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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)