Cosmology in nonrelativistic general covariant theory of gravity

0.19 MB·English

by  Anzhong Wang

Cosmology in nonrelativistic general covariant theory of gravity
Anzhong Wang∗ and Yumei Wu†
GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA
(Dated: January 26, 2011)
Horava and Melby-Thompson recently proposed a new version of the Horava-Lifshitz theory of
gravity,in which thespin-0 graviton is eliminated by introducing aNewtonian pre-potential ϕ and
a local U(1) gauge field A. In this paper, we first derive the corresponding Hamiltonian, super-
momentumconstraints,thedynamicalequations,andtheequationsforϕandA,inthepresenceof
matterfields. Then,weapplythetheorytocosmology,andobtainthemodifiedFriedmannequation
and the conservation law of energy, in addition to the equations for ϕ and A. When the spatial
curvatureisdifferentfromzero,termsbehavinglikedarkradiationandstiff-fluidexist,fromwhich,
amongotherpossibilities,bouncinguniversecanbeconstructed. Wealsostudylinearperturbations
1 of the FRW universe with any given spatial curvature k, and derive the most general formulas for
1 scalar perturbations. The vector and tensor perturbations are the same as those recently given by
0 oneofthepresentauthors[A.Wang,Phys. Rev. D82,124063(2010)]inthesetupofSotiriou,Visser
2 and Weinfurtner. Applyingthese formulas to theMinkowski background,we haveshown explicitly
thatthescalarandvectorperturbationsofthemetricindeedvanish,andtheonlyremainingmodes
n
are themassless spin-2gravitons.
a
J
PACSnumbers: 04.60.-m;98.80.Cq;98.80.-k;98.80.Bp
5
2
I. INTRODUCTION restored.” Such an anisotropy between time and space
]
h can be easily realized when one writes the metric in the
t Arnowitt-Deser-Misner (ADM) form [19],
- Recently, Horava proposed a quantum gravity theory
p [1],motivatedbytheLifshitztheoryinsolidstatephysics
e [2],forwhichthetheoryisoftenreferredtoastheHorava- ds2 = −N2c2dt2+gij dxi+Nidt dxj +Njdt ,
h
[ Lifshitz (HL) theory. It is non-relativistic and power- (cid:0) (i, j(cid:1)=(cid:0)1,2,3). ((cid:1)1.2)
counting ultraviolet (UV)-renormalizable, and was ex-
3 pected to recover general relativity (GR) in the infrared Under the rescaling (1.1) with z = d = 3, a condition
v we shall assume in the rest of this paper, the dynamical
(IR) limit. HL theory has attracted a great deal of at-
9 variables N, Ni and g scale, respectively, as,
tention due to its several remarkable features, such as ij
8
0 the divergence of its effective speed of light in the UV, N N, Ni ℓ−2Ni, g g . (1.3)
ij ij
2 whichcouldpotentiallyresolvethehorizonproblemwith- → → →
9. out invoking inflation [3]. Scale-invariant super-horizon The gauge symmetry of the system are the foliation-
0 curvatureperturbationscanalsobeproducedwithoutin- preserving diffeomorphisms Diff(M, ),
0 flation[4–11],anddarkmatteranddarkenergycanhave F
1 their geometric origins [12, 13]. Furthermore, bouncing t˜=t f(t), x˜i =xi ζi(t,x), (1.4)
− −
: universe can be easily constructed due to the high-order
v
derivative terms of the spacetime curvature [14–16]. For for which the dynamical variables change as
i
X detail, we refer readers to [17] and references therein.
δg = ζ + ζ +fg˙ ,
r TheHLtheoryisbasedontheperspectivethatLorentz ij ∇i j ∇j i ij
a symmetry should appear as an emergent symmetry at δNi = Nk iζk+ζk kNi+gikζ˙k+N˙if +Nif˙,
∇ ∇
long distances, but can be fundamentally absent at high δN = ζk N +N˙f +Nf˙, (1.5)
k
energies [18]. With this in mind, Horava considered sys- ∇
tems whose scaling at short distances exhibits a strong where f˙ df/dt, denotes the covariant derivative
i
anisotropy between space and time, with resp≡ect to the∇3-metric g , and N = g Nk, etc.
ij i ik
From these expressions one can see that the lapse func-
x ℓx, t ℓzt. (1.1) tionN andtheshiftvectorN playtheroleofgaugefields
→ → i
of the Diff(M, ) symmetry. Therefore, it is natural to
In (d+1)-dimensional spacetimes, in order for the the- assume that NFand N inherit the same dependence on
i
ory to be power-counting renormalizable,it requires z spacetime as the corresponding generators, in addition
≥
d. At low energies, the theory is expected to flow to to the fact that the dynamical variables g should in
ij
z = 1, whereby the Lorentz invariance is “accidentally general depend on both time and space, that is,
N =N(t), N =N (t,x), g =g (t,x), (1.6)
i i ij ij
∗Electronicaddress: anzhong˙[email protected] which is clearly preserved by the Diff(M, ), and often
F
†Electronicaddress: yumei˙[email protected] referred to as the projectability condition.
2
Due to the restricted diffeomorphisms (1.4), one more takeexactlyits relativisticvalue λ=1. Note thatinGR
degree of freedom appears in the gravitational sector - the spacetime diffeomorphism symmetry, Diff(M),
a spin-0 graviton. This is potentially dangerous, and
needs to decouple in the IR regime, in order to be con- x˜µ =xµ ζµ(t,x), (µ=0,1,2,3) (1.9)
−
sistent with observations. Unfortunately, it was shown
that this might not be the case. In particular, the spin- also forces λ = 1 and protects this value from quantum
0 mode is not stable in the original version of the HL corrections.
theory[1]aswellasintheSotiriou,VisserandWeinfurt- At short distances, the theory exhibits a high
ner (SVW) generalization [7, 8]. Note that in both of anisotropy between time and space. As a result, the UV
these two versions, it was all assumed the projectability behaviorofthetheoryisdramaticallyimporved. Atlong
condition. In addition, these instabilities are all found distances, the theory is driven to an IR regime, where it
in the Minkowski background. Recently, it was found shares many features with GR. In particular, under the
that the de Sitter spacetime is stable in the SVW setup influences of the relevant terms, the scaling is naturally
[17]. So, one can take the latter as its legitimate back- isotropic with the relativistic value z = 1. Moreover,
ground,similarto whathappened inthe massivegravity sincetheextendedsymmetryforcesλ=1,intheIRlimit
[20]. However, the strong coupling problem still exists the action will be dominated exactly by the Einstein-
[21, 22], althoughit might be circumvented by the Vain- Hilbert terms in the ADM decomposition [19].
shteinmechanism[23],asrecentlyshowedinthespherical Inthispaper,weinvestigatethisnewversionoftheHL
[24] and cosmological[22] cases. theory. Specifically, in Sec. II we firstgivea briefreview
Onthe other hand, givingup the projectability condi- of it, and then derive the corresponding Hamiltonian,
tion,thatis,assumingthatthelapsefunctionN depends super-momentum constraints, the dynamical equations,
on both time and spatial coordinates, Blas, Pujolas and and the field equations for the Newtonian pre-potential
Sibiryakov(BPS)[25]foundthatinclusionoftermsmade ϕ and the U(1) gauge field A, in the presence of mat-
of ai, ter fields. The potential used in this paper is the one
constructed by SVW [7], which represents the most gen-
a =∂ ln(N), (1.7)
i i eral form, subjected to the assumptions that it respects
can cure the instability of the Minkowski spacetime. By the parity andits highestorder ofthe spatialderivatives
properlychoosingthecouplingconstants,thestrongcou- is six, the minimal requirement to have the theory be
pling problem [26–29] can be also addressed [30]. How- power-counting renormalizable [1]. In Sec. III, we ap-
ever, a price to pay is the enormous number of indepen- ply the theory to cosmology, and obtain the modified
dent coupling constants: only the sixth-order derivative Friedmann equation and conservation law of energy, in
terms in the potential are more than 60 [29]. It should addition to the equations for ϕ and A. In Sec. IV we
be also noted that giving up the projectability condition study the linearperturbationsofthe FRW universewith
often causes the theory to suffer the inconsistence prob- any given spatial curvature k, and present the general
lem [31]. Kluson recently showed that the Hamiltonian formulasfor scalarperturbations. The vectorandtensor
formalism of the BPS model is very rich, and the corre- perturbations are the same as those given by one of the
sponding algebra of constraints is well-defined [32]. currentauthorsin[11]intheSVWsetup[7],becausethe
Tocuretheinstabilityproblem,anotherveryattractive gaugefieldandtheNewtonianpre-potentialhavenocon-
wayistoeliminatethespin-0gravitonfromthetheory,so tributionstotheseparts. Applyingtheseformulastothe
that the resulting one has as manygeneratorsper space- MinkowskibackgroundinSec. V,weshowexplicitlythat
time point as GR does. This is done recently by Horava the scalar and vector perturbations of the metric vanish
and Melby-Thompson(HMT) [33] (with the assumption identically. The only non-vanishing dynamical variables
of the projectability condition (1.6)) by extending the are the traceless and divergence-free tensor Hij, which
foliation-preserving-diffeomorphisms, Diff(M, ), to in- describesthemasslessspin-2graviton,asituationthatis
clude a local U(1) symmetry, F preciselythe sameasinGR.These resultsareconsistent
with the ones obtained earlier in [33]. In Sec. VI, we
U(1)⋉Diff(M, ). (1.8) present our main conclusions.
F
Effectively,thespatialdiffeomorphismsymmetriesofGR
arekeptintact,butitstimereparametrizationsymmetry
II. NON-RELATIVISITC GENERAL
is linearized and the corresponding algebrais contracted
COVARIANT HL THEORY
to a local gauge symmetry [34]. The restoration of gen-
eral covariance, characterized by Eq.(1.8), nicely main-
tains the special status of time, so that the anisotropic In orderto limit the spin-0 graviton,HMT introduced
scaling (1.1) with z >1 can still be realized. twonewfields,theU(1)gaugefieldAandtheNewtonian
A remarkable by-production of this “non-relativistic pre-potential ϕ, where in general both of them depend
general covariant” setup is that it forces the coupling on space and time,
constant λ, introduced originally to characterize the de-
viation of the kinetic part of the action from GR [1], to A=A t,xk , ϕ=ϕ t,xk . (2.1)
(cid:0) (cid:1) (cid:0) (cid:1)
3
Note that the notations used in this paper are slightly ofthe coupling constants. With this condition, takes
V
different from those adopted in [33] 1. Under the the simple form, L
Diff(M, ), these fields transfer as,
F =w2C Cij, (2.8)
V ij
δA = ζi∂ A+f˙A+fA˙, L
i
δϕ = fϕ˙ +ζi∂ ϕ, (2.2) wherew isacouplingconstant,andCij denotestheCot-
i
ton tensor, defined by
whileunderthelocalU(1),they,togetherwithg ,trans-
ij
1
fer as Cij =ǫikl Rj Rδj . (2.9)
∇k l − 4 l
δ A = α˙ Ni α, δ ϕ= α, (cid:16) (cid:17)
α i α
− ∇ − In [7], by assuming that the highest order derivatives
δ N = N α, δ g =0=δ N, (2.3)
α i ∇i α ij α are six and that the theory respects the parity, SVW
constructed the most general form of , given by
where α is the generatorofthe local U(1)gaugesymme- LV
try. For the detail, we refer readers to [33].
1
The total action is given by, LV = ζ2g0+g1R+ ζ2 g2R2+g3RijRij
1 (cid:0) (cid:1)
S = ζ2 dtd3xN√g LK −LV +Lϕ+LA +ζ4 g4R3+g5RRijRij +g6RjiRkjRik
Z (cid:16)+ζ12LM , (2.4) +ζ14 (cid:16)g7R∇2R+g8(∇iRjk) ∇iRjk ,(cid:17)(2.10)
(cid:19)
(cid:2) (cid:0) (cid:1)(cid:3)
where g =detgij, and where the coupling constants gs(s = 0,1,2,...8) are all
dimensionless. The relativistic limit in the IR requires
= K Kij K2, g = 1 and ζ2 = 1/(16πG) [7]. In this paper, we
K ij 1
L − −
shall be concerned only with this potential, and our for-
= ϕ ij 2K + ϕ ,
Lϕ G ij ∇i∇j mulas to be obtained below can be easily generalized to
A (cid:16) (cid:17) other forms of the potential, including the one given by
= 2Λ R . (2.5)
A g Eq.(2.8), and the f(R) term studied in [13].
L N −
(cid:16) (cid:17) VariationwithrespecttothelapsefunctionN(t)yields
Here the coupling constant Λg, acting like a 3- the Hamiltonian constraint,
dimensional cosmologicalconstant, has the dimension of
(length)−2. TheRicciandRiemanntermsallrefertothe
d3x√g + ϕ ij ϕ =8πG d3x√gJt,
three-metricgij,Kij istheextrinsiccurvature,and ij is LK LV − G ∇i∇j
the3-dimensional“generalized”Einsteintensor,defiGned, Z (cid:0) (cid:1) Z (2.11)
respectively, by where
1 δ(N )
Kij = ( g˙ij + iNj + jNi), Jt =2 LM . (2.12)
2N − ∇ ∇ δN
1
= R g R+Λ g . (2.6)
Gij ij − 2 ij g ij VariationwithrespecttotheshiftNi yieldsthesuper-
momentum constraint,
is the matter Lagrangian density, which in general
M
L
is a function of all the dynamical variables, U(1) gauge
πij ϕ ij =8πGJi, (2.13)
field, and the Newtonian prepotential, ∇j − G
(cid:16) (cid:17)
= N, N , g , ϕ, A; χ , (2.7) where the super-momentum πij and matter current Ji
M M i ij
L L
are defined as
where χ denotes colle(cid:0)ctively the matter fie(cid:1)lds. is an
V
arbitrary Diff(Σ)-invariant local scalar functionLal built πij δ(NLK) = Kij +Kgij,
out of the spatial metric, its Riemann tensor and spatial ≡ δg˙ −
ij
covariantderivatives,withouttheuseoftimederivatives. δ
In the original approach of Horava [1], the detailed bal- Ji N LM. (2.14)
≡ − δN
i
anceconditionwasimposed,inordertolimitthenumber
Similarly, variations of the action with respect to ϕ and
A yield,
1 In particular, we have ϕ = −νHMT, Kij = −KiHjMT, Λg = ij Kij + i jϕ =8πGJϕ, (2.15)
ΩHMT, Gij = ΘHijMT, where quantities with super indice G ∇ ∇
“HMT”aretheones usedin[33]. R=(cid:16)2Λg+8πGJA(cid:17), (2.16)
4
where 1g RkRl Rm 3 R2 kl ,
−2 ij l m k − ∇k∇l
Jϕ ≡−δδLϕM, JA ≡2δ(NδALM). (2.17) (F7)ij = 2∇i∇jh∇2R −2 ∇2R Ri(cid:0)j (cid:1) i
1
+( R(cid:0))( R(cid:1)) (cid:0)g ((cid:1) R)2+4 4R ,
On the other hand, variationwith respect to gij leads to ∇i ∇j − 2 ij ∇ ∇
the dynamical equations, (F ) = 4R 2Rk+h 2Rk i
8 ij ∇ ij −∇k ∇i∇ j ∇j∇ i
1 √g πij ϕ ij = 2 K2 ij +2KKij − ∇iRlk ∇(cid:0)jRkl −2 ∇kRil (∇k(cid:1)Rjl)
N√g − G ,t − 1
(cid:0)g ((cid:1)(cid:0)R )2(cid:1) 2 (cid:0) (cid:1)2Rkl , (2.21)
(cid:2)1 (cid:0) (cid:1)(cid:3) (cid:0) (cid:1) −2 ij ∇k lm − ∇k∇l∇
+ Nkπij 2πk(iNj)
N∇k − Fij = 1ϕ 2hK+ 2ϕ Rij(cid:0) 2 2Kj + (cid:1)ji ϕ Rik
1 h i (ϕ,1) 2 ∇ − k ∇ ∇k
+ ( + + )gij
K ϕ A n(cid:16) (cid:17) (cid:16) (cid:17)
2 L L L 2 2Ki + i ϕ Rjk
+Fij +Fij +Fij +8πGτij, (2.18) − k ∇ ∇k
ϕ A (cid:16) (cid:17)
2Λ R 2Kij + i jϕ ,
g
where K2 ij ≡KilKlj, f(ij) ≡(fij +fji)/2, and Fij = 1 −(cid:16)ϕ ik−2N(cid:17)j(cid:16)+ jϕ ∇ ∇ (cid:17)o
F(cid:0) ij (cid:1) 1 δ −√gLV = 8 gsζns(Fs)ij, (ϕ,2) 2∇k(cid:26) G2N(cid:16)i N ∇ (cid:17) 2Nk
≡ √g (cid:0) δgij (cid:1) Xs=0 +ϕGjk N +∇iϕ −ϕGij N +∇kϕ ,
3 (cid:16) (cid:17) (cid:16) (cid:17)(cid:27)
Fij = Fij , Fij = 1 2 (ifj)k 2fij fkl gij ,
ϕ (ϕ,n) (ϕ,3) 2 ∇k∇ ϕ −∇ ϕ − ∇k∇l ϕ
n=1
X n (cid:0) (cid:1) o(2.22)
1
Fij = ARij i j gij 2 A ,
A N − ∇ ∇ − ∇ where
h (cid:16) (cid:17) i (2.19)
1
fij = ϕ 2Kij + i jϕ 2K+ 2ϕ gij .
with n = (2,0, 2, 2, 4, 4, 4, 4, 4). The stress ϕ ∇ ∇ − 2 ∇
3-tensosr τij is de−fine−d as− − − − − (cid:26)(cid:16) (cid:17) (cid:16) (cid:17) ((cid:27)2.23)
τij = 2 δ √gLM , (2.20) The matter quantities (Jt, Ji, Jϕ, JA, τij) satisfy
√g (cid:0)δgij (cid:1) the conservation laws,
1 2N
andthe geometric3-tensors(Fs)ij andF(iϕj,n) aredefined d3x√g g˙klτkl− √g √gJt ,t+ N√kg √gJk ,t
as, Z (cid:20)
(cid:0) (cid:1) (cid:0) (cid:1)
A
(F ) = 1g , −2ϕ˙Jϕ− N√g (√gJA),t =0, (2.24)
0 ij −2 ij (cid:21)
1 Jk
1 kτ (√gJ ) ( N N )
(F1)ij = Rij − 2Rgij, ∇ ik− N√g i ,t− N ∇k i−∇i k
(F2)ij = 2(Rij −∇i∇j)R− 12gij R−4∇2 R, −NNi∇kJk+Jϕ∇iϕ− 2JNA∇iA=0. (2.25)
(F ) = 2R ( 3R )R(cid:0) 4 R2(cid:1)
3 ij ∇ ij − ∇i∇j − ij − ij
1 III. COSMOLOGICAL MODELS
+ g 3R Rkl+ 2R 2R2(cid:0), (cid:1)
ij kl
2 ∇ −
(F ) = 3(R (cid:0) )R2 1g R (cid:1)6 2 R2, The homogeneous and isotropic universe is described
4 ij ij −∇i∇j − 2 ij − ∇ by,
(F ) = (R + ) R Rkl +2(cid:0)R R2 (cid:1)
5 ij ij ∇i∇j kl ij N =1, Ni =0, gij =a2(t)γij, (3.1)
+∇2(RRij)−(cid:0)∇k[∇i(cid:1)(RRjk)(cid:0)+∇(cid:1)j(RRik)] where
1
g R 2 2 R Rkl δ
ij kl ij
−2 − ∇ γ = , (3.2)
2 k(cid:2)(cid:0)l RRkl (cid:1),(cid:0) (cid:1) ij 1+ 14kr2 2
− ∇ ∇
(F6)ij = 3 R3 ij +(cid:0) 23 ∇(cid:1)2(cid:3) R2 ij gwaituhger,2on≡txh2e+otyh2er+hza2nda(cid:0),nwdekca=n0s(cid:1),e±t 1. Using the U(1)
h
−(cid:0)∇k(cid:1)∇i R2 jk +(cid:0) ∇(cid:1)j R2 ik ϕ=0, (3.3)
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1) (cid:17)i
5
without loss of generality. Then, we find that When matter is not present, we have J = 0 = J .
ϕ A
Then, Eqs.(3.12) and (3.13) implies that Λ = 0 = k,
K = a2Hγ , R =2kγ , (3.4) g
ij − ij ij ij while Eq.(3.8) yields a(t) = eHt, where H Λ/3,
≡
where H =a˙/a. Thus, we obtain whichis the de Sitter spacetime. It is interesting to note
p
that the de Sitter space can be also obtained from ρ =
LK = −6H2, Lϕ =0, p=k =0 and JA =−Λg/(4πG), Jϕ =−3HΛg/(8πG).
3k It shouldbe noted that the energy conservation(3.11)
LA = 2A Λg− a2 , canbealsoobtainedfromEq.(2.24),whiletheconserva-
(cid:18) (cid:19) tionlawofmomentum,Eq.(2.25),issatisfiedidentically.
6kg 12β k2 24β k3
LV = 2Λ+ a21 + a41 + a62 , (3.5) rWadhieantioβn1βa2n6=d a0,stthiffe-flcouridre,srpeosnpdeicntgivetelyr.mDsuaecttolikteheapdraersk-
where Λ ζ2g /2, and ence of these terms, one can easily construct bouncing
0
≡ universe in the early epoch of the universe [14–16].
3g2+g3 9g4+3g5+g6 Inaddition,inderivingEq.(3.8)wefollowedtheusual
β , β . (3.6)
1 ≡ ζ2 2 ≡ ζ4 assumption that the whole FRW universe is homoge-
neous and isotropic. In [12], it was argued that such
The matter components are
an assumption might be too strong. If one relaxes the
Jt = 2ρ, Ji =0, τ =pg , (3.7) assumption and requires that only the observed patch
ij ij
− of our universe is homogeneous and isotropic, one can
where ρ and p are the total density and pressure of the introduce the notion of “dark matter as an integration
matter fields. Then the Hamiltonian constraint (2.11) constant” of the Hamiltonian constraint (2.11): ρ(t) in
reduces to the super-Hamiltonian constraint, Eqs.(3.8)and(3.11)canbereplacedbyρ(t)+ (t)inthe
E
observable patch, where (t) = const/a3 in the IR limit
(t)+ (t)=8πGJt(t), E
LK LV [12, 35]. Beyond the observable patch, is necessarily
E
inhomogeneous. In order to analyze perturbations onan
which leads to the modified Friedmann equation,
FRWbackground,oneneedstorestricttheperturbations
H2 g1k = 8πGρ+ Λ + 2β1k2 + 4β2k3. (3.8) to the observable patch, which then raises issues about
− a2 3 3 a4 a6 matching across the boundary of the observable patch.
Inourapproach,thebackgroundisahomogeneousFRW
From Eqs. (2.14) and (2.19) we also find that
spacetime, so that =0 in the background.
E
g k 2β k2 12β k3
Fij = Λ 1 + 1 + 2 gij,
− − a2 a4 a6
(cid:18) (cid:19) IV. COSMOLOGICAL SCALAR
πij = 2Hgij. (3.9) PERTURBATIONS
−
Hence, the dynamical equation (2.18) reduces to [8]
In this section, we consider linear scalar perturbation
a¨ 4πG 1 A k oftheFRWuniversestudiedinthelastsection. Weshall
= (ρ+3p)+ Λ Λ
a − 3 3 − 2 g− a2 closely follow [8] and use the notations adopted there
(cid:18) (cid:19) without further explanations. However, in order to have
2β k2 8β k3
1 2 . (3.10) thepresentpaperasindependentaspossible,itisdifficult
− a4 − a6
to avoidrepeating the same materials,althoughwe shall
Similar to GR, the super-momentum constraint (2.13) try to limit it to its minimum.
is then satisfied identically, since Ji = 0 and, from In the quasi-longitudinal gauge [8],
Eq. (3.6), ~ πij πij = 0, where ~ denotes the co-
∇j ≡ |j ∇i φ=0=E, (4.1)
variant derivative with respect to γ . Using Eqs. (3.8)
ij
and (3.10), it follows that in the FRW background the the metric scalar perturbations are given by
matter satisfies the conservation law,
ds2 = a2 dη2+2B dxidη+(1 2ψ)γ dxidxj .
|i ij
ρ˙+3H(ρ+p)=AJ . (3.11) − −
ϕ (4.2)
(cid:2) (cid:3)
Thus,due to the interactionbetweenthe gaugefield and
The gauge-invaraintquantities are now given by
the fluid, its energy in general is not conserved.
Ontheotherhand,Eqs.(2.15)and(2.16)yield,respec- Φ= B+B′, Ψ=ψ B, (4.3)
tively, H −H
where a′/aandaprimedenotestheordinaryderiva-
k 8πG H≡
H Λ = J , (3.12) tive with respect to η. Using the U(1) gauge, we can
g− a2 − 3 ϕ
(cid:18) (cid:19) further set
3k
a2 −Λg =4πGJA. (3.13) δϕ=0, (4.4)
6
so that = 0. Following [8], we use quantities with is given by
ϕ
L
over-barsastheonescalculatedinthebackground. Then
1 1
we find that ψ′′+2 ψ′ ψ γijδF + ~2B′+2 ~2B
ij
H −F − 6 3 ∇ H∇
K = a γ +a B +(ψ′+2 ψ)γ , A¯ (cid:16) (cid:17)
ij − H ij |ij H ij + ~2 3Λga2+6k ψ
Rij = 2kγij +ψ|ij (cid:2)+∇~2ψγij, (cid:3) 31a(cid:16)∇ − (cid:17)
6 2 4 2~2+3Λga2 3k δA=4πGa2δ , (4.10)
= H + H ~2B+3ψ′ , − 6a ∇ − P
LK − a2 a2 ∇ (cid:16) (cid:17)
2 (cid:16) 3k (cid:17)2A¯ where δFij = gsζnsδ(Fs)ij, with δ(Fs)ij given by
= ¯ + Λ δA ~2ψ+3kψ , Eq. (A1) in [8], and
LA LA a" g− a2 − a2 ∇ # P
(cid:16) (cid:17) (cid:16) (cid:17) g k 2β k2 12β k3
LV = L¯V + a42 g1+ 4βa21k ∇~2+3k ψ F = a2(cid:18)−Λ− a12 + a14 + a62 (cid:19),
(cid:18) (cid:19)(cid:16) (cid:17) 1
48β k2 δτij = (δ +2p¯ψ) γij +Π|hiji , (4.11)
+ 2 ~2+3k ψ a2 P
a6 ∇ h i
24g k (cid:16) (cid:17) wheretheangledbracketsonindicesdefinethetrace-free
+ 7 ~2 ~2+3k ψ, (4.5)
ζ4a6 ∇ ∇ part:
(cid:16) (cid:17)
1
wbyheEreq.L(¯3V.5)d,enanotdesBthe po~te,nwtiiatlhof~thbeebinagcktghreoucnodvagriivaennt f|hiji ≡f|ij − 3γijf|k|k. (4.12)
|i i i
≡ ∇ ∇
derivative with respect to γ and ~2 γij~ ~ . The trace-free part of the dynamical equations is
ij i j
∇ ≡ ∇ ∇
To first-order the Hamiltonian constraint (2.11) takes
1
the form, B′ +2 B +δF δA A¯ψ
|hiji H |hiji hiji − a − |hiji
= 8πGa2Π(cid:0) . (cid:1) (4.13)
√γd3x ~2+3k ψ (2−3ξ)H ~2B+3ψ′ − |hiji
" ∇ − 2 ∇ Eqs.(4.10)and(4.13)generalizetheGRijperturbedfield
Z (cid:16) (cid:17) (cid:16) (cid:17)
2β 6β k 3g equations [36].
2k 1 + 2 + 7 ~2 ~2+3k ψ The perturbed parts of the conservation laws (2.24)
− a2 a4 ζ4a4∇ ∇
(cid:16) (cid:17)(cid:16) (cid:17) and (2.25) give
4πGa2δµ =0, (4.6)
−
i √γd3x δµ′+3 (δ +δµ) 3(ρ¯+p¯)ψ′
which is the same as that given in the SVW setup [8], ( H P −
whereδµ δJt/2. Eq.(4.6)representsageneralization Z
≡−
of the Poissonequation of GR [36]. + 1 a3J¯ ′δA+A¯ a3 δA 3A¯ψ ′ =0,
To the first-order the supermomentum constraint 2a4" A − #)
(2.13) takes the form, (cid:16) (cid:17) (cid:16) (cid:0) (cid:1)(cid:17)
(4.14)
2ψ′−2kB =8πGaq, (4.7) q′+3 q aδ 2a ~2+3k Π+ 1J¯AδA=0,
H − P − 3 ∇ 2
which is the generalization of the GR 0i constraint [36], (cid:16) (cid:17) (4.15)
where δJi a−2q|i.
Ontheo≡therhand,thelinearizedequations(2.15)and whereJ¯A andJ¯ϕ aregivenbyEqs.(3.12)and(3.13). The
(2.16)fortheNewtonianpre-potentialandthegaugefield energy conservation equation is an integrated general-
reduce, respectively, to, ization of the GR energy equation, and the momentum
equation generalizes the GR momentum equation [36].
k
Λ ~2B+3 ψ′+2 ψ
g− a2 ∇ H
(cid:16) + 2aH2 (cid:17)∇~h2ψ+3 2(cid:0)k−a2Λg ψ(cid:1)i =8πGaδJϕ, (4.8) V. STABILITSYPAOCFETTIHMEEMINKOWSKI
~2ψ+3khψ =2πGa(cid:0)2δJ . (cid:1) i (4.9)
A
∇ It can be shown that the Minkowski spacetime,
The linearly perturbed dynamical equations require
a=1, A¯=ϕ¯=k=0, (5.1)
the calculations of the perturbed (F ) of Eq. (2.21),
s ij
which were given by Eq.(A1) in [8]. To avoid repeating, is a solution of the HMT theory, provided that
we shall not write them down here, and refer readers di-
rectly to that paper. Then, we find that the trace part Λ =Λ=J¯ =J¯ =ρ¯=p¯=0. (5.2)
g A ϕ
7
Then,thelinearizedHamiltonianconstraint(4.6)andthe Motivated by these remarkable features, in this paper
field equation (4.8) for δϕ are satisfied identically, while we have studied the theory in some detail by assuming
Eqs.(4.7) and (4.9) yield the presence of matter fields. The potential of the ac-
tion has been taken to be the one constructed by SVW
ψ˙ =0=∂2ψ, (5.3) [7], which represents the most general potential, which
respects the parity and its highest order of the spatial
where ∂2 δij∂i∂j. These are the same as the ones derivatives is six. We have first derived the Hamiltonian
≡
obtained in GR, and lead to ψ = 0 with proper bound- andsuper-momentumconstraints,given,respectively,by
ary conditions. It is interesting to note that in GR the Eqs.(2.11)and(2.13),andthenthefieldequations(2.15)
equation∂2ψ =0isobtainedfromthelocalHamiltonian and (2.16), respectively, for the Newtonian pre-potential
constraint,while in the presentsetup it is obtained from ϕ and the local U(1) gauge field A. The dynamical
the variation of the gauge field A. From this analysis, equations are given by Eq.(2.18), while the conservation
one can see clearly the reason why A is needed in order lawsofenergyandmomentumaregiven,respectively,by
to eliminate the sprin-0 graviton. On the other hand, Eqs.(2.24) and (2.25).
the trace and traceless parts of the dynamical equations
Applying the above generalformulasto cosmology,we
(4.10) and (4.13) yield
have obtained the general modified Friedmann equation
B˙ =δA. (5.4) (3.8) and the equation (3.10) for the accelerationa¨. It is
remarkable that these equations give precisely the con-
Using the U(1)gaugefreedom(2.3), withoutloss ofgen- servation law of energy, which takes the same form as
erality, we can set thatgiveninGRandcanbe alsoobtainedfromthe con-
servationlaw (2.24), despite the fact that J and J are
ϕ A
δA=0. (5.5) non-vanishing,andgiven,respectively,byEqs.(3.12)and
(3.13). Whenthespatialcurvatureisdifferentfromzero,
Note that this gauge choice is consistent with our quasi- termsactingasdarkradiationandstiff-fluidarepresent,
longitudinal gauge φ = E = 0 [8], because under this and bouncing universe can be easily constructed from
U(1) gauge transformation, E and φ remain the same, these terms.
as one can see from Eq.(2.3). Then, Eq.(5.4) yields B =
We have also studied the scalar perturbations of the
B(x),andthegauge-invariantquantitiesΨandΦdefined
FRW universe with any given spatial curvature, and
by Eq.(4.3) are zero,
the linearized Hamiltonian, momentum constraints, the
equations for the Newtonian pre-potential ϕ and the
Ψ=Φ=0. (5.6)
gaugefieldA,thetraceandtracelesspartsofthedynam-
Therefore, the scalar perturbations of the metric van- icalequationsaregiven,respectively,by Eqs.(4.6), (4.7),
ish identically in the Minkowski background. Hence, (4.8), (4.9), (4.10), and (4.13), while the conservation
the spin-0gravitonis completely eliminated inthe HMT lawsofenergyandmomentumaregiven,respectively,by
setup [33]. Eqs.(4.14) and (4.15).
It should be noted that δϕ is undetermined in the ApplyingtheseformulastotheMinkowskibackground,
presentcase. However,sinceϕ¯=0,itisquitereasonable we have shown explicitly that the metric scalar pertur-
to assume that its linear perturbation also vanishes in bations vanish identically, that is, the spin-0 graviton
the Minkowski background. appearing in all the previous versions of the HL theory
is eliminated in the current HMT setup.
VI. CONCLUSIONS Since the Newtonian pre-potential ϕ and the gauge
field A have no contributions to the vector and tensor
perturbations,thecorrespondinglinearperturbationsare
Recently, Horava and Melby-Thompson [33] proposed
given precisely by the same equations as those recently
a new version of the HL theory of gravity, in which the
presentedin [11] inthe SVW setup. In particular,in the
spin-0 graviton,appearing in all the previous versions of
Minkowskibackgroundvector perturbations also vanish,
the HLtheory,is eliminatedbyintroducingaNewtonian
althoughthe tensor perturbations in generaldo not [11].
pre-potential ϕ and a local U(1) gauge field A. Due to
Thesetwonon-vanishingcomponentsrepresentthemass-
such an elimination, the dynamical coupling constant λ,
lessspin-2gravitons,whichareexactlythesameasthose
which characterizes the deviation of the kinetic part of
found in GR.
the action from that of the Einstein-Hilbert, is forced to
take its relativistic value λ = 1. As a result, the theory
in the IR regime exhibit many features that are quite Acknowledgements: The work of AW was supported
similar to those given in GR. in part by DOE Grant, DE-FG02-10ER41692.
8
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