Tiberiu Harko
email: harko@hkucc.hku.hk

Zoltán Kovács
email: zkovacs@mpifr-bonn.mpg.de

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

Thin accretion disk signatures in dynamical Chern-Simons modified gravity

Keywords: numbers: 04.50.Kd, 04.70.Bw, 97.10.Gz

A promising extension of general relativity is Chern-Simons (CS) modified gravity, in which the Einstein-Hilbert action is modified by adding a parity-violating CS term, which couples to gravity via a scalar field. In this work, we consider the interesting, yet relatively unexplored, dynamical formulation of CS modified gravity, where the CS coupling field is treated as a dynamical field, endowed with its own stress-energy tensor and evolution equation. We consider the possibility of observationally testing dynamical CS modified gravity by using the accretion disk properties around slowly-rotating black holes. The energy flux, temperature distribution, the emission spectrum as well as the energy conversion efficiency are obtained, and compared to the standard general relativistic Kerr solution. It is shown that the Kerr black hole provide a more efficient engine for the transformation of the energy of the accreting mass into radiation than their slowly-rotating counterparts in CS modified gravity. Specific signatures appear in the electromagnetic spectrum, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations of the emission spectra from accretion disks.

I. INTRODUCTION

Recently, modified theories of gravity have received a considerable amount of attention mainly motivated by the problems of dark energy (see [1] for reviews) and dark matter [2], and from quantum gravity. A promising extension of general relativity is Chern-Simons (CS) modified gravity [3][4][5], in which the Einstein-Hilbert action is modified by adding a parity-violating CS term, which couples to gravity via a scalar field. It is interesting to note that the CS correction introduces a means to enhance parity violation through a pure curvature term, as opposed to through the matter term, as is usually considered in general relativity. In fact, CS modified gravity can be obtained explicitly from superstring theory, where the CS term in the Lagrangian density is essential due to the Green-Schwarz anomaly-canceling mechanism, upon four-dimensional compactification [6]. Two formulations of CS modified gravity exist as independent theories, namely, the nondynamical formulation and the dynamical formulation (see [5] for an excellent recent review). In the former, the CS scalar is an a priori prescribed function, where its effective evolution equation reduces to a differential constraint on the space of allowed solutions; in the latter, the CS is treated as a dynamical field, possessing an effective stress-energy tensor and an evolution equation. The majority of the work, up to date, has considered the nondynamical formulation [7][8][9][10], whereas the dynamical formulation remains mostly unexplored territory.

Relative to rotating black hole spacetimes, several solutions in the nondynamical formulation were found in CS modified gravity [8][9][10]. The first solutions were found by Alexander and Yunes [8,9], using a far-field approximation (where the field point distance is considered to be much larger than the black hole mass). The second rotating black hole solution was found by Konno et al [10], using a small slow-rotation approximation, where the spin angular momentum is assumed to be much smaller than the black hole mass. However, it is interesting to note that recently, using the dynamical formulation of CS modified gravity, spinning black hole solutions in the slow-rotation approximation have been obtained [11,12].

An interesting feature of CS modified gravity is that it has a characteristic observational signature, which could allow to discriminate an effect of this theory from other phenomena. However, most of the tests of CS modified gravity to date have been performed with astrophysical observations and concern the non-dynamical framework. In particular, it was found that the CS modified theory predicts an anomalous precession effect [14], which was tested [15] with LAGEOS [START_REF] Ciufolini | Gravitomagnetism[END_REF]. Another constraint on the non-dynamical theory was proposed in [17], where it was considered that the CS correction could be used to explain the flat rotation curves of galaxies. However, in [18] a bound was placed on the non-dynamical model with a canonical CS scalar that is eleven orders of magnitude stronger than the Solar System one, using double binary pulsar data. Recently, using the dynamical formulation of CS modified gravity, a stringent constraint was placed Confidential: not for distribution. Submitted to IOP Publishing for peer review 8 March 2010 on the coupling parameter associated to the dynamical coupling of the scalar field [19].

In this work, we further extend the constraints placed on the dynamical formulation of CS gravity by using the observational signatures of thin disk properties around rotating black holes. In the context of stationary axisymmetric spacetimes, the mass accretion around rotating black holes was studied in general relativity for the first time in [START_REF] Novikov | Black Holes[END_REF], by extending the theory of non-relativistic accretion [START_REF] Shakura | [END_REF]. The radiation emitted by the disk surface was also studied under the assumption that black body radiation would emerge from the disk in thermodynamical equilibrium [22,23]. More recently, the emissivity properties of the accretion disks were investigated for exotic central objects, such as wormholes [24], and non-rotating or rotating quark, boson or fermion stars, brane-world black holes or gravastars [25][26][27][28][29][30][31][32][33][34].

Thus, it is the purpose of the present paper to study the thin accretion disk models for slowly-rotating black holes in the dynamical formulation of CS modified theories of gravity, and carry out an analysis of the properties of the radiation emerging from the surface of the disk. As compared to the standard general relativistic case, significant differences appear in the energy flux and electromagnetic spectrum for CS slowly-rotating black holes, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations of the emission spectra from accretion disks.

The present paper is organized as follows. In Sec. II, we review the dynamical formulation of CS modified gravity, and present the Yunes-Pretorius (YP) slowly-rotating solution found in [12]. In Sec. III, we review the formalism and the physical properties of the thin disk accretion onto compact objects, for stationary axisymmetric spacetimes. In Sec. IV, we analyze the basic properties of matter forming a thin accretion disk in slowly-rotating black hole spacetimes in CS modified gravity. We discuss and conclude our results in Sec. V. Throughout this work, we use a system of units so that c

= G = h = k B = 1,
where k B is Boltzmann's constant.

II. DYNAMICAL CHERN-SIMONS MODIFIED GRAVITY

In this Section, we write down the field equations of the Chern-Simons gravity, and present the Yunes-Pretorius (YP) slowly-rotating solution found in [12].

A. Field equations of Chern-Simons theory

Consider the dynamical Chern-Simons modified gravity theory provided by the action in the form

S = S EH + S CS + S ϑ + S mat . (1)
The first term is the standard Einstein-Hilbert action

S EH = κ d 4 x √ -gR , (2)
where κ -1 = 16πG and R is the Ricci scalar. The second term defined as

S CS = α 4 d 4 x √ -g ϑ * R R , (3)
is the Chern-Simons correction; the third term

S ϑ = - β 2 d 4 x √ -g [g µν (∇ µ ϑ) (∇ ν ϑ) + 2V (ϑ)] , (4)
is the scalar field term. The matter action is given by

S mat = d 4 x √ -gL mat , (5)
where L mat the matter Lagrangian.

The parameters α and β are dimensional coupling constants; the CS coupling field, ϑ, is a function of spacetime that parameterizes deformations from GR [12]; ∇ µ is the covariant derivative associated with the metric tensor g µν ; and the quantity * RR is the Pontryagin density defined as

* RR = * R τ σ µν R σ τ µν , (6)
where the dual Riemann tensor is given by * R τ σ µν = 1 2

µναβ R τ σαβ , with µναβ the 4-dimensional Levi-Civita tensor.

Varying the action S with respect to the metric g µν one obtains the gravitational field equation given by

G µν + α κ C µν = 1 2κ T mat µν + T ϑ µν , (7)
where G µν is the Einstein tensor, and C µν is the cotton tensor defined as

C µν = ∇ σ ϑ σαβ(µ ∇ β R ν) α + ∇ σ ∇ α ϑ * R α(µν)σ . ( 8
)
The total stress-energy tensor is split into the matter term T µν mat , and the scalar field contribution T µν ϑ , which is provided by the following relationship

T ϑ µν = β (∇ µ ϑ) (∇ ν ϑ) - 1 2 g µν (∇ µ ϑ) (∇ ν ϑ) -g µν V (ϑ) . (9)
Varying the action with respect to the scalar field ϑ, one obtains the equation of motion for the Chern-Simons coupling term, given by

β ∇ µ ∇ µ ϑ = β dV dϑ - α 4 * R R . (10)
Note that the evolution of the CS coupling is not only governed by its stress-energy tensor, but also by the curvature of spacetime. In the nondynamical formulation of CS modified gravity the constraint β = 0 is considered, while in the dynamical framework, β is allowed to be arbitrary, so that Eq. ( 10) is now the evolution equation for the CS coupling field.

Considering the diffeomorphism invariance of the matter part of the action, we have ∇ µ T µν mat = 0, and taking into account the Bianchi identities, i.e., ∇ µ G µν = 0, provides the following conservation law

∇ µ C µν = - 1 8 (∇ ν ϑ) * R R. (11)

B. Rotating black hole solutions in Chern-Simons model

In this paper, we consider the Yunes-Pretorius (YP) slowly-rotating solution found in [12], where the CS correction provides an effective reduction of the framedragging around a black hole, in comparison with that of the Kerr solution. We will not analyze the Konno, Matsuyama and Tanda (KMT) approximation [11], since the YP solution is taken to second order in the rotation parameter, and therefore gives more accurate results.

The Yunes-Pretorius (YP) approximation method employs two schemes [12], namely, a small-coupling approximation and a slow-rotation approximation. In particular, the small-coupling scheme treats the CS modification as a small deformation of general relativity. The slow-rotation scheme expands the background perturbations in powers of the Kerr rotation parameter a, and the background metric is formalized via the Hartle-Thorne approximation [13]. We refer the reader to [12] for details, and present the final metric given by

ds 2 = -1 - 2M r + 2a 2 M r 3 cos 2 θ dt 2 + 1 - 2M r -1 1 + a 2 r 2 cos 2 θ -1 - 2M r -1 dr 2 - 4Ma r sin 2 θ - 10 8 ξa r 4 1 + 12M 7r + 27M 2 10r 2 sin 2 θ dtdφ + r 2 + a 2 cos 2 θ dθ 2 + r 2 sin 2 θ + a 2 sin 2 θ 1 + 2M r sin 2 θ dφ 2 , (12)
with ξ = α 2 /(κβ).

In the following, we compare the properties of the metric given by Eq. ( 12) with the standard general relativistic Kerr metric, respectively, which in the equatorial approximation can be written as [22]

ds 2 = -DA -1 dt 2 + r 2 A (dφ -ωdt) 2 + D -1 dr 2 + dz 2 , (13)
where the coordinate z = r cos θ was used to replace θ, and

A = 1 + a 2 * x -4 + 2a 2 * x -6 and D = 1 -2x -2 + a 2 * x -4
, respectively. The dimensionless coordinate x is defined as x = r/M , and the spin parameter a * is defined as a * = J/M 2 = a/M .

III. THERMAL EQUILIBRIUM RADIATION PROPERTIES OF THIN ACCRETION DISKS IN STATIONARY AXISYMMETRIC SPACETIMES

A. Stationary and axially symmetric spacetimes

The physical properties and the electromagnetic radiation characteristics of particles moving in circular orbits around general relativistic bodies are determined by the geometry of the spacetime around the compact object. For a stationary and axially symmetric geometry the metric is given in a general form by ds 2 = g tt dt 2 + 2g tφ dtdφ + g rr dr 2 + g θθ dθ 2 + g φφ dφ 2 .

(14) In the equatorial approximation, which is the case of interest for our analysis, the metric functions g tt , g tφ , g rr , g θθ and g φφ only depend on the radial coordinate r, i.e., |θ -π/2|

1. To compute the relevant physical quantities of thin accretion disks, we determine first the radial dependence of the angular velocity Ω, of the specific energy E, and of the specific angular momentum L of particles moving in circular orbits in a stationary and axially symmetric geometry through the geodesic equations. The latter take the following form [24]

dt dτ = Eg φφ + Lg tφ g 2 tφ -g tt g φφ , (15)
dφ dτ = - Eg tφ + Lg tt g 2 tφ -g tt g φφ , (16)
g rr dr dτ 2 = -1 + E 2 g φφ + 2 E Lg tφ + L 2 g tt g 2 tφ -g tt g φφ . (17)
From Eq. ( 17) one can introduce an effective potential term as

V ef f (r) = -1 + E 2 g φφ + 2 E Lg tφ + L 2 g tt g 2 tφ -g tt g φφ . ( 18
)
For stable circular orbits in the equatorial plane the following conditions must hold: V ef f (r) = 0 and V ef f, r (r) = 0, where the comma in the subscript denotes a derivative with respect to the radial coordinate r. These conditions provide the specific energy, the specific angular momentum and the angular velocity of particles moving in circular orbits for the case of spinning general relativistic compact spheres, given by

E = - g tt + g tφ Ω -g tt -2g tφ Ω -g φφ Ω 2 , (19)
L = g tφ + g φφ Ω -g tt -2g tφ Ω -g φφ Ω 2 , (20)
Ω = dφ dt = -g tφ,r + (g tφ,r ) 2 -g tt,r g φφ,r g φφ,r . (21)
The marginally stable orbit around the central object can be determined from the further condition V ef f, rr (r) = 0, which provides the following important relationship

0 = (g 2 tφ -g tt g φφ )V ef f,rr = E 2 g φφ,rr + 2 E Lg tφ,rr + L 2 g tt,rr -(g 2 tφ -g tt g φφ ) ,rr , (22)
where g 2 tφ -g tt g φφ (appearing as a cofactor in the metric determinant) never vanishes. By inserting Eqs. ( 19)-( 21) into Eq. ( 22) and solving this equation for r, we obtain the radii of the marginally stable orbits, once the metric coefficients g tt , g tφ and g φφ are explicitly given.

B. Physical properties of thin accretion disks

For the thin accretion disk, it is assumed that its vertical size is negligible, as compared to its horizontal extension, i.e, the disk height H, defined by the maximum half thickness of the disk, is always much smaller than the characteristic radius r of the disk, H r. The thin disk has an inner edge at the marginally stable orbit of the compact object potential, and the accreting plasma has a Keplerian motion in higher orbits.

In steady state accretion disk models, the mass accretion rate Ṁ0 is assumed to be a constant that does not change with time. The radiation flux F emitted by the surface of the accretion disk can be derived from the conservation equations for the mass, energy and angular momentum, respectively. Then the radiant energy F (r) over the disk is expressed in terms of the specific energy, of the angular momentum, and of the angular velocity of the particles orbiting in the disk [START_REF] Novikov | Black Holes[END_REF]22],

F (r) = - Ṁ0 4π √ -g Ω ,r ( E -Ω L) 2 r rms ( E -Ω L) L ,r dr , ( 23
)
where Ṁ0 is the mass accretion rate, measuring the rate at which the rest mass of the particles flows inward through the disk with respect to the coordinate time t and r ms is the marginally stable orbit obtained from Eq. (22). Another important characteristics of the mass accretion process is the efficiency with which the central object converts rest mass into outgoing radiation. This quantity is defined as the ratio of the rate of the radiation energy of photons, escaping from the disk surface to infinity, and the rate at which mass-energy is transported to the central compact general relativistic object, both measured at infinity [START_REF] Novikov | Black Holes[END_REF]22]. If all the emitted photons can escape to infinity, the efficiency is given in terms of the specific energy measured at the marginally stable orbit r ms , = 1 -E ms .

(

) 24
For Schwarzschild black holes the efficiency is about 6%, whether the photon capture by the black hole is considered, or not. Ignoring the capture of radiation by the black hole, is found to be 42% for rapidly rotating black holes, whereas the efficiency is 40% with photon capture in the Kerr potential [23].

The accreting matter in the steady-state thin disk model is supposed to be in thermodynamical equilibrium. Therefore the radiation emitted by the disk surface can be considered as a perfect black body radiation, where the energy flux is given by F (r) = σT 4 (r) (σ is the Stefan-Boltzmann constant), and the observed luminosity L (ν) has a redshifted black body spectrum [28]:

L (ν) = 4πd 2 I (ν) = 8 π cos γ r f ri 2π 0 ν 3 e rdφdr exp (ν e /T ) -1 . ( 25
)
Here d is the distance to the source, I(ν) is the thermal energy flux radiated by the disk, γ is the disk inclination angle, and r i and r f indicate the position of the inner and outer edge of the disk, respectively. We take r i = r ms and r f → ∞, since we expect the flux over the disk surface vanishes at r → ∞ for any kind of general relativistic compact object geometry. The emitted frequency is given by ν e = ν(1 + z), and the redshift factor can be written as

1 + z = 1 + Ωr sin φ sin γ -g tt -2Ωg tφ -Ω 2 g φφ , (26)
where we have neglected the light bending [35,36].

IV. ELECTROMAGNETIC SIGNATURES OF ACCRETION DISKS AROUND SLOWLY-ROTATING BLACK HOLES IN DYNAMICAL CHERN-SIMONS GRAVITY

Close to the equatorial plane of the slowly-rotating black holes, one can introduce the coordinate z = r cos θ describing "the height above the equatorial plane" and write the metrics given by Eq. ( 12) in the form

ds 2 = -f (r)dt 2 -4 Ma r [1 + h 1 (r)]dtdφ + 1 + h 2 (r) f (r) dr 2 +r 2 [1 + h 3 (r)]dφ 2 + dz 2 , ( 27
)
where f (r) = 1 -2M/r is the Schwarzschild form factor, and

h 1 (r) = - 5 16
ξ Mr 3 1 + 12 7 M r + 27 10
M 2 r 2 , (28)
h 2 (r) = - a 2 r 2 f (r) , (29)
h 3 (r) = a 2 r 2 1 + 2M r (30)
for the YP metric.

A. Constants of motion

If we insert the metric components of Eq. ( 27) into the expressions ( 19)-( 21) of the specific energy, of the specific angular momentum, and of the angular velocity, we obtain

E = f + 2Mar -1 Ω(1 + h 1 ) f + 4Mar -1 Ω(1 + h 1 ) -r 2 Ω 2 (1 + h 3 ) , (31)
L = -2Mar -1 (1 + h 1 ) + r 2 Ω(1 + h 3 ) f + 4Mar -1 Ω(1 + h 1 ) -r 2 Ω 2 (1 + h 3 ) , (32)
Ω = - Ma 2r 3 H 1 (r) ± H 1 (r) + r 3 Ma 2 H 2 (r) , (33)
with

H 1 (r) = 1 + h 1 -rh 1,r H 2 (r) , H 2 (r) = 1 2 (1 + h 3 + rh 3,r ) .
As Eqs. ( 31)- (33) show, the constants of motion for the particles orbiting in the equatorial plane depend only on the metric functions f (r), h 1 (r) and h 3 (r), respectively. The coupling constant ξ of the CS gravity appears only in h 1 (r). Since h 3 (r) is non zero, it reduces the value of g φφ for any r outside the marginally stable orbit. However, the decrease in g φφ,r is proportional to h 3,r ∼ a 2 r -3 , and it causes only a small variation in the radial distribution of the angular velocity. The specific energy and angular momentum, depending on g φφ , decreases only slightly in amplitude as well. The non-vanishing functions h 1 (r), h 2 (r) and h 3 (r) give a negligible contribution to the volume element

√ -g CS = (1 + h 2 ) 4M 2 a 2 (1 + h 1 ) 2 r 2 -2Mr + r 2 (1 + h 3 ) ,
as compared to the case of the equatorial approximation for slowly rotating general relativistic black holes,

√ -g GR = 4M 2 a 2 r 2 -2Mr + r 2 .
As a result, the properties of particles orbiting the equatorial plane of slowly-rotating black holes in the standard general relativistic theory and in CS modified gravity are essentially the same. The only difference is in the location of the marginally stable orbits, which are strongly affected by the coupling. Since the inner edge of a thin accretion disk is supposed to be at the radius r ms , the radial profile of the energy flux radiated over the disk surface can indicate the differences in the mass accretion processes in the general relativistic theory, and in its CS type modification, respectively.

B. Flux and temperature distribution

In Figs. 1 we present the flux distribution for the slowly-rotating Kerr black holes, and for the slowly rotating YP solution, respectively. We consider the mass accretion driven by black holes with a total mass of M = 10 6 M , and with a mass accretion rate of Ṁ0 = 10 -12 M /year. In units of the Eddington accretion rate ṀEdd = 1.5 × 10 17 (M/M ) g/s we have Ṁ0 = 4.22 × 10 -10 ṀEdd . The spin parameter a * , runs from 0.1 to 0.4, whereas the coupling constant of the CS gravity is set to ξ = 28M 4 , 56M 4 , 112M 4 and 168M 4 , respectively.

The plots show that the energy flux profiles of the disks in the CS modified gravity models deviate from the slowly-rotating general relativistic Kerr black hole case. For the smallest values of ξ, the inner edge of the accretion disk is located at somewhat higher radius than the inner edge of the disk around the Kerr black hole (the location of the marginally stable orbits can be found in Table I, respectively, considered below). As the quantities E, L and Ω in the flux integral (23) are still close for the CS model of gravity to those for the general relativistic case, for the lower boundary r ms,GR < r ms,CS , the integral gives lower flux values. Thus, the maximum of the integrated flux is smaller in the CS modified theory of gravity, and it decreases further as we increase the coupling constant. The effect of the coupling becomes more important as the black holes are rotating faster: for higher values of the spin parameter the same increase in the coupling constants produces considerably lower flux values, and shifts the marginally stable orbit to somewhat higher radii, as compared to the case of the very slow rotation (like, for example, in the cases with a * = 0.1 and a * = 0.4).

Similar features can be found in Fig. 2, where we plot the temperature profiles of the disk. Although the differences here are not so large, since the temperature is proportional to F 1/4 , the CS approximations of the slowlyrotating black hole metrics can still be discriminated from the standard general relativistic case.

C. Disk spectra and conversion efficiency

In Fig. 3, we present the spectral energy distribution of the disk radiation around the slowly-rotating black holes for the general relativistic case, and for the CS modified gravity.

The plots show that with the increase of the coupling constants of the CS gravity, the cut-off frequency of the spectra decreases, from its value corresponding to the Kerr black hole, to lower frequencies of the order of 10 14 Hz. Similarly to the case of the flux profiles, the effects of the CS coupling on the spectral cut-off are stronger for black holes rotating faster than for very slowly-rotating black holes. For the radiation of the accretion disks around black holes this means that the CS theory produces rather similar disk spectra as in standard general relativity, even that with increasing coupling constants the radial distributions of the fluxes differ to some extent for these two theories (see the top left hand plot in Figs. 1, and 3, respectively).

In Table I we also present the conversion efficiency of the accreting mass into radiation for the case when the photon capture by the rotating black hole is ignored. The variation of as a function of ξ/M 4 is presented in Fig. 4.

The value of measures the efficiency of energy generating mechanism by mass accretion. The amount of energy released by matter leaving the accretion disk and falling down the black hole is the binding energy E(r)| r=rms of the black hole potential.

Table I shows that is always higher for rotating general relativistic black holes than for their counterparts in CS modified gravity. As the Kerr black holes spin up, the accreted mass-radiation conversion efficiency raises from about 6%, the characteristic value of the mass accretion of the static black holes, to 7.5%. This feature is much more moderate for the rotating black holes in the CS theory: with increasing rotational velocity, also increases; however, the rate of this increase becomes smaller for stronger CS coupling. However, these values show that the Kerr black holes provide a more efficient engine for the transformation of the energy of the accreting mass into radiation than their slowly-rotating counterparts in the modified CS theory of gravity, no matter what approximation is used.

V. DISCUSSIONS AND FINAL REMARKS

In the present paper we have considered the basic physical properties of matter forming a thin accretion disk in slowly-rotating black hole spacetimes in the context of the dynamical formulation of CS modified theories of gravity. The physical parameters of the disk -energy flux, temperature distribution and emission spectrum profiles -have been explicitly obtained for several values of the coupling constant for the YP solution. Due to the differences in the spacetime structure, the CS black holes present some very important differences with respect to the disk properties, as compared to the standard general relativistic Kerr case. We have also shown that the Kerr black holes also provide a more efficient engine for the transformation of the energy of the accreting mass into radiation than their slowly-rotating counterparts in the modified CS theory of gravity.

It is generally expected that most of the astrophysical The inner edge of the accretion disk and the efficiency for slowly-rotating black holes in general relativity and in the CS modified theory of gravity with the YP approximation. The lines where the value of ξ is not defined correspond to the general relativistic case.

objects grow substantially in mass via accretion. Recent observations suggest that around most of the active galactic nuclei (AGN's) or black hole candidates there exist gas clouds surrounding the central far object, and an associated accretion disk, on a variety of scales from a tenth of a parsec to a few hundred parsecs [37]. These clouds are assumed to form a geometrically and optically thick torus (or warped disk), which absorbs most of the ultraviolet radiation and the soft x-rays. The most powerful evidence for the existence of super massive black holes comes from the very long baseline interferometry (VLBI) imaging of molecular H 2 O masers in the active galaxy NGC 4258 [38]. This imaging, produced by Doppler shift measurements assuming Keplerian motion of the masering source, has allowed a quite accurate estimation of the central mass, which has been found to be a 3.6 × 10 7 M super massive dark object, within 0.13 parsecs. Hence, important astrophysical information can be obtained from the observation of the motion of the gas streams in the gravitational field of compact objects.

The flux and the emission spectrum of the accretion disks around compact objects satisfy some simple scaling relations, with respect to the simple scaling transformation of the accretion rate and mass. In order to analyze the scaling properties of the physical parameters of the accretion disks we introduce the scale invariant dimensionless coordinate x = r/M . Then the functions h 1 (x) and h 3 (x) given by Eqs. ( 28) and (30) depend only on the scale invariant dimensionless spin parameter a * = a/M , and they have no explicit dependence on the mass M . As a consequence, in Eqs. ( 19)-( 21), only the specific angular momentum and the rotational frequency have an explicit dependence on M , in the form L ∝ M and Ω ∝ M -1 , whereas E does depend only on a * . For any rescaling M 2 = αM 1 of the mass of the star we obtain x 2 = α -1 x 1 , and the relations

E (M 2 ; x 2 ) = E(M 1 )(x 1 ), L (M 2 ; x 2 ) = α L (M 1 ; x 1 ) , (34) and Ω
(M 2 ; x 2 ) = α -1 Ω (M 1 ; x 1 ) , (35)
respectively. For any rescaling Ṁ

) 0 = β Ṁ (1) (2
of the accretion rate, these relations give

F M 2 , Ṁ (2)
0

(x 2 ) = β α 2 F M 1 , Ṁ (1) 0
(x 1 ) , (36)
as the scaling relation of the flux integral given by F

max = 4 × 10 25 erg s -1 cm -2 . The scaling law of the temperature is T → Ṁ 1/4 0 / √ M T . Due to the temperature scaling, the maximum value of the spectrum increases, but the relative positions of the different spectral curves does not change.

The determination of the accretion rate for an astrophysical object can give a strong evidence for the existence of a surface of the object. A model in which Sgr A*, the 3.7 × 10 6 M super massive black hole candidate at the Galactic center, may be a compact object with a thermally emitting surface was considered in [39]. Given the very low quiescent luminosity of Sgr A* in the nearinfrared, the existence of a hard surface, even in the limit in which the radius approaches the horizon, places a severe constraint on the steady mass accretion rate onto the source, Ṁ ≤ 10 -12 M yr -1 . This limit is well below the minimum accretion rate needed to power the observed submillimeter luminosity of Sgr A*, Ṁ ≥ 10 -10 M yr. Thus, from the determination of the accretion rate it follows that Sgr A* does not have a surface, that is, it must have an event horizon. Therefore the study of the accretion processes by compact objects is a powerful indicator of their physical nature. However, up to now, the observational results have confirmed the predictions of general relativity mainly in a qualitative way. With the present observational precision one cannot distinguish between the different classes of compact/exotic objects that appear in the theoretical framework of general relativity [29].

However, important technological developments may allow one to image black holes and other compact objects directly [39]. In principle, detailed measurements of the size and shape of the silhouette could yield information about the mass and spin of the central object, and provide invaluable information on the nature of the accretion flows in low luminosity galactic nuclei.

The spectrum in black hole systems can be dominated by the disc emission [40]. Recently, the RXTE satellite has provided a large number of data of the X-ray observations of the accretion flows in galactic binary systems. there is also a huge increase of the radio data for these systems. The behavior of the spectrum in such systems is consistent with the existence of a last stable orbit, and such data can be used to estimate the black hole spin. At high luminosities these systems can also show very different spectra [40]. Changes in the spectra of the disks are driven by a changing geometry. Presently there exists an enormous amount of data from the X-ray binary systems which can be used to test this assumption. Therefore the study of accretion processes in Low Mass X-ray Binaries with well constrained thermal spectra could also lead to the possibility of discriminating between the various extensions of standard general relativity.

Hence the study of the accretion processes by compact objects is a powerful indicator of their physical nature. Since the energy flux, the temperature distribution of the disk, the spectrum of the emitted black body radiation, as well as the conversion efficiency show, in the case of the Chern-Simons theory vacuum solutions, significant differences as compared to the general relativistic case, the determination of these observational quantities could discriminate, at least in principle, between standard general relativity and Chern-Simons gravity, and constrain the parameters of the model.

4 FIG. 1 :

 41 FIG.1:The energy flux from accretion disks around slowly-rotating black holes for different spin parameters in the general relativity, and in the modified CS theory of gravity with the YP approximation. The coupling constant ξ is running from 28M 4 to 168M4 . The spin parameters are set to a * = 0.1 (upper left hand plot), a * = 0.2 (upper right hand plot), a * = 0.3 (lower left hand plot) and a * = 0.4 (lower right hand plot), respectively. The total black hole mass is 10 6 M and the mass accretion rate is 10 -12 M /year=4.22 × 10 -10 ṀEdd .

4 FIG. 2 :

 42 FIG.2:The temperature distribution over the accretion disks around slowly-rotating black holes for different spin parameters in general relativity and in the modified CS theory of gravity with the YP approximation. The values of M , a * and Ṁ0 are the same as in Fig.1.

FIG. 3 :

 3 FIG.3:The accretion disks spectra for slowly-rotating black holes with different spin parameters in general relativity and in the modified CS theory of gravity with the YP approximation. The values of M , a * and Ṁ0 are the same as in Fig.1.

5

 5

TABLE I :

 I FIG. 4: The conversion efficiency as a function of the CS coupling parameter ξ/M 4 for different values of the spin parameter.

				6.6 6.8 7		a * =0.1 a * =0.2 a * =0.3 a * =0.4
			ε [%]	6.2 6.4
				6
				5.8
				.6
				20	40	60	80	100 120 140 160
						ξ / M 4
	a * ξ/M 4 rin/M
	0.1	-	5.6739 0.0606
		28 5.7936 0.0599
		56 5.8952 0.0592
		112 6.0762 0.0581
		168 6.2250 0.0571
	0.2	-	5.3315 0.0646
		28 5.6122 0.0626
		56 5.8226 0.0611
		112 6.1405 0.0588
		168 6.3807 0.0572
	0.3	-	4.9818 0.0694
		28 5.4662 0.0653
		56 5.7744 0.0628
		112 6.1951 0.0595
		168 6.5025 0.0573
	0.4	-	4.6182 0.0751
		28 5.3546 0.0679
		56 5.7456 0.0643
		112 6.2550 0.0601
		168 6.6100 0.0574

Acknowledgments

We would like to thank the two anonymous referees whose comments and suggestions helped us to significantly improve the manuscript. The work of T. H. was supported by the General Research Fund grant number HKU 701808P of the government of the Hong Kong Special Administrative Region.

Eq. (23). Then the temperature scales like

0

(x 1 ) .

(37) For the maximum of the luminosity L we have ν max ∝ T , which gives L (ν max ) ∝ ν 3 max . As the frequency scales like the temperature, we obtain that

0

0

with

On the other hand, the flux is proportional to the accretion rate Ṁ0 , and therefore an increase in the accretion rate leads to a linear increase in the radiation emission flux from the disk. For a simultaneous scaling of both the accretion rate Ṁ0 and of the mass of the black hole M , the maximum of the flux scales as F → Ṁ0 /M 2 F , with all the other characteristics of the flux unchanged. Thus, if for a black hole mass with mass M 1 = 10 6 M and by considering a mass accretion rate of Ṁ

ṀEdd , in the case a * = 0.1 and ξ = 28M 4 , the maximum of the flux is F


S Carloni
email: carloni@ieec.uab.es

E Elizalde
email: elizalde@ieec.fcr.es

P J Silva
email: psilva@ifae.es

J Silva

An analysis of the phase space of HoavaLifshitz cosmologies

Keywords: numbers: 04.50.-h, 98.80.Jk, 45.30.+s, 04.60.Bc, 95.36.+x

Using the dynamical system approach, properties of cosmological models based on the Hořava-Lifshitz gravity are systematically studied. In particular, the cosmological phase space of the Hořava-Lifshitz model is characterized. The analysis allows to compare some key physical consequences of the imposition (or not) of detailed balance. A result of the investigation is that in the detailed balance case one of the attractors in the theory corresponds to an oscillatory behavior. Such oscillations can be associated to a bouncing universe, as previously described by Brandenberger, and will prevent a possible evolution towards a de Sitter universe. Other results obtained show that the cosmological models generated by Hořava-Lifshitz gravity without the detailed balance assumption have indeed the potential to describe the transition between the Friedmann and the dark energy eras. The whole analysis leads to the plausible conclusion that a cosmology compatible with the present observations of the universe can be achieved only if the detailed balance condition is broken.

Introduction

Recently, Hořava made a proposal for an ultraviolet completion of general relativity (GR) normally referred to as the Hořava-Lifshitz (HL) theory [START_REF] Hořava | Quantum Gravity at a Lifshitz Point[END_REF]. (The name is due to Hořava's initial inspiration on the Lifshitz theory in solid state physics.) The salient characteristic of the HL proposal is that it seems to be renormalizable, at least at the level of power counting. This ultraviolet behavior is obtained by introducing irrelevant operators that explicitly break Lorentz invariance but ameliorate the ultraviolet divergences. On the other hand, Lorentz invariance is expected to be recovered at low energies, as an accidental symmetry of the theory.

Originally, the HL proposal came with the possibility of imposing or not the socalled projectability condition and the detailed balance condition. The first condition is related to the space-time dependence of the lapse function, N , which characterizes a canonical 3 + 1 decomposition of the metric field g, while the second is a restriction on the form of the potential terms which may appear in the Lagrangian that leads to simplifications since it reduces the final number of couplings. Notice therefore that we have, in principle, four different incarnations of this proposal.

Since its publication, the HL theory has been the object of an exhaustive research regarding its different properties and implications to space-time physics. In particular, a lot of attention has been paid to its internal consistency, to how to define the infrared limit, its compatibility with GR, and the potential application of the results obtained to cosmology.

Presently-as this article is being written-the consistency status of the theory is not completely clear, nor its low energy limit and, hence, how GR is recovered at the different regimes. In fact, for the non-projectable version, although the low energy limit has been found [START_REF] Afshordi | Cuscuton and low energy limit of Hořava-Lifshitz gravity[END_REF], there seem to remain important problems, regarding strongly coupled features that may preclude any type of perturbation approach [START_REF] Charmousis | Strong coupling in Horava gravity[END_REF][START_REF] Blas | On the extra mode and inconsistency of Hořava Gravity[END_REF], while the corresponding symplectic structure and systems of constraints do not seem to be well defined [START_REF] Kobakhidze | On the infrared limit of Hořava's gravity with the global Hamiltonian constraint[END_REF][START_REF] Kocharyan | Is nonrelativistic gravity possible?[END_REF][START_REF] Mukohyama | Dark matter as integration constant in Hořava-Lifshitz gravity[END_REF]. Also, imposing detailed balance leads to a cosmological constant with the wrong sign, from what we should expect to be at odds with cosmological observations [START_REF] Hořava | Quantum Gravity at a Lifshitz Point[END_REF][START_REF] Kiritsis | Hořava-Lifshitz Cosmology[END_REF]. In comparison, the projectable version seems to be less problematic, since the above listed problems can in principle be evaded by the non-local form of the Hamiltonian constraint [START_REF] Kocharyan | Is nonrelativistic gravity possible?[END_REF]. Also, if detail balance is not imposed a richer phenomenology seems to appear, where cosmological applications may lead to new results in inflation, bouncing cosmology, dark matter, and dark energy (see, for example, [START_REF] Kiritsis | Hořava-Lifshitz Cosmology[END_REF][START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF][START_REF] Sotiriou | Phenomenologically viable Lorentz-violating quantum gravity[END_REF][START_REF] Nastase | On IR solutions in Hořava gravity theories[END_REF][START_REF] Brandenberger | Matter bounce in Hořava-Lifshitz cosmology[END_REF][START_REF] Wang | Thermodynamics and classification of cosmological models in the Horava-Lifshitz theory of gravity[END_REF][START_REF] Calcagni | Cosmology of the Lifshitz universe[END_REF][START_REF] Piao | Primordial perturbation in Hořava-Lifshitz cosmology[END_REF][START_REF] Park | A Test of Horava Gravity: The Dark Energy[END_REF][START_REF] Appignani | The Cosmological Constant and Horava-Lifshitz Gravity[END_REF][START_REF] Cai | Non-singular cosmology in a model of non-relativistic gravity[END_REF]).

At this point it is important to investigate the key aspects of the theory, which may help in clarifying the status of the different HL proposals as plausible candidates of a quantum theory of gravity. By now, it is clear that the study of the consistency of the theory is a complicated problem that will still take a lot of effort and time to ascertain and that will involve detailed research of the renormalization properties of the theory and, most probably, of its non-perturbative nature (see e.g. [START_REF] Orlando | On the Perturbative Expansion around a Lifshitz Point[END_REF][START_REF] Visser | Lorentz symmetry breaking as a quantum field theory regulator[END_REF]). On the other hand, another important requirement the theory should fulfill is its implication with phenomenology and, in particular, with cosmology. In this last regard there are already papers on some of the consequences of these new scenarios ‡; however, no systematic study of the possible form of the cosmological relevant solutions of the theory, either for the detailed balance case or for a general potential, has appeared yet.

In the present work, motivated by the above facts, we investigate the space of all possible classical solutions of the HL theory that are relevant for cosmology, i.e. we study the cosmological phase space of the HL model. To address this involved issue, we will borrow techniques from the field of dynamical systems that are frequently used on more canonical studies applied to diverse types of cosmologies. As a matter of fact, dynamical system techniques in cosmology have been used for many years. In our work, we will consider the dynamical system approach to cosmology as formulated by Collins and applied by Wainwright and Ellis [START_REF]Dynamical Systems in Cosmology[END_REF]. Such approach has the advantage of relying on dynamical system variables which are directly related to cosmological observables-like the matter density parameter-and of being relatively easy to apply to very complicated cosmological models. During the past years this method has been able to unfold some very interesting properties of Bianchi universes [START_REF]Dynamical Systems in Cosmology[END_REF], as well as of scalar-tensor and higher-order gravity cosmological models (for some detailed examples, see [START_REF] Carloni | Cosmological dynamics of scalartensor gravity[END_REF][START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | Some remarks on the dynamical systems approach to fourth order gravity[END_REF][START_REF] Abdelwahab | Cosmological dynamics of exponential gravity[END_REF]).

As we will see, when applied to HL cosmologies these techniques will be able to reveal some interesting results. In particular, when detailed balance is imposed, although one is able to prove that the corresponding cosmology could-given suitable initial conditions-behave like a Friedmann one, it turns out that it will eventually evolve towards an oscillatory behavior or either recollapse, leaving no space for a darkenergy era. When detailed balance is broken, however, the situation changes: given the proper initial conditions and values of the parameters the cosmology undergoes an almost Friedmann era(s) to evolve towards a de Sitter era which can be associated to dark energy. In this sense the plausible consequence is that a cosmology compatible with the present observations of the universe can be achieved only if the detailed balance condition is broken. In addition to that, we will show that the information on the cosmological relevance of the values and signs of the parameters in the theory helps to uncover properties of the general theory. For example, from the relative sign of the Hubble and matter terms in the cosmological equations, we will be able to deduce that the spin zero modes of the theory can be excluded.

The paper is organized as follows. In Sect. 2 we present the Lagrangian of the HL theory for the two cases of either imposing or not the detailed balance constraint, where the minimal potential defined in [START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF] for the non-detailed-balance case will be used. The projectability condition does not affect our analysis, since for the cosmological ansatz the shift N is only time dependent. In Sects. 3 and 4 we present the specific analysis for the theory with detailed balance and for the one without detailed balance, respectively. ‡ See [START_REF] Wang | Thermodynamics and classification of cosmological models in the Horava-Lifshitz theory of gravity[END_REF][START_REF] Leon | Phase-space analysis of Horava-Lifshitz cosmology[END_REF] for some general results of this type based on different assumptions and technics.

We will be able to characterize the classical phase space, discussing its structure in depth, in particular all the fixed points and their nature as repellers or attractors, in each of the cases. Finally, in Sect. 5 we summarize the results obtained, giving some perspectives for further work.

Hořava-Lifshitz gravity

The following is a short overview on the minimal set of definitions needed to understand the dynamical system analysis of the forthcoming sections. In the HL theory, the dynamical variables are defined to be the laps N , the shift N i and the space metric g ij , Latin indices running from 0 to 3. The space-time metric is defined using the ADM construction, as

ds 2 = -N 2 dt 2 + g ij (dx i + N i dt)(dx j + N j dt) , (1)
where N i = g ij N j as usual. The action S is written in terms of geometric objects, characteristics of the ADM slicing of space-time, like the 3d-covariant derivative ∇ i , the spatial curvature tensor R ijkl , and the extrinsic curvature K ij . They are defined as follows,

R i jkl = W i jl,k -W i jk,l + W m jl W i km -W n jk W i lm , (2)
where W i jl are the Christoffel symbols (symmetric in the lower indices), given by

W i jl = 1 2
g im (g jm,l + g ml,j -g jl,m ) .

The Ricci tensor is obtained by contracting the first and the third indices

R ij = g kl R ikjl and R = g ij R ij , (4)
while the extrinsic curvature is defined as

K ij = 1 2N (-ġij + ∇ i N j + ∇ j N i ) , (5)
where the dot stands for time derivative.

In terms of the above tensor fields, the HL action can be written as

S = dt dx 3 N √ g (L kinetic -L potential + L matter ) , (6)
being the kinetic term universally given by

L kinetic = α(K ij K ij -λK 2 ) , (7)
with α and λ playing the role of coupling constants. The potential term is, in principle, a generic function of R ijkl and ∇ i . Here we will work with two different types of potentials: first, the detailed balance potential and, second, the potential defined in [START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF] (the SVW case), that corresponds to the more general potential with dimensionless kinetic couplings (see the original article for clarifications.). Finally, the matter term corresponds to the coupling of the matter fields to gravity.

The potential for the detailed balance case is

L potential-detail = βC ij C ij + γ ijk R il ∇ j R l k + ζR ij R ij + ηR 2 + δR + σ , (8) where √ gC ij = ikl ∇ k (R j l -1/4Rg j l
) and the Greek letters define the different coupling constants. We have closely followed the notation of [START_REF] Kiritsis | Hořava-Lifshitz Cosmology[END_REF] for simplicity. We stress that only three out of the whole set of coupling constants above are independent, due to the detailed balance constraint. Then, it is not difficult to see that, if we want the theory to make contact with GR in the infrared regime, we have to do the following identifications:

α = c 16πG N , λ = 1 , δ = 1 16πcG N , σ = -2Λ 16πcG N , (9)
with all other couplings being equal to zero. To get the above results, we also have defined a new time coordinate x 0 = ct, where (c, G N , Λ) are the velocity of light, Newton's constant, and the cosmological constant, respectively. From this point onwards, we will keep c = 1, so that our time coordinate is actually t.

The potential of the SVW case is

L potential-SV W = g 8 ∇ i R jk ∇ i R jk + g 7 R∇ 2 R + g 6 R i j R j k R k i + g 5 R(R jk R jk ) + g 4 R 3 (10) + g 3 R jk R jk + g 2 R 2 + g 1 R + g 0 ,
where we have modified the original notation to better accommodate our formulation of the universal kinetic term [START_REF] Kocharyan | Is nonrelativistic gravity possible?[END_REF], therefore our coupling constants are not all dimensionless.

As in the detailed balance case, in order to recover GR at low energies we get similar equations to those in [START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF]. Just notice we need to identify δ ⇔ g 1 and σ ⇔ g 0 , while all the other couplings in the potential should go to zero. At this point, other phenomenological and theoretical considerations may help to constraint the range of values that the different couplings should take. For example, in [START_REF] Bogdanos | Perturbative instabilities in Hořava gravity[END_REF] it was found that ghost instabilities are present if λ ∈ (1/3, 1), the cosmological constant is negative for the detailed balance potential, α > 0, etc. Here, we will constraint as little as possible the different ranges of values on each coupling constant to see how much information comes out of the dynamical system approach. Then, we will add this information to the constraints arising from other considerations, to finally obtain the most promising form of the potential. In particular, we will take λ different from 1/3 (corresponding to the scale invariant case) as the only limitation on its range.

To start our studies on cosmological applications, we have to consider a FLRW ansatz on the 4D metric. Due to the homogeneity and simple time dependence of this type of ansatz, only a subset of the coupling constants plays a role in the dynamics, in any case. Also, as we already said, N is a function of time only and, therefore, all issues related to the projectability condition turn out to be irrelevant §. Then, we set, [START_REF] Nastase | On IR solutions in Hořava gravity theories[END_REF] § Recall that we are interested in the classical phase space and its structure. On the other hand, the dynamics of fields propagating in the above solutions are determined by the specific incarnation of the HL theory chosen, where projectability is important.

N -→ N (t) , N i -→ 0 , g ij -→ a(t)γ ij , (
where γ ij is a maximally symmetric 3D metric, of constant curvature R = 6k, with k = (-1, 0, 1).

The inclusion of matter content in the theory in its general form has not been worked out yet. There are various discussions and suggestions (see for example [START_REF] Kiritsis | Hořava-Lifshitz Cosmology[END_REF]), but this is still an open subject. An interesting approach is based on adding some new degrees of freedom to the framework, so that the theory gains in diffeomorphism invariance. Then, minimal coupling to matter is invoked, as in GR, although new couplings to the extra degrees of freedom could also be introduced (see [START_REF] Blas | On the extra mode and inconsistency of Hořava Gravity[END_REF][START_REF] Germani | Relativistic Quantum Gravity at a Lifshitz point[END_REF][START_REF] Nojiri | Covariant Hořava-like renormalizable gravity and its FRW cosmology[END_REF] for this type of constructions). Here, we will follow previous discussions where a very general approach is taken. We will add to the gravity field equations a cosmological stress-energy tensor, such that in the low-energy limit we recover the usual GR formulation. This tensor is a hydrodynamical approximation with two quantities, ρ, p, corresponding to the density and pressure, and where both matter fields are related by the usual equation of state p = wρ. Since one of our goals is to investigate the relation between HL and dark energy (cosmic acceleration), we will only consider here w > 0, so that no dark energy is introduced by hand in the model.

Hořava-Lifshitz cosmology: detailed balance case

From the discussion of the previous section, we can write the relevant field equations on a FLRW ansatz. We found, after some trivial algebra, that the system can be written as follows

α(3λ -1) Ḣ + α(3λ -1)H 2 = - 1 6 (1 + 3w)ρ + Aα(1 -3λ) k 2 a 4 + Aα(3λ -1)Λ 2 , (12a) 3α(3λ -1)H 2 + 6Aα(3λ -1)Λ k a 2 = ρ + 3Aα(3λ -1) k 2 a 4 + 3Aα(3λ -1)Λ 2 , ( 12b
) ρ + 3(w + 1)Hρ = 0 , (12c)
where we have chosen to leave the parameters α, λ, Λ explicit because of their cosmological relevance, and we have defined the variable A = -ζ α(1-3λ) 2 which is always positive, since ζ ≤ 0 owing to the detailed balance constraint. Notice that, already at this level, only a subset of all the initially defined couplings (α, λ, Λ, A), plays a role in the cosmology.

Dynamical analysis

In order to analyze the phase space of this cosmological model, let us define the variables

Ω = ρ 3αH 2 , z = AΛ 2 H 2 , K = 2Λ kA a 2 H 2 , C = k 2 A a 4 H 2 , ( 13
)
with the cosmic time N = ln[a(t)].

The cosmological equations (12a)-(12c) are then equivalent to the system

Ω = 1 + 3w 3λ -1 Ω 2 + Ω(2C -2z -3w -1) , (14a)
z = -2z 2 + z 2 + 2C + 1 + 3w 3λ -1 Ω , (14b)
K = K 2C -2z + 1 + 3w 3λ -1 Ω , (14c)
C = 2C 2 -C 2 + 2z - 1 + 3w 3λ -1 Ω , (14d)
with the Gauss constraint

1 + K -z -C + Ω 1 -3λ = 0 , (15)
where the prime denotes the derivative with respect to N . The [START_REF] Piao | Primordial perturbation in Hořava-Lifshitz cosmology[END_REF] allows to eliminate one of the equations. If we choose to eliminate the one for Ω we obtain

z = z[3 + C + K -3z -3w(C -K + z -1)] , (16a)
C = C [C + K -3z -3w(C -K + z -1) -1] , (16b)
K = K [1 + C + K -3z -3w(C -K + z -1)] . (16c)
This system presents three invariant submanifolds z = 0, K = 0, C = 0 which, by definition, cannot be crossed by any orbit. This implies that no global attractor can exist in this type of HL cosmology. Also, the structure of ( 15) reveals that the system is non-compact and asymptotic analysis will be required in order to complete the study of the phase space. Due to its dimensionality it is not easy to draw a plot of the phase space. Naturally, since invariant submanifolds are present, one could think to give a plot of orbits belonging to them, however the physical meaning of such manifold is not clear. For example, the invariant submanifold C = 0 might be associated at first glance to flat cosmology, however from the definitions [START_REF] Wang | Thermodynamics and classification of cosmological models in the Horava-Lifshitz theory of gravity[END_REF], one can see that this would be the case only for K = 0, so that the rest of C = 0 has no real physical meaning. Similar arguments can be given for z = 0 and K = 0 submanifolds. As consequence only in orbits in the phase space bulk correspond to physical evolutions for the system and we have describe them without the aid of graphics .

Finite analysis.

Let us start with the finite analysis. The finite fixed points can be found setting z , C , K in (16a)-(16c) to be zero and solving the corresponding algebraic equations. The results are shown in Table 1. The solutions associated to the fixed point can be derived from the Raychaudhuri equation

Ḣ = 1 2 [3z -C -K -3 + 3w(C -K + z -1)] H 2 , (17)
and the results are shown in Tab. 1, too. As one can see there, we have a general Friedmann solution which depends on the barotropic factor w of standard matter, a

Of course one could consider the Ω = 0 invariant submanifold which could be of interest, but in our setting this would require the derivation of a new system of the type (16a)-(16c) in which another variable is eliminated. Since here we are mainly interested in the role of the HL corrections, drawing Ω = 0 will not add much to the understanding of the dynamics of the cosmology. pure radiation like solution, a Milne universe, and an exponential solution. Note that the Friedmann solution is generated when the term related to the spatial curvature k is dominant. Instead, when the effective Hořava radiation is dominant, the corresponding cosmology presents a "radiation-dominated like" solution, as expected from the form of the corresponding terms in (12a)-(12c). Substitution into Eqs. (12a)-(12c) reveals that this is actually a vacuum solution and that the exponential solution corresponds to a an oscillating evolution with period T = 2π 3λ-1 Λ α |ζ| . Such solution can be connected to the scenario proposed in [START_REF] Brandenberger | Matter bounce in Hořava-Lifshitz cosmology[END_REF]. The Milne solution, instead, is not an actual solution of the system ¶.

The stability of the fixed points can be determined by evaluating the eigenvalues of the Jacobian matrix associated with the system (16a)-(16c), as prescribed by the Hartman-Grobman Theorem [START_REF] Hartman | Ordinary Differential Equations[END_REF]. The results can be found in Table 1. We can observe that the thermodynamical properties of matter influence the stability of the Friedmann point A and of the point B. This means that, if 0 < w < 1/3, the typical completely finite orbit implies an initial radiation-like behavior that evolves towards a Friedmann or a Milne behavior (or both), to eventually approach an oscillating state. Instead, if 1/3 < w < 1, A is a source so that the typical orbit will start with a Friedmann evolution and evolve towards a radiation-like or a Milne evolution before converging to an oscillatory behavior. In both these scenarios we do not find any transition to a dark energy era.

In fact, in terms of the phase space one can characterize this transition as the fact that the orbits will cross the plane

C + K -3z -3(C -K + z -1)w + 1 = 0 , (18)
but none of the finite orbits described above can actually achieve that. This conclusion was in some sense expected, since it was already noticed [START_REF] Nastase | On IR solutions in Hořava gravity theories[END_REF] that in this case the cosmological term seems to have the wrong sign when the detailed balance condition is imposed.

Asymptotic analysis

Owing to the fact that the dynamical system (3.1) is non-compact, there could be features in the asymptotic regime which are non trivial for the global dynamics. Thus, in order to complete the analysis of the phase space we will now extend our study using the Poincaré projection method [START_REF] Lefschetz | Differential Equations: Geometric Theory[END_REF]. This method is based on a phase space coordinate change

C = ρ sin θ cos φ, K = ρ sin θ sin φ, z = ρ cos θ , (19)
where

ρ = r 1 + r , (20)
¶ However this does not constitute a real problem because, as we will see, this point is always unstable and there is no orbit which can reach it. Table 1. Finite fixed points of the system (16a)-(16c) and their associated solutions.

Point Coordinates Solution Energy density Stability

[Ω, z, C, K] A [3λ -1, 0, 0, 0] a = a 0 (t -t 0 ) 2 3(1+w) ρ = ρ 0 (t -t 0 ) -2 saddle 0 ≤ w ≤ 1/3 repeller 1/3 ≤ w ≤ 1 B [0, 0, 1, 0] a = a 0 (t -t 0 ) 1 2 ρ = 0 repeller 0 ≤ w ≤ 1/3 saddle 1/3 ≤ w ≤ 1 C [0, 0, 0, -1] a = a 0 (t -t 0 ) ρ = 0 saddle D [0, 1, 0, 0] a = a 0 e τ (t-t 0 ) ρ = 0 attractor τ = i |ζ| α Λ 3λ-1 θ ∈ [0, π], and φ ∈ [0, 2π].
In this way, the limit ρ → 1 corresponds to r → ∞, and one can study the asymptotics of the system. Performing the transformation (19), the system (16a) becomes, in the limit ρ → 1,

ρ → 3(1 + w) cos θ -sin θ [(1 -3w) cos φ + (1 + 3w) sin φ] , (21a)
θ → 1 2 sin(2θ)[3 + cos(2φ)] , (21b)
φ → sin(2φ) .

Notice that the radial equation does not contain the radial coordinate, so that the fixed points can be obtained using the angular equations only. Setting φ = 0 and θ = 0, we obtain the fixed points which are listed in Table 2. The solutions associated with these points can be found with the method described in [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF]. The corresponding results are also given in Table 2. The stability of these points is studied by analyzing first the stability of the angular coordinates and then deducing, from the sign of Eq. (21a), the stability on the radial direction [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF]. Ensuing results are given in Table 3. The presence of these additional points greatly enriches the dynamics of the cosmology. The new points represent a Gauss function evolution that is characterized by a very fast expansion and a similarly fast recollapse. If unstable, such points could represent-at least in principle-an Point Coordinates Solution (φ, θ)

E 1 [0, 0] a = a 0 e 3 2 (w+1)(t-t 0 ) 2 E 2 [π, π] a = a 0 e 3 2 (w+1)(t-t 0 ) 2 F 1 [0, π/2] a = a 0 e 1 4 (1-3w)(t-t 0 ) 2 F 2 [π, π/2] a = a 0 e 1 4 (1-3w)(t-t 0 ) 2 G 1 [π/2, π/2] a = a 0 e 1 4 (1+3w)(t-t 0 ) 2 G 2 [3π/2, π/2] a = a 0 e 1 4 (1+3w)(t-t 0 ) 2 Table 2.
Asymptotic fixed points of the system (16a)-(16c), their associated solutions, and stability.

inflationary or dark energy era. However, none of the orbits connecting these points can be considered of much cosmological relevance, because the ones that involve the asymptotic fixed points with the right stability behavior do not contain a transient Friedmann era.

Hořava-Lifshitz cosmology: no detailed balance case (SVW potential)

If we do not impose detailed balance to hold, the cosmological equations in presence of matter (assumed to be a barotropic fluid) can be written as (we closely follow the notation of [START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF])

1 - 3ξ 2 Ḣ + H 2 + 1 2 κ 2 ρ(1 + 3w) - χ 1 6 + χ 3 k 2 6a 4 + χ 4 k 3a 6 = 0 , (22a) 1 - 3ξ 2 H 2 - χ 2 k 6a 2 -κ 2 ρ - χ 1 6 - χ 3 k 2 6a 4 - χ 4 k 6a 6 = 0 , ( 22b
) ρ + 3(w + 1)Hρ = 0 , (22c)
where

κ 2 = 1 6α , ( 23a
)
χ 1 = g 0 α 3 6 , (23b)
χ 2 = -6g 1 α 2 > 0 , (23c)
χ 3 = 12α (3g 2 + g 3 ) , (23d)

Point Eigenvalues

ρ Stability

E 1 2i √ 2, -2i √ 2 ρ > 0 always attractive center E 2 2i √ 2, -2i √ 2 ρ < 0 always repulsive center F 1 2 √ 2, -2 √ 2 ρ > 0 0 ≤ w < 1 3 , ρ < 0 1 3 < w ≤ 1 saddle F 2 2 √ 2, -2 √ 2 ρ > 0 1 3 < w ≤ 1, ρ < 0 0 ≤ w < 1 3 , saddle G 1 [2i, -2i] ρ < 0 always repulsive center G 2 [2i, -2i] ρ > 0 always attractive center χ 4 = 24 (9g 4 + 3g 5 + g 6 ) , (23e)
and we have defined ξ = 1 -λ assuming also that ξ = 2/3. Notice that taking ξ > 2/3 would imply that the energy density in the Friedmann equation has a negative sign. In fact, this range of the parameter corresponds to spin zero modes of the theory, which can be excluded at the cosmological level and are related to unwanted ghost modes. Also the sign of the term χ 1 determines the sign of an (effective) cosmological constant in the model and χ 2 can be always taken to be negative [START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF]. Comparing the system above with the one in (12a)-(12c) one can note that, as pointed out in [START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF], there is not much difference between the two cases. For example, apart from the values of the constants, (12b) and (22b) differ only by the term associated with χ 4 . However, as we are going to show, the associated cosmological dynamics will be non-trivially changed.

Dynamical analysis

Let us define the variables

Ω = κ 2 ρ H 2 , x= k 2 χ 3 6a 4 H 2 , y = kχ 4 6a 6 H 2 , z = χ 1 6H 2 , K = kχ 2 6a 2 H 2 , (24)
and the cosmic time N = ln[a(t)]. The cosmological equations (22a)-(22c) are then equivalent to the system

Ω = 4(z -x -2y) 2 -3ξ -1 -3w Ω - 2(1 + 3w) 2 -3ξ Ω 2 , (25a) x
= 2x 2 -3ξ [-4y + 2z -(1 + 3w)Ω -2] - 4x 2 2 -3ξ , (25b) y
= 2y 2 -3ξ [-2x + 2z -(1 + 3w)Ω -4] - 8y 2 2 -3ξ , (25c)
z = 2z 2 -3ξ [-2x -4y -(1 + 3w)Ω + 2] + 4z 2 2 -3ξ , (25d)
K = - 2K 2 -3ξ [2x + 4y -2z + (1 + 3w)Ω] , (25e)
with the Gauss constraint

K + x + y + z + Ω + 1 - 3 2 ξ = 0, (26)
where the prime denotes derivation with respect to N . As before, the constraint ( 15) allows to eliminate one of the equations. If we choose to eliminate the one for Ω, we obtain

x = 2(1 -3w)x 2 2 -3ξ + x 2 -3ξ [2K(3w + 1) -6(w -1)y -6(w + 1)z -(1 -3w)(2 -3ξ)] , (27a)
y = - 6(w -1)y 2 2 -3ξ + y 2 -3ξ [2K(3w + 1) + 2(1 -3w)x -6(w + 1)z + 3(w -1)(2 -3ξ)] , (27b) z
= - 6(w + 1)z 2 2 -3ξ + z 2 -3ξ [2K(3w + 1) + 2(1 -3w)x -6(w -1)y + 3(w + 1)(2 -3ξ)] , (27c)
K = 2K 2 (3w + 1) 2 -3ξ + K 2 -3ξ [2(1 -3w)x -6(w -1)y -6(w + 1)z + (3w + 1)(2 -3ξ)]. (27d)
As expected, the new degrees of freedom, associated with additional terms present in this case, result in an additional dimension for the phase space. The system above possesses four invariant submanifolds, namely x = 0, y = 0, z = 0, K = 0. This implies that, also in this case, no global attractor can exist. As before, the structure of ( 26) is such that the system is non-compact and this will require an asymptotic analysis to be performed. Also in this case, the dimensionality of the phase space and the nature of the invariant submanifolds is such that we cannot picture clearly the phase space. Thus we will not be able to use graphics to show our results.

Finite analysis

The finite fixed points are found by setting x , y, z , K = 0 in (27a)-(27d) and solving the resulting system of algebraic equations. The results are shown in Table 4. Note that the presence of a fixed point depends on the exact sign of the constant χ i , as well as on the value of ξ. For example, given the fact that the variable Ω is defined positive, the fixed point A, can exist only if its coordinates are non negative, and this happens for ξ < 2 3 only. The solutions associated to the fixed point can be derived from the Raychaudhuri equation

Ḣ = - H 2 4 -6ξ [2K + 2x + 6y -6z -6w(x + y + z -1 -K) -9(w + 1)ξ + 6] . (28)
The corresponding results are shown in 2-3ξ > 0, otherwise it is associated with oscillations. In other words, if one wants standard matter to interact with HL gravity in the standard way (gravity makes matter to attract itself), then the de Sitter solution is only present if χ 1 > 0, i.e., if the cosmological constant has the right sign, as expected.

In the same way as in the previous case, the stability of the fixed points can be determined by evaluating the eigenvalues of the Jacobian matrix associated with the system (16a), as prescribed by the Hartman-Grobman theorem [START_REF] Hartman | Ordinary Differential Equations[END_REF]. The results can be also found in Table 4. The stabilities of the fixed points in this model are different from the corresponding ones in the previous case although, again, the only stable finite fixed point continues to be the de Sitter one.

As before we can characterize the transition form decelerated and accelerate expansion as the crossing of the hyperplane

0 = 2K + 2x + 6y -6z -6w(x + y + z -1 -K) -3(1 + 3w)ξ + 2 . (29)
Unfortunately, due to the higher dimensionality of the phase space, the dynamics of this model are not as easy to extract as the ones in the previous paragraph. However in the general structure of the equations there is no feature which prevents the existence of an orbit connecting the unstable Friedmann phase with the de Sitter attractor. This means that HL cosmologies without detailed balance can actually admit a transition between Friedmann evolution and a dark energy era.

Asymptotic analysis

Let us now look at the asymptotic analysis of the dynamical system (25a)-(25e) which, like (3.1), is non-compact. We will use again the Poincaré projection method [START_REF] Lefschetz | Differential Equations: Geometric Theory[END_REF]. Let us perform the phase space coordinate change x = ρ cos φ sin θ sin ψ, y = ρ sin θ sin φ sin ψ, z = cos θ sin ψ, K = ρ cos ψ , [START_REF] Germani | Relativistic Quantum Gravity at a Lifshitz point[END_REF] where again

ρ = r 1 + r , (31)
Point Coordinates Solution Energy density Stability [Ω, x, y, z, K]

A 1 2 (2 -3ξ), 0, 0, 0, 0 a = a 0 (t -t 0 ) 2 3(1+w) ρ = ρ 0 (t -t 0 ) -2 saddle B 0, 1 2 (2 -3ξ), 0, 0, 0 a = a 0 (t -t 0 ) 1 2 ρ = 0 saddle C 0, 0, 0, 0, 1 2 (3ξ -2) a = a 0 (t -t 0 ) ρ = 0 saddle D 0, 0, 0, 1 2 (2 -3ξ), 0 a = a 0 e τ (t-t 0 ) ρ = 0 attractor E 0, 0, 1 2 (2 -3ξ), 0, 0 a = a 0 (t -t 0 ) 1 3 ρ = 0 repeller τ = α 1 3(2-3ξ)
Table 4. Finite fixed points of the system (27a)-(27d), their associated solutions and stability.

θ ∈ [0, π], and φ ∈ [0, 2π]. Performing the transformation (30), the system (27a)-(27d) becomes now, in the limit ρ → 1,

ρ → 2 2 -3ξ {sin θ[(1 -3w) cos φ + 3(1 -w) sin φ] -3(w + 1) cos θ} sin ψ - 2(3w + 1) 2 -3ξ cos ψ , (0)
θ → 1 2 sin(2θ)[cos(2φ) -5] , (1)
φ → sin(2φ) , (2)
ψ → - 1 2 sin(2ψ) 2 cos 2 θ + (cos(2φ) -3) sin 2 θ . (33)
Also here the radial equation does not contain the radial coordinate, so that the fixed points can be obtained using just the angular equations. Setting φ = 0, θ = 0 and ψ = 0, we obtain the fixed points listed, together with their solutions, in Table 5. Their stability behavior is given in Table 6.

Discussion and conclusions

In this paper we have used dynamical system techniques to analyze the non vacuum cosmology of Horâva-Lifshitz gravity both in the presence and in the absence of detailed balance. Our analysis has allowed to both gain an understanding of the qualitative

Point Coordinates Solution [φ, θ.ψ] F 1 [0, 0, 0] a = a 0 e 1 2 (1+3w)(t-t 0 ) 2 F 2 [π, π, 0] a = a 0 e 1 2 (w+1)(t-t 0 ) 2 G 1 [0, 0, π/2] a = a 0 e 3 2 (w+1)(t-t 0 ) 2 G 2 [0, π, π/2] a = a 0 e 3 2 (w+1)(t-t 0 ) 2 H 1 [π/2, π/2, π/2] a = a 0 e 3 2 (w+1)(t-t 0 ) 2 H 2 [π/2, π/2, 3π/2] a = a 0 e 3 2 (w+1)(t-t 0 ) 2 I 1 [0, π/2, π/2] a = a 0 e 1 2 (1-3w)(t-t 0 ) 2 I 2 [π, π/2, π/2] a = a 0 e 1 2 (1-3w)(t-t 0 ) 2 Table 5
. Asymptotic fixed points of the system (27a)-(27d) and their associated solutions.

behavior of the cosmology and to obtain interesting new constraints on the parameters of the model. In the first case the phase space exhibits four finite fixed points, three of which represent physical solutions of the system. Although one of these points is associated to an unstable classical Friedmann solution that could be certainly useful to model the nucleosynthesis and the structure formation periods, our analysis does not reveal any useful fixed point which could model an inflationary or dark energy phase. It has been proposed that, because of the changes in the value of the speed of light contained in the theory, the absence of an explicit inflationary phase might not be such a serious shortcoming of the theory [START_REF] Kiritsis | Hořava-Lifshitz Cosmology[END_REF], although at first glance it is difficult in this setting to produce a dark energy era. A more conclusive analysis of this issue, however, will require a complete numerical study, which is left to future work.

An interesting result of our investigation is that one of the attractors in the theory corresponds to an oscillatory behavior. Such oscillations can be associated with a bouncing universe, which can be connected to the analysis in [START_REF] Brandenberger | Matter bounce in Hořava-Lifshitz cosmology[END_REF]. Other attractors are associated with a recollapsing solution in the form of a Gauss function. These solutions, which also appear in the context of GR plus a scalar field, higher-order and scalar- tensor gravities, could-if proven to be unstable-model some kind of superinflationary phase whose properties ought to be carefully investigated. However, most of the orbits that contain these points will either evolve towards recollapse or to the oscillatory attractor. In other words, when detailed balance is imposed, although one is able to prove that the corresponding cosmology could, given suitable initial conditions, behave like a Friedmann one, it turns out that it will eventually evolve towards an oscillatory behavior or either recollapse, leaving no space for a dark-energy era.

Maybe the most important result in this paper is related to the HL cosmology without detailed balance. In this case, in fact, the additional freedom in the values of the parameters allows the existence of cosmic histories which contain a Friedmann era and evolve towards a dark energy one. This follows because the phase space contains a fixed point associated to the standard Friedmann solution which is unstable and another one which can be associated to a de Sitter solution which is an attractor. The last point can then model a dark energy era. However, the existence of these fixed points is only a necessary condition: because of the presence of invariant manifolds and the constraints on the parameters only a subset of the phase space and the parameter space will realize this scenario.

On the other hand the fact that the fixed points all lie on invariant submanifolds guarantee that such orbits can exist. Unfortunately the high dimension of the phase space makes it quite hard to perform any qualitative analysis. Therefore only numerical methods will allow the investigation of the details of these orbits. Notwithstanding these problems, we feel that it is safe to conclude that a cosmology compatible with the present observations can be obtained, in the HL framework, only if the detailed balance is broken. Such result makes this type of HL gravity a very promising phenomenological model for both the study of dark energy and quantum gravity.

Table 3 .

 3 Stability of the asymptotic fixed points of the system (16a)-(16c).

  Table[START_REF] Blas | On the extra mode and inconsistency of Hořava Gravity[END_REF]. The new terms in the system (22a)-(22c) induce a new fixed point characterized by a behavior t 1/3 which corresponds to the domination of a new cosmic component which goes like a -6 . Substitution into Eqs. (12a)-(12c) reveals that, in this case, only the Friedmann solution and the exponential solution yield identities. Specifically, the exponential solution represents a de Sitter solution if

	χ 1

Table 6 .

 6 Stability of the asymptotic fixed points of the system (27a)-(27d).

Acknowledgments

This work was partially funded by Ministerio de Educación y Ciencia, Spain, projects CICYT-FEDER-FPA2005-02211 and FIS2006-02842, CSIC under the I3P program, and AGAUR, contract 2005SGR-00790 and grant 2009BE-100132. SC was funded by Generalitat de Catalunya through the Beatriu de Pinós contract 2007BP-B1 00136. EE's research was performed while on leave at Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA.