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arXiv:1001.4102v3 [hep-th] 5 Jul 2010
Modified F(R) Hoˇ rava-Lifshitz gravity: a way to accelerating FRW cosmology
Masud Chaichian1,2, Shin’ichi Nojiri3, Sergei D. Odintsov4,5∗, Markku Oksanen1, and Anca Tureanu1,2
1Department of Physics, University of Helsinki, P.O. Box 64, FI-00014 Helsinki, Finland
2Helsinki Institute of Physics, P.O. Box 64, FI-00014 Helsinki, Finland
3Department of Physics, Nagoya University, Nagoya 464-8602, Japan
4Instituci` o Catalana de Recerca i Estudis Avan¸ cats (ICREA), Barcelona
5Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB,
Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain
We propose a general approach for the construction of modified gravity which is invariant under
foliation-preserving diffeomorphisms. Special attention is paid to the formulation of modified F(R)
Hoˇ rava-Lifshitz gravity (FRHL), whose Hamiltonian structure is studied. It is demonstrated that
the spatially-flat FRW equations of FRHL are consistent with the constraint equations. The analysis
of de Sitter solutions for several versions of FRHL indicates that the unification of the early-time
inflation with the late-time acceleration is possible. It is shown that a special choice of parameters
for FRHL leads to the same spatially-flat FRW equations as in the case of traditional F(R)-gravity.
Finally, an essentially most general modified Hoˇ rava-Lifshitz gravity is proposed, motivated by its
fully diffeomorphism-invariant counterpart, with the restriction that the action does not contain
derivatives higher than the second order with respect to the time coordinate.
PACS numbers: 11.10.Ef, 95.36.+x, 98.80.Cq, 04.50.Kd, 11.25.-w
I. INTRODUCTION
Recent observational data clearly indicates that our universe is currently expanding with an accelerating rate, ap-
parently due to Dark Energy. The early universe has also undergone a period of accelerated expansion (inflation).
The modified gravity approach (for a general review, see [1]) suggests that such accelerated expansion is caused by a
modification of gravity at the early/late-time universe. A number of modified theories of gravity, which successfully
describe the unification of early-time inflation with late-time acceleration and which are cosmologically and observa-
tionally viable, has been proposed (for a review, see [1]). Despite some indications [2] that such alternative theories
of gravity may emerge from string/M-theory, they are still mostly phenomenological theories that are not yet related
to a fundamental theory.
Recently the so-called Hoˇ rava-Lifshitz quantum gravity [3] has been proposed. This theory appears to be power-
counting renormalizable in 3+1 dimensions. One of the key elements of such a formulation is to abandon the local
Lorentz invariance so that it is restored as an approximate symmetry at low energies. Despite its partial success as
a candidate for a fundamental theory of gravity, there are a number of unresolved problems (see refs. [4–9]) related
with the detailed balance and the projectability conditions (see section II for definitions), strong couplings, an extra
propagating degree of freedom and the GR (infrared) limit, the relation with other modified theories of gravity etc.
Moreover, study of the spatially-flat FRW cosmology in the Hoˇ rava-Lifshitz gravity indicates that its background
cosmology [10] is almost the same as in the usual GR, although an effective dark matter could appear as a kind of a
constant of integration in the Hoˇ rava-Lifshitz gravity [15]. Hence, it seems that there is no natural way (without extra
fields) to obtain an accelerating universe from Hoˇ rava-Lifshitz gravity, let alone a unified description of the early-time
inflation with the late-time acceleration. Therefore it is natural to search for a generalization of the Hoˇ rava-Lifshitz
theory that could be easily related to a traditional modified theory of gravity. On the one hand, it may be very useful
for the study of the low-energy limit of such a generalized Hoˇ rava-Lifshitz theory due to the fact that a number of
modified theories of gravity are cosmologically viable and pass the local tests. On the other hand, it is expected that
such a generalized Hoˇ rava-Lifshitz gravity may have a much richer cosmological structure, including the possibility of
a unification of the early-time inflation with the late-time acceleration. Finally, within a more general theory one may
hope to formulate the dynamical scenario for the Lorentz symmetry violation/restoration caused by the expansion of
∗Also at Tomsk State Pedagogical University
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the universe.
In the present work we propose such a general modified Hoˇ rava-Lifshitz gravity. We mainly consider modified
F(R) Hoˇ rava-Lifshitz gravity which is shown to coincide with the traditional F(R)-gravity on the spatially-flat FRW
background for a special choice of parameters. Another limit of our model leads to the degenerate F(R) Hoˇ rava-Lifshitz
gravity proposed in ref. [11]. The Hamiltonian analysis of the modified F(R) Hoˇ rava-Lifshitz theory is presented. The
preliminary investigation of the FRW equations for models from this class indicates a rich cosmological structure and
a natural possibility for the unification of the early-time inflation with the Dark Energy epoch. Finally, we propose
the most general modification of Hoˇ rava-Lifshitz-like theory of gravity. Our formulation ensures that the spatially-flat
FRW cosmology of any modified Hoˇ rava-Lifshitz gravity (for a special choice of parameters) coincides with the one
of its traditional modified gravity counterpart.
II. MODIFIED F(R) HOˇ RAVA-LIFSHITZ GRAVITY
In this section we propose a new extended action for F(R) Hoˇ rava-Lifshitz gravity. The FRW equations for this
theory are also formulated. The action of the standard F(R)-gravity is given by
SF(R)=
?
d4x√−gF(R). (1)
Here F is a function of the scalar curvature R. By using the ADM decomposition [12] (for reviews and mathematical
background see [13, 14]), we can write the metric in the following form:
ds2= −N2dt2+ g(3)
ij
?dxi+ Nidt??dxj+ Njdt?,i = 1,2,3. (2)
Here N is called the lapse variable and Ni’s are the shift variables. Then the scalar curvature R has the following
form:
R = KijKij− K2+ R(3)+ 2∇µ(nµ∇νnν− nν∇νnµ)
g(3)N. Here R(3)is the three-dimensional scalar curvature defined by the metric g(3)
extrinsic curvature defined by
(3)
and√−g =
?
ij
and Kij is the
Kij=
1
2N
?
˙ g(3)
ij− ∇(3)
iNj− ∇(3)
jNi
?
,K = Ki
i. (4)
nµis a unit vector perpendicular to the three-dimensional hypersurface Σtdefined by t = constant and ∇(3)
the covariant derivative on the hypersurface Σt.
Recently an extension of F(R)-gravity to a Hoˇ rava-Lifshitz type theory [3] has been proposed [11], by introducing
the action
i
expresses
SFHL(R)=
?
d4x
?
g(3)NF(RHL),RHL≡ KijKij− λK2− EijGijklEkl. (5)
Here λ is a real constant in the “generalized De Witt metric” or “super-metric” (“metric of the space of metric”),
Gijkl=1
2
?
g(3)ikg(3)jl+ g(3)ilg(3)jk?
− λg(3)ijg(3)kl, (6)
defined on the three-dimensional hypersurface Σt, Eijcan be defined by the so called detailed balance condition by
using an action W[g(3)
kl] on the hypersurface Σt
?
g(3)Eij=δW[g(3)
kl]
δgij
, (7)
and the inverse of Gijklis written as
Gijkl=1
2
?
g(3)
ikg(3)
jl+ g(3)
ilg(3)
jk
?
−˜λg(3)
ijg(3)
kl,
˜λ =
λ
3λ − 1.(8)
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Acknowledgments
This researchhas been supported in part by MEC (Spain) project FIS2006-02842and AGAUR(Catalonia) 2009SGR-
994 (SDO), by Global COE Program of Nagoya University (G07) provided by the Ministry of Education, Culture,
Sports, Science & Technology (SN). M. O. is supported by the Finnish Cultural Foundation. The support of the
Academy of Finland under the Projects No. 121720 and 127626 is greatly acknowledged.
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