shows the function D(X), from Note that, since the damping force always dominates
which we can see that the condition (2.26) holds when when η 0, even the instability develops, it always
| | ≪
occur within the period, η <η <η , and will be finally
1 2
D(X ) = 2MB12 r4+3 3
m 36 d+b3/2 k12 −
h (cid:0) (cid:1)
2
(cid:0) 9 (cid:1) 27H2 2 It should be noted that c cannot be too closed to zero. Oth-
+ r6+ r2+ 0,(2.28) ψ
2 2|cψ|2MB2! ≥ ethrwaniske,TChhoemreanskSoovtirraioduiaftoiornpowiinltlinimgpooustetsheivsetroeucos.nstraints. We

5
D(X)
2
1.8
F
1.6 G; k2/M2=0.01,k4/M4=0.01
A B
1.4 G; k2/M2=0.1,k4/M4=0.1
A B
1.2 G; k2/M2=1.0,k4/M4=1.0
A B
1
0 Xm X 0.8
0.6
− X
m D(X ) 0.4
m
0.2
0
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5
Hη
FIG. 2: The functions F ≡ F2/(4H2) and G≡−ω2/H2 for
k
FIG. 1: The function D(X) defined by Eq.(2.28). Note that dhiaffsebreenentscehnoticteosoonfeki2n/aMllA2thaencdaske4s/.MWB4h,ewnhke2r/eMω20a≡nd|ckψ4|/kM/H4
only the half plane X > 0 is valid, as one can see from A B
aresmall, F −G=0hastwosolutions η1 and η2 where ηc <
Eq.(2.25). η1 < η2 < 0, as shown by the doted line. When k2/MA2 and
k4/M4 arelarge, F isalwaysgreater thanG,andF−G=0
B
has no solution, as shown by the dash-dot line. The dashed
stabilized by , where ηc < η1 η2 <0 [as can be seen linerepresentsthecasewhereF−G=0hasonlyonesolution.
F ≤
from Fig. 2], where η are the two real and positive
1,2
roots of D(X) = 0. Thus, for a given time interval of k2M−2=0.01, k4M−4=0.01, ω=1.0
A B 0
interest, if the instability happens in a sufficient short 100
periodandwillnotgrowmuch,beforethedampingforce
ψk 0
takes over, then such an instability is still acceptable.
−100
However, the following results show that this is possible −20 −15 −10 −5 0
Hη
othnalyt for modes of short-scales. In fact, it can be shown 2x 105 k2MA−2=0.0001, k4MB−4=0.0001, ω0=1.0
ψk 0
∆η ≡ η2−η1 −−240 −35 −30 −25 −20 −15 −10 −5 0
= 2MA2r2 r4+3 cos 2θ+π , (2.35) x 1014 k2MA−2=0.000001, k4HMηB−4=0.000001, ω0=1.0
k2 3 6 1
r (cid:18) (cid:19)
ψk 0
where θ [π/2, π] and is defined as
∈ −1
−80 −70 −60 −50 −40 −30 −20 −10 0
Hη
1 9 27H2
cosθ = r6+ r2+ .
− r4+3 3/2 2 2|cψ|2MB2! Fk2I/GM.32:aTnhdekm4/eMtri4capreertvuerrbyastmioanllψ,kso,itnhathteFc−asGes=wh0ehreasbtowtho
(2.36) A B
Clearly,tohav(cid:0)eareal(cid:1)θ,thedenominatorofEq.(2.36)has solutions η1 and η2 where ηc <η1 <η2 <0, as shown in Fig.
2. It can be seen that ψ oscillates with almost a constant
to be greater or at least equal to the nominator, which k
amplitude when Hη ≪ 0, and then increases a bit, before it
is equivalent to M > Λ , where Λ is defined
B stable stable decaysrapidly to zero, where we had chosen c1 =c2.
in Eq.(2.29). Since ∆η k−2, one can see that, for any
∝
givenξ,g andH,∆η whenk2 0. Thus,to limit
i
→∞ →
the instability completely for anyk,one needs to require that ω2 c 2k2 as Hη 0−. Then, the asymptot-
ψ
that the condition Eq.(2.29) hold strictly. ical solut→ion−o|f E|q.(2.11) sat→isfies the equation,
Fig. 3showsthecasewhereM >Λ withafinite
B stable
and non-zero k, from which we can see that the mode is ψ ′′ 2ψ ′ c 2k2ψ =0, (2.37)
oscillating all the way down to Hη = (10), and then k − η k −| ψ| k
−O
grows a little bit, before it starts to decay. Since the
decayingrateis verylarge(inverselyproportionalto η, which has the general solution [19],
−
as one can see from Eq.(2.16) where = 2/η), it dies
away rapidly afterwards. F − ψk = c1 z 1 ez+c2 z+1 e−z,
−
Our above analytical analysis is further supported by (3ξ 2)z
B = (cid:0) − (cid:1) c ez (cid:0) c e−(cid:1)z , (2.38)
the following numerical calculations. Let us first notice k ξk2 1 − 2
(cid:16) (cid:17)
6
k2M−2=1.0, k4M−4=0.0, ω=1.0
A B 0
1 2
0.9 ω0=1.0 ψk 0
0.8 ω0=0.5 −−240 −35 −30 −25 −20 −15 −10 −5 0
0.7 ω0=0.2, k2M−2=0.1, k4HMη−4=0.0, ω=1.0
A B 0
0.6 2
ψk0.5 ψk 0
0.4 −2
−40 −35 −30 −25 −20 −15 −10 −5 0
0.3 k2M−2=0.01, k4HMη−4=0.0, ω=1.0
A B 0
0.2 2
0.1 ψk 0
0 −2
−8 −7 −6 −5 −4 −3 −2 −1 0 −40 −35 −30 −25 −20 −15 −10 −5 0
Hη Hη
FIG. 4: The metric perturbation ψ , defined by Eq.(2.7) in FIG. 5: The function ψ , defined by Eq.(2.7) for different
k k
the quasi-longitudinal gauge for different choices of ω0, with choices of k2/MA2 with ω0=1 and k4/MB4 =0.
k2/M2 = 0 and k4/M4 = 0. In all the three cases we have
A B
set c1 =c2.
k2M−2=1.0, k4M−4=0.0, ω=0.1
A B 0
2
where z c kη = (c k/H)e−Ht. Clearly, they are
allfinite≡as|Hψη| 0−−(o|rψt| ). In particular, B 0 ψk 0
k
and ψ c →c in the IR→l∞imit Hη 0−. Note→the −2
k 2 1 −80 −70 −60 −50 −40 −30 −20 −10 0
slitdiffer→enceb−etweenthetwoconstants→c definedhere Hη
1,2 k2M−2=0.1, k4M−4=0.0, ω=0.1
andtheonesusedin[19]. Fig. 4showsthefunctionψ (η) A B 0
k 10
with different choices of ω [ c k/H], from which we
0 ≡ | ψ| ψk 0
can see that the larger ω is, the faster ψ decays. That
0 k −10
is,small-scalemodesalwaysdecayfasterthanlarge-scale −80 −70 −60 −50 −40 −30 −20 −10 0
Hη
ones. k2M−2=0.01, k4M−4=0.0, ω=0.1
A B 0
To study the effects of the high-order curvatures, we 2
gradually turn on the fourth- and sixth- order correc- ψk 0
tions. In particular, Fig. 5 shows the case where
−2
ω0 ≡ |cψ|k/H = 1 and k4/MB4 = 0, with three differ- −80 −70 −60 −50 −H40η −30 −20 −10 0
ent values of the suppressed scale, M . From there one
A
can see that, when M is small, the perturbation oscil-
A FIG. 6: The function ψ , defined by Eq.(2.7) for different
k
lates many times before it starts to decay. As MA in- choices of k2/MA2 with ω0=0.1 and k4/MB4 =0.
creases, the oscillating times becomes less and less. The
same characteristics persist even for small values of ω ,
0
asshownby Figs. 6and7. The effects ofthe sixth-order
k2M−2=1.0, k4M−4=0.0, ω=0.01
term k4/MB4 are shown in Fig. 8. 2 A B 0
In all the cases, the perturbations will finally decay
ψk 0
exponentially for any given k, as the damping force is
independent of k and +2iω (η) 2ω F1 as −2
k 0 −200 −150 −100 −50 0
η 0−, since (0−) =F . Therefo≃re,Ffo−r any ≫given k Hη
th→e perturbatioFns always∞decay exponentially as η 0− k2MA−2=0.1, k4MB−4=0.0, ω0=0.01
2
≃
or (t ).
→∞ ψk 0
−2
−200 −150 −100 −50 0
Hη
III. STRONG COUPLING k2M−2=0.01, k4M−4=0.0, ω=0.01
A B 0
200
To understand the strong coupling problem, we shall ψk 0
restrict ourselves mainly to the perturbations in the −200
−200 −150 −100 −50 0
Minkowski background, because such obtained results Hη
can be easily generalized to the de Sitter background
[25]. Suchastudyalsohelpsustoclarifysomedifferences FIG. 7: The function ψ , defined by Eq.(2.7) for different
k
regarding to the strength of strong couplings, obtained choices of k2/MA2 with ω0=0.01 and k4/MB4 =0.
recentlyin[25–27,31]. Inaddition,thetreatmentinthis
7
k2M−2=0.01, k4M−4=1.0, ω=0.01 which are allowed by the gauge freedom (2.3), where α
A B 0
10 and β are arbitrary constants. Choosing
ψk 0
ζˆ
−10 ζ = , β = c α, (3.6)
−50 −40 −30 Hη −20 −10 0 Mpl cψ 1/2α | ψ|
k2M−2=0.01, k4M−4=0.1, ω=0.01 | |
A B 0
2 one finds that S(2) given by Eq.(3.2) can be written as,
ψk 0
−2
−50 −40 −30 Hη −20 −10 0 S(2) = dtˆd3xˆ ζˆ∗2+ ∂ˆζˆ 2 , (3.7)
2 k2MA−2=0.01, k4MB−4=0.01, ω0=0.01 Z (cid:16) (cid:0) (cid:1) (cid:17)
where ζˆ∗ dζˆ/dtˆ, while the cubic action S(3) takes the
ψk 0 ≡
form,
−2
−50 −40 −30 −20 −10 0
Hη 2
1 2 c 2 2 ∂ˆ∂ˆ
S(3) = dtˆd3xˆ | ψ| ζˆ ∂ˆζˆ +ζˆ i jζˆ∗
cFhIoGic.e8s:ofTkh4e/MfuB4ncwtiiotnh ωψ0k,=d0e.fi0n1eadnbdyk2E/qM.(A22.7=)0f.o0r1.different ΛS4C Z ∂ˆi ( 3 (cid:0) (cid:17) ∆ˆ !
ζˆ∗∂ˆζˆ ζˆ∗ 1 2 c 2 ζˆζˆ∗2 , (3.8)
−3 i ∆ˆ − − | ψ| )
(cid:16) (cid:17)
section can be applied to the HL theory both with and
without the projectabilitycondition. So, the conclusions where
obtained in this section are applicable to both cases.
2
When the background is Minkowski, without loss of ΛSC Mpl cψ 5/2α. (3.9)
≡ 3 | |
generality, we consider the metric perturbations [26],
Clearly, if one chooses α c −5/2, one finds that Λ
N =1, g =e2ζ(t,x)δ , N =∂ β(t,x), (3.1) ∝ | ψ| SC
ij ij i i willremainfinite whenc 0. Inthe followingwe shall
ψ
→
choose α=3 c −5/2/2, so that Λ =M .
which lead, respectively, to the following second- and ψ SC pl
| |
third-order actions, Requiring that the quadratic action S(2) be invariant
under the rescaling [41],
S(2) = Mp2lZ dtd3x −cζ˙2ψ2 + ∂ζ 2!, (3.2) tˆ→s−γ1tˆ, xˆ→s−γ2xˆ, ζˆ→sγ3ζˆ, (3.10)
(cid:0) (cid:1)
wefindthatγ =γ =γ . Withoutlossofgenerality,we
3 2c2 +1 1 2 3
S(3) = M2 dtd3x ζ ∂ζ 2 ψ ζζ˙2 can always choose γi = 1 (i = 1,2,3), so that Eq.(3.10)
plZ ( − (cid:16) 2c4ψ (cid:17) is identical to the relativistic scaling. Then, it can be
(cid:0) (cid:1) shown that all the terms in the cubic action (3.8) scale
2 ζ˙∂ ζ∂iζ˙+ 3ζ ∂i∂jζ˙ 2 , (3.3) ass1, whichmeans thatthese terms areirrelevantin the
−c4ψ i ∆ c4ψ (cid:18) ∆ (cid:19) ) low energy limit, but diverge in the UV, so they are not
renormalizable [41]. This indicates that the perturba-
where ∆ ∂i∂ , and tions break down when the coupling coefficients greatly
i
≡
exceed units. To calculate these coefficients, let us con-
1
∆β = ζ˙. (3.4) sider a process at the energy scale E, then we find that
−c2ψ allthetermsinthe cubicactionhasthe samemagnitude
as E, for example,
Note that the above expressions can be easily obtained
tfrhoemgrtahdeileinmtitteηrm→, a∞ia,iw,hinetrreoηduiscethdeinco[u3p0l]i,nwghciocnhstsahnotulodf dtˆd3xˆζˆ ∂ˆζˆ 2 ≃E. (3.11)
Z
notbeconfusedwiththeconformaltime,usedinthelast (cid:0) (cid:1)
section. In addition, in the Minkowski background the Since the action is dimensionless, all the coefficients in
conformal time is identical to the cosmic time t. Then, (3.8)musthavethedimensionE−1. Writingtheminthe
comparing Eqs.(3.1) and (2.7), we find that ζ = ψ and form,
−
β =a2B to the linear order of perturbations for a=1.
Toconsiderthestrongcouplingproblem,wefirstwrite λˆi
λ = , (3.12)
the quadratic action S(2) in a canonical form with unity i Λi
coupling constants. This can be done by the coordinate
transformations, where λˆi is a dimensionless parameter of order one, we
findthatthelowestscaleofΛ ’sisgivenbythelastthree
i
t=αtˆ, xi =βxˆi, (3.5) termsinEq.(3.8)andisofthe orderofΛ . Translating
SC
8
E E A. MB <MA
Inthiscase,wehaveM =M . Then,wecanseethat
∗ B
Λ
k the sixth-orderderivativetermwilldominate the fourth-
M order one. If we consider a process at the momentum
* scale k > M , then the first and last terms in S(2) will
B
be domi∼nant. Using the coordinate transformation (3.5)
and the rescaling of ζ = γζˆ, we first transform these
terms to the ones with unit coefficients. It can be shown
Λk M∗ that this can be realized by choosing
M M
α= Bβ3, γ = B c 1/2, (3.16)
ψ
c M | |
ψ pl
| |
(b) where β is arbitrary. Then, we obtain that
(a)
FIG. 9: Theenergy scales: (a) Λk <∼M∗;and (b) Λk >∼M∗. S(2) = dtˆd3xˆ ζˆ∗2+β4MB4 ∂ˆζˆ 2
Z M4β(cid:16)2 (cid:0) (cid:1)
B ζˆ∂ˆ4ζˆ+ζˆ∂ˆ6ζˆ , (3.17)
it back to the coordinates t and x, the corresponding − M2
A
(cid:17)
energy and momentum scales are,
and
Λ
Λ = SC c 5/2M ,
ω ψ pl 1 2
α ≃| | S(3) = dtˆd3xˆ β4M4ζˆ ∂ˆζˆ
Λk = ΛSC cψ 3/2Mpl, (3.13) Λ(SBC) Z ( B (cid:16) (cid:17)
β ≃| | 3 ∂ˆi
1 2 c 2 ζˆζˆ∗2 2ζˆ∗∂ˆζˆ ζˆ∗
which areconsistentwith the results obtainedin [30, 31] −2 − | ψ| − i ∆ˆ
(cid:16) (cid:17)
byusingtheStu¨ckelbergtrick(Seealso[25]),butslightly ∂ˆ∂ˆ 2
different from that given in [26]. +3ζˆ i jζˆ∗ +... , (3.18)
As c 0, these scales vanish, indicating that strong ∆ˆ ! )
ψ
→
coupling happens when c is very small. For processes
ψ
with momentum k > Λ , the problem becomes strong with
k
coupling, and non-lin∼ear effects are important and must M
betakenintoaccount. Mukohyamarecentlyshowedthat Λ(SBC) = Mpl |cψ|3/2. (3.19)
these effects make the spin-0 graviton finally decoupled, B
andtherelativisticlimitξ 0intheIRexistsforspher- The “...” in S(3) represents the cubic terms coming from
→
ically symmetric, static, vacuum spacetimes [40]. the high-order derivative corrections, such as f ζ2∂4ζ
1
It must be noted that the above analysis is valid only andf ζ2∂4ζ, where f andf are independent of c and
for M >Λ [cf. Fig.9(a)]. If 2 1 2 ψ
∗ k functions of the coupling constants g only. As a result,
∼ i
the limit, c 0, of these terms always finite, and have
M∗ <Λk, (3.14) nocontribuψtio→nstothestrongcouplingproblem. Infact,
∼
it can be shown that these terms are either relevant or
whichis the precisecondition for the BPSmechanismto
marginal (cf. the following analysis.). So, in the fol-
work [cf. Eq.(1.2)], then the high-order derivative terms
lowing, without loss of generality we shall ignore them.
becomeimportantbeforethestrongcouplingenergyscale
Then, under the re-scaling,
Λ reaches,and the above analysis is no longer valid [cf.
k
Fig.9(b)]. Including the high order derivatives, one finds
t s−3t, x s−1x, ζ s0ζ, (3.20)
that the quadratic action becomes, → → →
thefirstandlasttermsintheright-handsideofEq.(3.17)
ζ˙2 1
S(2) = M2 dtd3x + ∂ζ 2 ζ∂4ζ are unchanged, while the second and third terms scale
plZ −c2ψ − MA2 like s−4 and s−2, respectively. Therefore, these terms
(cid:0) (cid:1) arerelevantandsuper-renormalizable. Similarly,thefirst
1
+ ζ∂6ζ . (3.15) term in the cubic action S(3) of Eq.(3.18) scales as s−4,
M4
B (cid:19) while all the rest scales as s0, that is, the term ζˆ ∂ˆζˆ 2 is
Depending onwhether M <M orM >M , the low relevant, while the rest, the second, third and fourth in
B A B A
energy behavior will be different. In the following, let us S(3), are all marginal and strictly renormalizable(cid:0). T(cid:1)hus,
consider them separately. as the energyscale of the systemchanges,the amplitude
9
of these latter terms remain the same. That is, these Then, if we consider processes at the energy scale E, we
terms are equally important at all scales of energy, pro- findthat dtˆd3xˆζˆζˆ∗2 E1/4,sothatthe secondtermin
≃
vided that the condition (3.14) holds. Since they are all S(3) is suppressed by,
suppressed by the dimensionless quantity Λ(B), we can R
SC 4
see that in the present case the problem becomes strong Λ = Mpl c 6. (3.27)
coupling when cψ is very small, unless Λ(SBC) > 1, which ωˆ (cid:18)MA(cid:19) | ψ|
is equivalent to, ∼
Itcanbe shownthatthe thirdandfourthterms aresup-
M <M c 3/2. (3.21) pressed by the same factor. Transforming it back to the
B ∼ pl| ψ| (t,xi)-coordinates, we find that the energy and momen-
tum are suppressed, respectively, by
B. MB >MA 4
Λ M c 7/4
ωˆ pl ψ
Λ = = | | M ,
ω A
When MB > MA, we have M∗ = MA. Then, the α MA !
fourth-orderderivative term in the quadratic action S(2) 2
(Λ )1/2 M c 3/2
given by Eq.(3.15) will dominate the sixth-order term. ωˆ pl ψ
Λ = = | | M . (3.28)
k A
Then, the rescaling (3.5) and ζ =γζˆwith β MA !
α= 1 , β = 1 , γ = MA c 1/2, (3.22) Then, the condition (3.14) implies that MA <
|cψ|MA MA Mpl | ψ| Mpl|cψ|3/2, which, together with Eq.(3.21), can be wri∼t-
ten as
will bring the quadratic action (3.15) to the form,
M <M c 3/2. (3.29)
∗ pl ψ
S(2) = dtˆd3xˆ ζˆ∗2+ ∂ˆζˆ 2 ∼ | |
IfonetakestheMinkowskispacetimeasthelegitimate
Z (cid:16) (cid:0) 4(cid:1) background, as shown in [11, 21], it is not stable in the
M
ζˆ∂ˆ4ζˆ+ A ζˆ∂ˆ6ζˆ , (3.23) SVW setup, and one would require that the instability
− (cid:18)MB(cid:19) ! should not show up within the age of the universe,
while the cubic action takes the form, H
c < 0. (3.30)
ψ
| |∼ M∗
1 2
S(3) = ΛS(AC) Z dtˆd3xˆ(|cψ|2ζˆ(cid:16)∂ˆζˆ(cid:17) BimPpSliefosuMnd∗t<ha(t10t0hims,)t−o1g,eothreerqwuiivtahletnhtetcoondition (3.29),
3 ∂ˆi ∼
−2 1−2|cψ|2 ζˆζˆ∗2−2ζˆ∗∂ˆiζˆ∆ˆ ζˆ∗ ξ < H0 2/5 10−24. (3.31)
(cid:16) ∂ˆ∂ˆ 2(cid:17) | |∼(cid:18)M∗(cid:19) ≃
+3ζˆ i jζˆ∗ +... , (3.24)
∆ˆ ! ) Clearly, this raises the fine-tuning problem, as a natural
valueofξ inthe UVisexpectedtobeorderofone[4]. It
where is unclear by which kind of mechanism it can be driven
soclosedtoitsrelativisticvalueξ =0[31](Seealso[25]).
M
Λ(A) pl c 3/2. (3.25) Following [25], it can be easily generalized the above
SC ≡ MA | ψ| studies to the de Sitter background, and similar conclu-
sionswillbeobtained: (a)WhenM >Λ =M c 3/2,
Steimrmilsaroftothtehofosremgsivfen(gin)ζE2∂q.4(ζ3.a1n8)d, fth(eg“).ζ..2”∂4aζr,ewcuhbicihc thetheorybecomesstrongcoupling∗fo∼rprkocessesplw|itψh|en-
1 s 2 s ergies E Λ [See Eq.(3.13)]. (b) When M < Λ , the
are all finite in the limit ξ →0, so they are irrelevant to strong co≃uplinkg problem does not exist. How∗ev∼er,kif one
the strong coupling problem.
considers the studies of perturbations given in Sec. II as
Then, we find that the first and third terms in the
in the current universe, namely, H = H , then the sta-
right-hand side of Eq.(3.23) are unchanged, under the 0
bility conditionisthatofEq.(2.34), whichis the sameas
rescaling,
Eq.(3.30). Hence, the results obtained above also apply
t s−2t, x s−1x, ζ s1/2ζ, (3.26) to the de Sitter background with H = H0. Therefore,
→ → → it is concluded that the mechanism, M Λ , of solv-
∗ k
≤
forwhichthefirsttermintheright-handsideofthecubic ing the strong coupling problem invented in [30, 31] for
actionS(3) givenbyEq.(3.24)scalesass−3/2. Thus,this the HL theory without projectability condition, cannot be
term is relevant and super-renoralizable. The second, applied to the case with projectability condition.
third and fourth terms, on the other hand, are all scale Thus, in the SVW setup one may choose the de Sitter
ass1/2,sotheyareallirrelevantandnon-renormalizable. spacetime as its legitimate backgroundin order to avoid
10
the instability problem. In order to have a reasonable that, once nonlinear effects are taken into account, the
UVcutoffscaleM , wherenowM mustsatisfythe con- separation of scalar, vector and tensor become impossi-
∗ ∗
ditions, ble. This is well-knownin GR when we consider second-
order perturbations, where all the sectors of the first-
H
M c 3/2 <M < 0 , (3.32) order perturbations become the sources of the second-
pl ψ ∗
| | ∼ ∼ cψ orderones[50]. Takingtheseintoaccount,letusconsider
| |
the non-perturbative solutions of the type,
its IR limit has to be very closed to, if not precisely at,
the GR fixed point. Of course, with such a choice, the N = a(η), N =a2(η)n (t,x),
i i
theory is strong coupling. This will not be a problem, if
the relativisticlimit canbe obtainedafter the non-linear gij = a2(η)e−2ψ(t,x)δij. (4.2)
effectsaretakenintoaccount. Inthesphericallysymmet-
After simple but tedious calculations [cf. Appendix A],
ric, static, vacuum spacetimes, Mukohyama showedthat
wefindthefollowingexactsolutionsofthecorresponding
this is indeed the case [40]. In the following section, we
HL equations with a non-zero cosmologicalconstant Λ,
shallpresentaclassofexactsolutionsofthetheory,from
which one can show clearly that the relativistic limit ex-
3(2 3ξ) 1/2 1
ists, and the limited spacetime is exactly the (rotating)
a(η) = − ,
de Sitter spacetime of GR. − 2Λ η
(cid:18) (cid:19)
ψ = ψ , n =n ( y,x,0), (4.3)
0 i 0
−
IV. NON-PERTURBATIVE COSMOLOGICAL where ψ andn aretwointegrationconstants. Without
0 0
SOLUTIONS loss of generality, one can set ψ = 0 by rescaling of the
0
coordinatesxi (andredefinitionoftheconstantn ). The
0
In the DGP model of branes [48], Newtonian approx- constantn ,ontheotherhand,representstherotationof
0
imations lead to a Friedmann equation with a constant thespacetime,andcannotbegaugedaway,ascanbeseen
G˜ that is different from the Newtonian constant G by a fromthe followinganalysis. If one considersthe rotation
factor 4/3 [49], while the non-perturbative equation in as perturbations, one can see that it corresponds to the
the flat FRW universe with zero-cosmological constant sum of infinitely high order perturbation terms, some of
takes the form, which will become singular in the limit ξ 0, as it is
→
expectedfromtheanalysisgiveninthelastsection. But,
8πG
H2 = ρ m H, the analytical solutions themselves indeed have a finite
c
3 − and smoothy limit, ξ 0, as one can see from Eq.(4.3).
→
In particular, when ξ = 0, the above solutions reduce
where m is the graviton mass [44]. Clearly, when
c
to a rotating de Sitter spacetime. In fact, introducing
m 0, it reduces precisely to the Friedmann equa-
c
→ the cylindrical coordinates r and θ via the relations x=
tion in GR. This shows clearly that the spin-0 massive
rcos(θ) andy =rsin(θ), we find thatthe metric canbe
gravitondecouples,whenthe non-lineareffects aretaken
written in the form,
intoaccount,and,asaresult,thetheorysmoothlypasses
over to the GR limit.
1
In this section, we shall show that the same happens ds2 = dη2+dr2+dz2
ξ=0 ( Hη)2 −
here in the SVW setup, too. That is, when we do the (cid:12) − n
linearperturbationsofthedeSitterbackground,wehave (cid:12) +(dθ+n dη)2 , (4.4)
0
the strong coupling problem, as shown explicitly in the
o
lastsection. But,theexactlysolutionsofthetheoryhave
with H = 3/Λ, and n represents the angular velocity
asmoothyGRlimit. Asamatteroffact,thiscanalready 0
of the rotation.
be seen clearly if one simply looks at the corresponding p
Friedmann equation [19],
V. CONCLUSIONS
3 8πG 1
1 ξ H2 = ρ+ Λ. (4.1)
− 2 3 3
(cid:18) (cid:19) In this paper, we have considered two different issues
(The other independent equation is the well-known con- raised recently in the studies of the HL theory, the sta-
servationlawofenergyandmomentum, ρ˙+3H(ρ+p)= bility of background spacetime and strong coupling, by
0. Fordetail,see[11,13,21].) Fromtheaboveexpression payingmainattentionontheSVWsetup[21],whichrep-
we can see that replacing G in all the solutions obtained resents the most general HL theory with projectability
in GR by G˜ G/(1 3ξ/2), we shall obtain all the cos- condition. AlthoughtheMinkowskispacetimeisnotsta-
≡ −
mological (flat) solutions in the HL theory. ble in such a setup, the de Sitter spacetime is, due to
In the following, we shall go a little bit further, and two different kinds of effects: one is from the high-order
show that this is true also in the sense of non-linear per- derivatives of the spacetime curvature, and the other is
turbations of the de Sitter spacetime. Let us first note from the exponential expansion of the de Sitter space.