Charalampos Bogdanos
email: charalampos.bogdanos@th.u-psud.fr

Emmanuel N Saridakis
email: msaridak@phys.uoa.gr

Perturbative instabilities in Hoava gravity

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Introduction

Relativity is commonly assumed to be the foundation of every gravitational construction, especially in its cosmological application. However, since a quantum field theory of general relativity proves to be non-renormalizable, numerous attempts of its modification have appeared in the literature. Recently, a power-counting renormalizable, ultra-violet (UV) complete theory of gravity was proposed by Hořava in [START_REF] Horava | Quantum Criticality and Yang-Mills Gauge Theory[END_REF][START_REF] Horava | Membranes at Quantum Criticality[END_REF][START_REF] Horava | Quantum Gravity at a Lifshitz Point[END_REF][START_REF] Hořava | Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point[END_REF]. Although presenting General Relativity as an infrared (IR) fixed point, in the UV the theory possesses a fixed point with an anisotropic, Lifshitz scaling between time and space.

Due to these novel features, there has been a large amount of effort in examining and extending the properties of the theory itself [START_REF] Volovik | z=3 Lifshitz-Horava model and Fermi-point scenario of emergent gravity[END_REF][START_REF] Cai | On the z=4 Horava-Lifshitz Gravity[END_REF][START_REF] Cai | Dynamical Scalar Degree of Freedom in Horava-Lifshitz Gravity[END_REF][START_REF] Orlando | On the Renormalizability of Horava-Lifshitz-type Gravities[END_REF][START_REF] Nishioka | Horava-Lifshitz Holography[END_REF][START_REF] Konoplya | Towards constraining of the Horava-Lifshitz gravities[END_REF][START_REF] Charmousis | Strong coupling in Horava gravity[END_REF][START_REF] Li | A Trouble with Hořava-Lifshitz Gravity[END_REF][START_REF] Visser | [END_REF][START_REF] Sotiriou | [END_REF][START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF][START_REF] Chen | Timelike Geodesic Motion in Horava-Lifshitz Spacetime[END_REF][START_REF] Chen | Field Theory at a Lifshitz Point[END_REF][START_REF] Shu | Stochastic Quantization of the Hořava Gravity[END_REF][START_REF] Dutta | Observational constraints on Horava-Lifshitz cosmology[END_REF]. Additionally, application of Hořava gravity as a cosmological framework gives rise to Hořava cosmology, which proves to lead to interesting behavior [START_REF] Calcagni | Cosmology of the Lifshitz universe[END_REF][START_REF] Kiritsis | Horava-Lifshitz Cosmology[END_REF]. In particular, one can examine specific solution subclasses [START_REF] Lu | Solutions to Horava Gravity[END_REF][START_REF] Nastase | On IR solutions in Horava gravity theories[END_REF][START_REF] Colgain | Dyonic solution of Horava-Lifshitz Gravity[END_REF][START_REF] Ghodsi | Toroidal solutions in Horava Gravity[END_REF][START_REF] Minamitsuji | Classification of cosmology with arbitrary matter in the Hořava-Lifshitz theory[END_REF][START_REF] Ghodsi | Extremal rotating solutions in Horava Gravity[END_REF], the perturbation spectrum [START_REF] Mukohyama | Scale-invariant cosmological perturbations from Horava-Lifshitz gravity without inflation[END_REF][START_REF] Piao | Primordial Perturbation in Horava-Lifshitz Cosmology[END_REF][START_REF] Gao | Cosmological Perturbations and Non-Gaussianities in Hořava-Lifshitz Gravity[END_REF][START_REF] Chen | Scale Invariant Power Spectrum in Hořava-Lifshitz Cosmology without Matter[END_REF][START_REF] Gao | Cosmological Perturbations in Hořava-Lifshitz Gravity[END_REF], the gravitational wave production [START_REF] Mukohyama | Phenomenological Aspects of Horava-Lifshitz Cosmology[END_REF][START_REF] Takahashi | Chiral Primordial Gravitational Waves from a Lifshitz Point[END_REF][START_REF] Koh | Relic gravitational wave spectrum, the trans-Planckian physics and Hořava-Lifshitz gravity[END_REF], the matter bounce [START_REF] Brandenberger | Matter Bounce in Horava-Lifshitz Cosmology[END_REF][START_REF] Brandenberger | Processing of Cosmological Perturbations in a Cyclic Cosmology[END_REF], the black hole properties [START_REF] Danielsson | Black holes in asymptotically Lifshitz spacetime[END_REF][START_REF] Cai | Topological Black Holes in Horava-Lifshitz Gravity[END_REF][START_REF] Myung | Thermodynamics of Hořava-Lifshitz black holes[END_REF][START_REF] Kehagias | The black hole and FRW geometries of non-relativistic gravity[END_REF][START_REF] Cai | Thermodynamics of Black Holes in Horava-Lifshitz Gravity[END_REF][START_REF] Mann | Lifshitz Topological Black Holes[END_REF][START_REF] Bertoldi | Black Holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent[END_REF][START_REF] Castillo | Entropy for Black Holes in the Deformed Horava-Lifshitz Gravity[END_REF][START_REF] Botta-Cantcheff | [END_REF], the dark energy phenomenology [START_REF] Saridakis | Horava-Lifshitz Dark Energy[END_REF][START_REF] Park | A Test of Horava Gravity: The Dark Energy[END_REF][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] etc.

However, investigations of various aspects of Hořava gravity have started to reveal some potentially troublesome features. Although some of them can be alleviated or removed once the detailed-balance condition is relaxed [START_REF] Li | A Trouble with Hořava-Lifshitz Gravity[END_REF][START_REF] Mukohyama | Dark matter as integration constant in Horava-Lifshitz gravity[END_REF][START_REF] Mukohyama | Caustic avoidance in Horava-Lifshitz gravity[END_REF], the most significant core remains unaffected. In [START_REF] Charmousis | Strong coupling in Horava gravity[END_REF] it was shown that the diffeomorphisminvariance breaking leads to an additional degree of freedom (known already in [START_REF] Horava | Membranes at Quantum Criticality[END_REF] as the longitudinal degree of freedom of metric perturbations), which is not needed to decouple in the IR, and thus General Relativity is not recovered at any scale. Similarly, in [START_REF] Blas | On the Extra Mode and Inconsistency of Horava Gravity[END_REF] it was shown that the explicit breaking of general covariance uncovers an extra scalar degree of freedom, with fast exponential instabilities at short distances and strong-coupling at extremely low cutoff scales. In addition, observational constraints may rule out the theory completely [START_REF] Calcagni | Detailed balance in Horava-Lifshitz gravity[END_REF]. These features led many authors to examine various forms of "modified" Hořava gravity, such as versions with full diffeomorphism invariance [START_REF] Germani | Relativistic Quantum Gravity at a Lifshitz Point[END_REF][START_REF] Nojiri | Covariant Horava-like renormalizable gravity and its FRW cosmology[END_REF], with deformed action and zero cosmological constant (and thus with Minkowski and not AdS IR limit) [START_REF] Myung | Thermodynamics of black holes in the deformed Hořava-Lifshitz gravity[END_REF][START_REF] Kim | Nonpropagation of scalar in the deformed Hořava-Lifshitz gravity[END_REF], with restored parity invariance beyond detailed-balance [START_REF] Visser | [END_REF][START_REF] Sotiriou | [END_REF][START_REF] Sotiriou | Quantum gravity without Lorentz invariance[END_REF], with extra derivative-terms [START_REF] Cai | Non-singular cosmology in a model of non-relativistic gravity[END_REF], and with "soft" detailed-balance breaking [START_REF] Park | The Black Hole and Cosmological Solutions in IR modified Horava Gravity[END_REF].

On the other hand, most works on the cosmological application of Hořava gravity focus straightaway on its IR limit, as it is realistically expected to happen, without examining the running of the theory towards this regime. Indeed, in the original papers of Hořava the running behavior was revealed, but not the precise flow. It is interesting to point out that, up to this point, there is an amount of ambiguity concerning the IR behavior of the theory. The presence of additional degrees of freedom appears to be in some sense background-dependent. Although strong coupling arises once the perturbations around a flat Minkowski spacetime are considered [START_REF] Charmousis | Strong coupling in Horava gravity[END_REF], the theory seems to be well behaved when perturbing a cosmological background [START_REF] Gao | Cosmological Perturbations in Hořava-Lifshitz Gravity[END_REF]. It is therefore non-trivial to trace internal consistency problems of the theory directly at the IR limit, at least in the perturbative level, even though it is already known that non-perturbative solutions such as the Schwarzschild black hole are not recovered in the context of the conventional, detailed-balance-preserving Hořava gravity [START_REF] Lu | Solutions to Horava Gravity[END_REF].

In the present work we are interested in performing a detailed investigation of the gravitational perturbations of Hořava gravity, using it as a tool to examine its consistency. We study both scalar and tensor sectors, around a Minkowski background. We find that instabilities do appear in the whole range of parameter values, between which the theory is supposed to flow in order to provide a consistent UV completion. Such instabilities seem to exclude any viable flow of the theory within the range needed to provide power-counting renormalizability. Breaking detailed balance also does not seem to improve the situation. Additionally, these results are not affected by the above mentioned ambiguity about the existence or not of additional degrees of freedom at the IR. Finally, we argue that the IR limit behavior may actually be the least of the problems.

In summary, the gravitational perturbation investigation reveals that Hořava gravity, in its present form, suffers from instabilities and fine-tunings that seem to originate from its deep and implicit features and thus they cannot be overcome by simple tricks such as analytic continuation. The paper is organized as follows: In section 2 we extract the scalar and tensor perturbations under the ADM diffeomorphism group in Minkowski spacetime, setting the relevant degrees of freedom and fixing the gauge. In section 3 we discuss about the emerging instabilities, the fine-tunings and the possible causality problems. Finally, section 4 is devoted to a summary and discussion of the obtained results.

Gravitational perturbations in Minkowski background

One of the most decisive tests for the reliability of a gravitational theory is the examination of its perturbations. This investigation reveals some of the deep features of the theory, and additionally, one may hope to eventually extract possible observational signatures. Thus, the study of gravitational perturbations of Hořava gravity will be the basic tool of the present work.

Variables and gauge transformations

We consider coordinate transformations of the form x µ → xµ = x µ + ξ µ . Under this transformation the metric-perturbation around a given background changes as

δg µν = δg µν -∇ µ ξ ν -∇ ν ξ µ .
As it is known, the perturbation analysis, especially its primordial phase, is sensitive to the background evolution. A usual choice for cosmological applications [START_REF] Calcagni | Cosmology of the Lifshitz universe[END_REF][START_REF] Kiritsis | Horava-Lifshitz Cosmology[END_REF], is to impose

N (0) = 1, N (0) i = 0, g (0) ij = a 2 (t)δ ij
in the ADM ("foliationpreserving") formalism. Although we will soon perform perturbations around a flat background, we keep for the moment the explicit dependence on the scale factor for the sake of generality. The background metric is given by

ds 2 (0) = (g (0) ij N i(0) N j(0) -N 2 )dt 2 + 2g (0) ij N i(0) dtdx j + g (0) ij dx i dx j = a 2 (n) -dn 2 + δ ij dx i dx j .
(2.1)

From now on the 0-coordinate denotes the conformal time n, defined through dt 2 /a 2 = dn 2 . Therefore, the general perturbations of the metric around this background read:

δg 00 = -2a 2 φ (2.2)
δg 0i = a 2 ∂ i B + a 2 Q i (2.3)
δg ij = a 2 h ij -a 2 (∂ i W j + ∂ j W i ) -2a 2 ψδ ij + 2a 2 ∂ i ∂ j E.
(2.4)

The vector modes are assumed to be transverse, that is

∂ i W i = ∂ i Q i = 0,
while the tensor mode is forced to be transverse and traceless,

∂ i h ij = δ ij h ij = 0. Finally, the Christoffel symbols read Γ 0 00 = H, Γ i 0j = Hδ i j , Γ 0 ij = Hδ ij , (2.5)
where H is the conformal Hubble parameter, H = 1

a(n) da(n)
dn . In the following, we concentrate on the flat geometry, and since there is now no distinction between coordinate and conformal time we restore t as denoting the time coordinate.

The spatial part of the gauge transformation vector is defined as

ξ i = ∂ i ξ+ζ i with ζ i = ζ i , so that ξ i = a 2 (∂ i ξ + ζ i ).
Anticipating the application to the Hořava gravity framework, where only transformations of the time coordinate of this sort are allowed, we assume that ξ 0 = ξ 0 (t). Using these transformation rules, we obtain the following expressions for the gauge transformation of each mode (note that ξ 0 = -a 2 ξ 0 ):

• Scalars ψ = ψ + Hξ 0 (2.6) φ = φ -Hξ 0 -ξ0 (2.7) B = B -ξ (2.8) Ẽ = E -ξ (2.9) • Vectors Qi = Q i -ζi (2.10) Wi = W i + ζ i (2.11) • Tensor hij = h ij , (2.12)
where a dot denotes the coordinate time-derivative. Lastly, note that since we restrict ourselves to the flat case, in the following we set H = 0. Let us now discuss about the gauge fixing, which is required for the action derivation and the determination of the physical degrees of freedom. The projectability condition of Hořava gravity [START_REF] Horava | Quantum Gravity at a Lifshitz Point[END_REF] requires that the perturbation of the lapse-function N depends only on time, thus φ ≡ φ(t). This allows us to "gauge away" the φperturbation, that is to set φ = 0 imposing φ = ξ0 . Similarly, we fix B = 0 from setting B = ξ. Finally, setting Q i = ζi we eliminate the Q i degree of freedom. Therefore, the remaining degrees of freedom are ψ, E, W i and h ij .

In summary, in the aforementioned gauge we obtain δN = δN i = 0 (2.13)

δ ij = h ij -2ψδ ij + 2∂ i ∂ j E -(∂ i W j + ∂ j W i ). (2.14)
Note that since only perturbations imposed on the "same-time" spatial hypersurface are allowed, this is equivalent to a synchronous gauge choice.

Perturbed Action

In Hořava gravity the gravitational action can be decomposed into a kinetic and a potential part as S g = S K + S V , where

S K = 2 κ 2 dtd 3 x √ gN K ij K ij -λK 2 , (2.15)
with

K ij = 1 2N ( ġij -∇ i N j -∇ j N i ) , (2.16)
the extrinsic curvature and κ a constant with mass dimension -1.

One of the subjects that have led to discussion in the literature is the imposition of detailed balance [START_REF] Horava | Quantum Gravity at a Lifshitz Point[END_REF], which apart from reducing the possible terms in the action, it allows for a quantum inheritance principle [START_REF] Horava | Quantum Criticality and Yang-Mills Gauge Theory[END_REF] (the D + 1 dimensional theory acquires the renormalization properties of the D-dimensional one). In order to proceed and write explicitly the potential term for the moment we impose detailed balance too. However, in order to avoid possible accidental artifacts of this, possibly ambiguous, condition in the obtained results, later on we will extend our analysis beyond detailed balance.

Under detailed balance we can write

S V = dtd 3 x √ gN - κ 2 2w 4 C ij C ij + κ 2 µ 2w 2 ijk √ g R il ∇ j R l k - κ 2 µ 2 8 R ij R ij + κ 2 µ 2 8(1 -3λ) 1 -4λ 4 R 2 + ΛR -3Λ 2 .
(2.17)

In the above expression, the Cotton tensor of the spatial hypersurface is given by

C ij = ikl √ g ∇ k R j l - 1 4 Rδ j l , (2.18)
and the covariant derivatives are defined with respect to the spatial metric g ij . ijk is the totally antisymmetric unit tensor, Λ is a negative constant which is related to the cosmological constant in the IR limit, and the variables w and µ are constants with mass dimensions 0 and 1, respectively. Finally, λ is a dimensionless constant, which incorporates the running behavior of Hořava gravity. Although a priori arbitrary, it is now known that the important variation-regime is between 1/3 and 1, with the former corresponding to the UV limit and the later to the IR one.

We will now perturb the Hořava gravitational action up to second order. At this point, we should mention that varying the action once we have substituted our metric ansatz for the perturbations is equivalent to deriving the Hamilton and momentum constraints along with the remaining equations of motion for an arbitrary metric perturbation and then imposing the ansatz. We choose the former approach here for reasons of simplicity. After non-trivial but straightforward calculations that are presented in Appendix A, for the perturbed kinetic part (2.15) we obtain δS (2)

K = dtd 3 x 2 κ 2 1 4 ḣij ḣij + (1 -3λ) 3 ψ2 -2 ψ∇ 2 Ė + (1 -λ) Ė∇ 4 Ė , (2.19)
while for the perturbed potential part (2.17) we acquire δS

(2)

V = dtd 3 x κ 2 8w 4 h ij ∇ 6 h ij + κ 2 µ 8w 2 ijk h il ∂ j ∇ 4 h l k - κ 2 µ 2 32 h ij ∇ 4 h ij + κ 2 µ 2 Λ 32(1 -3λ) h ij ∇ 2 h ij - κ 2 µ 2 (1 -λ) 4(1 -3λ) ψ∇ 4 ψ - κ 2 µ 2 Λ 4(1 -3λ) ψ∇ 2 ψ + 27κ 2 µ 2 Λ 2 16(1 -3λ) ψ 2 - 9κ 2 µ 2 Λ 2 8(1 -3λ) ψ∇ 2 E + 3κ 2 µ 2 Λ 2 16(1 -3λ) E∇ 4 E . (2.20)

Scalar perturbations

As can be observed from (2.19),(2.20) the action for scalar perturbations includes the two modes ψ and E and it is written as δS

S = dtd 3 x 2(1 -3λ) κ 2 3 ψ2 -2 ψ∇ 2 Ė + 2(1 -λ) κ 2 Ė∇ 4 Ė - κ 2 µ 2 (1 -λ) 4(1 -3λ) ψ∇ 4 ψ - κ 2 µ 2 Λ 4(1 -3λ) ψ∇ 2 ψ + 27κ 2 µ 2 Λ 2 16(1 -3λ) ψ 2 - 9κ 2 µ 2 Λ 2 8(1 -3λ) ψ∇ 2 E + 3κ 2 µ 2 Λ 2 16(1 -3λ) E∇ 4 E . (2.21) (2)
Varying it with respect to E and ψ we obtain the equations of motion:

8 κ 2 Ë + κ 2 µ 2 (1 -λ) 2 (1 -3λ) ∇ 2 ψ + κ 2 µ 2 Λ 2 (1 -3λ) ψ = 0 (2.22) 8 κ 2 1 -3λ 1 -λ ψ - 9κ 2 µ 2 Λ 2 4 (1 -λ) (1 -3λ) ψ + 3κ 2 µ 2 Λ 2 4 (1 -λ) (1 -3λ) ∇ 2 E + κ 2 µ 2 (1 -λ) 2 (1 -3λ) ∇ 4 ψ + κ 2 µ 2 Λ 2 (1 -3λ) ∇ 2 ψ = 0. (2.23)
As can be seen these two equations are coupled, not allowing for a straightforward stability investigation. However, we can still acquire information about the stability of the configuration by studying it at high and low momenta. Taking the IR limit of (2.22), (2.23), that is considering their low-k behavior, they reduce to

8 κ 2 Ë + κ 2 µ 2 Λ 2 (1 -3λ) ψ = 0 (2.24) 8 κ 2 1 -3λ 1 -λ ψ - 9κ 2 µ 2 Λ 2 4 (1 -λ) (1 -3λ) ψ = 0. (2.25)
Thus, the second equation is decoupled, acting as a low-momentum equation of motion for the scalar field ψ.

A straightforward observation from (2.25) is that it leads to a ghost-like behavior in the IR limit whenever 1 3 < λ < 1. Inverting the overall sign of the Lagrangian is not going to help, since as we will promptly see, the time derivative of the tensor perturbation has the opposite sign. In particular, (2.25)

leads to ψ - 9κ 4 µ 2 Λ 2 32 (1 -3λ) 2 ψ = 0, (2.26)
where

ψ(t, x) = d 3 k (2π) 3 2 ψk (t)e ik•x . (2.27)
Therefore, we acquire the following dispersion relation

ω 2 ≡ m 2 = - 9κ 4 µ 2 Λ 2 32 (1 -3λ) 2 < 0, (2.28)
which induces instabilities at the IR, regardless of the λ-value and of the sign of the cosmological constant. Finally, combination of both (2.24),(2.25) fixes E against ψ, and in particular it leads to

ψ = - 9 2 Λ 1 -3λ E. (2.29)
Thus, we are led to only one effective degree of freedom, or in other words the system has been diagonalized. Now, for high k, (2.22), (2.23) reduce to

8 κ 2 Ë + κ 2 µ 2 (1 -λ) 2 (1 -3λ) ∇ 2 ψ = 0 (2.30) 8 κ 2 1 -3λ 1 -λ ψ + κ 2 µ 2 (1 -λ) 2 (1 -3λ) ∇ 4 ψ = 0. (2.31)
Therefore, the ghost-like coefficient in the time derivative part remains and (2.31) yields a high-k dispersion relation of the form

ω 2 ≡ κ 4 µ 2 16 1 -λ 1 -3λ 2 k 4 . (2.32)
In general the homogeneous system of equations (2.22),(2.23) leads to only one active degree of freedom (one of the fields should be defined in terms of the other, in order to have a non-trivial solution) and a dispersion relation of the form

64 k 4 1 -3λ 1 -λ ω 4 + 18µ 2 Λ 2 (1 -λ) (1 -3λ) + 4µ 2 Λ 1 -3λ k 2 - 4µ 2 (1 -λ) 1 -3λ k 4 ω 2 + 3κ 4 µ 4 Λ 2 8 (1 -λ) (1 -3λ) 2 Λ + (λ -1) k 2 k 2 = 0.
(2.33)

Therefore, the physical requirement of obtaining a positive solution for ω 2 will lead to restrictions on the various parameters of the theory.

Tensor perturbations

Let us now examine the tensor perturbations. Their action can be extracted from (2.19),(2.20) and it reads:

δS (2) T = dtd 3 x 1 2κ 2 ḣij ḣij + κ 2 µ 2 Λ 32(1 -3λ) h ij ∇ 2 h ij + κ 2 8w 4 h ij ∇ 6 h ij + κ 2 µ 8w 2 ijk h il ∂ j ∇ 4 h l k - κ 2 µ 2 32 h ij ∇ 4 h ij . (2.34)
Therefore, the graviton equation of motion writes as ḧij -

κ 4 µ 2 Λ 16(1 -3λ) ∇ 2 h ij - κ 4 4w 4 ∇ 6 h ij - κ 4 µ 4w 2 ilk ∂ l ∇ 4 h j k + κ 4 µ 2 16 ∇ 4 h ij = 0. (2.35)
Assuming graviton propagation along the x 3 direction, that is k i = k i = (0, 0, k), the h ij can be written as usual in terms of polarization components as

h ij = h ij =   h + h × 0 h × -h + 0 0 0 0   . (2.36)
Using this parametrization we derive the two equations for the different polarizations

ḧ+ - κ 4 4w 4 ∇ 6 h + + κ 4 µ 4w 2 ∂ 3 ∇ 4 h × + κ 4 µ 2 16 ∇ 4 h + - κ 4 µ 2 Λ 16(1 -3λ) ∇ 2 h + = 0, (2.37)
ḧ× -

κ 4 4w 4 ∇ 6 h × - κ 4 µ 4w 2 ∂ 3 ∇ 4 h + + κ 4 µ 2 16 ∇ 4 h × - κ 4 µ 2 Λ 16(1 -3λ) ∇ 2 h × = 0, (2.38)
and thus we identify the light speed from the low k regime as

c 2 = κ 4 µ 2 Λ 16(1 -3λ) . (2.39)
A significant observation is that the two polarization modes are coupled due to the ijk term, which arises from the Cotton tensor. Thus, it is more convenient to shift to the Left-Right base defining

h L ≡ 1 √ 2 (h + + ih × ), (2.40) h R ≡ 1 √ 2 (h + -ih × ). ( 2

.41)

In this case, and after Fourier transforming we obtain the decoupled equations of motion (2.37),(2.38):

-

ω 2 hL + c 2 k 2 hL + κ 4 µ 2 16 k 4 hL + κ 4 µ 4w 2 k 5 hL + κ 4 4w 4 k 6 hL = 0, (2.42) -ω 2 hR + c 2 k 2 hR + κ 4 µ 2 16 k 4 hR - κ 4 µ 4w 2 k 5 hR + κ 4 4w 4 k 6 hR = 0, (2.43)
an equation system that accepts a non-trivial solution only if the corresponding determinant is zero, which leads to the dispersion relation

ω 2 = c 2 k 2 + κ 4 µ 2 16 k 4 ± κ 4 µ 4w 2 k 5 + κ 4 4w 4 k 6 .
(2.44)

Therefore, we can identify the effective light speed as

c 2 s (k) = c 2 + κ 4 µ 2 16 k 2 ± κ 4 µ 4w 2 k 3 + κ 4 4w 4 k 4 = c 2 1 + (1 -3λ) Λ k 2 1 ± 2 w 2 µ k 2 .
(2.45)

In expressions (2.44),(2.45) the plus and minus branches correspond to Left-handed and Right-handed mode respectively. Now, since the signs of the k 5 -term are different in the Left-handed and Righthanded modes, they propagate with a different speed, leading to a rotation of the polarization plane, and the rotation angle can be calculated by

δχ = f i (ω R -ω L )dt, (2.46)
where ω R and ω L represent the frequency of the Right-handed and Left-handed mode respectively and the subscripts "i" and "f" denotes the initial and final moment of the propagation. Note that due to this rotation, the spectrum of tensor perturbations would be expected to be suppressed at high energy scales, as was shown in a different context in [START_REF] Cai | Probing Noncommutativity with Inflationary Gravitational Waves[END_REF].

Beyond Detailed Balance

In the above analysis we were restricted to the detailed-balance condition, which constraints the terms in the potential part of the action. Since we desire to examine the general features of Hořava gravity, and thus avoiding possible accidental artifacts of this condition, in this subsection we extend the investigation beyond detailed balance.

As a demonstration, and without loss of generality, we consider a detailedbalance-breaking term of the form ∇ i R jk ∇ i R jk . This term induces in the action a second order term which reads:

∇ i δR jk ∇ i δR jk = 1 4 ∂ i ∇ 2 h jk 2 + (∂ i ∂ j ∂ k ψ) 2 + 5 ∂ i ∇ 2 ψ 2 .
(2.47)

Thus, the corresponding contribution to the action will be δS (2) new = η dtd 3 x -

1 4 h ij ∇ 6 h ij -6ψ∇ 6 ψ , (2.48)
where η is an additional parameter. It is straightforward to calculate the modifications that S

(2)

new brings to the dispersion relations for scalar and tensor perturbations obtained above (expressions (2.32) and (2.44) respectively). The extended dispersion relations read:

• Scalar perturbations (UV-behavior):

ω 2 ∼ κ 2 (1 -λ) 2 16 (1 -3λ) 2 k 4 - 3κ 2 (1 -λ) 2 (1 -3λ) ηk 6
(2.49)

• Tensor perturbations:

ω 2 = c 2 k 2 + κ 4 µ 2 16 k 4 ± κ 4 µ 4w 2 k 5 + κ 4 4w 4 - κ 2 η 2 k 6 .
(2.50)

As was expected, the new, detailed-balanced-breaking term, modifies mainly the UV regime of the theory.

Instabilities, fine-tunings and super-luminal propagation

In the previous section we investigated the gravitational perturbations of Hořava gravity. Such a study is crucial in order to examine the consistency of the theory itself.

In the present section we analyze the obtained results, and focus on the problematic features. Let us first discuss about the scalar perturbations. As was mentioned above, (2.25),(2.31) imply that the regime 1 3 < λ < 1 leads to instabilities. Unfortunately, this is exactly the flow-interval of λ-parameter between the UV and IR regimes. This unstable behavior of the scalar mode at 1 3 < λ < 1, has been already observed in the literature. In [START_REF] Lu | Solutions to Horava Gravity[END_REF] it was argued that it could be amended by imposing an analytic continuation of the form µ → iµ , w 2 → -iw 2 .

(3.1)

This could also allow for a positive cosmological constant solution. Unfortunately, performing the analytic continuation (3.1), we straightforwardly see that the UV behavior is spoiled (see (2.32)) and thus instabilities re-emerge at high energies.

Finally, as we will see later on, such an analytic continuation would induce similar problems in the tensor sector, too. Without analytic continuation, the only physically interesting case that remains, allowing for a possible flow towards General Relativity (at λ = 1) is the regime λ ≥ 1 (since the region λ ≤ 1 3 is disconnected). Even in this case though, we cannot evade the instability coming from the negative mass term, whose sign is independent of both λ and Λ. IR instabilities persist in this regime as long as we have a nonvanishing cosmological constant. Variations of the theory with Λ = 0 thus seem to be favored from such a viewpoint [START_REF] Kehagias | The black hole and FRW geometries of non-relativistic gravity[END_REF]. However, as we will see later on, the Λ = 0 subclass of the original version of Hořava gravity leads to phenomenological problems concerning the light-speed definition.

We now turn to the tensor sector. Here again, from (2.44) we see that if we desire a well-behaved UV regime we cannot impose the analytic continuation (3.1). Thus, consistency with the scalar-sector results means restriction to the λ ≥ 1 regime.

Let us now proceed relaxing the detailed-balance condition, that is analyzing the results of subsection 2.5, according to which the behavior of detailed-balance Hořava gravity is modified mainly at the UV. A first and crucial observation is that the ghost instability of the scalar mode arises from the kinetic term of the action and thus the breaking of detailed balance, which affects the potential term, will not alter the aforementioned scalar-instabilities results concerning the exclusion of 1 3 < λ < 1 regime. In the physical λ ≥ 1 region, the scalar dispersion relation (2.49) remains well-behaved in the UV, provided η is negative. Additionally, for negative η the UV behavior of the tensor perturbations is not affected. Finally, it is interesting to notice that if η is sufficiently negative, then analytic continuation would not bring problems to the tensor sector (as can be seen imposing (3.1) in (2.50)). Unfortunately, since analytic continuation cannot cure the instabilities of the scalar-sector, this possibility does not have any further physical utility. However, it can offer an additional indication that a consistent and well-behaved Hořava gravity should be sought beyond detailed balance.

In summary, we see that the combined scalar and tensor perturbation analysis excludes the regime 1 3 < λ < 1, since the involved ghost instabilities cannot be removed. Thus, we conclude that the allowed and physically interesting interval of Hořava gravity is that with λ ≥ 1, and even in this case, the problem of IR instabilities persists. However, it seems unlikely that a RG flow in this regime will render the theory power-counting renormalizable, thus negating its initial motivation as a possible UV completion of gravity. We mention that this result is valid independently of the imposition of the detailed-balance condition, and this is the reason we did not perform an investigation beyond detailed balance in full generality, examining more such terms. It is also complementary to already known works, concerning the presence of the additional scalar modes and the way they may defer the theory from becoming equivalent to GR at the IR [START_REF] Charmousis | Strong coupling in Horava gravity[END_REF][START_REF] Blas | On the Extra Mode and Inconsistency of Horava Gravity[END_REF][START_REF] Myung | Propagations of massive graviton in the deformed Hořava-Lifshitz gravity[END_REF][START_REF] Kobakhidze | On the infrared limit of Horava's gravity with the global Hamiltonian constraint[END_REF]. This approach is however less ambiguous, since in the regime we are discussing the new degree of freedom is definitely present. There is also no need to invoke strong coupling arguments, i.e. bring matter into the picture [START_REF] Charmousis | Strong coupling in Horava gravity[END_REF].

Let us now discuss a different, but equally significant problem, that is related to some basic observable consequences of Hořava gravity. For simplicity we restrict ourselves to the detailed-balance, but this discussion is independent of that. In expression (2.45) we provide the effective light speed, comparing to the standard light speed c 2 given by (2.39). If we desire the correction term to be of next-toleading order, then Λ must be large in general. However, as it is known the effective cosmological constant in a universe governed by Hořava gravity is [START_REF] Calcagni | Cosmology of the Lifshitz universe[END_REF][START_REF] Kiritsis | Horava-Lifshitz Cosmology[END_REF]:

|Λ ef f | = κ 4 µ 2 16(1 -3λ) 2 Λ 2 = c 2 |1 -3λ| |Λ|, (3.2)
where the second expression is obtained using (2.39). Thus, in general Λ should be very small. The aforementioned contradiction could be resolved by the addition of a suitable positive constant in an introduced matter sector. However, having in mind the physical values of the aforementioned quantities, this would lead to an incredible fine-tuning problem. In other words, the cosmological constant problem does remain in Hořava gravity, which is strange since this theory is constructed to incorporate the underlying and fundamental gravitational features. Furthermore, the fact that its solution would demand the contribution of the matter-sector is even more problematic. Finally, we mention that the aforementioned light-speed definition makes the Λ = 0 theory problematic and this was already mentioned in [START_REF] Calcagni | Cosmology of the Lifshitz universe[END_REF][START_REF] Kiritsis | Horava-Lifshitz Cosmology[END_REF]. Therefore, one cannot easily set Λ = 0 in order to cure the instabilities as described above, and thus the aforementioned instability results seem to be robust. We close this section by referring to another potential problem. According to (2.45), in the physical case λ ≥ 1 the effective light-speed is sub-luminal if Λ > 0, while it is super-luminal if Λ < 0. However, in the regime λ ≥ 1, Λ must be indeed negative in order to assure for a well-defined light speed [START_REF] Calcagni | Cosmology of the Lifshitz universe[END_REF][START_REF] Kiritsis | Horava-Lifshitz Cosmology[END_REF]. Therefore, we observe a possible causality violation. At first it could be stated that such a violation is not a surprise, since Hořava gravity violates relativity and thus the light cone is not a bound anymore. However, it is the classical causality that is violated, and therefore, one could still construct extensions where both energy condition and causality would be restored. For such extensions one could use the formalism and ideas of reggeon field theory [START_REF] Abarbanel | Reggeon Field Theory: Formulation and Use[END_REF][START_REF] Moshe | Recent Developments In Reggeon Field Theory[END_REF] and of holographic correspondence used in studies of AdS/CFT [START_REF] Janik | Reggeon exchange from AdS/CFT[END_REF][START_REF] De Boer | Brownian motion in AdS/CFT[END_REF][START_REF] Son | Thermal Noise and Stochastic Strings in AdS/CFT[END_REF][START_REF] Giecold | Stochastic trailing string and Langevin dynamics from AdS/CFT[END_REF]. It seems that Hořava gravity could correspond to a subclass of models where non-relativistic (super)-conformality, required in AdS/CFT [START_REF] Nakayama | Index for Non-relativistic Superconformal Field Theories[END_REF], can be acquired. Definitely, causality in Hořava gravity is a subject that requires further and thorough investigation.

Conclusions

In this work we have investigated the gravitational perturbations of Hořava gravity, with and without the detailed-balance condition. Setting the relevant degrees of freedom and fixing the gauge we have extracted the scalar and tensor dispersion relations and the corresponding effective speeds of propagation. As we have seen, the scalar sector is plagued by instabilities in the 1 3 < λ < 1 regime, which cannot be cured by analytic continuation. Thus, the only physical regime, that additionally allows for an IR limit towards General Relativity, is the λ ≥ 1 one. This result is in line with previous treatments discussing potential problems in recovering General Relativity as an IR limit of the theory. We stress though that in our case there is no ambiguity about the role of the additional scalar degree of freedom. Since we are dealing with λ away from unity, the scalar mode is certainly present and no ambiguity arises. Regardless of possible strong coupling issues occurring at λ = 1, we see that perturbative instabilities arise long before we reach the IR limit. The ability of Hořava gravity to fulfill its RG flow between the UV and IR, seems to be jeopardized. This λ-restriction also leads to a significant fine-tuning problem, which is related to the value of the effective cosmological constant in the universe. This fact casts additional doubt on Hořava gravity, since any theory that desires to incorporate some of the fundamental gravitational features, should provide some route to alleviate the cosmological constant problem.

Additionally, in the present form of Hořava gravity, one could have causality violation, which would be either a signal of problematic behavior, or a novel property that originates from the relativity abandonment. Clearly, a detailed examination of this subject is crucial in revealing the underlying structure of the theory.

Although the treatment of the present work is robust at the usual perturbative framework, we would like to close by commenting on the possible limitations of our procedure. The presence of a ghost-like scalar mode may be rendered ambiguous, unless the entire set of perturbations of the metric is taken into account. In particular, if there is mixing between different types of perturbations (that is if the "orthogonality" assumption that forbids the mixing between scalar and tensor modes is not valid), it may lead to the appearance of "fictitious" scalar ghosts, such as the conformal ghost in General Relativity (see also [START_REF] Charmousis | Stealth Acceleration and Modified Gravity[END_REF] and [START_REF] Garriga | Gravity in the brane-world[END_REF]). Furthermore, it could be important to include backreactions of the higher spatial curvature terms to the geometry, since the presence of non-linear corrections may lead to non-trivial modifications, as was recently pointed out in [START_REF] Mukohyama | Caustic avoidance in Horava-Lifshitz gravity[END_REF]. Both these non-trivial extensions could provide additional information on Hořava gravity and deserve further inquiry. in the literature. In [START_REF] Wang | Linear perturbations of cosmological models in the Horava-Lifshitz theory of gravity without detailed balance[END_REF] the authors claimed that a more general breaking of the detailed balance can cure the instabilities. However, as we showed in our analysis, the problematic behavior is not related to the strength of the detailed-balance breaking but it seems to be a deeper constituent of the theory, at least in its basic version (with or without detailed-balance) examined in the present work. These results were verified by many other authors, and led to generalizations of the basic Hořava gravity version in order to cure the instabilities. For instance, in [START_REF] Blas | A healthy extension of Horava gravity[END_REF][START_REF] Koyama | Pathological behaviour of the scalar graviton in Hořava-Lifshitz gravity[END_REF][START_REF] Kiritsis | Spherically symmetric solutions in modified Horava-Lifshitz gravity[END_REF] it was argued that the cause of the instabilities is the projectability condition itself, and thus a new non-projectable version of Hořava gravity was formulated. But as it was shown in [START_REF] Papazoglou | Strong coupling in extended Horava-Lifshitz gravity[END_REF][START_REF] Henneaux | A dynamical inconsistency of Horava gravity[END_REF], even in such an extension it is ambiguous if the strong coupling problem can be avoided (see also [START_REF] Blas | Comment on 'Strong coupling in extended Horava-Lifshitz gravity[END_REF]). In the same lines, some other authors claimed that in order to avoid the instabilities, one should study suitably modified versions of Hořava gravity [START_REF] Chen | Power spectra of scalar and tensor modes in modified Horava-Lifshitz gravity[END_REF] (see also [START_REF] Park | Remarks on the Scalar Graviton Decoupling and Consistency of Hořava Gravity[END_REF] for a different approach). The evolution of the literature after the appearance of our work seems to verify our basic result about the problematic features of the basic version of Hořava gravity. Clearly, the subject is still open in extended Hořava-like gravitational theories and deserves further investigation, but this is the center of interest of separate works.

A. Perturbations of kinetic and potential parts of Hořava action

A.1 Perturbation of S K

In order to obtain the perturbed kinetic and potential terms that constitute the Hořava gravitational action, we need to extract the perturbations of the basic geometrical quantities. Following the formalism of subsection 2.1, the perturbed spatial Christoffel symbols read

δΓ i jk = 1 2 ∂ j h i k + ∂ k h i j -∂ i h jk -∂ l ∂ k W i -δ i k ∂ j ψ + δ i j ∂ k ψ -δ jk ∂ i ψ + ∂ k ∂ j ∂ i E, (1.1)
and thus the perturbation of the Riemann tensor is δR i jkl = δΓ i jl,k -δΓ i jk,l , while that of the Ricci tensor is

δR ij = - 1 2 ∇ 2 h ij + ∂ i ∂ j ψ + δ ij ∇ 2 ψ, (1.2) with ∇ 2 ≡ ∂ i ∂ i . Therefore, the Ricci-scalar perturbation δR = δ(g ij R ij ) = δg ij R (0) ij + g ij(0) δR ij (since R (0) ij = 0 and g ij(0) = δ ij ) writes as δR = 4∇ 2 ψ. (1.3)
Since the extrinsic curvature of the background spacetime is zero, the only second-order contributions we may have will be of the form δK ij δK ij and (δK) 2 .

Higher order contributions arising from the √ gN will not enter here and this coefficient evaluated at zeroth-order is equal to one. We thus calculate

δK ij = 1 2 (δ ġij -∂ i δN j -∂ j δN i ) = 1 2 ḣij -2 ψδ ij + 2∂ i ∂ j Ė -∂ i Ẇj -∂ j Ẇi . (1.4)
Therefore, we also determine δK

= δ(g ij K ij ) = δg ij K (0) ij + g ij(0) δK ij = δ ij δK ij , that is δK = -3 ψ + ∇ 2 Ė. (1.5)
Assembling everything, the perturbed kinetic part of the action, up to second order, writes δS (2)

K = dtd 3 x 2 κ 2 1 4 ḣij ḣij + 1 2 ∂ i Ẇj ∂ i Ẇ j + (1 -3λ) 3 ψ2 -2 ψ∇ 2 Ė + (1 -λ) Ė∇ 4 Ė , (1.6
) where we have also used the transversality of fields and integration by parts to eliminate irrelevant terms and remove derivatives. Thus, switching off the W i mode, and concentrating only on scalar and tensor perturbations, the perturbed kinetic part of the action (2.15), up to second order, yields the expression (2.19).

A.2 Perturbation of S V

Let us now examine the perturbation of potential term (2.17). Note that again all quadratic terms must yield only contributions of the form δRδR etc, since these tensors vanish for a flat background. The only terms that do not follow this rule are the fifth and sixth. The fifth is first order in R, so the second-order perturbation will be of the form δ( √ g)δR, while the sixth only receives contributions at second order from the determinant √ g. We will calculate each quadratic expression in turn.

For the first term we need to evaluate the perturbed Cotton tensor

δC ij = ikl g (0) ∂ k δR j l - 1 4 δRδ j l = - 1 2 ikl ∂ k ∇ 2 h j l , (1.7)
which allows to write for the first four-terms:

δC ij δC ij = - 1 4 h ij ∇ 6 h ij (1.8) ijk δR il ∂ j δR l k = 1 4 ijk ∇ 2 h il ∂ j ∇ 2 h l k
(1.9)

δR ij δR ij = 1 4 ∇ 2 h ij 2 + 6ψ∇ 4 ψ, (1.10)
together with the square of (1.3). Concerning the perturbation of the determinant part, we write:

√ gNR = N (0) g (0) + 1 2 g (0)

δg -1 8

1 g (0) 3 δg 2 + . . .

(1.11)

× δ ij R (1) ij + δ ij R (2) ij -δg ij R (1) ij (1.12) = δ ij R (1) ij + δ ij R (2) ij -δg ij R (1) ij + 1 2 δgδ ij R (1) ij , (1.13)
since the perturbation of N is δN = 0 and N (0) = 1. Note that the combination

δ ij R (1)
ij is just δR, and thus the second-order perturbation becomes

( √ gNR) (2) = δ ij R (2) ij -δg ij R (1) ij + 1 2 δgδR (1.14)
where, to first order, δg = g (0) g ij(0) δg ij = δ ij δg ij = -6ψ + 2∇ 2 E. Thus the third term in (1.14) becomes

1 2 δgδR = -12ψ∇ 2 ψ + 4∇ 2 E∇ 2 ψ. (1.15)
Additionally, using integration by parts and imposing transversality, we obtain

δg ij R (1) ij = - 1 2 h ij ∇ 2 h ij -8ψ∇ 2 ψ + 4ψ∇ 4 E. ( 1
δ ij Γ (1)l jk Γ (1)k il = 1 4 h ij ∇ 2 h ij + ψ∇ 2 ψ + 2ψ∇ 4 E -E∇ 6 E, (1.21)
we conclude that the Ricci scalar can be expressed as √ g (0) δg = -3ψ + ∇ 2 E and (δ √ g) (2) = -1 8 δg 2 = -9 2 ψ 2 + 3ψ∇ 2 E -1 2 (∇ 2 E) 2 , the sixth term in the perturbed action reads

δ ij R (2) ij = δ ij Γ (2)l ij,l -δ ij Γ (2)l il,l + δ ij Γ (1)l lk Γ (1)k ij -δ ij Γ (
(δ √ g) (2) N (0) = - 9 2 ψ 2 + 3ψ∇ 2 E - 1 2 (∇ 2 E) 2 .
(1.24)

Finally, assembling the aforementioned terms, we conclude that the perturbed potential part of the action is (2.20).

2 = 3ψ∇ 2 ψ

 22 i k + ∂ k δg i j -∂ i δg jk (1δg il (∂ j δg lk + ∂ k δg jl -∂ l δg jk ) + 2ψ∇ 4 E -E∇ 6 E (1.20)

  ij ∇ 2 h ij + 2ψ∇ 2 ψ. (1.22)Assembling(1.15),(1.16),(1.22) we find that the fifth term in the perturbed action reduces to ( √ gNR)(2) = 1 4 h ij ∇ 2 h ij -2ψ∇ 2 ψ.

	Lastly, using δ	1)l jk Γ h (1.23) (1)k il √ g = 1 = -1 4 2

Acknowledgments

We would like to thank Y.F. Cai, C. Charmousis, S. Mukohyama and A. Padilla for useful discussions. We also thank K. Sfetsos for fruitful comments on the earlier version of this paper. C.B. is supported by the CNRS and the Université de Paris-Sud XI.

Note added

After the present work was pre-printed and after its submission to the journal, the subject of possible instabilities in Hořava gravity has gained a significant attention


Maurice H P M Van Putten

Extended black hole cosmologies in de Sitter space

We generalize the superposition principle for time-symmetric initial data of black hole spacetimes to (anti-)de Sitter cosmologies in terms of an eigenvalue problem ∆ g φ = 1 8 (R g -2Λ)φ for a conformal scale φ applied to a a metric g ij with constant three-curvature R g . Here, R g = 0, 2 in the Brill-Lindquist and, respectively, Misner construction of multihole solutions for Λ = 0. For de Sitter and anti-de Sitter cosmologies, we express the result for R g = 0 in incomplete elliptic functions. The topology of a black hole in de Sitter space can be extended into an infinite tower of universes, across the turning points at the black hole and cosmological event horizons. Superposition introduces binary black holes for small separations and binary universes for separations large relative to the cosmological event horizon. The evolution of the metric can be described by hyperbolic system equations with curvature-driven lapse function, of alternating sign at successive cosmologies. The computational problem of interacting black hole-universes is conceivably of interest to early cosmology when Λ was large and black holes were of mass < 1 3 Λ -1/2 , here facilitated by a metric which is singularity-free and smooth everywhere on real coordinate space.

Introduction

The multihole solutions of Brill-Lindquist and Misner (Misner 1963;Brill & Lindquist 1964;[START_REF] Cook | Initial data for numerical relativity[END_REF]) arise out of a superposition principle in the Hamiltonian constraint of the Einstein equations for time-symmetric initial data. Here, the Schwarzschild line-element in coordinates (r, θ, ϕ) is transformed into an isotropic line-element described by a conformal factor φ(r), giving rise to a conformally flat spacetime in vacuum. This construction is remarkable, in allowing for multihole solutions with different extended topologies. To a black hole binary, the Brill-Lindquist solution attributes a three-sheet topology, whereas the Misner solution attributes a two-sheet topology to the same.

The embedding of black holes in (anti-)de Sitter space is well-known for a single black hole with one-sheet topology. De Sitter space is of interest in view of considerable observational evidence for a small but distinctly positive cosmological constant in the ΛCDM model (de Bernardis et al. 2000;Hanany et al. 2000) with an expected improved uncertainty by the recently launched Planck satellite (ESA-SCI 2005). The stability and thermodynamics of the cosmological event horizon in the black hole-de Sitter space is relevant to its potential role to early cosmology (Davies 1987;Chambers 1997).

Here, we consider the problem of extending space beyond the cosmological event horizon. We approach this problem in terms of the turning points at extrema of the circumference in the one black hole-de Sitter space. The result opens the possibility for novel topologies on scales that reach beyond the visible universe. Space of finite volume further facilites representation in a metric that is smooth everywhere. This is of direct interest to introducing spectral methods for calculating wave-templates for gravitational-wave observatories LIGO and Virgo as they are gradually improving their sensitivity.

To begin, we recal the Brill-Lindquist line-element of a single Schwarzschild black hole Brill & Lindquist (1964); [START_REF] Cook | Initial data for numerical relativity[END_REF]. The event horizon of the black hole represents a turning point in the embedding in two asymptotically flat sheets,

ds 2 = -tanh 2 (λ/2)dt 2 + 4E 2 cosh 4 (λ/2)ds 2 D , (1)
over the donut ds 2 D = dλ 2 + dx 2 1-x 2 + (1 -x 2 )dϕ 2 with -∞ < λ < ∞.
Here, we transform the spherical coordinates (ρ, θ, ϕ), x = cos θ of Brill-Lindquist by ρ = 1 2 Ee λ . The Möbius invariance λ ↔ -λ comes with opposite signs in the lapse fuction N = tanh(λ/2) on either sheet, wherein the horizon surface corresponds to the simple zero N = 0. By Liouville's theorem, spacetime singularities are inherent to any black hole spacetime which is asymptotically regular. They can be moved away into the complex plane as (1) illustrates by mapping ρ = -1 2 E to λ = πi (mod 2π), whereby the metric is analytic everywhere for all real and finite λ, wherein λ = ±∞ represent coordinate singularities associated with asymptotic infinity on each sheet.

The black hole singularity is not directly accessible by observation, whether in real or complex coordinate space. Even an observer in free fall onto a black hole never reaches the central singularity when considered in real coordinates, as in the Schwarzschild line element. Upon approaching the event horizon, time-at-infinity, t, becomes arbitrarily large, which signals evaporation of the black hole by Hawking radiation. The observer hereby traces a shrinking event horizon, never to penetrate it, during which time the black hole singularity diminishces in strength. Note that this result only uses evaporation of a black hole in a finite time-at-infinity, regardless of the details of the luminosity of Hawking radiation. Hawking radiation hereby introduces invariance to the cosmic censorship conjecture, by treating singularities in real and complex coordinate space on equal footing. The same arguments shows that (1) defines a non-traversable wormhole.

In de Sitter space, coordinate singularities associated with the infinite extend of space are avoided. The result for multihole solutions in de Sitter space, then, is a metric which is everywhere nonsingular and finite on the coordinate cover of the visible universe with no need for compactification. The cosmological event horizon represents an additional length scale, which can be of interest in considering black hole binaries with large separations. It introduces an additional turning point which opens a window for novel large scale extensions beyond.

In §2, we derive the formalism for constructing multihole initial data in (anti-)de Sitter space. In §3 we give some illustrative examples for binary black holes and binary cosmologies. In §4, we propose a hyperbolic system of equations for their evolution based on the 3+1 Hamiltonian equations of motion [START_REF] Arnowitt | Gravitation: an introduction to current research[END_REF], where hyperbolicity generally facilitates stable numerical implementation (e.g., van Putten & Eardley (1996); Nagy, Ortiz, & Reula (2004); Calabrese, Hinder & Husa (2006) and references therein) by ensuring a real dispersion relation and hence stability whenever the Courant-Friedrichs-Lewy condition (Courant, Friedrichs & Lewy 1967) is satisfied. An outlook is included in §5.

A superposition principle in (anti-)de Sitter space

The line-element of a Schwarzschild black hole of mass m in a de Sitter space with cosmological constant Λ is

ds 2 = -(1 -2m/r - 1 3 Λr 2 )dt 2 + dr 2 1 -2m/r -1 3 Λr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 .
(2)

When Λm 2 < 1 9 , a black hole of mass m can exist within the larger cosmological event horizon defined by Λ with a black hole horizon at coordinate radius r = r 1 and an event horizon at r = r 2 > r 1 corresponding to the roots of the redshift factor N = 0, N 2 = 1 -2m/r -1 3 Λr 2 , in (2). The roots are shown in Fig.

(1) with the bifurfaction point at Λm 2 = 1 9 , including the non-physical real root r 3 < 0. The line-element (2) can be transformed to an isotropic line-element

ds 2 = -N 2 dt 2 + φ 4 (dρ 2 + ρ 2 dθ 2 + ρ 2 sin 2 θdϕ 2 ) (3)
according to the conditions

± dr 1 -2m/r -1 3 Λr 2 = φ 2 dρ, r = φ 2 ρ. (4)
For Λ > 0 the three real roots r 3 < 0 < r 1 < r 2 of N satisfy r 1 = 2m(1+ ), r 2,3 = ± 3 Λ (1∓δ), where = 4 3 Λm 2 +O( 2 ) denotes the horizon surface of the black hole and δ = m Λ 3 +O(δ 2 ). as a function of Λm 2 > 0. They define the coordinate location of the horizon of the black hole (r 1 ) and of the cosmological event horizon (r 2 ). The roots bifurcate into a pair of real roots at Λm 2 = 1/9. The maximum mass of a black hole in a de Sitter space is hereby 1/3Λ -1/2 (Podolský 1999). Shown is further the numerical root r 3 < 0.

The solution to (4) can be expressed following a logarithmic transformation dλ = dρ/ρ. For Λ > 0, the Möbius transformation t = 2m/r (with no reference to time intended) gives

λ = t 1 t 2 - t t 2 dτ τ 2 -τ 3 -4Λm 2 /3 = 2 F (π/2, m 1 ) -F (φ, m 1 ) √ t 1 -t 3 (5)
in terms of the three roots

t i = 2m/r i (t 3 < 0 < t 2 < t 1 ), m 1 = t 1 -t 2 t 1 -t 3 , sin φ = (t 1 -t 3 )(t-t 2 ) (t 1 -t 2 )(t-t 3 )
, where F (φ, m 1 ) denotes the incomplete elliptic function of the first kind (Abramowitz & Stegun 1968). For Λ < 0, the only real root is r 2 . We proceed with r = 6m 1-12v ,

λ = v v 2 dP 4P 3 -P/12 + 1/216 + Λm 2 /12 , (6)
giving rise to the Weierstrass elliptic function v(λ) = P (λ, 1/12, -1/216-Λm 2 /12) (Abramowitz & Stegun 1968).

In the resulting conformally flat approach, the resulting extension of de Sitter black hole spacetime is shown in Figs. 23). We note that the singular limit Λ = 0 reduces to the familiar expressions (e.g. Abrahams & Price (1996))

λ = ln 1 m [r -m + √ r 2 -2mr] , r = ρ 1 + m 2ρ 2 , ρ = 1 4 √ r + √ r -2m 2 . (7)
We interpret the result as follows.

Theorem. The superposition principle for time-symmetric data in the Hamiltonian energy constraint generalizes to (anti-)de Sitter space for eigenfunctions of the Laplace operator of a scaled metric with constant 3-curvature.

Proof. We recall the conformal decomposition of the Ricci tensor for

h ij = φ 4 g ij , whereby (3) R ij (h) = (3) R ij (g) + C ij with C ij = -2φ -1 [D i D j φ + g ij ∆φ] + 2φ -2 [3D i φD j φ - g ij D p φD p φ].
Following a reduction of the Ricci tensor from four to three dimensions, the Einstein equations for time-symmetric data give

(3) R(h) = (3) R(g) -8φ -1 ∆ g φ = 2Λ. ( 8
)
If the scaled metric g ij produces a constant curvature R g , the φ are eigenfunctions of the associated Laplace operator,

∆ g φ = 1 8 (R g -2Λ) φ. (9)
We can construct solutions by superposition -For a single black hole in de Sitter space, the scale φ is periodic in the radial coordinate -∞ < λ < ∞, representing a "tower" of cosmologies by successive continuations at the turning points defined by the black hole and cosmological event horizons. The results shown are computed for m = 1 and Λ = 0.001, and may be contrasted with φ = 1 -e -λ 1/2 in singular limit Λ = 0. µ i > 0, Σµ i = 1, of different eigenfunctions φ i to the same eigenvalue, e.g., those translated in any one of the homogeneous directions. The Brill-Lindquist case corresponds to the flat metric g ij = δ ij with R g = 0 and three homogeneous directions, whereas the Misner case corresponds to the donut metric of (1) with R g = 2 and one homogeneous direction, both with Λ = 0. In case of the former, we have the coordinate density

φ = Σµ i φ i , (10)
φ 2 p = r ρ , ρ = (x 1 -p 1 ) 2 + (x 2 -p 2 ) 2 + (x 3 -p 3 ) 2 , (11)
associated with black holes at p (and r = 0 in the corresponding Schwarzschild coordinates), where the singular limit Λ → 0 recovers φ = 1 + m 2ρ . 2. The eigenvalue problem (10) suggests an association to a boson field with frequency ω = 1 2 √ Λ in the tangent bundle of de Sitter space. However, this association does not reflect scale invariance in the Riemann tensor, in that it depends only on the log of φ.

A cosmological event horizon at finite distance alters the spectrum of eigenfunctions. Whereas the eigenfunctions r -(l+1) P l (x) to ∆φ = 0 give convergence and divergence at asymptotic infinity on the two sheet embedding of the Schwarzschild black hole in the Brill-Lindquist case Λ = 0, the eigenfunctions of ∆φ = -1 4 Λφ for Λ > 0 are periodic across the black hole and cosmological event horizons in Figs. 2-3 and are hereby necessarily bounded and regular everywhere.

A cosmological event horizon also alters the Hilbert space of radiation states at large distances from a black hole, and hence the details of Hawking radiation (Kanti, Grain & Barrau 2005;[START_REF] Zeng | [END_REF]. This is particularly pertinent during inflation, when Λ is large. The evaporation time of black holes should remain finite, however, whereby the strong form of the cosmic censorship conjecture described in the introduction should continue to hold whenever m 2 Λ < 1 9 . A construction similar to (5-6) may be pursued for the Misner two-sheet embedding of two black holes, based on eigenfunctions ∆ D φ = 1 4 (1 -a 2 Λ)φ, where D refers to the donut line-element in (1) and a refers to the Misner length scale. A detailed consideration, however, falls outside of the present discussion. It may also be generized to extra dimensions on the basis of the Schwarzschild-de Sitter solution (Tangherlini 1963;Gao 2004). In 4+1, for example, the transformation (5-6) produces a trigonometric expression.

Extended black hole-de Sitter cosmologies

For 0 < m 2 Λ < 1 9 , we construct a few binary black hole spacetimes with different separations as shown in Fig. (4). For small separations, the results are very similar to a Brill-Lindquist spacetime inside the cosmological event horizon surrounding the binary. For large separations, however, the cosmological event horizon splits into two, leading to a binary of two universes. Here, the second universe lives in the "tower" of the first, on the sheet beyond its cosmological event horizon.

The dynamical evolution of an extended black hole-de Sitter universe with multiple black holes and cosmological event horizons offers a new route to gravitational radiation from the early universe. Here, the waves are generated by the dynamics of primordial black holes, as well as by multipole moments of cosmological event horizons. The spectrum of relic waves may hereby be extended in the infrared, below which what may be expected from mergers of primordial black holes alone. It poses a novel computational challenge for numerical relativity, i.e., to compute these relic waves at the present epoch from an initial distribution of primordial black hole-de Sitter universes at the prior to or at the onset of inflation.

Hyperbolicity with curvature-driven lapse

Spacetime can be described by a foliation in spacelike hypersurfaces, given by a 3+1 decomposition in terms of h ij (Thorne, Price & McDonald 1986)

ds 2 = -N 2 dt 2 + h ij dx i + β i dt)(dx j + β j dt (12)
with lapse function N and shift functions β i . The N and β i govern the flow of coordinates in the foliation of hypersurfaces of constant coordinate time t. In what follows, we shall restrict our attention to normal coordinates described by vanishing shift functions, β i = 0.

In the Hamiltonian equations of motion with vanishing shift function, the three-metric

h ij evolves according to ∂ t h ij = -2NK ij = -NK n i h nj -h im NK m j .
Integration over finite time-intervals ∆t gives the product

h ij (t i ) = Π i e -K m i N ∆t h mn (t 0 )e -K n j N ∆t , (13)
We propose time-evolution with curvature-driven lapse function, i.e.,

D t N = -K, D t K ij = -D ij N + K ij N, (14)
where

D ij = D i D j -R ij and K = KK ij -2K m i K jm .
The gauge condition D t N = -K is curvature-driven and is distinct from the product of curvature and lapse function in the harmonic slicing condition ∂ t N = -N 2 K in Eqs. ( 69)-(77) of Abrahams et al. (1997); see also Brown (2008) for a recent review. . Shown are isocurves of constant J = |∂φ/∂ λ + 1/2|. For small separations, the result is a black hole binary inside a slightly deformed cosmological event horizon J = 0 (thick dark line). As the separation increases and becomes large, a binary of two universes forms, wherein the second lives on a sheet adjacent to the first and beyond its cosmological event horizon corresonding to a dashed sheet in Fig.

(

3) (right below).

There is an infinity of black hole and cosmological event horizons at exponentially increasing coordinate distances and within the two black holes (not shown).

Theorem. The 3+1 Hamiltonian evolution equations are hyperbolic with curvaturedriven lapse function ∂ t N = -K.

Proof. It suffices to consider the problem of small amplitude wave-motion about flat spacetime, e.g., the asymptotically flat region with h ij = δ ij with N = ±1 at large distances. Here, we have

∂ t N = -K, ∂ 2 t h ij = -2R ij + 2D i D j N, ∂ 2 t K = ∆K. ( 15
)
We recall that [START_REF] Wald | General Relativity[END_REF])

R ij = - 1 2 ∆δh ij + 1 2 ∂ i ∂ e δh ej + 1 2 ∂ j ∂ e δh ei ( 16
)
where hij = δh ij -1 2 δ ij δh, where δh = h ij δh ij refers to the trace of the metric perturbations. Small amplitude harmonic perturbations about the flat metric are given by δh ij ∼ ĥij e -iωt e ik i x i . Conservation of momentum,

D i K ij = D j K, the lapse condition -iωN = -K, δh ij = -2iω -1
Kij and k i Kij = k j K, give rise to

∂ i ∂ e hej → k i k e ĥej - 1 2 k i k j δh = iω -1 (-2k i k e Kej + k i k j K) = -iω -1 k i k j K. ( 17
)
We then have

Rij -∂ i ∂ j N = 1 2 k 2 ĥij -iω -1 k i k j K + iω -1 k i k j K = 1 2 k 2 ĥij , (18)
whereby ∂ t h ij = -2R ij + 2D i D j N gives rise to the dispersion relation

ω 2 = k 2 . ( 19
)
It follows that all small amplitude metric perturbations propagate along the light cone, which completes the proof. 2

Clearly, the system ( 15) is asymptotically stable, as metric perturbations become small at arbitrarily large distances. Asymptotic wave-motion is commonly studied in the socalled transverse traceless gauge, or harmonic coordinates-neither of these two coordinate conditions are used here.

Numerical integration of (13-14) on the donut (λ, θ, ϕ) can be pursued using a conformal decomposition to bring out invariance of the aforementioned C ij with respect to scaling in φ. To this end, we may set φ = e η , whereby

C ij = -2[D i D j η + g ij ∆η] + 4[D i ηD j η -g ij D p ηD p η]. (20)
It follows that

∂ t K ij = F ij with F ij = -N (2K m i K jm -KK ij ) + N (R ij + C ij ) -D i D j N + Ω k ij ∂ k N , where Ω k ij = 2φ -1 g ke (g ie ∂ j φ + g je ∂ i φ -g ij ∂ e φ) = 2g ke (g ie ∂ j η + g je ∂ i η -g ij ∂ e η).
A closed system is obtained by choosing an equation for the conformal factor. A common choice is φ = h 1/12 in terms of the determinant h of the three metric h ij . For outgoing radiation, note that φ = 1 up to including first order in the wave-amplitude, whereby φ = 1 tracks the merger phase of black hole coalescence (here with overlapping horizon surfaces) and ringdown in collapse to a single black hole. A slight variation is to insist g ij ∂ t g ij = 0. The complete hyperbolic system for numerical integration using normal coordinates (

β i = 0) hereby becomes          ∂ t N = -K, ∂ t η = -1 6 NK, ∂ t g ij = -2N K ij -1 3 g ij Kij , ∂ t K ij = F ij (η, g ij , N), (21)
where Kij = e -4η K ij .

The cosmological event horizon λ ± λ 2 represents a turning point in (4), representing an extremum of the radius ρφ 2 defined by the Neumann condition

dη dλ = - 1 2 . ( 22
)
As such, (22) defines the cosmological event horizon (λ 2 (θ, ϕ), θ, ϕ) following superpositions (10). In dynamical evolutions, it can be used to define apparent horizon surfaces. A proper condition of lapse function N is that it preserves a simple zero across, i.e., the initial condition

N = d log(ρφ 2 ) dλ . (23)

Conclusions

We have extended the superposition principle for the Hamiltonian energy constraint to (anti-)de Sitter cosmologies with cosmological constant Λ in terms of an eigenvalue problem for the Laplace operator on a metric with constant curvature.

A positive cosmological constant Λ is of increasing relevance to our study of cosmological spacetimes in view of its role in the early universe in view of observational evidence for a small positive value at the present epoch.

For one black hole, the conformal scale φ is periodic, representing the extension of the black hole-de Sitter space to an infinite tower of universes. Superposition of two eigenfunctions produces a binary black hole-de Sitter universe with one (adjacent) cosmological event horizon when the separation is small relative to the cosmological scale 1/ √ Λ. However, it produces a binary of universes when their separation is large described by two adjacent cosmological event horizons.

The results demonstrate extended topologies of black hole-de Sitter cosmologies, wherein the metric is everywhere smooth, as the physical singularities inherent to black holes are moved away into the complex plane and space is given a finite physical extent. By virtue of a finite evaporation time of black holes by Hawking radiation, there is a strong cosmic censorship conjecture by which singularities are protected from direct observation, regardless whether they are located in real or complex coordinate space.

The dynamical interaction of a binary of two universes ( 4) is conceivably of interest to the early universe and the ensuing generation of primordial gravitational waves from an initial distribution of primordial black hole-de Sitter universes, when Λ was large at the onset of inflation. An extension of our approach to extra dimensions is readily given, which is conceivably of interest to dynamical evolution of our four-dimensional spacetime within a spacetime of extra dimensions. For computational purposes, we give a new hyperbolic formulation of the equations of motion based on a curvature-driven lapse function.

  Fig. 1.-Shown are the three real roots of the redshift factor in the Schwarzschild-de Sitter space

Fig

  Fig.2.-For a single black hole in de Sitter space, the scale φ is periodic in the radial coordinate

  Fig. 3.-The extension of black hole-de Sitter cosmology of a single black hole-de Sitter spacetime can be realized by joining the eigenfunction solutions at successive turning points (solid-to-dashed transitions), representing black hole and cosmological event horizons (H i , E i ). The results shown are computed for m = 1 and Λ = 0.001.

  Fig. 4.-Initial data of binaries constructed by the generalized superposition principle starting from zero separation (left top). Shown are isocurves of constant J = |∂φ/∂ λ + 1/2|. For small separations, the result is a black hole binary inside a slightly deformed cosmological event horizon J = 0 (thick dark line). As the separation increases and becomes large, a binary of two universes forms, wherein the second lives on a sheet adjacent to the first and beyond its cosmological event horizon corresonding to a dashed sheet in Fig.(3) (right below). There is an infinity of black hole and cosmological event horizons at exponentially increasing coordinate distances and within the two black holes (not shown).

This preprint was prepared with the AAS L A T E X macros v5.0.

Acknowledgment. The author gratefully acknowledges stimulating discussions with A. Spallicci, M. Volkov, G. Barles, members of the Fédération Denis Poisson, and AEI of the Max Planck Institute, where some of the work was performed. This work is supported, in part, by Le Studium IAS of the Université d'Orléans. The author thanks the referees for their constructive comments.


H Cruz Sanchez

S Nourizadeh
email: nourizas@loria.fr

Y Q Song

MPIGate : Multi Protocol Interface and Gateway

We present MPIGate, a multiprotocol gateway and interface for assisted living applications. This project aims to facilitate the development of applications thanks to the abstraction of sensing data and the facility of access to read and write functions over the devices plugged to the network. The contribution includes modularity and a middleware layer offering modularity and quality of service. The proposed unified syntax of messages facilitates interconnection with other systems, networks and devices.

Introduction

The interest of using Information Technologies (TI) to provide services in assisted living context is growing in these days. In industry/literature we can find several projects aiming to bring services facilitating patients suffering a chronic disease or any kind of physical dependency stay at home during long-period treatments. Professionals in health and public entities share this interest and they are working together to propose a solution in a long-term basis to face the growth of dependent population. Elderly people percentage and diseases associated to age are growing as well: living longer increases probability of suffering chronic diseases and/or dependency [START_REF] Nourizadeh | A Distributed Elderly Healthcare System[END_REF]. In this context, we point out the interest of having a system incorporating existing infrastructure (in domotics, sensor networks, actimetry systems) in a home living environment. Until now, the industries in these fields have developed systems for a very specific technology/market. In the existing systems we can identify an individual activity of each of these industries based on their (proprietary/specific) technologies and it is difficult to imagine a migration of these systems to a unified//homogeneous system. This contribution describes MPIGate (Multi Protocol Interface and Gateway for Telecare, Environment Monitoring and Control), a technical solution for system interoperability in an assisted living context by using a middleware offering modularity and quality of service. We propose a gateway, which can be used in a home assisted living system to integrate multi-platform systems in a unified framework. Information coming from different systems can be processed and used for having a real state of home environment, human activity, etc. The gateway uses existing and standard industry infrastructure incorporating wireless sensor networks to complete and propose new sensor and mobile applications (Figure 1).

Figure 1. MPIGate in an assisted living network.

The paper is organized as follows. In Section 2 we describe architecture of the proposition. In Section 3 we discuss the importance of the QoS in the field of health care and the aspects we considered during the project design. Finally we present the on going work and future tests of MPIGate.

Architecture

!"# "$%$#&'(()&*'+# ,-'%'&'(#$./%-$&*'+## 0123#4#567!))#4#!(8)%''%9#4#:6;#4#<%9)-+)%=# >'+;78-$*'+#6+%)-?$&)# @/)-#6+%)-?$&)# A./%-$&*'+# "$%$#B-'&)//6+7# C8)-D#E$+$7)-# F)&8-6%D#$+G#/8B)-H6/6'+# Figure 2. MPIGate components.
In this section we describe the MPIGate architecture from the functional point of view.

Configuration interface: is an interface to configure different parameters of the system. By using this interface we can provide and modify necessary information to process the information in Data processing layer.

User interface: provides an Application Programming Interface (API) allowing the programmer of applications to read each sensor of the system. The API allows sending queries to update the sensing values if this is needed. A Graphical Interface is also included to visualize the state of each sensor and actuator, their historical information and the reports generated by the system. Abstraction: this layer hides the heterogeneity of the system for the users, by providing communication facilities between the interface layer and the other parts of system. The user doesn't need any prior knowledge of the protocols behind the system, greatly facilitating the application developers' task. Data processing: this layer is the main part of the system. It processes the information collected from the network and save the results in the database. Query manager: receives the query from user interface, creates the query and sends them to the network. The query might be a command to control an actuator or a request to read a sensor data.

Protocol abstraction: provides necessary information about each communication protocol used in the network. Security and supervision: it is the responsible of providing the security of the system. Data collection: collects the data from the network and sends them to the data processing layer. It implements the reading daemons or sniffers for KNX, Bluetooth and IEEE 802.15.4 over the peripherals of MPIGate. The syntax of the messages has been defined in an internal report of INRIA. It is defined as follows:

Node Id, Location, Date, NameOfVariable(i), ValueOfVariable(i), …

Types of variable i can be environmental, biometric, equipment state… 3. Discussion An important point considered in developing this concept is Quality of Service (QoS) requirements like availability and response time. As the healthcare industry is turning to information technology to help solve its business issues, specially provide to quality patient care services, it is important to develop QoS specification in distributed health information systems [START_REF] Ray | Quality of service management in health care organizations: a casestudy[END_REF]. Unlike traditional systems, many non-end-to-end missioncritical applications envisioned for healthcare systems, which are complex systems, because in these systems, we can find different subsystems, such as, wireless sensor networks, LANs, software platforms, home automation systems, mobile systems, Internet… that require different QoS requirements on the system, and these requirements pose unprecedented challenges in the area of QoS support in healthcare systems. Any failure or lack of performance on the system which could not be tackled in a reasonable delay may have some damageable consequences on the solutions' acceptance and development potential: the confidence is a basic and elementary factor of acceptance or reject, such incident could also generate a psychological defiance towards ICT's in general and towards such innovative assistance and monitoring services. For these raisons and for providing real-time data collecting, we use a context-aware middleware [START_REF] Nourizadeh | CodaQ -A Context-aware and QoSaware Middleware for Activity Monitoring[END_REF] to give a dynamic and intelligent QoS to the system. It provides context data and takes into account QoS requirements of the applications. Measurements on a test bed have been carried out showing the good performance of our design. This context-aware middleware is a bridge between the healthcare applications and the sensor networks, to guarantee QoS needs, for supervision of the system and for easy configuration and installation of the sensors and actuators.

Conclusions

A prototype of MPIGate including a simple application for telemetry is under development. Practical tests are overseen on the Assisted Living Environment platform at LORIA (www.infositu.loria.fr). The development to include other protocols libraries is in study.


Jamila Ben Slimane
email: jamilabs07@yahoo.fr

Ye-Qiong Song
email: song@loria.fr

Anis Koubâa
email: akoubaa@dei.isep.ipp.pt

Mounir Frikha
email: m.frikha@supcom.rnu.tn

Hugo Cruz-Sanchez
email: hugo.cruzsanchez@loria.fr

PMCMTP's implementation (in nesC/TinyOS2.x) and Testbed for its operation validation

Keywords: WSN, PMCMTP, Testbed

PMCMTP is a Prioritized Multi-Channel Multi-Time slot MAC protocol that the authors have proposed for allowing to simultaneous use of several frequency channels. This protocol is designed for UWB of IEEE802.15.4a but it can also be used over IEEE802.15.4. In this paper, we design and implement a testbed of this protocol to demonstrate its practical implementability. Due to the unavailability of UWB transceiver, the testbed has been performed using classic 2.4GHz WSN transceivers. To reduce the complexity of resource sharing, the global network is composed of a set of Personal Area Networks (PANs) or cells. So, the PMCMTPs experiments are performed for a single PAN and two PANs.

I. INTRODUCTION

Wireless sensor networks may consist of a collection of spatially distributed autonomous sensor nodes characterized by limited memory, processing capability and battery power supply. Such networks represent an emerging technology with wide range of potential applications such as medical systems, environment monitoring, military applications, etc. An important fact that can not be neglected in WSNs is the need of quality-of-service (QoS) support. According to novel application requirements, QoS constraints become more and more critical in terms of end-to-end delay and data throughput. Also, due to energetic constraints at node level, energy saving remains the most challenging issue. Both IEEE 802.15.4 and its recent amendment IEEE 802.15.4a [START_REF]IEEE Standard for Information Technology, Amendment to IEEE Std 802[END_REF] standards allow dynamic channel allocation and use of multiple channels available at their physical layers but their MAC protocols are designed only for single channel. Also, sensor's transceivers such as CC2420 used by current WSN hardware (MICAZ, TelosB, and CMU FireFly), provide multiple channels, and as shown in [START_REF] Wang | Spatial channel reuse in wireless sensor networks[END_REF] and [3], channel switch latency of CC2420 transceiver is short (just about 200µs). So this gives the possibility to simultaneous use of multi-channels, greatly increasing data throughput. The idea of multi-channel MAC protocols is not new in the wireless network research community. Recently some MAC layer multi-channel protocols have been proposed to improve This paper has been accepted for publication in the proceedings of WSN Workshop that had been held in conjunction with NTMS 2011. network performance in WSNs [START_REF] Kim | Y-MAC: An Energy-Efficient Multi-channel MAC Protocol for Dense Wireless Sensor Networks[END_REF]- [START_REF] Zhiwu | A Dynamic Multi-radio Multi-channel MAC Protocol for Wireless Sensor Networks[END_REF]. But, the majority of the proposed protocols were tested and evaluated by simulations, using personal or academic or commercial network simulator, and only protocols proposed in [START_REF] Hieu | A Practical Multi-Channel Media Access Control Protocol for Wireless Sensor Networks[END_REF] and [START_REF] Zhiwu | A Dynamic Multi-radio Multi-channel MAC Protocol for Wireless Sensor Networks[END_REF] were validated and evaluated by testbed deployed in a realistic setting.

In this paper, we present the implementation of a Prioritized Multi-Channel Multi-Time slot media access control Protocol (PMCMTP) [START_REF] Ben Slimane | A Prioritized Multi-Channel Multi-Time slot MAC Protocol For Large-Scale Wireless Sensor Networks[END_REF] for WSNs. We, experimentally, validate the PMCMTP by using a realistic testbed setting.

II. RELATED WORK

The WSNs research is currently gaining an exponentially increasing interest from both academia and industry. New application scenarios, MAC, routing and physical layer optimizing protocols and algorithms are being developed by the research community. The ultimate test for the validity of these research activities is to implement the various scenarios, algorithms and protocols on a testbed and deploy it in a realistic setting. Several testbeds have been designed for the experimentation of WSNs [START_REF] Hieu | A Practical Multi-Channel Media Access Control Protocol for Wireless Sensor Networks[END_REF], [START_REF] Zhiwu | A Dynamic Multi-radio Multi-channel MAC Protocol for Wireless Sensor Networks[END_REF], [START_REF] Werner-Allen | MoteLab: a wireless sensor network testbed[END_REF]- [START_REF] Baccour | A TestBed for the Evaluation of Link Quality Estimators in Wireless Sensor Networks[END_REF]. We can distinguish two class of testbeds: the first for general WSNs tests and the second for specific WSNs tests. MoteLab [START_REF] Werner-Allen | MoteLab: a wireless sensor network testbed[END_REF] is a web-based WSN testbed consisting of a set of MicaZ motes connected to a central server. Its goal is to allow users to evaluate WSN applications without manually re-programming and redeploying the nodes into the physical environment. Emulab [START_REF] Johnson | Mobile Emulab: a robotic wireless and sensor network testbed[END_REF] is a remotely accessible mobile and wireless sensor testbed. It can be used to study network topologies, mobility effects on protocols, test algorithms, and mobile applications. So, motes are deployed in mobile robots and users can create experiments through a web interface and schedule events to control the robots movement. The goal of such category of WSNs testbeds is to allow a large community of users (with different research goals) to share access to the testbed platform. So, resource are shared between users and it can not be available any time also testing environment is not manageable which. Such testebeds are useful for general WSNs tests by not for a specific WSNs tests. For a specific tests, in [START_REF] Baccour | A TestBed for the Evaluation of Link Quality Estimators in Wireless Sensor Networks[END_REF], the authors proposed their own testbed for the performance evaluation of link quality estimators. In [START_REF] Hieu | A Practical Multi-Channel Media Access Control Protocol for Wireless Sensor Networks[END_REF] and [START_REF] Zhiwu | A Dynamic Multi-radio Multi-channel MAC Protocol for Wireless Sensor Networks[END_REF], the proposed WSNs tesbeds targeted multi-channel WSN MAC protocols, to test respectively a multi-radio multi-channel MAC protocol and a mono-radio multi-channel MAC protocol. In this paper, we propose a WSN testbed to test the validity of our mono-radio multi-time slots multi-channel MAC protocol (PMCMTP). Our paper makes the following main contributions: -First, we implemented the PMCMTP in the nesC [13] programming language for MicaZ [14] motes with TinyOS-2.x. [15] -Second, we deployed a WSN testbed to ensure the experimental validation of the proposed protocol, The rest of the paper is organized as follows: In Section 3, we present the system model. Section 4 gives an overview of the PMCMTP protocol. Section 5 gives a brief overview of the implementation of the PMCMTP on the MicaZ motes. In Section 6, we validate and demonstrate the effective operation of the PMCMTP and evaluate its performance by analyzing and commenting some experimental results.

III. SYSTEM MODEL

We have proposed in [START_REF] Ben Slimane | A Three-Tiered Architecture for Large-Scale Wireless Hospital Sensor Networks[END_REF] the WHSN (Wireless Hospital Sensor Network) for an application in hospital (medical monitoring of patients and management of doctors). The WHSN is a three-tiered network, using UWB sensors in the first and second network levels and WiFi technology for the third tier (See Figure 1). The different network tiers are as follows:

1) First tier or the BSN (Body Sensor Network): a star network composed of one coordinator and a set of biosensors that ensure physiological measurements and the medical monitoring of patient. 2) Second tier or the PAN (Personal Area Network): a hexagonal cell of sensors organized in a full mesh topology including one PAN's coordinator, several mobile BSNs coordinators (one coordinator per BSN) and several routers. 3) Third tier or the Cellular Mesh WiFi Network: based on UWB/Wifi technologies, is chosen to the third level to have finally a three-tier hierarchical cellular network. The detailed description of the global network architecture is out of the scope of this paper, so for more details, one can refer to [START_REF] Ben Slimane | A Three-Tiered Architecture for Large-Scale Wireless Hospital Sensor Networks[END_REF]. We focus in this paper to the second tier or the PAN. So, we consider a set of PANs organized on hexagonal cells to facilitate the spectrum resource allocation. Each PAN is composed by one coordinator and a set of router and/or source nodes. A source node can act as a BSN coordinator or a simple node. To provide time guarantee to deliver data frames, beacon-enabled mode is used. The format of the superframe is defined by the PAN Coordinator. The superframe, corresponding to the Beacon Interval (BI), is defined by the time between two consecutive beacons, and includes an active period and, optionally, a following inactive period. The active period, corresponding to the Superframe Duration (SD), is divided into 16 equally sized time slots, during which data transmission is allowed. For a global network of N c PANs, each PAN coordinator is characterized by its superframe duration {P AN i = (SD i , BI i )} 1≤i≤Nc as shown in Figure 2. We define BI maj and SD min as respectively the major cycle (the least common multiple of all PANs BI) and the elementary active cycle (the least common denominator of all PANs SD).

IV. PMCMTP FOR IR UWB SENSOR NETWORKS

In [START_REF] Ben Slimane | Control and data channels allocation for Large-Scale UWB-based WSNs[END_REF], we have proposed a channel allocation scheme to statically assign one control channel to each PAN to support the control traffic and dynamically share the set of available data channels between active PANs to support data traffic. The detailed description of the proposed channels (Control and data channels) sharing between PANs is out of the scope of this paper, so for more details, one can refer to [START_REF] Ben Slimane | Control and data channels allocation for Large-Scale UWB-based WSNs[END_REF]. So, let us assume that each PAN permanently dispose a control channel to ensure control traffic and a variable set of data channels to ensure data traffic. In this section, we focus on the detail of the PMCMTP [START_REF] Kim | Y-MAC: An Energy-Efficient Multi-channel MAC Protocol for Dense Wireless Sensor Networks[END_REF] for logical channels and time slots allocation inside each active PAN. Similar to [START_REF] Pedreiras | FTT-Ethernet: A Flexible Real-Time Communication Protocol That Supports Dynamic QoS Management on Ethernet-Based Systems[END_REF], a key concept in PMCMTP is the elementary active cycle, which is composed by two consecutive active periods, the first for synchronization and collection of resource requests, and the second for the Request Scheduling Algorithm (RSA) process, the reception of the second beacon and the allowable data communications (See Figure 3).

Inside each PAN, according to the PMCMTP, the PAN coordinator collects all the resource allocation requests of its sources nodes. Then, according to the available spectrum resource, it tries to allocate available time slots per available channel of frequencies in response to the collected requests. Finally, concerned sensors can start their data communication. The principle of PMCMTP is based on the three following phases:

1) Synchronization and Request Transmission: Transmission requests phase must precede each PAN's data communication active period. This phase is divided into two substeps :

(i) the first step consists in PAN synchronization: by listening to the beacon frame, PAN members adjust their wake-up clocks, (ii) the second step consists in collecting all transmission requests from PAN's members. This step represents a set of equal short time slots, during which, the PAN coordinator is listening to the requests of PAN's member. So, just following the reception of the first beacon frame, each PAN's member waits for its own time slot in order to send its request packets. During this phase, the allocated control channel is used. 2) Channels/time slots allocation : According to the RSA algorithm, after reception of all transmission requests, the PAN coordinator tries to schedule them according to their assigned priority. Once the list of requests is scheduled, the PAN coordinator tries to launch the phase of time slots and channels allocation. For each request, it tries to find the earliest available time slots per channel to assign it to the suitable request. At the end of the process of time slots and channels allocation, the PAN coordinator registers a trace of the requests not served in its queue in order to analyze them during the next cycle. Then, it inserts into the next beacon frame the necessary information (Request identifier, index of allocated channel, index of the first allocated time slot, number of allocated time slots, address of the request's sender and of flow destination) of the served requests. 3) Data communication: After listening to the second beacon frame, PAN's members can have a feedback of their transmission requests. Each concerned sensor switches to the suitable channel at the suitable time slot and begins sending or receiving data frames during the reserved duration. Communication protocols and techniques are often tested and evaluated using simulation techniques. However, the use of realistic testbed setting for verification and evaluation in particular for WSNs becomes a necessity. In this part, we focus on the PMCMTP implementation in the nesC programming language for MICAz motes with TinyOS-2.x for the use in a real-world testbed setting to justify its operation validity in a real environment. The hardware used for the implementation of the PMCMTP protocol is the Crossbow MICAz motes, operating in the 2.4 GHz ISM band. The MIB510 [19] programming board was used to program the motes. The PMCMTP's implementation is composed of two components CoordiatorC and DeviceC respectively implemented at the coordinator and device motes and an interface P M CM T P .

A. Component CoordinatorC

The component CoordinatorC implements, for the coordinator mote, the MAC layer functions that will interact with upper and lower layers. This component ensures all tasks supported by the coordinator including the synchronization, the data sending, the reception of data and/or resource requests and all the treatment concerning the PMCMTP process. According to the Figure 5 illustrating the graph of the coordinator component, this component is implemented as a set of component modules written in nesC language. These components are connected to each other using adequate interfaces.

B. Component DeviceC

The component DeviceC implements, for the device motes, the MAC layer functions that will interact with upper and lower layers. This component ensures all tasks supported by a device including the synchronization with the coordinator, the data and/or resource requests sending and the reception of data. According to the Figure 6 illustrating the graph of the Device component, this component is wired to other components of TinyOS library via the adequate interfaces.

C. Interface PMCMTP

The interface PMCMTP defines required commands and hardware events to ensure the best execution of the RSA algorithm process. Given that the CoordinatorC uses and provides the interface PMCMTP it implements both hardware events and commands.

VI. VALIDATION OF PMCMTP OPERATION

A WSN testbed consists of a set of sensor nodes deployed in a controlled environment. It is designed to support experimental research in a real-world setting. It provides to researchers a way to test their protocols, algorithms, network issues and applications. For the testbed, a Sensor Network Analyzer SNA, consisting of a Jennic [20] mote, as hardware, and a laptop installing the SNA software is used. The SNA can be configured to scan a set of channel of frequencies by selecting the channels on which we want to perform the scan and its time duration. The functionality of channels scan, shows details of all devices detected using at least one of the selected channels over a specified period of time. For more details about devices and activity on a specific channel, we can select the channel on which we want to capture traffic. Two scenarios are driven to test the PMCMTP operation.

1) Test of the multi-channel access feasibility for a PAN composed by 5 motes: The scenario used to test the feasibility of PMCMTP operation is the following:

• The network is composed of one PAN coordinator and four devices as shown in Figure 7.

• The channel N • 26 is allocated for control traffic (e.g., network synchronization and resource requests sending and reception). -For the first time slot, Device (0x0001) sends, over channel N • 26, to the coordinator a resource request to communicate with Device (0x0002),

-For the third time slot, Device (0x0003) sends, over channel N • 26, to the coordinator a resource request to communicate with Device (0x0004). Experimental results

• Channels scan: We have launched a channel scan operation with a list of three channels (N • 11, N • 15 and N • 26) and for a duration of 1 sec per channel. According to Table I and Figure 8, we note that effectively: The channel N • 26 is used by the coordinator to send beacon frames and by devices (0x0001 and 0x0003) to send their resource requests (See Figure 8 the PAN coordinator sends the first beacon to synchronize its network, the source nodes according to their id, if they want, they can send a resource request to their coordinator to reserve resource (temporal and spectral resource according to demanded and available resource), just before the end of the resource transmission phase, the PAN coordinator launches the RSA process to allocate resource in response to collected resource requests, at the end of the resource transmission phase, the PAN coordinator sends the second beacon including the necessary information of served resource requests. According to the Table II, Figure 10, Figure 11 and Figure 12, we note that effectively:

VII. CONCLUSION

In this paper, we have presented the implementation details of the PMCMTP multi-channel multi-time slots MAC protocol in nesC/TinyOS environment for the Crossbow MICAz motes. Two scenarios with respectively one PAN of five motes and two PANs with five motes each one, were used to test, validate and demonstrate the effective operation of the proposed MAC protocol. This set of experiments shows the feasibility of the Multi-channel multi-time slots access mechanism, proving that it can be used in real environments. However, the evaluation of the performance of the PMCMTP, via a realistic testbed setting, remains very encouraged that we will tackle in next work.

Fig. 1 .

 1 Fig. 1. Network architecture

Fig. 2 .

 2 Fig. 2. Network configuration

Fig. 4 .

 4 Fig. 4. PMCMTP process during an elementary active cycle

Fig. 5 .

 5 Fig. 5. Coordinator -Component Graph

Fig. 6 .

 6 Fig. 6. Device -Component Graph

•

  The channels N • 11 and N • 15 are allocated for data communication traffic. • During the Request Transmission phase:

Fig. 7 .

 7 Fig. 7. Network organization for the first scenario

  (a)). The channel N • 11 is used by device (0x0001) to send data frames (Figure 8(b)). The channel N • 15 is used by device (0x0003) to send data frames (Figure 8(c)); • Packet timeline: By analyzing the scan of the control channel (i.e., channel N • 26) over time, we can verify the control part of the PMCMTP process. According to Figure 9, we note that, using the channel N • 26, periodically:

Fig. 8 .Fig. 9 .

 89 Fig. 8. Scan of channels N • 26 , 11 and 15

17 Fig. 10 .Fig. 11 .

 171011 Fig. 10. Scan of data channels N • 11 and 17

Fig. 12 .

 12 Fig. 12. Scan of control channels N • 23 and 26