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a r X i v : 1 0 0 3 . 3 9 2 5 v 4 [ h e p - t h ] 6 J u n 2 0 1 2
Modiﬁed ﬁrst-order Hoˇrava-Lifshitz gravity: Hamiltonian analysis of the generaltheory and accelerating FRW cosmology in power-law
F
(
R
)
model
Sante Carloni
1
, Masud Chaichian
2
,
3
, Shin’ichi Nojiri
4
,Sergei D. Odintsov
1
,
5a
, Markku Oksanen
2
, and Anca Tureanu
2
,
3
1
Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB,Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain
2
Department of Physics, University of Helsinki, P.O. Box 64, FI-00014 Helsinki, Finland
3
Helsinki Institute of Physics, P.O. Box 64, FI-00014 Helsinki, Finland
4
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
5
Instituci`o Catalana de Recerca i Estudis Avan¸cats (ICREA), Barcelona
We propose the most general modiﬁed ﬁrst-order Hoˇrava-Lifshitz gravity, whose action does notcontain time derivatives higher than the second order. The Hamiltonian structure of this theory isstudied in all the details in the case of the spatially-ﬂat FRW space-time, demonstrating many of the features of the general theory. It is shown that, with some plausible assumptions, including theprojectability of the lapse function, this model is consistent. As a large class of such theories, themodiﬁed Hoˇrava-Lifshitz
F
(
R
) gravity is introduced. The study of its ultraviolet properties showsthat its
z
= 3 version seems to be renormalizable in the same way as the srcinal Hoˇrava-Lifshitzproposal. The Hamiltonian analysis of the modiﬁed Hoˇrava-Lifshitz
F
(
R
) gravity shows that it is ingeneral a consistent theory. The
F
(
R
) gravity action is also studied in the ﬁxed-gauge form, wherethe appearance of a scalar ﬁeld is particularly illustrative. Then the spatially-ﬂat FRW cosmologyfor this
F
(
R
) gravity is investigated. It is shown that a special choice of parameters for this theoryleads to the same equations of motion as in the case of traditional
F
(
R
) gravity. Nevertheless, thecosmological structure of the modiﬁed Hoˇrava-Lifshitz
F
(
R
) gravity turns out to be much richerthan for its traditional counterpart. The emergence of multiple de Sitter solutions indicates to thepossibility of uniﬁcation of early-time inﬂation with late-time acceleration within the same model.Power-law
F
(
R
) theories are also investigated in detail. It is analytically shown that they havea quite rich cosmological structure: early/late-time cosmic acceleration of quintessence, as wellas of phantom types. Also it is demonstrated that all the four known types of ﬁnite-time futuresingularities may occur in the power-law Hoˇrava-Lifshitz
F
(
R
) gravity. Finally, a covariant proposalfor (renormalizable)
F
(
R
) gravity within the Hoˇrava-Lifshitz spirit is presented.
PACS numbers: 11.10.Ef, 95.36.+x, 98.80.Cq, 04.50.Kd, 11.25.-w
a
Also at Tomsk State Pedagogical University
2
I. INTRODUCTION
Recently, it has become clear that our universe has not only undergone the period of early-time accelerated expansion(inﬂation), but also is currently in the so-called late-time accelerating epoch (dark energy era). An extremely powerfulway to describe the early-time inﬂation and the late-time acceleration in a uniﬁed manner is modiﬁed gravity. Thisapproach does not require the introduction of new dark components like inﬂaton and dark energy. The uniﬁeddescription of inﬂation and dark energy is achieved by modifying the gravitational action at the very early universe aswell as at the very late times (for a review of such models, see [1]). A number of viable modiﬁed gravity theories hasbeen suggested. Despite some indications to possible connection with string/M-theory [2], such theories remain to bemainly phenomenological. It is a challenge to investigate their srcin from some (not yet constructed) fundamentalquantum gravity theory.Among the recent attempts to construct a consistent theory of quantum gravity much attention has been paid tothe quite remarkable Hoˇrava-Lifshitz quantum gravity [3], which appears to be power-counting renormalizable in fourdimensions. In this theory the local Lorentz invariance is abandoned, but it is restored as an approximate symmetryat low energies. Despite its partial success as a candidate for fundamental theory of gravity, there are a number of unresolved problems related to the detailed balance and projectability conditions, consistency, its general relativity(GR) limit, realistic cosmological applications, the relation to other modiﬁed gravities, etc. Due to the fact that itsspatially-ﬂat FRW cosmology [4] is almost the same as in GR, it is diﬃcult to obtain a uniﬁed description of theearly-time inﬂation with the late-time acceleration in the standard Hoˇrava-Lifshitz gravity.Recently the modiﬁed Hoˇrava-Lifshitz
F
(
R
) gravity has been proposed [5]. Such a modiﬁcation may be easily relatedwith the traditional modiﬁed gravity approach, but turns out to be much richer in terms of the possible cosmologicalsolutions. For instance, the uniﬁcation of inﬂation with dark energy seems to be possible in such Hoˇrava-Lifshitzgravity due to the presence of multiple de Sitter solutions. Moreover, on the one hand, there is the hope that thegeneralization of Hoˇrava-Lifshitz gravity may lead to new classes of renormalizable quantum gravity. On the otherhand, one may hope to formulate the dynamical scenario for the Lorentz symmetry violation/restoration, caused bythe expansion of the universe, in terms of such generalized theory.In the present work (section II) we propose the most general modiﬁed ﬁrst-order Hoˇrava-Lifshitz-like theory, withouthigher derivative terms which are normally responsible for the presence of ghosts. The general form of the action inthe spatially-ﬂat FRW space-time is found, and the Hamiltonian structure of the action is analyzed in section III.As a speciﬁc example of such a ﬁrst-order action we introduce the modiﬁed Hoˇrava-Lifshitz
F
(
R
) theory which ismore general than the model of ref. [5]. Nevertheless, its spatially-ﬂat FRW cosmology turns out to be the same as forthe model [5] (this is not the case for black hole solutions, etc). Therefore it also coincides with the conventional
F
(
R
)spatially-ﬂat cosmology for a speciﬁc choice of the parameters. The ultraviolet structure of the new Hoˇrava-Lifshitz
F
(
R
) gravity is carefully investigated. It is shown that such models can have very nice ultraviolet behaviour at
z
= 2.Moreover, for
z
= 3 a big class of renormalizable models is suggested (section II). The Hamiltonian analysis of themodiﬁed Hoˇrava-Lifshitz
F
(
R
) gravity is presented in section IV. The ﬁxed gauge modiﬁed Hoˇrava-Lifshitz
F
(
R
)gravity is analyzed in section V.Section VI is devoted to the investigation of spatially-ﬂat FRW cosmology for power-law
F
(
R
) gravity. The generalequation for the de Sitter solutions is obtained. It acquires an extremely simple form for a special choice of parameters,when de Sitter solutions are roots of the equation
F
= 0. The existence of multiple de Sitter solutions indicates theprincipal possibility of attaining the uniﬁcation of the early-time inﬂation with the late-time acceleration in themodiﬁed Hoˇrava-Lifshitz
F
(
R
) gravity. The reconstruction technique is developed for the study of analytical andaccelerating FRW cosmologies in power-law models. A number of explicit analytical solutions are presented. It isshown by explicit examples that some of the quintessence/phantom-like cosmologies may develop the future ﬁnite-timesingularity of all the known four types, precisely in the same way as for traditional dark energy models. The possiblecuring of such singularities could be achieved in a similar way as in the case of traditional modiﬁed gravity. Someremarks about small corrections to the Newton law are made in section VII. A summary and outlook are given in thelast section VIII. In the appendix A we propose a covariant
F
(
R
) gravity that is quite similar to the correspondingHoˇrava-Lifshitz version but remains to be a covariant theory. It seems that it could also be made renormalizable.
3
II. GENERAL ACTION FOR HOˇRAVA-LIFSHITZ-LIKE GRAVITY AND RENORMALIZABILITY
In this section we propose the essentially most general Hoˇrava-Lifshitz-like gravity action, which does not containderivatives with respect to the time coordinate higher than the second order. Its ultraviolet properties are discussed.By using the Arnowitt-Deser-Misner (ADM) decomposition [6] (for reviews and mathematical background, see [7]),
one can write the metric of space-time in the following form:d
s
2
=
−
N
2
d
t
2
+
g
(3)
ij
d
x
i
+
N
i
d
t
d
x
j
+
N
j
d
t
, i,j
= 1
,
2
,
3
.
(1)Here
N
is called the lapse variable and
N
i
is the shift 3-vector. Then the scalar curvature
R
has the following form:
R
=
K
ij
K
ij
−
K
2
+
R
(3)
+ 2
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
)
.
(2)Here
R
(3)
is the three-dimensional scalar curvature deﬁned by the metric
g
(3)
ij
and
K
ij
is the extrinsic curvature deﬁnedby
K
ij
= 12
N
˙
g
(3)
ij
−∇
(3)
i
N
j
−∇
(3)
j
N
i
, K
=
K
ii
.
(3)
n
µ
is the unit vector perpendicular to the three-dimensional space-like hypersurface Σ
t
deﬁned by
t
= constantand
∇
(3)
i
is the covariant derivative on the hypersurface Σ
t
. From the determinant of the metric (1) one obtains
√ −
g
=
g
(3)
N
.For general Hoˇrava-Lifshitz-like gravity models, we do not require the full diﬀeomorphism-invariance, but onlyinvariance under “foliation-preserving” diﬀeomorphisms:
δx
i
=
ζ
i
(
t,
x
)
, δt
=
f
(
t
)
.
(4)Therefore, there are many invariants or covariant quantities made from the metric, in particular
K
,
K
ij
,
∇
(3)
i
K
jk
,
···
,
∇
(3)
i
1
∇
(3)
i
2
···∇
(3)
i
n
K
jk
,
···
,
R
(3)
,
R
(3)
ij
,
R
(3)
ijkl
,
∇
(3)
i
R
(3)
jklm
,
···
,
∇
(3)
i
1
∇
(3)
i
2
···∇
(3)
i
n
R
(3)
jklm
,
···
,
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
),
···
,etc. Then the general consistent action composed of invariants that are constructed from such covariant quantities,
S
gHL
=
d
4
x
g
(3)
NF
g
(3)
ij
,K,K
ij
,
∇
(3)
i
K
jk
,
···
,
∇
(3)
i
1
∇
(3)
i
2
···∇
(3)
i
n
K
jk
,
···
,R
(3)
,R
(3)
ij
,R
(3)
ijkl
,
∇
(3)
i
R
(3)
jklm
,
···
,
∇
(3)
i
1
∇
(3)
i
2
···∇
(3)
i
n
R
(3)
jklm
,
···
,
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
)
,
(5)could be a rather general action for the generalized Hoˇrava-Lifshitz gravity. Note that one can also include the(cosmological) constant in the above action. Here it has been assumed that the action does not contain derivativeshigher than the second order with respect to the time coordinate
t
. In the usual
F
(
R
) gravity, there appears theextra scalar mode, since the equations given by the variation over the metric tensor contain the fourth derivative. Byassuming that the action does not contain derivatives higher than the second order with respect to the time coordinate
t
, we can avoid more extra modes in addition to the only one scalar mode which appears in the usual
F
(
R
) gravity.For example, if we consider the action containing the terms like(
∇
µ
∇
µ
)
n
+1
R
(3)
,
(
∇
ρ
∇
ρ
)
n
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
)
,
(6)the equations given by the variation over the metric tensor contain the ﬁfth or higher derivatives (for a review of Hamiltonian structure of higher derivative modiﬁed gravity, see [8]). If we deﬁne new ﬁelds recursively
χ
(
m
+1)
R
=
∇
µ
∇
µ
χ
(
m
)
R
, χ
(0)
R
=
R
(3)
, χ
(
m
+1)
n
=
∇
µ
∇
µ
χ
(
m
)
n
, χ
(0)
n
=
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
)
,
(7)the equations can be rewritten so that only second derivatives appear. The scalar ﬁelds in (7), however, often becomeghost ﬁelds that generate states of negative norm. Thus, we only consider actions of the form given by (5) in thispaper.In the Hoˇrava-Lifshitz-type gravity, we assume that
N
can only depend on the time coordinate
t
, which is calledthe
projectability condition
. The reason is that the Hoˇrava-Lifshitz gravity does not have the full diﬀeomorphism-invariance, but is invariant only under the foliation-preserving diﬀeomorphisms (4). If
N
depended on the spatialcoordinates, we could not ﬁx
N
to be unity (
N
= 1) by using the foliation-preserving diﬀeomorphisms. Moreover, there
4are strong reasons to suspect that the non-projectable version of the Hoˇrava-Lifshitz gravity is generally inconsistent[9]. Therefore we prefer to assume that
N
is projectable.In the FRW space-time with the ﬂat spatial part and the non-trivial lapse
N
(
t
),d
s
2
=
−
N
(
t
)
2
d
t
2
+
a
(
t
)
23
i
=1
d
x
i
2
,
(8)we ﬁndΓ
000
=˙
N N ,
Γ
0
ij
=
a
2
H N
2
δ
ij
,
Γ
ij
0
=
Hδ
ij
other Γ
µνρ
= 0
,K
ij
=
a
2
H N δ
ij
,
∇
(3)
i
= 0
, R
(3)
ijkl
= 0
,
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
) = 3
a
3
N
dd
t
a
3
H N
,
(9)where
H
=
˙
aa
is the Hubble parameter. Then one gets
g
(3)
ij
=
a
2
δ
ij
, K
= 3
H N , K
ij
K
ij
= 3
H N
2
,
∇
(3)
i
K
jk
=
···
=
∇
(3)
i
1
∇
(3)
i
2
···∇
(3)
i
n
K
jk
=
···
= 0
,R
(3)
=
R
(3)
ij
=
R
(3)
ijkl
=
∇
(3)
i
R
jklm
=
···
=
∇
(3)
i
1
∇
(3)
i
2
···∇
(3)
i
n
R
(3)
jklm
=
···
= 0
,
(10)and since
F
must be a scalar under the spatial rotation, the action (5) reduces to
S
gHL
=
d
4
x
g
(3)
NF
H N ,
3
a
3
N
dd
t
a
3
H N
.
(11)Therefore, if we consider the FRW cosmology, the function
F
should depend on only two variables,
H
N
and
3
a
3
N
dd
t
a
3
H N
.As a speciﬁc example of the above general theory, we may consider the following modiﬁed Hoˇrava-Lifshitz
F
(
R
)gravity, whose action is given by
S
F
( ˜
R
)
= 12
κ
2
d
4
x
g
(3)
NF
( ˜
R
)
,
˜
R
≡
K
ij
K
ij
−
λK
2
+ 2
µ
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
)
−L
(3)
R
g
(3)
ij
.
(12)Here
λ
and
µ
are constants and
L
(3)
R
is a function of the three-dimensional metric
g
(3)
ij
and the covariant derivatives
∇
(3)
i
deﬁned by this metric. Note that this action (12) is more general than the one introduced in ref. [5] due to the
presence of the last term in ˜
R
. We normalize
F
( ˜
R
) or redeﬁne
κ
2
so that
F
′
(0) = 1
.
(13)In [3],
L
(3)
R
is chosen to be
L
(3)
R
g
(3)
ij
=
E
ij
G
ijkl
E
kl
,
(14)where
G
ijkl
is the “generalized De Witt metric” or “super-metric” (“metric of the space of metric”),
G
ijkl
= 12
g
(3)
ik
g
(3)
jl
+
g
(3)
il
g
(3)
jk
−
λg
(3)
ij
g
(3)
kl
,
(15)deﬁned on the three-dimensional hypersurface Σ
t
.
E
ij
can be deﬁned by the so called
detailed balance condition
byusing an action
W
[
g
(3)
kl
] on the hypersurface Σ
t
g
(3)
E
ij
=
δW
[
g
(3)
kl
]
δg
(3)
ij
,
(16)and the inverse of
G
ijkl
is written as
G
ijkl
= 12
g
(3)
ik
g
(3)
jl
+
g
(3)
il
g
(3)
jk
−
˜
λg
(3)
ij
g
(3)
kl
,
˜
λ
=
λ
3
λ
−
1
.
(17)
5The action
W
[
g
(3)
kl
] is assumed to be deﬁned by the metric and the covariant derivatives on the hypersurface Σ
t
. Thereis an anisotropy between space and time in the Hoˇrava-Lifshitz gravity. In the ultraviolet (high energy) region, thetime coordinate and the spatial coordinates are assumed to behave as
x
→
b
x
, t
→
b
z
t, z
= 2
,
3
,
···
,
(18)under the scale transformation. In [3],
W
[
g
(3)
kl
] is explicitly given for the case
z
= 2,
W
= 1
κ
2
W
d
3
x
g
(3)
R
(3)
−
2Λ
W
,
(19)and for the case
z
= 3,
W
= 1
w
2
Σ
t
ω
3
(Γ)
,
(20)where
ω
3
(Γ) = Tr
Γ
∧
dΓ + 23Γ
∧
Γ
∧
Γ
≡
ε
ijk
Γ
mil
∂
j
Γ
lkm
+ 23Γ
nil
Γ
ljm
Γ
mkn
d
3
x
.
(21)Here
κ
W
in (19) is a coupling constant of dimension
−
1
/
2 and
w
2
in (20) is a dimensionless coupling constant. Ageneral
E
ij
consist of all contributions to
W
up to the chosen value
z
. The srcinal motivation for the detailed balancecondition is its ability to simplify the quantum behaviour and renormalization properties of theories that respect it.Otherwise there is no a priori physical reason to restrict
L
(3)
R
to be deﬁned by (14). In the following we abandon thedetailed balance condition and consider
L
(3)
R
to have a more general form, since it is not always relevant even for therenomalizability problem.We now investigate the renormalizability and the unitarity of the model (12). For this purpose, by introducing anauxiliary ﬁeld
A
, we rewrite the action (12) in the following form:
S
F
(˜
R
)
= 12
κ
2
d
4
x
g
(3)
N
F
′
(
A
)( ˜
R
−
A
) +
F
(
A
)
.
(22)For simplicity, the following gauge condition is used:
N
= 1
, N
i
= 0
.
(23)Then one ﬁndsΓ
0
ij
=
−
12 ˙
g
(3)
ij
,
Γ
ij
0
= Γ
i
0
j
= 12
g
(3)
ik
˙
g
(3)
kj
,
Γ
ijk
= Γ
(3)
ijk
≡
12
g
(3)
il
g
(3)
lk,j
+
g
(3)
jl,k
−
g
(3)
jk,l
,
other components of Γ
µνρ
= 0
,
(24)and therefore(
n
µ
) = (1
,
0
,
0
,
0)
, K
ij
= 12 ˙
g
(3)
ij
,
∇
µ
(
n
µ
∇
ν
n
ν
−
n
ν
∇
ν
n
µ
) = 12
∂
0
g
(3)
ij
˙
g
(3)
ij
+ 14
g
(3)
ij
˙
g
(3)
ij
2
.
(25)We deﬁne a new ﬁeld by
ϕ
≡
13 ln
F
′
(
A
)
,
(26)which can be algebraically solved as
A
=
A
(
ϕ
), so that
ϕ
= 13 ln
F
′
(
A
(
ϕ
))
⇔
F
′
(
A
(
ϕ
)) = e
3
ϕ
.
(27)The spatial metric is redeﬁned as
g
(3)
ij
= e
−
ϕ
¯
g
(3)
ij
.
(28)

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