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Modified First-Order Horava-Lifshitz Gravity: Hamiltonian Analysis of the General Theory and Accelerating FRW Cosmology In Power-Law F (R) Model

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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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