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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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The action W[g(3)

The original motivation for the detailed balance condition 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 Eijto

be defined by (7). There is an anisotropy between space and time in the Hoˇ rava-Lifshitz gravity. In the ultraviolet

(high energy) region, the time coordinate and the spatial coordinates are assumed to behave as

kl] is assumed to be defined by the metric and the covariant derivatives on the hypersurface Σt.

x → bx ,t → bzt ,z = 2,3,··· ,(9)

under the scale transformation. In [3], W[g(3)

kl] is explicitly given for the case z = 2,

W =

1

κ2

W

?

d3x

?

g(3)(R − 2ΛW), (10)

and for the case z = 3,

W =

1

w2

?

Σt

ω3(Γ). (11)

Here κW in (10) is a coupling constant of dimension −1/2 and w2in (11) is the dimensionless coupling constant.

ω3(Γ) in (11) is given by

ω3(Γ) = Tr

?

Γ ∧ dΓ +2

3Γ ∧ Γ ∧ Γ

?

≡ εijk

?

Γm

il∂jΓl

km+2

3Γn

ilΓl

jmΓm

kn

?

d3x. (12)

A general Eijconsist of all contributions to W up to the chosen value z.

In the Hoˇ rava-Lifshitz-like F(R)-gravity, we assume that N can only depend on the time coordinate t, which is

called the projectability condition. The reason is that the Hoˇ rava-Lifshitz gravity does not have the full diffeomorphism

invariance, but is invariant only under “foliation-preserving” diffeomorphisms, i.e. under the transformations

δxi= ζi(t,x), δt = f(t). (13)

If N depended on the spatial coordinates, we could not fix N to be unity (N = 1) by using the foliation-preserving

diffeomorphisms. There exists a version of Hoˇ rava-Lifshitz gravity without the projectability condition, but it is

suspected to possess few additional consistency problems [5, 9]. Therefore we prefer to assume that N depends only

on the time coordinate t.

Let us consider the FRW universe with a flat spatial part,

ds2= −N2dt2+ a(t)2

?

i=1,2,3

?dxi?2.(14)

Then, it is clear from the explicit expressions in (10) and (11) that W[g(3)

assume since a non-vanishing ΛWgives a cosmological constant. Then one can obtain

kl] vanishes identically if ΛW= 0, which we

R =12H2

N2

+6

N

d

dt

?H

N

?

= −6H2

N

+

6

a3N

d

dt

?Ha3

N

?

,RHL=(3 − 9λ)H2

N2

. (15)

Here the Hubble rate H is defined by H ≡ ˙ a/a. In the case of the Einstein gravity, the second term in the last

expression for R becomes a total derivative:

?

d4x√−gR =

?

d4x a3N

?

−6H2

N

+

6

a3N

d

dt

?Ha3

N

??

=

?

d4x

?

−6H2a3+ 6d

dt

?Ha3

N

??

.(16)

Therefore, this term can be dropped in the Einstein gravity. The total derivative term comes from the last term

2∇µ(nµ∇νnν− nν∇νnµ) in (3), which is dropped in the usual Hoˇ rava-Lifshitz gravity. In the F(R)-gravity, however,

this term cannot be dropped due to the non-linearity. Then if we consider the FRW cosmology with the flat spatial

part, there is almost no qualitative difference between the Einstein gravity and the Hoˇ rava-Lifshitz gravity, except

that there could appear an effective dark matter as a kind of a constant of integration in the Hoˇ rava-Lifshitz gravity

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[15]. The effective dark matter appears since the constraint given by the variation over N becomes global in the

projectable Hoˇ rava-Lifshitz gravity.

Now we propose a new and very general Hoˇ rava-Lifshitz-like F(R)-gravity by

SF(˜ R)=

?

d4x

?

g(3)NF(˜R),

˜R ≡ KijKij− λK2+ 2µ∇µ(nµ∇νnν− nν∇νnµ) − EijGijklEkl. (17)

In the FRW universe with the flat spatial part,˜R has the following form:

˜R =(3 − 9λ)H2

N2

+

6µ

a3N

d

dt

?Ha3

N

?

=(3 − 9λ + 18µ)H2

N2

+6µ

N

d

dt

?H

N

?

. (18)

The case one obtains with the choice of parameters λ = µ = 1 corresponds to the usual F(R)-gravity as long as we

consider spatially-flat FRW cosmology, since˜R reduces to R in (15). On the other hand, in the case of µ = 0,˜R

reduces to RHLin (15) and therefore the action (17) becomes identical with the action (5) of the Hoˇ rava-Lifshitz-like

F(R)-gravity in [11]. Hence, the µ = 0 version corresponds to some degenerate limit of the above general F(R)

Hoˇ rava-Lifshitz gravity. We call this limit degenerate because it is very difficult (perhaps even impossible) to obtain

FRW equations when µ = 0 is set from the very begining. In our theory the FRW equations can be obtained quite

easily, and then µ = 0 is a simple limit.

For the action (17), the FRW equation given by the variation over g(3)

FRW space-time (14) and setting N = 1:

ij

has the following form after assuming the

0 = F

?˜R

?

− 2(1 − 3λ + 3µ)

?

˙H + 3H2?

F′?˜R

?

− 2(1 − 3λ)H

dF′?˜R

dt

?

+ 2µ

d2F′?˜R

dt2

?

+ p,(19)

where F′denotes the derivative of F with respect to its argument. Here, the matter contribution (the pressure p) is

included. On the other hand, the variation over N gives the global constraint:

0 =

?

d3x

F

?˜R

?

− 6

?

(1 − 3λ + 3µ)H2+ µ˙H

?

F′?˜R

?

+ 6µH

dF′?˜R

dt

?

− ρ

, (20)

after setting N = 1. Here ρ is the energy density of matter. Since N only depends on t, but does not depend on the

spatial coordinates, we only obtain the global constraint given by the integration. If the standard conservation law is

used,

0 = ˙ ρ + 3H (ρ + p) ,(21)

Eq. (19) can be integrated to give

0 = F

?˜R

?

− 6

?

(1 − 3λ + 3µ)H2+ µ˙H

?

F′?˜R

?

+ 6µH

dF′?˜R

dt

?

− ρ −C

a3. (22)

Here C is the integration constant. Using (20), one finds C = 0. In [15], however, it has been claimed that C need

not always vanish in a local region, since (20) needs to be satisfied in the whole universe. In the region C > 0, the

Ca−3term in (22) may be regarded as dark matter.

Note that Eq. (22) corresponds to the first FRW equation and (19) to the second one. Specifically, if we choose

λ = µ = 1 and C = 0, Eq. (22) reduces to

0 = F

?˜R

?˜R

?

?

− 6

?

?

H2+˙H

?

?

F′?˜R

F′?˜R

?

?

+ 6H

dF′?˜R

dt

?

?

− ρ

= F

− 6H2+˙H

+ 364H2˙H +¨H

?

F′′?˜R

?

− ρ, (23)

which is identical to the corresponding equation in the standard F(R)-gravity (see Eq. (2) in [16] where a reconstruc-

tion of the theory has been made).

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We should note that in the degenerate µ = 0 case [11], the action (17) or (5) does not contain any term with

second derivatives with respect to the coordinates, which appears in the usual F(R)-gravity. The existence of the

second derivatives in the usual F(R)-gravity induces the third and fourth derivatives in the FRW equation as in (19).

Due to such higher derivatives, there appears an extra scalar mode, which is often called the scalaron in the usual

F(R)-gravity. This scalar mode often affects the correction to the Newton law as well as other solar tests. Therefore,

such a scalar mode does not appear in the F(R) Hoˇ rava-Lifshitz gravity with µ = 0. Hence, we have formulated a

general Hoˇ rava-Lifshitz F(R)-gravity which describes the standard F(R)-gravity or its non-degenerate Hoˇ rava-Lifshitz

extension in a consistent way.

III. HAMILTONIAN FORMALISM

Let us present some elements of the Hamiltonian analysis of our proposal (for Hamiltonian analysis of constrained

systems, and their quantization, see [17]). By introducing two auxiliary fields A and B we can write the action (17)

into a form that is linear in˜R:

SF(˜ R)=

?

d4x

?

g(3)N

?

B(˜R − A) + F(A)

?

. (24)

Variation with respect to B yields˜R = A that can be inserted back into the action (24) in order to produce the

original action (17). The variation with respect to A yields B = F′(A).

First we rewrite˜R in (24) into a more explicit and useful form (see (17) for the definition of˜R). The unit normal

nµto the hypersurface Σtin space-time can be written in terms of the lapse and the shift vector as nµ= (n0,ni) =

?

normal can be written

1

N,−Ni

N

?

. The corresponding one-form is nµ = −N∇µt = (−N,0,0,0). The term in (17) that involves the unit

∇µ(nµ∇νnν− nν∇νnµ) = ∇µ(nµK) −1

Ng(3)ij∇(3)

i∇(3)

jN .(25)

Thus we can rewrite˜R as

˜R = KijGijklKkl+ 2µ∇µ(nµK) −2µ

N∇(3)

i∇(3)iN − EijGijklEkl. (26)

Introducing (26) into (24) and performing integrations by parts yields the action

SF(˜ R)=

?

dtd3x

?

g(3)?

N?B?KijGijklKkl− EijGijklEkl− A?+ F(A)?

−2µK

?

˙B − Ni∂iB

?

− 2µNg(3)ij∇(3)

i∇(3)

jB

?

,(27)

where the integral is taken over the union U of the t = constant hypersurfaces Σtwith t over some interval in R, and

we have written Nnµ∇µB =˙B − Ni∂iB. We assume that the boundary integrals on ∂U and ∂Σtvanish.

In the Hamiltonian formalism the field variables gij, N, Ni, A and B have the canonically conjugated momenta

πij, πN, πi, πAand πB, respectively. For the spatial metric and the field B we have the momenta

πij=

δSF(˜ R)

δ˙ gij

δSF(˜ R)

δ˙B

=

?

g(3)?

?

BGijklKkl−µ

Ng(3)ij?

˙B − Ni∂iB

??

,(28)

πB =

= −2µ

g(3)K .(29)

We assume µ ?= 0 so that the momentum (29) does not vanish. Because the action does not depend on the time

derivative of N, Nior A, the rest of the momenta form the set of primary constraints:

πN≈ 0,πi(x) ≈ 0,πA(x) ≈ 0. (30)

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We consider N to be projectable, i.e. N = N(t), and therefore also the momentum πN= πN(t) is constant on Σtfor

each t. The Poisson brackets are postulated in the form (equal time t is understood)

{g(3)

{N,πN} = 1,

{A(x),πA(y)} = δ(x − y),

ij(x),πkl(y)} =1

2

?δk

iδl

j+ δl

iδk

j

?δ(x − y),

iδ(x − y),

{B(x),πB(y)} = δ(x − y),

{Ni(x),πj(y)} = δj

(31)

with all the other Poisson brackets vanishing. We shall continue to omit the argument (x) of the fields when there is

no risk of confusion. In order to obtain the Hamiltonian, we first solve (28)–(29) for Kijand˙B,

Kij =

1

g(3)

?

?1

B

?

g(3)

ikg(3)

jlπkl−1

3g(3)

ijg(3)

klπkl

?

−

1

6µg(3)

ijπB

?

,

˙B = Ni∂iB −

N

?

jNi. Therefore both g(3)

3µg(3)

?

g(3)

ijπij+1 − 3λ

2µ

BπB

?

, (32)

and further obtain ˙ g(3)

primary constraints are needed. The Hamiltonian is then defined

ij= 2NKij+∇(3)

iNj+∇(3)

ijand B are dynamical variables and no more

H =

?

d3x

?

πij˙ g(3)

ij+ πB˙B

?

− L =

?

d3x?NH0+ NiHi?, (33)

where the Lagrangian L is defined by the action (27), SF(˜ R)=?dtL, and the so called Hamiltonian constraint and

1

?

+

?

Hi= −2∇(3)

= −2∂jπij− g(3)ij?

respectively. Again we assume that the boundary term resulting from an integration by parts vanishes. We define the

total Hamiltonian by

the momentum constraint are found to be

H0 =

g(3)

g(3)?

jπij+ g(3)ij∇(3)

?1

B

?

g(3)

ikg(3)

jlπijπkl−1

3

?

g(3)

ijπij?2?

−

1

3µg(3)

ijπijπB−1 − 3λ

12µ2Bπ2

B

?

B?EijGijklEkl+ A?− F(A) + 2µg(3)ij∇(3)

jBπB

2∂kg(3)

i∇(3)

jB

?

,

jl− ∂jg(3)

kl

?

πkl+ g(3)ij∂jBπB, (34)

HT = H + λNπN+

?

d3x?λiπi+ λAπA

?,(35)

where the primary constraints (30) are multiplied by the Lagrange multipliers λN, λi, λA. Note that there is no space

integral over the product λNπN since they depend only on the time coordinate t due to the projectability of N.

The primary constraints (30) have to be preserved under time evolution of the system:

˙ πN = {πN,HT} = −

˙ πi= {πi,HT} = −Hi,

˙ πA = {πA,HT} =

?

d3xH0,

?

g(3)N (−B + F′(A)) .(36)

Therefore we impose the secondary constraints:

Φ0 ≡

S(x) ≡ Hi(x) ≈ 0,

ΦA(x) ≡ B(x) − F′(A(x)) ≈ 0.

?

d3xH0≈ 0,

Φi

(37)

Here the Hamiltonian constraint Φ0is global and the other two, the momentum constraint Φi

ΦA(x), are local. It is convenient to introduce a globalized version of the momentum constraints Φi

S(x) and the constraint

S:

ΦS(ξi) ≡

?

d3xξiHi≈ 0,(38)

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where ξi,i = 1,2,3 are three arbitrary smearing functions — the choices ξi= δj

constraints Hjwhich in turn imply the smeared one.

The total Hamiltonian (35) can be written in terms of the constraints as

iδ(x − y) will produce the three local

HT= NΦ0+ ΦS(Ni) + λNπN+

?

d3x?λiπi+ λAπA

?.(39)

The consistency of the system requires that also the secondary constraints Φ0, ΦS(ξi) and ΦA(x) have to be

preserved under time evolution:

˙Φ0 = {Φ0,HT} = N{Φ0,Φ0} + {Φ0,ΦS(Ni)} +

˙ΦS(ξi) = {ΦS(ξi),HT} = N{ΦS(ξi),Φ0} + {ΦS(ξi),ΦS(Ni)} ≈ 0

˙ΦA(x) = {ΦA(x),HT} = N{ΦA(x),Φ0} + {ΦA(x),ΦS(Ni)} +

?

d3xλA(x){Φ0,πA(x)} ≈ 0,

?

d3yλA(y){ΦA(x),πA(y)} ≈ 0, (40)

where we have used the fact that the constraints πN and πihave strongly vanishing Poisson brackets with every

constraint. We need to calculate the rest of the algebra of the constraints under the Poisson bracket. The Poisson

brackets between the constraint ΦS(ξi) and the canonical variables are

{ΦS(ξi),B} = −ξi∂iB ,

{ΦS(ξi),πB} = −∂i

{ΦS(ξk),g(3)

{ΦS(ξk),πij} = −∂k

?ξiπB

?ξkπij?+ πik∂kξj+ πjk∂kξi,

?,

ij} = −ξk∂kg(3)

ij− g(3)

ik∂jξk− g(3)

jk∂iξk,

(41)

where ξi= g(3)ijξj, and trivially zero for A and πA,

{ΦS(ξi),A} = 0,{ΦS(ξi),πA} = 0. (42)

Thus ΦS(ξi) generates the spatial diffeomorphisms for the variables B,πB,g(3)

or functional constructed from these variables, and treates the variables A,πA as constants. By using this result

(41)–(42) we obtain the Poisson brackets for the constraints Φ0and ΦS(ξi):

ij,πij, and consequently for any function

{Φ0,Φ0} = 0,{ΦS(ξi),Φ0} = 0,{ΦS(ξi),ΦS(ηi)} = ΦS(ξj∂jηi− ηj∂jξi) ≈ 0.(43)

For the constraints πAand ΦA(x) the Poisson brackets that do not vanishing strongly are:

{πA(x),Φ0} = −

{Φ0,ΦA(x)} =

?

g(3)ΦA(x) ≈ 0,

1

?

{πA(x),ΦA(y)} = F′′(A(x))δ(x − y)

ijπij+1 − 3λ

2µ

3µg(3)

?

g(3)

BπB

?

,{ΦS(ξi),ΦA(x)} = −ξi∂iB .(44)

Thus, in order to satisfy the consistency conditions (40), we have to impose the tertiary constraint

Φter≡ Ni∂iB −

N

?

3µg(3)

?

g(3)

ijπij+1 − 3λ

2µ

BπB

?

− λAF′′(A) ≈ 0.(45)

Since F′′(A) = 0 would essentially reproduce the original projectable Hoˇ rava-Lifshitz gravity, we assume that F′′(A) ?=

0. The first two terms in (45), i.e. the expression for˙B in (32), does not vanish due to the established constraints

(30) and (37). Therefore (45) is a restriction on the Lagrange multiplier λA, and we can solve it from Φter= 0:

λA=

1

F′′(A)

?

Ni∂iB −

N

?

3µg(3)

?

g(3)

ijπij+1 − 3λ

2µ

BπB

??

.(46)

Introducing (46) into the Hamiltonian (39) ensures that now all the constraints of the system are consistent.

According to the Poisson brackets (43)–(44) between the constraints, we can set the second-class constraints πA(x)

and ΦA(x) to vanish strongly, and as a result turn the Hamiltonian constraint Φ0 and the momentum constraint

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8

ΦS(ξi) into first-class constraints. For this end, we replace the Poisson bracket with the Dirac bracket, which is given

by

{f(x),h(y)}DB= {f(x),h(y)} +

?

d3z

1

F′′(A(z))({f(x),πA(z)}{ΦA(z),h(y)} − {f(x),ΦA(z)}{πA(z),h(y)}) ,

(47)

where f and h are any functions of the canonical variables. Assuming we can solve the constraint ΦA(x) = 0, i.e.

B = F′(A), for A =˜A(B), where˜A is the inverse of the function F′, we can eliminate the variables A and πA. Thus

the final variables of the system are g(3)

non-dynamical multipliers. Then since every dynamical variable has a vanishing Poisson bracket with the constraint

πA, the Dirac bracket (47) reduces to the Poisson bracket,

ij,πij,B,πB. The lapse N and the shift vector Ni, together with λNand λi, are

{f(x),h(y)}DB= {f(x),h(y)}. (48)

Finally the total Hamiltonian is the sum of the first-class constraints

HT= NΦ0+ ΦS(Ni) + λNπN+

?

d3xλiπi. (49)

It defines the equations of motion for every function f(x) (or functional f) of the canonical variables

˙f(x) = {f(x),HT} = N{f(x),Φ0} + {f(x),ΦS(Ni)} + λN{f(x),πN} +

?

d3yλi(y){f(x),πi(y)}. (50)

We have calculated the Hamitonian (33)–(34) of the proposed modified Hoˇ rava-Lifshitz F(R)-gravity and established

the preservation of the primary constraints (30) by imposing the required secondary constraints (37), including the

Hamiltonian constraint and the momentum constraint. In order to ensure the consistency of the secondary constraints

we introduced the tertiary constraint (45) that was used to fix the Lagrange multiplier λAof the primary constraint

πA. Finally, we eliminated the pair of variables A,πA by imposing the second-class constraints πA and ΦA, and

introduced the Dirac bracket (47) that reduced to (48). The total Hamiltonian was obtained in its final form (49) as

a sum of the first-class constraints. We conclude that the proposed action (17) of this modified F(R) Hoˇ rava-Lifshitz

gravity, which obeys the projectability condition, defines a consistent theory. This conclusion agrees with the recent

analysis of our theory presented in ref. [18].

IV.FRW COSMOLOGY FOR SOME VERSIONS OF MODIFIED HOˇ RAVA-LIFSHITZ F(R)-GRAVITY.

This section is devoted to the study of the FRW Eqs. (19) and (20) which admit a de Sitter universe solution. We

now neglect the matter contribution by putting p = ρ = 0. Then by assuming H = H0, both of Eq. (19) and (20)

lead to the same equation

0 = F?3(1 − 3λ + 6µ)H2

0

?− 6(1 − 3λ + 3µ)H2

0F′?3(1 − 3λ + 6µ)H2

0

?,(51)

as long as the integration constant vanishes (C = 0) in Eq. (22).

First we consider the popular case that

F

?˜R

?

∝˜R + β˜R2.(52)

Then Eq. (51) gives

0 = H2

0

?1 − 3λ + 9β (1 − 3λ + 6µ)(1 − 3λ + 2µ)H2

0

?. (53)

In the case of usual F(R)-gravity, where λ = µ = 1 and therefore 1 − 3λ + 2µ = 0, there is only the trivial solution

H2

added. For our general case, however, there exists the non-trivial solution

0= 0, although the R2-term could generate the inflation when more gravitational terms, like RµνRµνetc., are

H2

0= −

1 − 3λ

β (1 − 3λ + 6µ)(1 − 3λ + 2µ),(54)

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9

as long as the r.h.s. of (54) is positive. If the magnitude of this non-trivial solution is small enough, this solution

might correspond to the accelerating expansion in the present universe. Hence, the R2-term may generate the late-

time acceleration. On the other hand, the above solution may serve as an inflationary solution for the early universe

(with the corresponding choice of parameters).

Instead of (52) one may consider the following model:

F

?˜R

?

∝˜R + β˜R2+ γ˜R3. (55)

Then Eq. (51) becomes

0 = H2

0

?

1 − 3λ + 9β (1 − 3λ + 6µ)(1 − 3λ + 2µ)H2

0+ 9γ (1 − 3λ + 6µ)2(5 − 15λ + 12µ)H4

0

?

, (56)

which has the following two non-trivial solutions,

H2

0= −

(1 − 3λ + 2µ)β

2(1 − 3λ + 6µ)(5 − 15λ+ 12µ)γ

?

1 ±

?

1 −4(1 − 3λ)(5 − 15λ + 12µ)γ

9(1 − 3λ + 2µ)2β2

?

, (57)

as long as the r.h.s. is real and positive. If

?????

4(1 − 3λ)(5 − 15λ + 12µ)γ

9(1 − 3λ + 2µ)2β2

?????≪ 1, (58)

one of the two solutions is much smaller than the other solution. Then one may regard that the larger solution

corresponds to the inflation in the early universe and the smaller one to the late-time acceleration, similarly to the

modified gravity model [19], where such unification has been first proposed. The fact that such two solutions are

connected could be demonstrated by numerical calculation. Note that some of the above models may possess the

future singularity in the same way as the usual F(R)-gravity. However, it would be possible to demonstrate that

adding terms wtih even higher derivatives might cure this singularity, similarly as the addition of the R2-term did in

the usual F(R)-gravity. Hence, we have suggested the qualitative possibility to unify the early-time inflation with the

late-time acceleration in the modified Hoˇ rava-Lifshitz F(R)-gravity.

V.MORE GENERAL ACTION

In the formulation of F(R) Hoˇ rava-Lifshitz-like gravity, we do not require full diffeomorphism-invariance, but

only invariance under “foliation-preserving” diffeomorphisms (13). Therefore there are many invariants or covariant

quantities made from the metric like K, Kij, ∇(3)

∇µ(nµ∇νnν− nν∇νnµ), ···, etc. Then the action composed of such invariants as

?

··· ,R(3),R(3)

iKjk, ···, ∇(3)

i1∇(3)

i2···∇(3)

inKjk, R(3), R(3)

ij, R(3)

ijkl, ∇(3)

iR(3)

jk, ···,

SgHL =d4x

?

g(3)NF

?

ijkl,∇(3)

g(3)

ij,K,Kij,∇(3)

iKjk,··· ,∇(3)

i1∇(3)

i2···∇(3)

inKjk,

ij,R(3)

iR(3)

jk,··· ,∇µ(nµ∇νnν− nν∇νnµ)

?

,(59)

could be a rather general action for the generalized Hoˇ rava-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 derivatives

higher than the second order with respect to the time coordinate t. In the usual F(R)-gravity, there appears the extra

scalar mode since the equations given by the variation over the metric tensor contain the fourth derivative. Now we

avoid such extra modes except the one scalar mode.

In the FRW space-time (14) with the flat spatial part and non-trivial N = N(t), we find

Γ0

00=

˙N

N,

Γ0

ij=a2H

N2δij,Γi

j0= Hδi

j

other Γµ

νρ= 0,

Kij=a2H

N

δij,∇(3)

i

= 0,R(3)

ijkl= 0,∇µ(nµ∇νnν− nν∇νnµ) =

3

a3N

d

dt

?a3H

N

?

. (60)

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10

Then one gets

g(3)

ij= a2δij,K =3H

N

,∇(3)

iKjk= ··· = ∇(3)

iR(3)

i1∇(3)

i2···∇(3)

inKjk= ··· = 0,

R(3)= R(3)

ij= R(3)

ijkl= ∇(3)

jk= ··· = 0, (61)

and since F must be a scalar under the spatial rotation, the action (59) reduces to

SgHL =

?

d4x

?

g(3)NF

?H

N,

3

a3N

d

dt

?a3H

N

??

. (62)

Therefore, if we consider the FRW cosmology, the function F should depend on only two variables,

?

following one:

H

Nand

3

a3N

d

dt

a3H

N

?

. For instance,˜R in (18) is given by this combination. As an illustrative example, we may consider the

F = f0

?KijKij− λK2?+ f1∇µ(nµ∇νnν− nν∇νnµ)2. (63)

Then in the FRW space-time (2), by the variation of the scale factor a, we obtain the following equation:

0 = 2f0(1 − 3λ)

?

H2+˙H

?

+ 3f1

?

27H4+ 54H2˙H + 15˙H2+ 18H¨H + 2...

H

?

. (64)

If we assume a de Sitter universe H = H0with constant H0, Eq. (64) reduces to

0 = 2f0(1 − 3λ)H2+ 81f1H4, (65)

which has the non-trivial solution

H2= −2f0(1 − 3λ)

81f1

, (66)

as long as the r.h.s. is positive. In the same way, a large class of modified Hoˇ rava-Lifshitz gravities may be constructed.

For instance, one can construct Hoˇ rava-Lifshitz-like generalizations of F(G)-gravity where the action is the Einstein-

Hilbert term plus a function F of the Gauss-Bonnet invariant G, non-local gravity, F(R,RµνRµν,RµναβRµναβ), etc.

It is remarkable that some special subclass of such Hoˇ rava-Lifshitz-like theories will have the same spatially-flat FRW

background dynamics as the corresponding traditional modified gravity.

VI. DISCUSSION

We have suggested a quite general approach for the modification of Hoˇ rava-Lifshitz gravity. We concentrated mainly

on the F(R)-gravity version. The consistency of its spatially-flat FRW field equations has been demonstrated. The

Hamiltonian and the corresponding constraints of the modified F(R) Hoˇ rava-Lifshitz gravity have been derived. It has

been shown that these constraints are consistent under the dynamics of the system, and that they do not constrain

the physical degrees of freedom too much. It is demonstrated that a degenerate subclass of the proposed general

modified F(R) Hoˇ rava-Lifshitz gravity corresponds to the earlier proposed F(R) extension of Hoˇ rava-Lifshitz gravity.

The preliminary study of FRW cosmology indicates a possibility to describe or even to unify the early-time inflation

with the late-time acceleration [20]. The motivation to consider such a theory is clear: it includes conventional F(R)-

gravity and Hoˇ rava-Lifshitz gravity as limiting cases. The former offers interesting cosmological solutions, while the

latter may hold the promise of UV-completeness.

Our proposal opens the bridge between the conventional modified gravity and its Hoˇ rava-Lifshitz counterpart.

Indeed, it is demonstrated that our model with a special choice of parameters (λ = µ = 1) leads to the same

spatially-flat FRW dynamics as its traditional counterpart, which is fully diffeomorphism-invariant. Moreover, we

eventually proposed the most general construction for a modified gravity that is invariant under foliation-preserving

diffeomorphisms. In this way, any traditional modified gravity has its counterpart, where the Lorentz symmetry is

broken. The explicit construction may be made using the results of Section V. Having in mind that a number of

traditional modified theories of gravity are cosmologically viable and pass the local tests, one can expect that it will

eventually be possible to realize any accelerating FRW cosmology in this modified Hoˇ rava-Lifshitz theory. This will

be studied elsewhere.

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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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