This article reviews basic construction and cosmological implications of a power-counting renormalizable theory of gravitation recently proposed by Horava. We explain that (i) at low energy this theory does not exactly recover general relativity but instead mimic general relativity plus dark matter; that (ii) higher spatial curvature terms allow bouncing and cyclic universes as regular solutions; and that (iii) the anisotropic scaling with the dynamical critical exponent z=3 solves the horizon problem and leads to scale-invariant cosmological perturbations even without inflation. We also comment on issues related to an extra scalar degree of freedom called scalar graviton. In particular, for spherically-symmetric, static, vacuum configurations we prove non-perturbative continuity of the lambda->1+0 limit, where lambda is a parameter in the kinetic action and general relativity has the value lambda=1. We also derive the condition under which linear instability of the scalar graviton does not show up.
We consider the electrostatic field of a point charge coupled to Horava-Lifshitz gravity and find an exact solution describing the space with a surplus (or deficit) solid angle. Although, theoretically in general relativity, a surplus angle is hardly to be obtained in the presence of ordinary matter with positive energy distribution, it seems natural in Horava-Lifshitz gravity. We present the sudden disappearance and reappearance of a star image as an astrophysical effect of a surplus angle. We also consider matter configurations of all possible power law behaviors coupled to Horava-Lifshitz g…
We study linear cosmological perturbations in the ``healthy extension'' of Horava-Lifshitz gravity which has recently been analyzed \cite{BPS2}. We find that there are two degrees of freedom for scalar metric fluctuations, but that one of them decouples in the infrared limit. Also, for appropriate choices of the parameters defining the Lagrangian, the extra mode can be made well-behaved even in the ultraviolet.
The stability of de Sitter spacetime in Horava-Lifshitz theory of gravity with projectability but without detailed balance condition is studied. It is found that, in contrast to the case of the Minkowski background, the spin-0 graviton now is stable for any given $ξ$, and free of ghost for $ξ\le 0$ in the infrared limit, where $ξ$ is the dynamical coupling constant.
Using the multiplicity of solutions for the projectable case of the covariant extension of Horava-Lifshitz Gravity, we show that an appropriate choice for the auxiliary field allows for an effective description of galaxy rotation curves. This description is based on static and spherically symmetric solutions of covariant Horava-Lifshitz Gravity and does not require Dark Matter.
We formulate Horava-Lifshitz cosmology with an additional scalar field that leads to an effective dark energy sector. We find that, due to the inherited features from the gravitational background, Horava-Lifshitz dark energy naturally presents very interesting behaviors, possessing a varying equation-of-state parameter, exhibiting phantom behavior and allowing for a realization of the phantom divide crossing. In addition, Horava-Lifshitz dark energy guarantees for a bounce at small scale factors and it may trigger the turnaround at large scale factors, leading naturally to cyclic cosmology.
This paper is devoted to the study of various aspects of projectable F(R) Horava-Lifshitz (HL) gravity. We show that some versions of F(R) HL gravity may have stable de Sitter solution and instable flat space solution. In this case, the problem of scalar graviton does not appear because flat space is not vacuum state. Generalizing the U(1) HL theory proposed in arXiv:1007.2410, we formulate U(1) extension of scalar theory and of F(R) Horava-Lifshitz gravity. The Hamiltonian approach for such the theory is developed in full detail. It is demonstrated that its Hamiltonian structure is the same a…
Hořava-Lifshitz gravity is a potentially UV complete theory with important implications for the very early universe. In particular, in the presence of spatial curvature it is possible to obtain a non-singular bouncing cosmology. The bounce is realized as a consequence of higher order spatial curvature terms in the gravitational action. Here, we extend the study of linear cosmological perturbations in Hořava-Lifshitz gravity coupled to matter in the case when spatial curvature is present. As in the case without spatial curvature, we find that there is no extra dynamical degree of freedom for sc…
In the non-relativistic theory of gravitation recently proposed by Horava, the Hamiltonian constraint is not a local equation satisfied at each spatial point but an equation integrated over a whole space. The global Hamiltonian constraint is less restrictive than its local version, and allows a richer set of solutions than in general relativity. We show that a component which behaves like pressureless dust emerges as an "integration constant" of dynamical equations and momentum constraint equations. Consequently, classical solutions to the infrared limit of Horava-Lifshitz gravity can mimic ge…
In this paper we investigate the physical implications of the dynamical scalar mode in pure Horava-Lifshitz gravity on cosmology. We find that it can produce a scale-invariant power spectrum in UV era if the detailed balance condition on the action is relaxed. This indicates that the physical scalar mode may seed the large scale structure and Horava-Lifshitz cosmology could be a qualified alternative to inflationary scenario.
Horava-Lifshitz gravity, a recent proposal for a UV-complete renormalizable gravity theory, may lead to a bouncing cosmology. In this note we argue that Horava-Lifshitz cosmology may yield a concrete realization of the matter bounce scenario, and thus give rise to an alternative to inflation for producing a scale-invariant spectrum of cosmological perturbations. In this scenario, quantum vacuum fluctuations exit the Hubble radius in the pre-bounce phase and the spectrum is transformed into a scale-invariant one on super-Hubble scales before the bounce because the long wavelength modes undergo …
We show that, assuming the dispersion relation proposed recently by Horava in the context of quantum gravity, radiation energy density exhibits a peculiar dependence on the scale factor; the radiation energy density decreases proportional to a^{-6}. This simple scaling can have an impact on cosmology. As an example, we show that the resultant baryon asymmetry as well as the stochastic gravity waves can be enhanced. We also discuss current observational constraint on the dispersion relation.
We construct Horava-Lifshitz gravities that are invariant under anisotropic Weyl scaling. This construction is based on an extension of the group of symmetries of healthy extended Horava-Lifshitz gravity and RFDiff invariant Horava-Lifshitz gravity. We find their Hamiltonian formulation and determine their constraint structure.
We review the effective field theory of modified gravity in which the Lagrangian involves three dimensional geometric quantities appearing in the 3+1 decomposition of space-time. On the flat isotropic cosmological background we expand a general action up to second order in the perturbations of geometric scalars, by taking into account spatial derivatives higher than two. Our analysis covers a wide range of gravitational theories-- including Horndeski theory/its recent generalizations and the projectable/non-projectable versions of Hořava-Lifshitz gravity. We derive the equations of motion for …
We review some general aspects of Horava-Lifshitz cosmology. Formulating it in its basic version, we extract the cosmological equations and we use observational data in order to constrain the parameters of the theory. Through a phase-space analysis we extract the late-time stable solutions, and we show that eternal expansion, and bouncing and cyclic behavior can arise naturally. Concerning the effective dark energy sector we show that it can describe the phantom phase without the use of a phantom field. However, performing a detailed perturbation analysis, we see that Horava-Lifshitz gravity i…
In this paper we formulate RFDiff invariant f(R) Horava-Lifshitz gravity and we show that it is related to the ghost condensation in the projectable version of Horava-Lifshitz gravity.
We investigate the Hamiltonian structure of linearized extended Hořava- Lifshitz gravity in a flat cosmological background following the Faddeev-Jackiw's Hamiltonian reduction formalism. The Hamiltonian structure of extended Hořava-Lifshitz gravity is similar to that of the projectable version of original Hořava-Lifshitz gravity, in which there is one primary constraint and so there are two physical degrees of freedom. We also find that extra scalar graviton mode in an inflationary background can be decoupled from the matter field in the infrared (IR) limit, but it is coupled to the matter fie…
We perform the phase space analysis in terms of the linearization technique in the Hořava-Lifshitz gravity with the softly broken detailed balance condition. It can be shown that the bouncing universe appears only for the positive spatial curvature of $k=+1$, and it is possible to obtain oscillating universe with the help of the negative dark radiation and the negative cosmological constant.
We perform a detailed phase-space analysis of Horava-Lifshitz cosmology, with and without the detailed-balance condition. Under detailed-balance we find that the universe can reach a bouncing-oscillatory state at late times, in which dark-energy, behaving as a simple cosmological constant, is dominant. In the case where the detailed-balance condition is relaxed, we find that the universe reaches an eternally expanding, dark-energy-dominated solution, with the oscillatory state preserving also a small probability. Although this analysis indicates that Horava-Lifshitz cosmology can be compatible…
We perform a fully nonlinear analysis of superhorizon perturbation in Hořava-Lifshitz gravity, based on the gradient expansion method. We present a concrete expression for the solution of gravity equations up to the second order in the gradient expansion, and prove that the solution can be extended to any order. The result provides yet another example for analogue of the Vainshtein effect: the nonlinear solution is regular in the limit $λ\to 1$ and recovers general relativity coupled to dark matter at low energy. Finally, we propose a definition of nonlinear curvature perturbation ${\cal R}$ i…
We consider linear perturbations about a homogeneous and isotropic cosmological background in the projectable version of Hořava-Lifshitz gravity. Starting from the action for cosmological perturbations, we identify the canonically normalized fluctuation variables. We find that - in contrast to what happens in the non-projectable version of the theory - the extra scalar cosmological perturbation mode is already dynamical at the level of linear perturbations. For values of the parameter $λ$ in the range $1/3 < λ< 1$, the extra mode is ghost-like, for values $1 < λ$ and $λ< 1/3$ it is tachyonic. …
We study the spectral tilt of primordial perturbations in Horava-Lifshitz cosmology. The uniform approximation, which is a generalization of the familiar Wentzel-Kramers-Brillouin (WKB) method, is employed to compute the spectral index both numerically and analytically in a closed form. We clarify how the spectral index depends on the inflation model and parameters in the modified dispersion relation.
We study the quasinormal modes of the massless scalar perturbation in the background of a deformed black hole in the Hořava-Lifshitz gravity with coupling constant $λ=1$. Our results show that the quasinormal frequencies depend on the parameter in the Hořava-Lifshitz gravity and the behavior of the quasinormal modes is different from those in the Reissner-Norstr\"{om} and Einstein-Born-Infeld black hole spacetimes. The absolute value of imaginary parts is smaller and the scalar perturbations decay more slowly in the deformed Hořava-Lifshitz black hole spacetime. This information can help us un…
We consider RFDiff invariant Horava-Lifshitz gravity action with additional Lagrange multiplier term that is a function of scalar curvature. We find its Hamiltonian formulation and we show that the constraint structure implies the same number of physical degrees of freedom as in General Relativity.
We study the possibility of standard model Higgs boson acting as an inflaton field in the framework of Horava-Lifshitz Gravity. Under this framework, we showed that it is possible for the Higgs field to produce right amount of inflation and generate scale invariant power spectrum with the correct experimental value. Thanks to the foliation preserving diffeomorphism and anisotropic space-time scaling, it essentially helps us to construct this model without the pre-existing inconsistency coming from cosmological and particle physics constraints. We do not need to introduce any non-minimal or hig…
Using the brick wall method, we study statistical entropy for spherically symmetric black holes in Hořava-Lifshitz gravity. In particular, a Lifshitz scalar field is considered in order to incorporate foliation preserving diffeomorphism which eventually gives a modified dispersion relation. Finally, we obtain the area law without UV cutoff for $z > 3$, and discuss some of consequences in connection with the generalized uncertainty principle.
We argue that the true nature of the renormalizability of Horava-Lifshitz gravity lies in the presence of higher order spatial derivatives and not in the anisotropic Lifshitz scaling of space and time. We discuss the possibility of constructing a higher order spatial derivatives model that has the same renormalization properties of Horava-Lifshitz gravity but that does not make use of the Lifshitz scaling. In addition, the state-of-the-art of the Lorentz symmetry restoration in Horava-Lifshitz-type theories of gravitation is reviewed.
We study the dynamical evolution of a massless scalar perturbation in the Hořava-Lifshitz black-hole spacetimes with the coupling constants $λ={1/3}$, $λ={1/2}$ and $λ=3$, respectively. Our calculation shows that, for the three cases, the scalar perturbations decay without any oscillation in which the decay rate imprints the parameter of the Hořava-Lifshitz black hole. The results are quite different from those in the Schwarzschild AdS black hole and can help us understand more about the Hořava-Lifshitz gravity.
We investigate the linear cosmological perturbations of Hořava-Lifshitz gravity in a FRW universe without any matter. Our results show that a new gauge invariant dynamical scalar mode emerges, due to the gauge transformation under the "foliation-preserving" diffeomorphism and "projectability condition", and it can produce a scale invariant power spectrum. In the infrared regime with $λ=1$, the dynamical scalar degree of freedom turns to be a non-dynamical one at the linear order level.
We derive the projectable version of Horava - Lifshitz gravity from the localisation of the Galilean symmetry. Specifically we provide a dynamical construction of the metric, from first principles, that reproduces the transformations of the physical variables - lapse, shift and spatial component of the metric. Also, the measure defining the action is reproduced. The geometrical basis of the Horava-Lifshitz gravity is thereby revealed which also elucidates its difference from the Newton-Cartan geometry - the spacetime of Newtonian gravity. The connection of Newton's gravity with Horava-Lifshitz…
At the present work, it is studied the extension of F (R) gravities to the new recently proposed theory of gravity, the so-called Horava-Lifshitz gravity, which provides a way to make the theory power counting renormalizable by breaking Lorentz invariance. It is showed that dark energy can be well explained in the frame of this extension, just in terms of gravity. It is also explored the possibility to unify inflation and late-time acceleration under the same mechanism, providing a natural explanation the accelerated expansion.
Using the dynamical system approach, properties of cosmological models based on the Horava-Lifshitz gravity are systematically studied. In particular, the cosmological phase space of the Horava-Lifshitz model is characterized. The analysis allows to compare some key physical consequences of the imposition (or not) of detailed balance. A result of the investigation is that in the detailed balance case one of the attractors in the theory corresponds to an oscillatory behavior. Such oscillations can be associated to a bouncing universe, as previously described by Brandenberger, and will prevent a…
Horava-Lifshitz gravity with "detailed balance" but without the projectability assumption is discussed. It is shown that detailed balance is quite efficient in limiting the proliferation of couplings in Horava-Lifshitz gravity, and that its implementation without the projectability assumption leads to a theory with sensible dynamics. However, the (bare) cosmological constant is restricted to be large and negative.
We use observational data from Type Ia Supernovae (SNIa), Baryon Acoustic Oscillations (BAO), and Cosmic Microwave Background (CMB), along with requirements of Big Bang Nucleosynthesis (BBN), to constrain the cosmological scenarios governed by Horava-Lifshitz gravity. We consider both the detailed and non-detailed balance versions of the gravitational sector, and we include the matter and radiation sectors. We conclude that the detailed-balance scenario cannot be ruled out from the observational point of view, however the corresponding likelihood contours impose tight constraints on the involv…
At the present paper, it is studied cosmological solutions and its stability in the frame of F(R) Horava-Lifshitz gravity. The perturbations around general spatially flat FRW solutions are analyzed and it is showed that the stability of those solutions will depend on the kind of theory, i.e. on the form of the action F(R), as well as on the parameters contained in any Horava-Lifshitz theory due to the breaking of Lorentz invariance. The (in)stability of a given cosmic solution can restrict the models and gives new observational predictions, and can give a natural explanation on the end of infl…
We derive the detailed balance condition as a solution to the Hamilton-Jacobi equation in the Horava-Lifshitz gravity. This result leads us to propose the existence of the d-dimensional quantum field theory on the future boundary of the (d+1)-dimensional Horava-Lifshitz gravity from the viewpoint of the holographic renormalization group. We also obtain a Ricci flow equation of the boundary theory as the holographic RG flow, which is the Hamilton equation in the bulk gravity, by tuning parameters in the theory.
We discuss the cosmology of recently proposed Horava-Lifshitz f(R) gravity. In particular, we derive the modified Hubble equation that reduces to the standard HL gravity case in appropriate limit. We show how the bounce solutions in this theory are modified due to nonlinear effect of f(R) gravity. We also show that in principle, the Universe in this set-up can show cyclic behavior.
We show that the Kerr metric does not exist as a fully rotating black hole solution to the modified Hořava-Lifshitz (HL) gravity with $Λ_W=0$ and $λ=1$ case. We perform it by showing that the Kerr metric does not satisfy full equations derived from the modified HL gravity.
A proposal for a power-counting renormalizable theory of quantum gravity at a Lifshitz point was recently put forth by Horava (arXiv:0901.3775), and has been since dubbed as Horava-Lifshitz gravity. The theory explicitly breaks Lorentz invariance, which introduces an apparent extra scalar degree of freedom. In this note, we show that the low energy limit of (non-projectible) Horava-Lifshitz gravity is uniquely given by the quadratic cuscuton model: a covariant scalar field theory with an infinite speed of sound and a quadratic potential, which is minimally coupled to Einstein gravity. This imp…
We examine the phase space of Hořava-Lifshitz cosmology for a wide range of self-interacting potentials for the scalar field under the detailed-balance condition and without imposing it, by means of the powerful method of $f$-devisers. A compactification approach is performed for the exponential potential and for potentials beyond the exponential one, extending the previous findings in the literature. By using this approach it is possible to describe the finite region of the phase space and the region where the phase-space variables becomes infinity. Furthermore, we present several results con…
In this paper we continue the study of the Hamiltonian formalism of the healthy extended Horava-Lifshitz gravity. We find the constraint structure of given theory and argue that this is the theory with the second class constraints. Then we discuss physical consequence of this result. We also apply the Batalin-Tyutin formalism of the conversion of the system with the second class constraints to the system with the first class constraints to the case of the healthy extended Horava-Lifshitz theory. As a result we find new theory of gravity with structure that is different from the standard formul…
In this short note, we are going to classify the cosmological evolutions in the recently proposed, nonrelativistic gravitational theory, the Hořava-Lifshitz (HL) theory. We consider the original HL theory (theory I), and the modified version obtained by an analytic continuation of parameters (theory II). We discuss the possiblilities of cosmological evolutions with arbitrary matter.
We use the Wetterich equation for foliated spacetimes to study the RG flow of projectable Horava-Lifshitz gravity coupled to n Lifshitz scalars. Using novel results for anisotropic heat kernels, the matter-induced beta functions for the gravitational couplings are computed explicitly. The RG flow exhibits an UV attractive anisotropic Gaussian fixed point where Newton's constant vanishes and the extra scalar mode decouples. This fixed point ensures that the theory is asymptotically free in the large-n expansion, indicating that projectable Horava-Lifshitz gravity is perturbatively renormalizabl…
We quantize the two-dimensional projectable Horava-Lifshitz gravity with a bi-local as well as space-like wormhole interaction. The resulting quantum Hamiltonian coincides with the one obtained through summing over all genus in the string field theory for two-dimensional causal dynamical triangulations. This implies that our wormhole interaction can be interpreted as a splitting or joining interaction of one-dimensional strings.
We investigate the connection between Gravity's Rainbow and Horava-Lifshitz gravity, since both theories incorporate a modification in the UltraViolet regime which improves their quantum behavior at the cost of the Lorentz invariance loss. In particular, extracting the Wheeler-De Witt equations of the two theories in the case of Friedmann-Lemaitre-Robertson-Walker and spherically symmetric geometries, we establish a correspondence that bridges them.
We show that certain three-dimensional Horava-Lifshitz gravity theories can be written as Chern-Simons gauge theories on various non-relativistic algebras. The algebras are specific extensions of the Bargmann, Newton-Hooke and Schroedinger algebra each of which has the Galilean algebra as a subalgebra. To show this we employ the fact that Horava-Lifshitz gravity corresponds to dynamical Newton-Cartan geometry. In particular, the extended Bargmann (Newton-Hooke) Chern-Simons theory corresponds to projectable Horava-Lifshitz gravity with a local U(1) gauge symmetry without (with) a cosmological …
We find topological (charged) black holes whose horizon has an arbitrary constant scalar curvature $2k$ in Hořava-Lifshitz theory. Without loss of generality, one may take $k=1,0$ and -1. The black hole solution is asymptotically AdS with a nonstandard asymptotic behavior. Using the Hamiltonian approach, we define a finite mass associated with the solution. We discuss the thermodynamics of the topological black holes and find that the black hole entropy has a logarithmic term in addition to an area term. We find a duality in Hawking temperature between topological black holes in Hořava-Lifshit…
Approximately one year ago Horava proposed a power-counting renormalizable theory of gravity which abandons local Lorentz invariance. The proposal has been received with growing interest and resulted in various different versions of Horava-Lifshitz gravity theories, involving a colourful potpourri of new terminology. In this proceedings contribution we first motivate and briefly overview the various different approaches, clarifying their differences and similarities. We then focus on a model referred to as projectable Horava-Lifshitz gravity and summarize the key results regarding its viabilit…
We investigate the propagation and evolution for a massless scalar field in the background of $λ=1/2$ Hořava-Lifshitz black hole with the condition of detailed balance. We fortunately obtain an exact solution for the Klein-Gordon equation. Then, we find an analytical expression for the greybody factor which is valid for any frequency; and also exactly show that the perturbation decays without any oscillation. All of these can help us to understand more about the Hořava-Lifshitz gravity.
We investigate slowly rotating black holes in the Hořava-Lifshitz (HL) gravity. For $Λ_W=0$ and $λ=1$, we find a slowly rotating black hole of the Kehagias-Sfetsos solution in asymptotically flat spacetimes. We discuss their thermodynamic properties by computing mass, temperature, angular momentum, and angular velocity on the horizon.
Causal Dynamical Triangulations (CDT) is a lattice theory where aspects of quantum gravity can be studied. Two-dimensional CDT can be solved analytically and the continuum (quantum) Hamiltonian obtained. In this article we show that this continuum Hamiltonian is the one obtained by quantizing two-dimensional projectable Horava-Lifshitz gravity.
We look for a Brans-Dicke type of generalization of the Horava-Lifshitz gravity. It is shown that such a generalization is possible within the detailed balance condition. Classically, the resulting theory reduces in the IR limit to the usual Brans-Dicke theory with a negative cosmological constant for certain values of parameters. We then consider homogeneous and isotropic cosmology and study the effects of the new terms appearing in the model.
This short note is devoted to the canonical analysis of the Horava-Lifshitz gravity with mixed derivative terms that was proposed in arXiv:1604.04215. We determine the algebra of constraints and we show that there is one additional scalar degree of freedom with respect to the non-projectable Horava-Lifshitz gravity.
In this paper we show that by considering a slightly modified Horava-Lifshitz gravity one can find self-tuning cosmological scenarios. Since in the proposed model the cosmological constant cancels out, then regardless of its value it cannot produce any gravitational effect. As a consequence this removes the gravitational effect of the vacuum energy on the expansion of the Universe. Accelerating scenarios, however, can be found through the presence of conventional and non-conventional fluids.
We introduce conformal traceless decomposition in Lagrange Multiplier modified RFDiff invariant Horava-Lifshitz gravity. We perform Hamiltonian analysis of given action and determine the action for the physical degrees of freedom.
In this paper we focus on the Halmiton-Jacobi method to determine several thermodynamic quantities such as the temperature, entropy and specific heat of two-dimensional Horava-Lifshitz black holes by using the generalized uncertainty principles (GUP). We also address the product of horizons, mainly concerning the event, Cauchy, cosmological and virtual horizons.
The anti--de Sitter/conformal field theory (AdS/CFT) correspondence is a powerful tool that promises to provide new insights toward a full understanding of field theories under extreme conditions, including but not limited to quark-gluon plasma, Fermi liquid and superconductor. In many such applications, one typically models the field theory with asymptotically AdS black holes. These black holes are subjected to stringy effects that might render them unstable. Hořava-Lifshitz gravity, in which space and time undergo different transformations, has attracted attentions due to its power-counting …
The occurrence of a bounce in the FRW cosmology requires modifications of General Relativity. An example of such a modification is the recently proposed Horava-Lifshitz theory of gravity, which includes a "dark radiation" term with a negative coefficient in the analog of the Friedmann equation. A modification of the HL gravity, relaxing the "detailed balance" condition, brings additional terms to the equations of motion, corresponding to stiff matter. This paper presents comparison of the phase structure of the original and modified Horava cosmology. Special attention is paid to the analysis o…
We consider holographic superconductors related to the Schwarzschild black hole in the low energy limit of Hořava-Lifshitz spacetime. The non-relativistic electromagnetic and scalar fields are introduced to construct a holographic superconductor model in Hořava-Lifshitz gravity and the results show that the $α_2$ term plays an important role, modifying the conductivity curve line by means of an attenuation the conductivity.
