Christian G Böhmer
email: c.boehmer@ucl.ac.uk

Giuseppe De Risi
email: giuseppe.derisi@ba.infn.it

Tiberiu Harko
email: harko@hkucc.hku.hk

Francisco S N Lobo
email: flobo@cii.fc.ul.pt

Classical tests of general relativity in brane world models

Keywords: numbers: 04.80.Cc, 04.50.+h, 04.80.-y

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The idea, proposed in [1], that our four-dimensional Universe might be a three-brane, embedded in a fivedimensional space-time (the bulk), has attracted considerable interest in the past few years. According to the brane world scenario, the physical fields (electromagnetic, Yang-Mills, etc) in our four-dimensional Universe are confined to the three-brane. These fields are assumed to arise as fluctuations of branes in string theories. Only gravity can freely propagate in both the brane and bulk space-times, with the gravitational selfcouplings not significantly modified. This model originated from the study of a single 3-brane embedded in five dimensions, with the 5D metric given by ds 2 = e -f (y) η µν dx µ dx ν + dy 2 , which due to the appearance of the warp factor f (y), could produce a large hierarchy between the scale of particle physics and gravity. Even if the fifth dimension is uncompactified, standard 4D gravity is reproduced on the brane. Hence this model allows the presence of large, or even infinite non-compact extra dimensions. Our brane is identified to a domain wall in a 5-dimensional anti-de Sitter space-time. Due to the correction terms coming from the extra dimensions, significant deviations from the Einstein theory occur in brane world models at very high energies [2,3]. Gravity is largely modified at the electro-weak scale 1 TeV. The cosmological and astrophysical implications of the brane world theories have been extensively investigated in the literature [4]. Gravitational collapse can also produce high energies, with the five dimensional effects playing an important role in the formation of black holes [5].

In standard general relativity the unique exterior space-time of a spherically symmetric object is described by the Schwarzschild metric. In the five dimensional brane world models, the high energy corrections to the energy density, together with the Weyl stresses from bulk gravitons, imply that on the brane the exterior metric of a static star is no longer the Schwarzschild metric [6]. The presence of the Weyl stresses also means that the matching conditions do not have a unique solution on the brane; the knowledge of the five-dimensional Weyl tensor is needed as a minimum condition for uniqueness. Static, spherically symmetric exterior vacuum solutions of the brane world models were first proposed in [6] and in [7]. The first of these solutions, obtained in [6], has the mathematical form of the Reissner-Nordstrom solution of standard general relativity, in which a tidal Weyl param-Confidential: not for distribution. Submitted to IOP Publishing for peer review 7 July 2010 eter plays the role of the electric charge of the general relativistic solution. The solution was obtained by imposing the null energy condition on the 3-brane for a bulk having non zero Weyl curvature, and it can be matched to the interior solution corresponding to a constant density brane world star. A second exterior solution, which also matches a constant density interior, was derived in [7].

Several classes of spherically symmetric solutions of the static gravitational field equations in the vacuum on the brane have been obtained in [8][9][10][11]. As a possible physical application of these solutions the behavior of the angular velocity v tg of the test particles in stable circular orbits has been considered [9][10][11]. The general form of the solution, together with two constants of integration, uniquely determines the rotational velocity of the particle. In the limit of large radial distances, and for a particular set of values of the integration constants the angular velocity tends to a constant value. This behavior is typical for massive particles (hydrogen clouds) outside galaxies, and is usually explained by postulating the existence of dark matter. The exact galactic metric, the dark radiation, the dark pressure and the lensing in the flat rotation curves region in the brane world scenario has been obtained in [11]. It is also interesting to note that the flat rotation curves can also be explained in f (R) modified theories of gravity, without the need of dark matter [12].

Furthermore, two families of analytic solutions of the spherically symmetric vacuum brane world model equations (with g tt = -1/g rr ), parameterized by the ADM mass and a PPN parameter β have been obtained in [13]. Non-singular black-hole solutions in the brane world model have been considered in [14], by relaxing the condition of the zero scalar curvature but retaining the null energy condition. The four-dimensional Gauss and Codazzi equations for an arbitrary static spherically symmetric star in a Randall-Sundrum type II brane world have been completely solved on the brane in [15]. The on-brane boundary can be used to determine the full 5dimensional space-time geometry. The procedure can be generalized to solid objects such as planets. A method to extend into the bulk asymptotically flat static spherically symmetric brane world metrics has been proposed in [16]. The exact integration of the field equations along the fifth coordinate was done by using the multipole (1/r) expansion. The results show that the shape of the horizon of the brane black hole solutions is very likely a flat "pancake" for astrophysical sources. The general solution to the trace of the 4-dimensional Einstein equations for static, spherically symmetric configurations has been used as a basis for finding a general class of black hole metrics, containing one arbitrary function g tt = A(r), which vanishes at some r = r h > 0 (the horizon radius) in [17]. Under certain reasonable restrictions, black hole metrics are found, with or without matter. Depending on the boundary conditions the metrics can be asymptotically flat, or have any other prescribed asymptotic. For a review of the black hole properties and of the lensing in the brane world models see [18]. Now, to be viable models, the proposed models need to pass the astrophysical and cosmological observational tests. There are several possibilities of observationally testing the brane world models at an astrophysical/cosmological scale, such as using the time delay of gamma ray bursts [19] or by using the luminosity distance-redshift relation for supernovae at higher redshifts [20].

In addition to these possibilities, we also mention work on the gravitational lensing in brane world models [21], on the role of the brane charge in orbital models of highfrequency quasiperiodic oscillations observed in neutron star binary systems [22], on the complete set of analytical solutions of the geodesic equation of massive test particles in higher dimensional spacetimes which can be applied to brane world models [23], and on brane world corrections to the charged rotating black holes, to the perturbations in the electromagnetic potential and test particle motion around brane world black holes [24]. The classical tests of general relativity, namely, light deflection, time delay and perihelion shift, have been analyzed, for gravitational theories with large non-compactified extradimensions, in the framework of the five-dimensional extension of the Kaluza-Klein theory, using an analogue of the four-dimensional Schwarzschild metric in [25]. Solar system data also imposes some strong constraints on Kaluza-Klein type theories. The existence of extradimensions and of the brane world models can also be tested via the gravitational radiation coming from primordial black holes, with masses of the order of the lunar mass, M ∼ 10 -7 M , which might have been produced when the temperature of the universe was around 1 TeV. If a significant fraction of the dark halo of our galaxy consists of these lunar mass black holes, a huge number of black hole binaries could exist. The detection of the gravitational waves from these binaries could confirm the existence of extra-dimensions [26].

It is the purpose of the present paper to consider the classical tests (perihelion precession, light bending and radar echo delay) of general relativity for static gravitational fields in the framework of brane world gravity. To do this we shall adopt for the geometry outside a compact, stellar type object (the Sun), specific static and spherically symmetric vacuum solutions in the context of brane worlds. As a first step in our study, we consider the classical tests of general relativity in arbitrary spherically symmetric spacetimes, and develop a general formalism that can be used for any given metric. In particular, we consider the motion of a particle (planet), and analyze the perihelion precession, and in addition to this, by considering the motion of a photon, we study the bending of light by massive astrophysical objects and the radar echo delay, respectively. Existing data on light-bending around the Sun, using long-baseline radio interferometry, ranging to Mars using the Viking lander, and the perihelion precession of Mercury, can all give significant and detectable Solar System constraints, associated with the brane world vacuum solutions. More precisely, the study of the classical general relativistic tests, constrain the parameters of the various solutions analyzed.

This paper is organized in the following manner. In Section II, we outline, for self-completeness and selfconsistency, the field equations in brane world models. In Sec. III, we consider the classical Solar System tests in general relativity, namely, the perihelion shift, the light deflection and the radar echo delay, for arbitrary spherically symmetric spacetimes. In Sec. IV, we analyze the classical Solar System tests for the case of various brane world vacuum solutions. We conclude and discuss our results in Sec. V.

II. GRAVITATIONAL FIELD EQUATIONS ON THE BRANE

We start by considering a 5D spacetime (the bulk), with a single 4D brane, on which matter is confined. The 4D brane world ( (4) M, g µν ) is located at a hypersurface B X A = 0 in the 5D bulk spacetime ( (5) M, g AB ), where the coordinates are described by X A , A = 0, 1, . . . , 4. The induced 4D coordinates on the brane are x µ , µ = 0, 1, 2, 3.

The action of the system is given by [2] S = S bulk + S brane ,

where 5) M - (5) g 1 2k 2

S bulk = (

5

(5) R + (5)

L m + Λ 5 d 5 X, (2)
and

S brane = (4) M -(5) g 1 k 2 5 K ± + L brane (g αβ , ψ) + λ b d 4 x, (3)
where k 2 5 = 8πG 5 is the 5D gravitational constant; (5) R and (5) L m are the 5D scalar curvature and the matter Lagrangian in the bulk, L brane (g αβ , ψ) is the 4D Lagrangian, which is given by a generic functional of the brane metric g αβ and of the matter fields ψ; K ± is the trace of the extrinsic curvature on either side of the brane; and Λ 5 and λ b (the constant brane tension) are the negative vacuum energy densities in the bulk and on the brane, respectively.

The energy-momentum tensor of bulk matter fields is defined as (5) TIJ ≡ -2 δ (5) L m δ (5) g IJ + (5) g IJ (5) L m ,

while T µν is the energy-momentum tensor localized on the brane and is given by

T µν ≡ -2 δL brane δg µν + g µν L brane . (5)
Thus, the Einstein field equations in the bulk are given by [2] (5) G IJ = k 2 5 (5) T IJ ,

with (5) T IJ = -Λ 5 (5) g IJ + δ(B) -λ b (5) g IJ + T IJ . (7) The delta function δ (B) denotes the localization of brane contribution. In the 5D spacetime a brane is a fixed point of the Z 2 symmetry. The basic equations on the brane are obtained by projections onto the brane world.

The induced 4D metric is g IJ = (5) g IJ -n I n J , where n I is the space-like unit vector field normal to the brane hypersurface (4) M . In the following we assume (5) L m = 0.

Assuming a metric of the form ds 2 = (n I n J + g IJ )dx I dx J , with n I dx I = dχ the unit normal to the χ = constant hypersurfaces and g IJ the induced metric on χ = constant hypersurfaces, the effective 4D gravitational equation on the brane (the Gauss equation), takes the form [2]:

G µν = -Λg µν + k 2 4 T µν + k 4 5 S µν -E µν , (8)
where S µν is the local quadratic energy-momentum correction

S µν = 1 12 T T µν - 1 4 T µ α T να + 1 24 g µν 3T αβ T αβ -T 2 , (9
) and E µν is the non-local effect from the free bulk gravitational field, the transmitted projection of the bulk Weyl tensor C IAJB , E IJ = C IAJB n A n B , with the property

E IJ → E µν δ µ I δ ν J
as χ → 0. We have also denoted k → 0 we recover standard general relativity [2].

The Einstein equation in the bulk and the Codazzi equation also imply the conservation of the energymomentum tensor of the matter on the brane, D ν T µ ν = 0, where D ν denotes the brane covariant derivative. Moreover, from the contracted Bianchi identities on the brane it follows that the projected Weyl tensor obeys the constraint

D ν E µ ν = k 4 5 D ν S µ ν .
The symmetry properties of E µν imply that in general we can decompose it irreducibly with respect to a chosen 4-velocity field u µ as

E µν = -k 4 U u µ u ν + 1 3 h µν + P µν + 2Q (µ u ν) , ( 10
) where k = k 5 /k 4 , h µν = g µν + u µ u ν projects orthogonal to u µ , the "dark radiation" term U = -k 4 E µν u µ u ν is a scalar, Q µ = k 4 h α µ E αβ is a spatial vector and P µν = -k 4 h (µ α h ν) β -1 3 h µν h αβ E αβ
is a spatial, symmetric and trace-free tensor.

In the case of the vacuum state we have ρ = p = 0, T µν ≡ 0, and consequently S µν = 0. Therefore the field equation describing a static brane takes the form

R µν = -E µν + Λg µν , (11)
with the trace R of the Ricci tensor R µν satisfying the condition R = R µ µ = 4Λ. In the vacuum case E µν satisfies the constraint D ν E µ ν = 0. In an inertial frame at any point on the brane we have u µ = δ µ 0 and h µν = diag(0 ,1,1,1). In a static vacuum Q µ = 0 and the constraint for E µν takes the form [7]

1 3 D µ U + 4 3 UA µ + D ν P µν + A ν P µν = 0, ( 12
)
where A µ = u ν D ν u µ is the 4-acceleration. In the static spherically symmetric case we may choose A µ = A(r)r µ and P µν = P (r) r µ r ν -1 3 h µν , where A(r) and P (r) (the "dark pressure" although the name dark anisotropic stress might be more appropriate) are some scalar functions of the radial distance r, and r µ is a unit radial vector [6].

In order to obtain results which are relevant to the Solar System dynamics, in the following we will restrict our study to the static and spherically symmetric metric given by

ds 2 = -e ν(r) dt 2 + e λ(r) dr 2 + r 2 dΩ 2 , (13)
where dΩ 2 = dθ 2 + sin 2 θdφ 2 . Here θ and φ are the standard coordinates on the sphere, t ∈ R and r ranges over an open interval (r min , r max ) so that -∞ ≤ r min ≤ r max ≤ ∞. We also assume that the functions ν(r) and λ(r) are strictly positive and (at least piecewise) differentiable on the interval (r min , r max ).

Then the gravitational field equations and the effective energy-momentum tensor conservation equation in the vacuum take the form [8][9][10]

-e -λ 1 r 2 - λ r + 1 r 2 = 48πG k 4 4 λ b U + Λ, (14)
e -λ ν r + 1 r 2 - 1 r 2 = 16πG k 4 4 λ b (U + 2P ) -Λ, (15)
e -λ ν + ν 2 2 + ν -λ r - ν λ 2 = 32πG k 4 4 λ b (U -P )-2Λ, (16)
ν = - U + 2P 2U + P - 6P r (2U + P ) , (17)
where we denoted = d/dr. Equation ( 14) can immediately be integrated to give

e -λ = 1 - C 1 r - 2GM U (r) r - Λ 3 r 2 , ( 18
)
where C 1 is an arbitrary constant of integration, and we denoted

M U (r) = 24π k 4 4 λ b r 0 r 2 U (r) dr. (19)
The function M U is the gravitational mass corresponding to the dark radiation term (the dark mass). By substituting ν given by Eq. ( 17) into Eq. ( 15) and with the use of Eq. ( 18) we obtain the following system of differential equations satisfied by the dark radiation term U , the dark pressure P and the dark mass M U , describing the vacuum gravitational field, exterior to a massive body, in the brane world model:

dU dr = - (2U + P ) 2GM + M U + α (U + 2P ) r 3 -2 3 Λr 2 r 2 1 -2GM r -MU r -Λ 3 r 2 -2 dP dr - 6P r , (20)
and

dM U dr = 3αr 2 U , (21)
respectively.

The system of equations ( 20) and ( 21) can be transformed to an autonomous system of differential equations by means of the transformations

q = 2GM r + M U r + Λ 3 r 2 , µ = 3αr 2 U + 3r 2 Λ , p = 3αr 2 P -3r 2 Λ, θ = ln r . ( 22
)
We may call µ and p the "reduced" dark radiation and pressure, respectively. With the use of the new variables given by Eqs. ( 22), Eqs. ( 20) and (21) become

dq dθ = µ -q , (23)
and

dµ dθ = - (2µ + p) q + 1 3 (µ + 2p) 1 -q -2 dp dθ + 2µ -2p , (24)
respectively Equations ( 20) and ( 21), or equivalently, ( 23) and ( 24) may be called the structure equations of the vacuum on the brane. In order to close this system an "equation of state" relating the reduced dark radiation and the dark pressure terms is needed. Generally, this equation of state is given in the form P = P (U ). In the new variables the functional relation between dark radiation and dark pressure takes the form p = 3α exp (2θ) P [µ/3α exp (2θ)]. We consider several specific solutions below in the context of the Solar System tests.

III. CLASSICAL TESTS OF GENERAL RELATIVITY IN ARBITRARY SPHERICALLY SYMMETRIC STATIC SPACE-TIMES

At the level of the Solar System there are three fundamental tests, which can provide important observational evidence for general relativity and its generalization, and for alternative theories of gravitation in flat space. These tests are the perihelion precession of Mercury, the deflection of light by the Sun and the radar echo delay observations. These tests have been used to successfully test the Schwarzschild solution of general relativity. In order to constrain the brane world models at the level of the Solar System, we first have to study these effects in spherically symmetric space-times with arbitrary metrics. In this Section, we develop a formalism that can be used for any given metric. This useful formalism was first used in the Solar System tests applied to a specific vacuum solution in Hořava-Lifshitz gravity [27].

A. The perihelion precession

The motion of a test particle in the gravitational field on the brane in the metric given by Eq. ( 13) can be derived from the variational principle δ e ν c 2 ṫ2 -e λ ṙ2 -r 2 θ2 + sin 2 θ φ2 ds = 0, (25) where the dot denotes d/ds. It may be verified that the orbit is planar, and hence we can set θ = π/2 without any loss of generality. Therefore we will use φ as the angular coordinate. Since neither t nor φ appear explicitly in Eq. ( 25), their conjugate momenta yield constants of motion,

e ν c 2 ṫ = E = constant, r 2 φ = L = constant. ( 26
)
The constant E is related to energy conservation while the constant L is related to angular momentum conservation.

The line element Eq. ( 13) provides the following equation of motion for r ṙ2 + e -λ r 2 φ2 = e -λ e ν c 2 ṫ2 -1 .

(

) 27
Substitution of ṫ and φ from Eqs. ( 26) yields the following relationship

ṙ2 + e -λ L 2 r 2 = e -λ E 2 c 2 e -ν -1 . ( 28
)
The change of variable r = 1/u and the substitution d/ds = Lu 2 d/dφ transforms Eq. ( 28) into the form

du dφ 2 + e -λ u 2 = 1 L 2 e -λ E 2 c 2 e -ν -1 . ( 29
)
By formally representing e -λ = 1 -f (u), we obtain

du dφ 2 + u 2 = f (u)u 2 + E 2 c 2 L 2 e -ν-λ - 1 L 2 e -λ ≡ G(u).
(30) By taking the derivative of the previous equation with respect to φ we find

d 2 u dφ 2 + u = F (u), (31)
where

F (u) = 1 2 dG(u) du . (32)
A circular orbit u = u 0 is given by the root of the equation u 0 = F (u 0 ). Any deviation δ = u -u 0 from a circular orbit must satisfy the equation

d 2 δ dφ 2 + 1 - dF du u=u0 δ = O δ 2 , ( 33
)
which is obtained by substituting u = u 0 + δ in Eq. (31). Therefore, to first order in δ, the trajectory is given by

δ = δ 0 cos 1 - dF du u=u0 φ + β , (34)
where δ 0 and β are constants of integration. The angles of the perihelia of the orbit are the angles for which r is minimum and hence u or δ is maximum. Therefore, the variation of the orbital angle from one perihelion to the next is

φ = 2π 1 -dF du u=u0 = 2π 1 -σ . ( 35
)
The quantity σ defined by the above equation is called the perihelion advance. It represents the rate of advance of the perihelion. As the planet advances through φ radians in its orbit, its perihelion advances through σφ radians. From Eq. ( 35), σ is given by

σ = 1 -1 - dF du u=u0 , (36)
or, for small (dF/du) u=u0 , by

σ = 1 2 dF du u=u0 . ( 37
)
For a complete rotation we have φ ≈ 2π(1 + σ), and the advance of the perihelion is δφ = φ -2π ≈ 2πσ. In order to be able to perform effective calculations of the perihelion precession we need to know the expression of L as a function of the orbit parameters. Let's consider the motion of a planet on a Keplerian ellipse with semi-axis a and b, where b = a √

1 -e 2 , and e is the eccentricity of the orbit. The surface area of the ellipse is πab. Since the elementary oriented surface area of the ellipse is d σ = ( r × d r) /2, the areolar velocity of the planet is

|d σ/dt| = | r × d r| /2 = r 2 (dφ/dt) /2 ≈ πa 2 √
1 -e 2 /T , where T is the period of the motion, which can be obtained from Kepler's third law as T 2 = 4π 2 a 3 /GM . In the small velocity limit ds ≈ cdt, and the conservation of the relativistic angular momentum gives r 2 dφ/dt = cL. Therefore we obtain L = 2πa 2 √

1 -e 2 /cT and 1

/L 2 = c 2 /GM a 1 -e 2 .
As a first application of the present formalism we consider the precession of the perihelion of a planet in the Schwarzschild geometry with e ν = e -λ = 1-2GM/c 2 r = 1 -2GM/c 2 u. Hence f (u) = 2GM/c 2 u. Since for this geometry ν + λ = 0, we obtain first

G(u) = 2GM c 2 u 3 + 1 L 2 E 2 c 2 -1 + 2GM c 2 L 2 u, (38)
and then

F (u) = 3 GM c 2 u 2 + GM c 2 L 2 . ( 39
)
The radius of the circular orbit u 0 is obtained as the solution of the equation

u 0 = 3 GM c 2 u 2 0 + GM c 2 L 2 , (40)
with the only physical solution given by

u 0 = 1 ± 1 -12G 2 M 2 /c 4 L 2 6GM/c 2 ≈ GM c 2 L 2 . (41)
Therefore

δφ = π dF du u=u0 = 6πGM c 2 a (1 -e 2 ) , (42)
which is the standard general relativistic result.

B. The deflection of light

In the absence of external forces a photon follows a null geodesic, ds 2 = 0. The affine parameter along the photon's path can be taken as an arbitrary quantity, and we denote again by a dot the derivatives with respect to the arbitrary affine parameter. There are two constants of motion, the energy E and the angular momentum L, given by Eqs. (26).

The equation of motion of the photon is

ṙ2 + e -λ r 2 φ2 = e ν-λ c 2 ṫ2 , (43)
which, with the use of the constants of motion can be written as

ṙ2 + e -λ L 2 r 2 = E 2 c 2 e -ν-λ . ( 44
)
The change of variable r = 1/u and the use of the conservation equations to eliminate the derivative with respect to the affine parameter leads to

du dφ 2 + u 2 = f (u)u 2 + 1 c 2 E 2 L 2 e -ν-λ ≡ P (u) . (45)
By taking the derivative of the previous equation with respect to φ we find

d 2 u dφ 2 + u = Q(u), (46)
where

Q(u) = 1 2 dP (u) du . ( 47
)
In the lowest approximation, in which the term of the right hand side of the equation ( 46) is neglected, the solution is a straight line,

u = cos φ R , ( 48
)
where R is the distance of the closest approach to the mass. In the next approximation Eq. ( 48) is used on the right-hand side of Eq. ( 46), to give a second order linear inhomogeneous equation of the form

d 2 u dφ 2 + u = Q cos φ R . (49)
with a general solution given by u = u (φ). The light ray comes in from infinity at the asymptotic angle φ = -(π/2 + ε) and goes out to infinity at an asymptotic angle φ = π/2 + ε. The angle ε is obtained as a solution of the equation u (π/2 + ε) = 0, and the total deflection angle of the light ray is δ = 2ε.

In the case of the Schwarzschild metric we have ν + λ = 0 and f (u) = 2GM/c 2 u, which gives P (u) = 2GM/c 2 u 3 and Q(u) = 3GM/c 2 u 2 , respectively. In the lowest approximation order from Eqs. ( 48) and ( 49) we obtain the second order linear equation

d 2 u dφ 2 + u = 3GM c 2 R 2 cos 2 φ = 3GM 2c 2 R 2 (1 + cos 2φ) , (50)
with the general solution given by

u = cos φ R + 3GM 2c 2 R 2 1 - 1 3 cos 2φ . ( 51
)
By substituting φ = π/2 + ε, u = 0 into Eq. ( 51) we obtain

ε = 2GM c 2 R , (52)
where we have used the relations cos (π/2 + ε) =sin ε, cos (π + 2ε) =cos 2ε, sin ε ≈ ε and cos 2ε ≈ 1. The total deflection angle in Schwarzschild geometry is δ = 2ε = 4GM/c 2 R.

C. Radar echo delay

A third Solar System test of general relativity is the radar echo delay [28]. The idea of this test is to measure the time required for radar signals to travel to an inner planet or satellite in two circumstances: a) when the signal passes very near the Sun and b) when the ray does not go near the Sun. The time of travel of light between two planets, situated far away from the Sun, is given by

T 0 = l2 -l1 dy c , (53)
where l 1 and l 2 are the distances of the planets to the Sun. If the light travels close to the Sun, the time travel is

T = l2 -l1 dy v = 1 c l2 -l1 e [λ(r)-ν(r)]/2 dy, (54)
where v = ce (ν-λ)/2 is the speed of light in the presence of the gravitational field. The time difference is

δT = T -T 0 = 1 c l2 -l1 e [λ(r)-ν(r)]/2 -1 dy. ( 55
) Since r = y 2 + R 2 , we have δT = 1 c l2 -l1 e h λ "√ y 2 +R 2 " -ν "√ y 2 +R 2 "i /2 -1 dy. (56) In the case of the Schwarzschild metric λ = -ν, exp (λ/2 -ν/2) = exp (-ν) = 1 -2GM/c 2 r -1 ≈ 1 +
2GM/c 2 r, and therefore

δT = 2GM c 3 l2 -l1 dy y 2 + R 2 = 2GM c 3 ln R 2 + l 2 2 + l 2 R 2 + l 2 1 -l 1 . (57) Since ln R 2 + l 2 2 + l 2 R 2 + l 2 1 -l 1 ≈ ln 4l 1 l 2 R 2 , (58)
where we have used the conditions R 2 /l 2 1 1 and R 2 /l 2 2 1, respectively, the time delay is given by

δT = 2GM c 3 ln 4l 1 l 2 R 2 . ( 59
)

IV. SOLAR SYSTEM TESTS OF SPECIFIC BRANE WORLD MODELS

The braneworld description of our universe entails a large extra dimension and a fundamental scale of gravity that might be lower by several orders of magnitude compared to the Planck scale [1]. It is known that the Einstein field equations in five dimensions admit more general spherically symmetric black holes on the brane than four-dimensional general relativity. Hence an interesting consequence of the brane world scenario is in the nature of spherically symmetric vacuum solutions to the brane gravitational field equations, which could represent black holes with properties quite distinct as compared to ordinary black holes in four dimensions. Such black holes are likely to have very diverse cosmological and astrophysical signatures. In the present Section we consider the Solar System test properties of several brane world black holes, which have been obtained by solving the vacuum gravitational field equations. There are many black hole type solutions on the brane, and in the following we analyze three particular examples. We will assume, for astrophysical constants, the values [28]: The first brane solution we consider is a solution of the vacuum field equations, obtained by Dadhich, Maartens, Papadopoulos and Rezania in [6], which represent the simplest generalization of the Schwarzschild solution of general relativity. We call this type of brane black hole as the DMPR black hole. The Solar System tests for the DMPR solutions were extensively analyzed in [29], but we use the general and novel formalism developed above as a consistency check. For this solution the metric tensor components are given by

M = 1.
e ν = e -λ = 1 - 2m r + Q r 2 , ( 61
)
where Q is the so-called tidal charge parameter. In the limit Q → 0 we recover the usual general relativistic case.

In terms of the general equations discussed in section II, this class of brane world spherical solution is characterized by a equation of state relating dark energy and pressure: P = -2U . The metric is asymptotically flat, with lim r→∞ exp (ν) = lim r→∞ exp (λ) = 1. There are two horizons, given by

r ± h = m ± m 2 -Q. ( 62
)
Both horizons lie inside the Schwarzschild horizon r s = 2m, 0 ≤ r - h ≤ r + h ≤ r s . In the brane world models there is also the possibility of a negative Q < 0, which leads to only one horizon r h+ lying outside the Schwarzschild horizon,

r h+ = m + m 2 + Q > r s . ( 63
)
In this case the horizon has a greater area than its general relativistic counterpart, so that bulk effects act to increase the entropy and decrease the temperature, and to strengthen the gravitational field outside the black hole.

Perihelion precession

For the DMPR black hole we have the following relevant functions:

f (u) = 2mu -Qu 2 , ( 64
)
G(u) = 2mu 3 -Qu 4 + E 2 c 2 L 2 - 1 L 2 + 2mu L 2 - Qu 2 L 2 , ( 65
)
and

F (u) = 3mu 2 -2Qu 3 + m L 2 - Qu L 2 , ( 66
)
respectively. u 0 can be obtained as solution of the algebraic equation

3mu 2 0 -u 0 + m L 2 = 2Qu 3 0 + Q L 2 u 0 , (67)
which to first order may be approximated to u 0 ≈ GM/(c 2 L 2 ), assuming that Q/L 2 1. We have used the following relationship m = GM/c 2 . Thus, δφ takes the form

δφ = 6πGM c 2 a(1 -e 2 ) - c 2 πQ GM a(1 -e 2 ) , (68)
which is consistent with the result outlined in [29]. The first term in Eq. ( 68) is the well-known general relativistic correction term for the perihelion precession, while the second term gives the correction due to the non-local effects arising from the Weyl curvature in the bulk. The observed value of the perihelion precession of the planet Mercury is δϕ Obs = 43.11 ± 0.21 arcsec per century [28]. The general relativistic formula for the precession, δϕ GR = 6πGM/c 2 a 1 -e 2 , gives, using numerical values given in (60), δϕ GR = 42.94 arcsec per century. Therefore, the difference ∆ϕ = δϕ Obs -δϕ GR = 0.17 ± 0.21 arcsec per century can be attributed to other effects. By assuming that ∆ϕ is entirely due to the modifications of the general relativistic Schwarzschild geometry as a result of the five dimensional bulk effects, the observational results impose the following general constraint on the bulk tidal parameter Q:

|Q| ≤ GM a 1 -e 2 πc 2 ∆ϕ . ( 69
)
With the use of the observational data for Mercury, Eq. ( 69) gives |Q| ≤ (5.17 ± 6.39) × 10 4 m 2 , or in the natural system of units, with c = = G = 1, |Q| ≤ (1.32 ± 1.63) × 10 30 MeV -2 .

Deflection of light

We have the following functions

P (u) = 2mu 3 -Qu 4 + E 2 c 2 L 2 , (70)
and

Q(u) = 3mu 2 -2Qu 3 . ( 71
)
Solving the differential equation

d 2 u dφ 2 + u = 3m R 2 cos 2 φ - 2Q R 3 cos 3 φ , ( 72
)
provides the following exact solution

u(φ) = cos φ R + 3m 2R 2 1 - 1 3 cos 2φ - Q 4R 3 9 4 cos φ - 1 4 cos 3φ + 3φ sin φ .( 73
)
By substituting φ = π/2 + ε, u = 0 into Eq. ( 73), yields the following total deflection angle

δφ = 2ε = 4GM c 2 R - 3πQ 4R 2 = 4GM c 2 R 1 - 3πQc 2 16GM R , (74)
which is in agreement with the results obtained in [30].

The best available data on light deflection by the Sun come from long baseline radio interferometry [31], which gives δφ LD = δφ

Radar echo delay

The delay can be evaluated from the integral in Eq. ( 56), with λ = -ν, exp (λ/2 -ν/2) = exp (λ), so that

exp (λ) = 1 - 2GM c 2 r + Q r 2 -1 ≈ 1 + 2GM c 2 r - Q r 2 , ( 75
) so that the time delay, given by Eq. ( 56), is readily integrated to yield

δT = 2GM c 3 ln R 2 + l 2 2 + l 2 R 2 + l 2 1 -l 1 - Q cR arctan l 2 R + arctan l 1 R . (76)
Using the approximations R 2 /l 2 1 1 and R 2 /l 2 2 1, the above expression reduces to

δT ≈ 2GM c 3 ln 4l 1 l 2 R 2 - πQ cR . (77)
Recently the measurements of the frequency shift of radio photons to and from the Cassini spacecraft as they passed near the Sun have greatly improved the observational constraints on the radio echo delay. For the time delay of the signals emitted on Earth, and which graze the Sun, one obtains ∆t RD = ∆t [32]. Therefore radar echo delay constrains the tidal charge of the DMPR brane world black hole as

(GR) RD (1 + ∆ RD ), with ∆ RD (1.1 ± 1.2) × 10 -5
|Q| ≤ 2GM R∆ RD ln 4l 1 l 2 /R 2 /πc 2 ≈ (1.83 ± 1.99) × 10 8 m 2 .
In natural units we have |Q| ≤ (4.66 ± 5.08) × 10 33 MeV -2 .

B. The CFM solution

Two families of analytic solutions in the brane world model, parameterized by the ADM mass and the PPN parameters β and γ, and which reduce to the Schwarzschild black hole for β = 1, have been found by Casadio,Fabbri and Mazzacurati in [13]. We denote the corresponding brane black hole solutions as the CFM black holes.

The first class of solutions is given by

e ν = 1 - 2m r , (78)
and

e λ = 1 -3m 2r 1 -2m r 1 -3m 2r 1 + 4 9 η , (79)
respectively, where η = γ -1 = 2 (β -1). This solution is characterized by an equation of state of the form

P U = 1 -3m 4r m 3r (80)
As in the Schwarzschild case the event horizon is located at r = r h = 2m. The solution is asymptotically flat, that is lim r→∞ e ν = e ν∞ = lim r→∞ e λ = e λ∞ = 1. We consider this case in the analysis outlined below as the CFM1 solution.

The second class of solutions corresponding to brane world black holes obtained in [13] has the metric tensor components given by

e ν =   η + 1 -2m r (1 + η) 1 + η   2 , (81)
and

e λ = 1 - 2m r (1 + η) -1 , (82)
respectively. The equation of state describing this solution is U = 0 The metric is asymptotically flat. In the case η > 0, the only singularity in the metric is at r = r 0 = 2m (1 + η), where all the curvature invariants are regular. r = r 0 is a turning point for all physical curves. For η < 0 the metric is singular at r = r h = 2m/ (1 -η) and at r 0 , with r h > r 0 , where r h defines the event horizon. We consider this case in the analysis outlined below as the CFM2 solution.

Perihelion precession

The function G(u) in Eq. ( 30) specializes, for the CFM1 metric and CFM2 metric in

G 1 (u) = 1 - (1 -2mu) 1 -3 2 1 + 4 9 η mu (1 -3 2 mu) u 2 + 1 L 2 (1 -2mu) 1 -3 2 1 + 4 9 η mu (1 -3 2 mu) × × E 2 c 2 1 (1 -2mu) -1 , (83)
and

G 2 (u) = 2m(1 + η)u 3 - 1 -2m(1 + η)u L 2 + E 2 c 2 L 2 (1 + η) 2 1 -2m(1 + η)u η + 1 -2m(1 + η)u 2 , (84)
respectively.

The GR limit is obtained for η = 0, which gives the expected value of the PPN parameter γ = 1. The function F (u) is quite complicated, but we are only interested in the first order corrections to the GR results. Since the terms m/L, η and E 2 /c 2 -1 are very small, we can expand F (u) up to second order, and keep only the term of order O(2) in the product of the infinitesimals. We get, for the CFM1 and CFM2 solutions, respectively:

F 1 (u) m L 2 + 3(1 + η 3 ) m u 2 , (85)
F 2 (u) m L 2 + 3(1 + η) m u 2 . ( 86
)
Both function can be written as

F 1,2 (u) ≡ m1,2 L2 1,2 + 3 m1,2 u 2 , (87)
where the tilded parameters are defined as:

m1 = (1 + η 3 )m; L2 1 = (1 + η 3 )L 2 , m2 = (1 + η)m; L2 1 = (1 + η + 1)L 2 .
This allows us to use directly the result obtained in GR, substituting the values of the tilded parameters. The resulting perihelion precession is:

δφ 1 6πGM c 2 a(1 -e 2 ) + 2πGM c 2 a(1 -e 2 ) η , (88)
δφ 2 6πGM c 2 a(1 -e 2 ) + 6πGM c 2 a(1 -e 2 ) η , (89)
where we have used the value of the Schwarzschild radius of the sun m = GM /c 2 , as mentioned above. As in the previous section, we consider the case of the planet Mercury and assume that the discrepancy between the predicted and the observed value is completely due to extradimension influence. We thus obtain a limit for the value of η and consequently for β. In the CFM2 case (which is the most constraining) we have the following restrictions:

|η| ≤ 0.004 ± 0.005 (90)
The η parameter for the CFM1 solution has a bound three times larger.

Deflection of light

The function P (u) in Eq. ( 45) for the CFM1 and the CFM2 metric takes the following respective forms:

P 1 (u) = 1 - (1 -2mu) 1 -3 2 1 + 4 9 η mu (1 -3 2 mu) u 2 + E 2 c 2 L 2 1 -3 2 1 + 4 9 η mu (1 -3 2 mu) , (91)
P 2 (u) = 2m(1 + η)u 3 + E 2 c 2 L 2 (1 + η) 2 × × 1 -2m(1 + η)u η + 1 -2m(1 + η)u 2 . ( 92
)
The function Q(u) of Eq. ( 46) can be expanded as:

Q 1 (u) - η 3
E 2 c 2 m L 2 -η E 2 c 2 m 2 L 2 u +3m 1 + η 3 - 3 4 η E 2 c 2 m 2 L 2 u 2 + O 3 (u) , (93)
and

Q 2 (u) -η E 2 c 2 m L 2 -3η E 2 c 2 m 2 L 2 u +3m 1 + η - η + 5 2 E 2 c 2 m 2 L 2 u 2 + O 3 (u) , (94)
for the CFM1 and CFM2 solutions, respectively. As stated before, at first order, we can take, for both cases, Q(u) = 0, so that the zeroth-order solution is

u (0) = cos(φ) R , (95)
which means the light rays travel on straight lines. Assuming, again, that the infinitesimals η and m/L are comparable, the first order equation is the same obtained in standard GR, namely:

u (1) + u (1) = 3m cos φ R 2 (96)
which gives, of course, the standard GR result

u (1) = m 2R 2 [3 -cos(2φ)] . (97)
At second order Eq. ( 46) can be written as

u (2) 1 + u (2) 1 = - η 3 E 2 c 2 m L 2 + 6 m L u (0) u (1) + η m L u (0) 2 = - η 3 E 2 c 2 m L 2 + 3m 2 R 3 cos(φ) [2 -cos(2φ)] + mη cos 2 φ R 2 , (98)
and

u (2) 2 + u (2) 2 = -η E 2 c 2 m L 2 + 6 m L u (0) u (1) + 3η m L u (0) 2 = η E 2 c 2 m L 2 + 3m 2 R 3 cos(φ) [2 -cos(2φ)] + 3mη cos 2 φ R 2 , (99)
respectively for the CFM1 and CFM2 solutions. The solutions are provided by

u (2) 1 = m 2 16R 3 12 cos 3 (φ) - 16 3 η R m cos 2 (φ) +21 cos(φ) + 60φ sin(φ) - η 3 E 2 c 2 m L 2 + 2 3 η m R 2 , (100)
and

u (2) 2 = m 2 16R 3 12 cos 3 (φ) -16η R m cos 2 (φ) +21 cos(φ) + 60φ sin(φ) -η E 2 c 2 m L 2 + 2η m R 2 ,( 101
)
respectively.

The deflection angle is obtained by the equation u(π/2 + ) = 0 with δφ = 2 . Assuming L = RE/c and substituting again m = GM /c 2 , we finally find:

δφ 1 = 4 GM c 2 R 1 + 15 16 π GM c 2 R + η 6 , (102)
δφ 2 = 4 GM c 2 R 1 + 15 16 π GM c 2 R + η 2 . ( 103
)
Comparison with data from the long baseline radio interferometry experiments [28], allow us, as in the previous section, to place an upper limit for η (and thus for β) which is, for the CFM2 solution:

|η| ≤ 0.0004 ± 0.0016 (104)
whilst, as before, the η parameter for the CFM1 solution has a bound three times larger.

Radar echo delay

The delay can be evaluated from the integral in Eq. ( 56). The integrand, in the case of the CMF1 and CFM2 metrics, is

exp λ-ν 2 1 + 2 η 3 mu + 4 -5 6 η m 2 u 2 + O(u 3 ) , exp λ-ν 2 1 + (2 + η)mu + 4 + 9 2 η m 2 u 2 + O(u 3 ) ,
respectively, so that the delay is given by

δT 1 = 2 GM c 3 log 4l 1 l 2 R 2 1 + η 6 + 2π GM c 2 R log -1 4l 1 l 2 R 2 , ( 105
)
and

δT 2 = 2 GM c 3 log 4l 1 l 2 R 2 1 + η 2 + 2π GM c 2 R log -1 4l 1 l 2 R 2 , (106)
These results are to be compared with data from the Cassini spacecraft [32] described in the previous section. We get a bound for η in the CFM2 model:

|η| ≤ 0.000021 ± 0.000024 (107)
the η parameter for the CFM1 solution has again a bound three times larger. So we can conclude that the CFM models add a correction to the relevant test quantities which are directly proportional to the PPN parameter that describes the solution themselves, plus additional terms quadratic in the ratio between the Schwarzschild radius and the typical dimension of the phenomenon.

C. The BMD solution

Several classes of brane world black hole solutions have been obtained by Bronnikov, Melnikov and Dehnen in [17] (for short the BMD black holes). A particular class of these models has the metric given by

e ν = 1 - 2Gm c 2 r 2/s , e λ = 1 - 2Gm c 2 r -2 , (108)
where s ∈ N. The metric is asymptotically flat, and at r = r h = 2Gm/c 2 these solutions have a double horizon.

It is also interesting to mention that the DMPR and CFM black holes, analyzed above, are part of the families of solutions classified by Bronnikov, Melnikov and Dehnen in [17].

Perihelion precession

For the BMD black hole solutions f (u) = (4Gm/c 2 )u-(2Gm/c 2 ) 2 u 2 and F (u) results in an involved expression. The equation u 0 = F (u 0 ) cannot be solved exactly. We therefore expand F (u) in 1/c 2 keeping only terms up to second order, which means up to terms 1/c 4 , and find

F (u) 2Gm c 2 L 2 s + 8G 2 m 2 c 4 L 2 s 2 u - 12G 2 m 2 c 4 L 2 s u + 6Gm c 2 u 2 - 8G 2 m 2 c 4 u 3 . (109)
Therefore u 0 = F (u 0 ) now becomes a cubic equation for u 0 . The physical solution in up to terms of the order (Gm/c 2 ) 4 is given by

u 0 = 2Gm c 2 L 2 s 1 + 8G 2 m 2 c 4 L 2 s 2 . ( 110
)
Hence, the perihelion precession is

δφ = 6πGm a 2 c 2 (1 -e 2 ) 1 + 4 + 6s -3s 2 3s 2 . ( 111
)
Comparison with observations yields

-0.004 < 4 + 6s -3s 2 3s 2 < 0.004 , (112)
which can be achieved by the following parameter choices -0.528 < s < -0.527 , or 2.519 < s < 2.536 .

(113) These inequalities place stringent constraints on the parameter s, and since s ∈ N, the restrictions strongly violate Solar System constrains. Thus, we can safely conclude that the BDM solution is incompatible with Mercury's perihelion precession.

Deflection of light

As above, the functional form of Q(u) is too complicated to solve the resulting second order differential equation. Therefore, we expand Q(u) in 1/c 2 and find

Q(u) 2Gm c 2 L 2 s - 2Gm c 2 L 2 + 6Gm c 2 u 2 - 8G 2 m 2 c 4 u 3 + 8G 2 m 2 c 4 L 2 s 2 - 12G 2 m 2 c 4 L 2 s + 4G 2 m 2 c 4 L 2 u , (114)
where the u 2 corresponds to classical general relativity and all other terms are contribution from the brane. Despite this complicated form of Q(u) one can solve analytically the differential equation

d 2 u dφ 2 + u = Q cos φ R , (115)
which results in a rather lengthy expression. However, by evaluating the solution u(φ = π/2+ε) = 0 and by assuming that ε is small, and performing a Taylor expansion, the deflection angle becomes

δ = 4GM c 2 R 2 -(1/s -1) R 2 L 2 , (116)
where the factor up front corresponds to the case of general relativity. Long baseline radio interferometry constrains the factor in the bracket to be ≤ 1.0017 and therefore s would have to be a very small real number to yield compatibility with Solar System tests. This parameter choice would also contradict the perihelion precession results discussed above. Thus, there is no need to further discuss the radar echo delay for this solution as it is already incompatible with Solar System constraints. Lastly, we would like to note that these conclusions also follow by considering the PPN parameters for the BMD solution, which are given by γ = 2 and β = 2 + 4/s 2 -2/s. Clearly, γ = 2 violates Solar System constrain while the minimum value for β is given by β min = 7/4 which is attained when s = 4, again incompatible with Solar System tests.

V. DISCUSSIONS AND FINAL REMARKS

The study of the classical tests of general relativity provides a very powerful method for constraining the allowed parameter space of brane world solutions, and to provide a deeper insight into the physical nature and properties of the corresponding spacetime metrics. Therefore, this opens the possibility of testing braneworld gravity by using astronomical and astrophysical observations at the Solar System scale. In the present paper, we have developed a general formalism that facilitates the analysis of any given metric and provides the basic theoretical tools necessary for the in depth comparison of the predictions of the brane world models with the observational/experimental results. In this context, the classical tests of general relativity (perihelion precession, deflection of light, and the radar echo delay) were considered for specific static and spherically symmetric vacuum solutions in brane world models.

First, we analyzed a solution of the vacuum field equations, obtained by Dadhich, Maartens, Papadopoulos and Rezania (DMPR) [6], which represents the simplest generalization of the Schwarzschild solution of general relativity. The tightest limit we get on the parameter Q came from the perihelion precession of Mercury, and gives |Q| ≤ (1.32 ± 1.63) × 10 30 MeV -2 , with other bounds about three orders of magnitude larger. These results represent a significant improvement of the results obtained in [29], where a method based on the first order approximation of the Hamilton-Jacobi result was used.

Second, we considered two families of analytic solutions, parameterized by the ADM mass and the PPN parameters β and γ, found by Casadio, Fabbri and Mazzacurati (CFM) [13]. These solutions can be physically seen as "dark pressure-dominated" solutions. Contrary to what happens in the DMPR case, here the most stringent bound came from the radar echo delay data of the Cassini experiment, bounding the relevant parameter η to be |η| ≤ 0.000021 ± 0.000024.

Thus, these first two models analyzed are not directly ruled out by Solar System tests. To get a value that is more "physically" meaningful, we can use the matching conditions between the interior and the exterior of a uniform density star in brane world, as derived in [7] to obtain the brane tension in terms of the relevant parameter of the different solutions. We get:

λ b ≥ 9GM 2 4π|Q|R 2 λ b ≥ 27M c 2 1 -3GM c 2 R 8πR 3 |η| λ b ≥ 9M c 2 η + 1 -2GM c 2 R (1 + η) 8πR 3 |eta|(1 + eta) (117)
for the DMPR, the CFM1 and the CFM2 solutions respectively. Interestingly enough, the most stringent bound comes from the DMPR solution (we are only interested in orders of magnitude here, so we will omit actual numerical values and errors): λ b ≥ 10 7 GeV 4 , while the CFM solutions gives λ b ≥ 10 4 GeV 4 . We do not have a definitive explanation for this, but we believe it is related to the fact that in the CFM solutions the dark radiation is basically negligible. Incidentally, the two different bounds obtained in the different CFM solutions lead to exactly the same bound on the brane tension, thus indicating that, although causally different, the two solutions behave in the same way for what concerns physical phenomena far away the central source.

In third place, we considered several classes of brane world black hole solutions obtained by Bronnikov, Melnikov and Dehnen [17] (BMD). The observational Solar System test place stringent constraints on the parameter space of the model, and these restrictions strongly violate Solar System constrains, so we can safely rule out these solutions as viable solutions.

Despite the fact of having a whole plethora of brane world vacuum solutions that pass the Solar System observational test a few remarks are in order, namely, one may basically consider two strategies of obtaining solutions on the brane. First, the bulk spacetime may be given, by solving the full 5-dimensional equations, and the geometry of the embedded brane is then deduced. Second, due to the complexity of the 5-dimensional equations, one may follow the strategy outlined in this paper, by considering the intrinsic geometry on the brane, which encompasses the imprint from the bulk, and consequently evolve the metric off the brane. In principle, the second procedure may provide a well-determined set of equations, with the brane setting the boundary data. However, determining the bulk geometry proves to be an extremely difficult endeavor.

supported by the RGC grant No. HKU 701808P of the government of the Hong Kong SAR. FSNL acknowledges partial financial support of the Fundação para a Ciência e

2 4 =

 4 8πG, with G the usual 4D gravitational constant. The 4D cosmological constant, Λ, and the 4D coupling constant, k 4 , are related by Λ = k 2 /6, respectively. In the limit λ -1 b

  LD ), with ∆ LD ≤ 0.0002 ± 0.0008, where δφ (GR) LD = 1.7510 arcsec. Therefore light deflection constrains the tidal parameter Q as |Q| ≤ 16GM R∆ LD /3πc 2 . By taking for R the value of the radius of the Sun in (60), the light deflection gives the constraint |Q| ≤ (6.97 ± 27.88) × 10 8 m 2 , or, in natural units, |Q| ≤ (1.78 ± 7.11) × 10 33 MeV -2 .

  989 × 10 30 kg, R = 6.955 × 10 8 m c = 2.998 × 10 8 m/s, G = 6.67 × 10 -11 m 3 kg -1 s -2 , a = 57.91 × 10 9 m,

	e = 0.205615	(60)

A. The DMPR brane world vacuum solution

Acknowledgments

We would like to thank Maurizio Gasperini for helpful comments on the manuscript. The work of TH is Tecnologia through the grants PTDC/FIS/102742/2008 and CERN/FP/109381/2009.


Masud Chaichian

Shin ' Ichi Nojiri

Sergei D Odintsov

Markku Oksanen

Anca Tureanu

Modified F (R) Hořava-Lifshitz gravity: a way to accelerating FRW cosmology

Keywords: numbers: 11.10.Ef, 95.36.+x, 98.80.Cq, 04.50.Kd, 11.25.-w

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) Hořava-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 Hořava-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.

I. INTRODUCTION

Recent observational data clearly indicates that our universe is currently expanding with an accelerating rate, apparently 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 observationally 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 Hořava-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. [4,5,6,7,8,9]) 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 Hořava-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 Hořava-Lifshitz gravity [START_REF] Mukohyama | [END_REF]. Hence, it seems that there is no natural way (without extra fields) to obtain an accelerating universe from Hořava-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 Hořava-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 Hořava-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 Hořava-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 the universe.

In the present work we propose such a general modified Hořava-Lifshitz gravity. We mainly consider modified F (R) Hořava-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) Hořava-Lifshitz gravity proposed in ref. [11]. The Hamiltonian analysis of the modified F (R) Hořava-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 Hořava-Lifshitz-like theory of gravity. Our formulation ensures that the spatially-flat FRW cosmology of any modified Hořava-Lifshitz gravity (for a special choice of parameters) coincides with the one of its traditional modified gravity counterpart.

II. MODIFIED F (R) HO ŘAVA-LIFSHITZ GRAVITY

In this section we propose a new extended action for F (R) Hořava-Lifshitz gravity. The FRW equations for this theory are also formulated. The action of the standard F (R)-gravity is given by

S F (R) = d 4 x √ -gF (R) . ( 1
)
Here F is a function of the scalar curvature R. By using the ADM decomposition [START_REF] Arnowitt | Gravitation: An Introduction to Current Research[END_REF] (for reviews and mathematical background see [START_REF] Wald | General Relativity[END_REF][START_REF] Gourgoulhon | [END_REF]), we can write the metric in the following form:

ds 2 = -N 2 dt 2 + g (3) ij dx i + N i dt dx j + N j dt , i = 1, 2, 3 . ( 2
)
Here N is called the lapse variable and N i 's are the shift variables. 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 ) ( 3 )
and √ -g = g (3) N . Here R (3) is the three-dimensional scalar curvature defined by the metric g

ij and K ij is the extrinsic curvature defined by

K ij = 1 2N ġ(3) ij -∇ (3)
i N j -∇

(3)

j N i , K = K i i . ( 4
)
n µ is a unit vector perpendicular to the three-dimensional hypersurface Σ t defined by t = constant and ∇

(3) i expresses the covariant derivative on the hypersurface Σ t .

Recently an extension of F (R)-gravity to a Hořava-Lifshitz type theory [3] has been proposed [11], by introducing the action

S FHL(R) = d 4 x g (3) NF (R HL ) , R HL ≡ K ij K ij -λK 2 -E ij G ijkl E kl . ( 5
)
Here λ is a real constant in the "generalized De Witt metric" or "super-metric" ("metric of the space of metric"),

G ijkl = 1 2 g (3)ik g (3)jl + g (3)il g (3)jk -λg (3)ij g (3)kl , (6)
defined on the three-dimensional hypersurface Σ t , E ij can be defined by the so called detailed balance condition by using an action W [g ) kl ] on the hypersurface Σ t

g (3) E ij = δW [g (3) kl ] δg ij , (7)
and the inverse of G ijkl is written as

G ijkl = 1 2 g (3)
ik g

(3)

jl + g (3) il g (3) jk -λg (3) ij g (3) kl , λ = λ 3λ -1 . ( 8
)
The action W [g ) kl ] is assumed to be defined by the metric and the covariant derivatives on the hypersurface Σ t . 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 E ij to be defined by (7). There is an anisotropy between space and time in the Hořava-Lifshitz gravity. In the ultraviolet (high energy) region, the time coordinate and the spatial coordinates are assumed to behave as

x → bx , t → b z t , z = 2, 3, • • • , (9)
under the scale transformation. In [3], W [g

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

W = 1 κ 2 W d 3 x g (3) (R -2Λ W ) , (10)
and for the case z = 3,

W = 1 w 2 Σt ω 3 (Γ) . (11)
Here κ W in ( 10) is a coupling constant of dimension -1/2 and w 2 in ( 11) is the dimensionless coupling constant. 11) is given by

ω 3 (Γ) in (
ω 3 (Γ) = Tr Γ ∧ dΓ + 2 3 Γ ∧ Γ ∧ Γ ≡ ε ijk Γ m il ∂ j Γ l km + 2 3 Γ n il Γ l jm Γ m kn d 3 x . (12)
A general E ij consist of all contributions to W up to the chosen value z.

In the Hořava-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 Hořava-Lifshitz gravity does not have the full diffeomorphism invariance, but is invariant only under "foliation-preserving" diffeomorphisms, i.e. under the transformations

δx i = ζ i (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 Hořava-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,

ds 2 = -N 2 dt 2 + a(t) 2 i=1,2,3 dx i 2 . ( 14
)
Then, it is clear from the explicit expressions in (10) and ( 11) that W [g

kl ] vanishes identically if Λ W = 0, which we assume since a non-vanishing Λ W gives a cosmological constant. Then one can obtain

R = 12H 2 N 2 + 6 N d dt H N = - 6H 2 N + 6 a 3 N d dt Ha 3 N , R HL = (3 -9λ) H 2 N 2 . ( 15
)
Here the Hubble rate H is defined by H ≡ ȧ/a. In the case of the Einstein gravity, the second term in the last expression for R becomes a total derivative:

d 4 x √ -gR = d 4 x a 3 N - 6H 2 N + 6 a 3 N d dt Ha 3 N = d 4 x -6H 2 a 3 + 6 d dt Ha 3 N . ( 16
)
Therefore, this term can be dropped in the Einstein gravity. The total derivative term comes from the last term 3), which is dropped in the usual Hořava-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 Hořava-Lifshitz gravity, except that there could appear an effective dark matter as a kind of a constant of integration in the Hořava-Lifshitz gravity [START_REF] Mukohyama | [END_REF]. The effective dark matter appears since the constraint given by the variation over N becomes global in the projectable Hořava-Lifshitz gravity. Now we propose a new and very general Hořava-Lifshitz-like F (R)-gravity by

2∇ µ (n µ ∇ ν n ν -n ν ∇ ν n µ ) in (
S F ( R) = d 4 x g (3) NF ( R) , R ≡ K ij K ij -λK 2 + 2µ∇ µ (n µ ∇ ν n ν -n ν ∇ ν n µ ) -E ij G ijkl E kl . (17)
In the FRW universe with the flat spatial part, R has the following form:

R = (3 -9λ) H 2 N 2 + 6µ a 3 N d dt Ha 3 N = (3 -9λ + 18µ) H 2 N 2 + 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 [START_REF] Mukohyama | [END_REF]. On the other hand, in the case of µ = 0, R reduces to R HL in [START_REF] Mukohyama | [END_REF] and therefore the action ( 17) becomes identical with the action ( 5) of the Hořava-Lifshitz-like F (R)-gravity in [11]. Hence, the µ = 0 version corresponds to some degenerate limit of the above general F (R) Hořava-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 [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF], the FRW equation given by the variation over g

(3)

ij has the following form after assuming the FRW space-time ( 14) and setting N = 1:

0 = F R -2 (1 -3λ + 3µ) Ḣ + 3H 2 F R -2 (1 -3λ) H dF R dt + 2µ d 2 F R dt 2 + 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 = d 3 x   F R -6 (1 -3λ + 3µ) H 2 + µ Ḣ 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) H 2 + µ Ḣ F R + 6µH dF R dt -ρ - C a 3 . ( 22
)
Here C is the integration constant. Using (20), one finds C = 0. In [START_REF] Mukohyama | [END_REF], 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 -3 term 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 -6 H 2 + Ḣ F R + 6H dF R dt -ρ = F R -6 H 2 + Ḣ F R + 36 4H 2 Ḣ + Ḧ F R -ρ , (23)
which is identical to the corresponding equation in the standard F (R)-gravity (see Eq. ( 2) in [16] where a reconstruction of the theory has been made).

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 [START_REF] Nojiri | [END_REF]. 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) Hořava-Lifshitz gravity with µ = 0. Hence, we have formulated a general Hořava-Lifshitz F (R)-gravity which describes the standard F (R)-gravity or its non-degenerate Hořava-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 [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF]). By introducing two auxiliary fields A and B we can write the action ( 17) into a form that is linear in R:

S F ( R) = d 4 x 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 [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF]. The variation with respect to A yields B = F (A).

First we rewrite R in ( 24) into a more explicit and useful form (see [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF] for the definition of R). The unit normal n µ to the hypersurface Σ t in space-time can be written in terms of the lapse and the shift vector as

n µ = (n 0 , n i ) = 1 N , -N i N .
The corresponding one-form is n µ = -N ∇ µ t = (-N, 0, 0, 0). The term in ( 17) that involves the unit normal can be written

∇ µ (n µ ∇ ν n ν -n ν ∇ ν n µ ) = ∇ µ (n µ K) - 1 N g (3)ij ∇ (3) i ∇ (3) j N . (25)
Thus we can rewrite R as

R = K ij G ijkl K kl + 2µ∇ µ (n µ K) - 2µ N ∇ (3) i ∇ (3)i N -E ij G ijkl E kl . ( 26
)
Introducing ( 26) into ( 24) and performing integrations by parts yields the action

S F ( R) = dtd 3 x g (3) N B K ij G ijkl K kl -E ij G ijkl E kl -A + F (A) -2µK Ḃ -N i ∂ i B -2µN g (3)ij ∇ (3) i ∇ (3) j B , (27)
where the integral is taken over the union U of the t = constant hypersurfaces Σ t with t over some interval in R, and we have written

Nn µ ∇ µ B = Ḃ -N i ∂ i B.
We assume that the boundary integrals on ∂U and ∂Σ t vanish.

In the Hamiltonian formalism the field variables g ij , N , N i , A and B have the canonically conjugated momenta π ij , π N , π i , π A and π B , respectively. For the spatial metric and the field B we have the momenta

π ij = δS F ( R) δ ġij = g (3) BG ijkl K kl - µ N g (3)ij Ḃ -N i ∂ i B , (28)
π B = δS F ( R) δ Ḃ = -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 , N i or A, the rest of the momenta form the set of primary constraints:

π N ≈ 0 , π i (x) ≈ 0 , π A (x) ≈ 0 . (30)
Here the Hamiltonian constraint Φ 0 is global and the other two, the momentum constraint Φ i S (x) and the constraint Φ A (x), are local. It is convenient to introduce a globalized version of the momentum constraints Φ i S :

Φ S (ξ i ) ≡ d 3 xξ i H i ≈ 0 , (38)
where ξ i , i = 1, 2, 3 are three arbitrary smearing functions -the choices ξ i = δ j i δ(x -y) will produce the three local constraints H j which in turn imply the smeared one.

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

H T = N Φ 0 + Φ S (N i ) + λ N π N + d 3 x λ i π i + λ A π A . (39)
The consistency of the system requires that also the secondary constraints Φ 0 , Φ S (ξ i ) and Φ A (x) have to be preserved under time evolution:

Φ0 = {Φ 0 , H T } = N {Φ 0 , Φ 0 } + {Φ 0 , Φ S (N i )} + d 3 xλ A (x){Φ 0 , π A (x)} ≈ 0 , ΦS (ξ i ) = {Φ S (ξ i ), H T } = N {Φ S (ξ i ), Φ 0 } + {Φ S (ξ i ), Φ S (N i )} ≈ 0 ΦA (x) = {Φ A (x), H T } = N {Φ A (x), Φ 0 } + {Φ A (x), Φ S (N i )} + d 3 yλ A (y){Φ A (x), π A (y)} ≈ 0 , (40)
where we have used the fact that the constraints π N and π i have 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 Φ S (ξ i ) and the canonical variables are

{Φ S (ξ i ), B} = -ξ i ∂ i B , {Φ S (ξ i ), π B } = -∂ i ξ i π B , {Φ S (ξ k ), g (3) ij } = -ξ k ∂ k g (3) ij -g (3) ik ∂ j ξ k -g (3) jk ∂ i ξ k , {Φ S (ξ k ), π ij } = -∂ k ξ k π ij + π ik ∂ k ξ j + π jk ∂ k ξ i , (41)
where ξ i = g (3)ij ξ j , and trivially zero for A and π A ,

{Φ S (ξ i ), A} = 0 , {Φ S (ξ i ), π A } = 0 . ( 42
)
Thus Φ S (ξ i ) generates the spatial diffeomorphisms for the variables B, π B , g

ij , π ij , and consequently for any function or functional constructed from these variables, and treates the variables A, π A as constants. By using this result (41)-( 42) we obtain the Poisson brackets for the constraints Φ 0 and Φ S (ξ i ):

{Φ 0 , Φ 0 } = 0 , {Φ S (ξ i ), Φ 0 } = 0 , {Φ S (ξ i ), Φ S (η i )} = Φ S (ξ j ∂ j η i -η j ∂ j ξ i ) ≈ 0 . ( 43
)
For the constraints π A and Φ A (x) the Poisson brackets that do not vanishing strongly are:

{π A (x), Φ 0 } = -g (3) Φ A (x) ≈ 0 , {π A (x), Φ A (y)} = F (A(x))δ(x -y) {Φ 0 , Φ A (x)} = 1 3µ g (3) g (3) ij π ij + 1 -3λ 2µ Bπ B , {Φ S (ξ i ), Φ A (x)} = -ξ i ∂ i B . (44)
Thus, in order to satisfy the consistency conditions (40), we have to impose the tertiary constraint

Φ ter ≡ N i ∂ i B - N 3µ g (3) g (3) ij π ij + 1 -3λ 2µ Bπ B -λ A F (A) ≈ 0 . ( 45
)
Since F (A) = 0 would essentially reproduce the original projectable Hořava-Lifshitz gravity, we assume that F (A) = 0. The first two terms in (45), i.e. the expression for Ḃ in (32), does not vanish due to the established constraints ( 30) and (37). Therefore ( 45) is a restriction on the Lagrange multiplier λ A , and we can solve it from Φ ter = 0:

λ A = 1 F (A) N i ∂ i B - N 3µ g (3) g (3) ij π ij + 1 -3λ 2µ Bπ B . (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 π A (x) and Φ A (x) to vanish strongly, and as a result turn the Hamiltonian constraint Φ 0 and the momentum constraint Φ S (ξ i ) 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)} + d 3 z 1 F (A(z)) ({f (x), π A (z)}{Φ A (z), h(y)} -{f (x), Φ A (z)}{π A (z), h(y)}) , (47)
where f and h are any functions of the canonical variables. Assuming we can solve the constraint Φ A (x) = 0, i.e. B = F (A), for A = Ã(B), where à is the inverse of the function F , we can eliminate the variables A and π A . Thus the final variables of the system are g (48)

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

H T = N Φ 0 + Φ S (N i ) + λ N π N + d 3 xλ i π i . ( 49
)
It defines the equations of motion for every function f (x) (or functional f ) of the canonical variables

ḟ (x) = {f (x), H T } = N {f (x), Φ 0 } + {f (x), Φ S (N i )} + λ N {f (x), π N } + d 3 yλ i (y){f (x), π i (y)} . ( 50
)
We have calculated the Hamitonian ( 33)-( 34) of the proposed modified Hořava-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 λ A of the primary constraint π A . Finally, we eliminated the pair of variables A, π A by imposing the second-class constraints π A and Φ A , 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 [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF] of this modified F (R) Hořava-Lifshitz gravity, which obeys the projectability condition, defines a consistent theory. This conclusion agrees with the recent analysis of our theory presented in ref. [START_REF] Klusoň | [END_REF].

IV. FRW COSMOLOGY FOR SOME VERSIONS OF MODIFIED HO ŘAVA-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 = H 0 , both of Eq. ( 19) and ( 20) lead to the same equation

0 = F 3 (1 -3λ + 6µ) H 2 0 -6 (1 -3λ + 3µ) H 2 0 F 3 (1 -3λ + 6µ) H 2 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 = H 2 0 1 -3λ + 9β (1 -3λ + 6µ) (1 -3λ + 2µ) H 2 0 .

(53)

Our proposal opens the bridge between the conventional modified gravity and its Hořava-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 foliationpreserving 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 Hořava-Lifshitz theory. This will be studied elsewhere.

( 3 )

 3 ij , π ij , B, π B . The lapse N and the shift vector N i , together with λ N and λ i , are non-dynamical multipliers. Then since every dynamical variable has a vanishing Poisson bracket with the constraint π A , the Dirac bracket (47) reduces to the Poisson bracket, {f (x), h(y)} DB = {f (x), h(y)} .

Acknowledgments

This research has been supported in part by MEC (Spain) project FIS2006-02842 and 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.

We consider N to be projectable, i.e. N = N (t), and therefore also the momentum π N = π N (t) is constant on Σ t for each t. The Poisson brackets are postulated in the form (equal time t is understood)

{N, π N } = 1 , {N i (x), π j (y)} = δ j i δ(x -y) , {A(x), π A (y)} = δ(x -y) , {B(x), π B (y)} = δ(x -y) , (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 K ij and Ḃ,

1 B g

(3)

ik g

(3)

jl π kl -

kl π kl -

and further obtain ġ(3

ij and B are dynamical variables and no more primary constraints are needed. The Hamiltonian is then defined

where the Lagrangian L is defined by the action ( 27), S F ( R) = dtL, and the so called Hamiltonian constraint and the momentum constraint are found to be

ik g

(3)

j B ,

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

where the primary constraints (30) are multiplied by the Lagrange multipliers λ N , λ i , λ A . Note that there is no space integral over the product λ N π N 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:

Therefore we impose the secondary constraints:

In the case of usual F (R)-gravity, where λ = µ = 1 and therefore 1-3λ +2µ = 0, there is only the trivial solution H 2 0 = 0, although the R 2 -term could generate the inflation when more gravitational terms, like R µν R µν etc., are added. For our general case, however, there exists the non-trivial solution

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 R 2 -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:

Then Eq. ( 51) becomes

which has the following two non-trivial solutions,

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

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 [START_REF] Nojiri | [END_REF], 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 R 2 -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 Hořava-Lifshitz F (R)-gravity.

V. MORE GENERAL ACTION

In the formulation of F (R) Hořava-Lifshitz-like gravity, we do not require full diffeomorphism-invariance, but only invariance under "foliation-preserving" diffeomorphisms [START_REF] Wald | General Relativity[END_REF]. Therefore there are many invariants or covariant quantities made from the metric like K, K ij , ∇

Then the action composed of such invariants as

i1 ∇

(3)

ijkl , ∇

i R

(3)

could be a rather general action for the generalized Hořava-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

Then one gets

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

Therefore, if we consider the FRW cosmology, the function F should depend on only two variables, H N and 18) is given by this combination. As an illustrative example, we may consider the following one:

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

If we assume a de Sitter universe H = H 0 with constant H 0 , Eq. ( 64) reduces to

which has the non-trivial solution

as long as the r.h.s. is positive. In the same way, a large class of modified Hořava-Lifshitz gravities may be constructed. For instance, one can construct Hořava-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, R µν R µν , R µναβ R µναβ ), etc. It is remarkable that some special subclass of such Hořava-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 Hořava-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) Hořava-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) Hořava-Lifshitz gravity corresponds to the earlier proposed F (R) extension of Hořava-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 Hořava-Lifshitz gravity as limiting cases. The former offers interesting cosmological solutions, while the latter may hold the promise of UV-completeness.