PowerLaw EntropyCorrected Holographic Dark Energy in HořavaLifshitz Cosmology with GrandaOliveros Cutoff
Abstract
Abstract: In this paper, we study the Power Law Entropy Corrected Holographic Dark Energy (PLECHDE) model in the framework of a nonflat Universe and of HořavaLifshitz cosmology with infrared cutoff given by recently proposed GrandaOliveros cutoff, which contains one term proportional to the Hubble parameter squared and one term proportional to the first time derivative of the Hubble parameter . Moreover, this cutoff is characterized by two constant parameters, and . For the two cases corresponding to noninteracting and interacting DE and Dark Matter (DM), we derive the evolutionary form of the energy density of DE , the Equation of State (EoS) parameter of DE and the deceleration parameter . Using the parametrization of the EoS parameter , we obtain the expressions of the two parameters and . We also study the statefinder parameters , the snap and lerk cosmographic parameters and the squared speed of the sound . We also calculate the values of the quantities we study for different values of the running parameter and for different set of values of and .
Keywords: Dark Energy; HořavaLifshitz; GrandaOliveros cutoff.
1 Introduction
Recent cosmological and astrophysical data obtained with observations done thanks to the Supernova Cosmology Project, the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellites, the Sloan Digital Sky Survey (SDSS) and Xray experiments (Riess et al.,, 1998; Astier et al.,, 2006; de Bernardis et al.,, 2000; Bennett et al.,, 2003; Spergel et al.,, 2003; Ade et al.,, 2013; Tegmark et al.,, 2004; Allen et al.,, 2004; Abazajian et al.,, 2004; Seljak et al.,, 2005) give clear indications that the observable present day Universe is experiencing a phase of expansion with accelerated rate, which is practically the expansion with accelerated rate which the Universe undergoes, with the first happened during the inflationary period. The present day cosmic acceleration is one of the biggest challenges in the understanding of the standard models of gravity and particle physics.
Three main different classes of models have been suggested and well studied till now with the aim to give a proper explanation to the accelerated expansion of the present day observable Universe:

the Cosmological Constant (CC) model;

Dark Energy (DE) models;

Theories of Modified Gravity models.
The first and also the simplest candidate introduced with the aim to explain the present day observed accelerated expansion of the Universe is the Cosmological Constant , which can be considered as an extra term added to Einstein’s equations. One of the main features of the Cosmological Constant is that it has an Equation of State (EoS) parameter exactly equal to , i.e. . According to what we know thanks to the Quantum Field Theory (QFT), a cutoff at the Planck (or at the electroweak scale) leads to the production of a Cosmological Constant which is of the order of (or ), respectively, times bigger than the value we are able to observe. The fact that we still do not have a fundamental symmetry which is able to put the precise value of the Cosmological Constant to exactly zero (i.e., ) or, instead, to a very small value (i.e., ) produces the socalled Cosmological Constant problem, also known as fine tuning problem.
Moreover, it is also wellknown that the Cosmological Constat model is affected by another problem, which is the Cosmic Coincidence problem (Copeland et al.,, 2006). The Cosmic Coincidence problem states that the DM and the vacuum energy are almost equal at the present epoch of the Universe even if they had an independent evolution and they had an evolution starting from different mass scales. Many proposals have been suggested till now with the purpose and the hope to obtain an explanation to the Cosmic Coincidence problem (Leon & Saridakis,, 2010; Berger & Shojae,, 2006; Jimenez, & Maroto,, 2009; del Campo et al.,, 2009; Griest,, 2002; Zhang,, 2005; Jamil & Rahaman,, 2009; Jamil et al.,, 2010).
The second class of models which are suggested and widely studied with the aim to give a plausible explanation to the present day accelerated expansion of the Universe considers Dark Energy (DE) models.
The observational evidences of the cosmic accelerated expansion imply that, if, on cosmological scale, the theory of Einstein’s General Relativity (GR) is valid, we must have that the present day observable Universe must have as dominant component an unknown missing energy component which has some particular features, in particular: 1) its pressure must be sufficiently negative if it wants to be able to produce the rate of accelerated expansion of the Universe we are able to observe and 2) it must not be clustered on cosmological scales (i.e. on large scale length). The present day observed cosmic accelerated expansion of the Universe can be described, in relativistic cosmology, introducing a perfect fluid with energy density and pressure satisfying the following condition: (which implies that the pressure must be negative in order the condition is satisfied). This kind of fluid with a sufficient negative pressure in order to satisfy the condition is referred as Dark Energy (DE). The fact that the relation must be satisfied leads to the fact that the EoS parameter (defined as the ratio of the pressure and the energy density ) satisfies the following condition: . Instead, from an observational point of view, it is still a challenging task to constrain its exact value. The fundamental theory which can explain the microscopic physics of DE is still not known up to now, for this reason scientists continue to reconstruct and suggest different possible models which are mainly based on its macroscopic behavior.
Furthermore, recent cosmological experiments and observations have clearly indicated that the largest part of the total energy density of the present Universe is contained in the two Dark Sectors (Peiris,, 2003), i.e. DE and DM, which represent, respectively, the 68.3 and the 26.8 of the total energy density of the present day observable Universe. We also know that the Baryonic Matter (BM) we are able to observe with our scientific instruments contributes only for approximately the of the total energy density of the present day Universe. Moreover, we have that the contribution produced by the radiation term to the total cosmic energy density can be safely considered practically negligible, i.e we have that .
Other different candidates introduced and suggested for the DE problem are given by the dynamical DE scenarios with a time dependent EoS parameter , then not anymore constant. According to analysis of the available SNe Ia observational data, it has been derived that timevarying DE models lead to a better fit compared with a model with Cosmological Constant . There are two main different categories suggested for dynamical DE scenarios: (i) scalar fields models, which include kessence (ArmendarizPicon et al.,, 2000; Chiba et al.,, 2000; ArmendarizPicon et al.,, 1999), quintessence (Ratra & Peebles,, 1988; Wetterich,, 1988; Zlatev,, 1999), tachyon (Sen,, 2002; Padmanabhan,, 2002; Padmanabhan & Choudhury,, 2002), phantom (Caldwell,, 2002; Nojiri & Odintsov,, 2003a; Chimento & Lazkoz,, 2003; Boisseau et al.,, 2000), dilaton (ArkaniHamed et al.,, 2004a; Gaperini et al.,, 2002; Piazza & Tsujikawa,, 2004) and quintom field (Anisimov,, 2005; Elizalde, et al.,, 2004; Cai et al.,, 2007; Zhao & Zhang,, 2006), (ii) interacting DE models, which include for example Chaplygin gas (Kamenshchik et al.,, 2001; Bento et al.,, 2002; Setare,, 2007c) and Agegraphic DE (ADE) models (Wei & Cai,, 2008; Cai,, 2007).
The complete description of the DE features and nature must come from a consistent theory of Quantum Gravity (QG).
Unfortunately, we still do not have a complete and widely accepted theory of Quantum Gravity (QG) and then some approximations for this theory can be made: some examples are given by the Loop Quantum Gravity (LQG) and String Theory.
The third and last class of models proposed in order to give an explanation to the present day accelerated expansion of Universe involves extended theories of gravity, which correspond to a modification of the action of the gravitational fields. Some of the most famous and studied models of Modified Gravity are the modified gravity model (with being the torsion scalar), braneworld models, the modified gravity model (with being the GaussBonnet invariant defined as , being the Ricci curvature tensor and being the Riemann curvature tensor), the modified gravity model (with being the Ricci scalar curvature), the DvaliGabadadzePorrati (DGP) model, the modified gravity model, the DiracBornInfeld (DBI) model and the BransDicke model (Jawad et al.,, 2013c; Myrzakulov,, 2012; Alvarenga et al.,, 2013; Dvali et al.,, 2000; Capozziello,, 2002; Starobinsky,, 1980; Jawad et al.,, 2013b; Freese & Lewis,, 2002; ArkaniHamed et al.,, 2004b; Nojiri & Odintov,, 2003b; Abdalla & Odintsov,, 2005; Capozziello et al.,, 2006; Appleby & Battye,, 2007; Easson,, 2004; Aghmohammadi et al.,, 2010; Pasqua & Chattopadhyay,, 2013b; Bengochea & Ferraro,, 2009; Li et al.,, 2011; Karami & Abdolmaleki,, 2012; Bamba et al.,, 2011; Deffayet et al.,, 2002; Sahni & Shtanov,, 2003; Ovalle et al.,, 2013).
Using the holographic principle which was recently introduced by Fischler Susskind (Fischler & Susskind,, 1998; Susskind,, 1995; Hsu,, 2004; Huang & Li,, 2004), a model dubbed as Holographic DE (HDE) model has been recently proposed in the paper of Li (Li,, 2004). The HDE model is one of the most famous and studied candidate of DE (Myung & Seo,, 2009; ’t Hooft,, 2006; Lidsey, & Huston,, 2007; Kinney & Tzirakis,, 2008; Hořava & Minic,, 2000; Setare,, 2007b; Wang et al.,, 2005; Guberina et al.,, 2005; Gong,, 2004; Sheykhi, & Jamil, 2011; Sheykhi,, 2009; Elizalde et al.,, 2005; Zhang,, 2006; Setare & Saridakis,, 2009; Wang et al.,, 2008; Sheykhi,, 2010; Karami & Fehri,, 2010a; Saridakis,, 2008a, b).
It is wellknown that the holographic principle assumes a fundamental role in both black hole and string
theories. It was recently demonstrated in the work of Cohen et al. (Cohen et al.,, 1999) that, in the framework of the QFT, the UV cutoff, which is indicated with , is related to the IR cutoff, which is given by , due to the limitations produced by the formation of a black hole. If the vacuum energy density produced by the UV cutoff is given by the relation , then we have that the total energy density of a given size must be less or at least equal to the mass corresponding to the systemsize black hole, i.e. we must have that:
(1) 
which implies that:
(2) 
where represents the reduced Planck mass and represents the Newton’s gravitational constant. If the largest possible cutoff of the system is that one which is able to saturate the inequality given in Eq. (2), we derive the following expression for the energy density of the HDE model:
(3) 
where represents a dimensionless constant parameter. It has been obtained that, in the case of a Universe that is not flat (i.e. for a value of the curvature parameter which is different from zero) the value of such constant is given by while for a flat Universe (i.e. when the curvature parameter is equal to zero), we have that the value of is given by (Li et al.,, 2009).
The expression of the energy density of the HDE model can be also obtain using a different approach (Guberina et al.,, 2007). It must be here underlined that the black hole entropy has an important role in the derivation of the HDE energy density . In fact, we know that the derivation of the HDE energy density strongly depends on the entropyarea relation given, in Einstein’s gravity, by the relation (where gives the area of the black hole horizon). According to the laws of the thermodynamics of black holes (Bekenstein,, 1973; Hawking,, 1976), a maximum value of the entropy in a box with a dimension of (which is also referred as BekensteinHawking entropy bound), is given by the relation , which goes as the area of the box (given approximatively by the expression ) rather than the volume of the box (which is given by ). Moreover, for macroscopic systems having some selfgravitation effects which cannot be ignored, we have that the expression of the Bekenstein entropy bound (which is indicated with ) can be obtained multiplying the energy , given by the relation , and the linear size of the system. If we impose that the Bekenstein entropy bound must be smaller than the BekensteinHawking entropy (i.e., if we impose that , which implies that ), we obtain the same result obtained from energy bound arguments, i.e. we obtain that .
Using the holographic principle, Cohen et al. (Cohen et al.,, 1999) recently proposed that the vacuum energy density must be proportional to the Hubble parameter . In this particular model, both the finetuning and coincidence problems can be solved, but it is still not possible to give a reasonable explanation to the present day cosmic accelerated expansion of the Universe since the effective Equation of State (EoS) parameter for such vacuum energy is equal to zero, then it is different from what it is requested for the HDE model. In a recent paper, Li (Li,, 2004) suggested that the future event horizon of the Universe can be used as possible IR cutoff. This DE model not only has a reasonable value for the DE energy density but it also leads to an accelerated solution for the cosmological expansion.
Jamil et al. (Jamil et al.,, 2009b) studied the EoS parameter of the HDE model choosing a Newton’s gravitational constant which is not constant but it is time dependent, i.e. we have ; furthermore, they obtained that the EoS parameter can be significantly modified when the lowredshift limit is considered.
Chen et al. (Chen et al.,, 2009) studied the HDE model in order to obtain an inflationary epoch in the early evolutionary stages of our Universe. The HDE model was recently considered in other works with different IR cutoffs, for example the Hubble horizon, the particle horizon and the future event horizon (Sadjadi & Jamil,, 2011; Karami & Fehri,, 2010b; Jamil et al.,, 2009a, 2011; Jamil & Sheykhi,, 2011; Wang et al.,, 2006). Moreover, correspondences between some scalar field models and the HDE model have been recently proposed (Chattopadhyay & Debnath,, 2009; Karami et al.,, 2011), while in other works, the HDE model was accurately studied in different modified gravity theories, like for example scalartensor gravity, , DGP model, braneworld and BransDicke cosmology (Feng & Zhang,, 2009; Wei,, 2009; Bisabr,, 2009; Nozari & Rashidi,, 2010, 2009; Karami & Khaledian,, 2011; Setare,, 2007a; Setare & Jamil,, 2010c).
Different HDE models have also been constrained and tested by using different astronomical and cosmological observations (Eqnvist et al.,, 2005; Zhang,, 2009; Micheletti,, 2010; Kao et al.,, 2005; Feng et al.,, 2005; Shen at al.,, 2005) and also thanks to the anthropic principle (Huang & Li,, 2005). It is also known that the HDE model fits well cosmological data obtained using the data obtained from observations of SNeIa and CMB radiation anisotropies (Wang et al.,, 2007; Wu et al.,, 2008; Lu et al.,, 2010; Zhang,, 2010).
The definition of the entropyarea relation can be modified considering quantum effects which are motivated from the Loop Quantum Gravity (LQG). The relation entropyarea has an interesting modification (correction), i.e. the powerlaw correction (Das et al.,, 2008; Radicella & Pavon,, 2010) which arises in dealing with the entanglement of quantum fields in and out the horizon.
The powerlaw corrected entropyarea relation has the following specific form (Das et al.,, 2008; Radicella & Pavon,, 2010):
(4) 
where the term is given by the following powerlaw relation:
(5) 
with and indicating two constant parameters, being the UV cutoff at the horizon and being a fractional power which depends on the amount of mixing of ground and excited states. For a large horizon area (i.e. for ), the contribution given by the term to the entropy can be considered practically negligible and, therefore, the mixed state entanglement entropy asymptotically approaches the ground state (BekensteinHawking) entropy.
Another useful way to write the expression of the entropy area relation for the powerlaw corrected entropy is given by the following relation:
(6) 
with representing a dimensionless constant parameter and the term is a constant which is defined as follows:
(7) 
where the term indicates the crossover scale. Moreover, we have that the quantity gives the area of the horizon (with the term indicating the radius of the horizon). The second term present in Eq. (6) gives the powerlaw correction to the entropyarea law. In order the entropy is a welldefined quantity, we need to have that the parameter is positive defined, i.e. we must have that the condition must be satisfied. Motivated by the relation defined in Eq. (6), a new version of HDE (also known with the name of PowerLaw EntropyCorrected HDE (PLECHDE) model) was recently introduced as follows:
(8) 
with being a positive dimensionless parameter and begin a positive exponent.
In the limiting case of (or, equivalently, for ), Eq. (8) reduces to the wellknown expression of the HDE energy density. The correction term present in Eq. (8) is of the same order of to the first one only when assumes a very small value. Then, at the very early evolutionary phases and stages of our Universe history (i.e., when the Universe underwent the inflationary phase), the contribution of the correction term in the PLECHDE energy density can be safely considered relevant but, when the Universe became larger, the PLECHDE energy density reduced to the ordinary HDE energy density. Therefore, PLECHDE model can be also considered as a model of entropic cosmology which is able to unify the earlytime inflation and latetime cosmic acceleration of the Universe.
In some recent works, Hořava (Hořava,, 2009b, 2010, c) recently introduced a new theory of gravity which is renormalizable with higher spatial derivatives in four dimensions. This theory leads to the Einstein’s gravity (i.e.
to GR) with nonvanishing value of the Cosmological Constant in the infrared (IR) limit and it also have some improved behaviors and features in the ultraviolet (UV) regime. Hořava gravity can be also considered similar to a scalar field theory previously proposed by Lifshitz (Lifshitz,, 1949) in which we have that the temporal dimension has a weight equal to three if the space dimension has a weight of one. For this reason, the gravity theory proposed by Hořava is also known with the name of HořavaLifshitz gravity. The HořavaLifshitz gravity has been extended and studied in detail in some papers, like for example (Cai et al.,, 2009; Germani et al.,, 2009; Klusoň,, 2009; Afshordi,, 2009; Bogdanos,, 2010; Myung,, 2009; Alexandre et al.,, 2004; Blas et al.,, 2010), and it has been applied as a possible cosmological framework of our Universe (Kiritsis & Kofinas,, 2009; Wang & Wu,, 2009; Sotiriou et al.,, 2009; Carloni et al.,, 2010; Bakas et al.,, 2010; Mukohyama et al.,, 2009; Cai & Zhang,, 2009; Gao et al.,, 2010; Kim et al.,, 2009; Dutta & Saridakis,, 2010; Greenwald et al.,, 2010; Kiritsis & Kofinas,, 2010; Lü et al.,, 2009; Cai et al.,, 2009).
Furthermore, HořavaLifshitz theory is not Lorentz invariant (with the exception of the IR limit), test particles do not follow geodesics, it is nonrelativistic and we also have that the speed of light diverges in the UV limit. We have four different versions of HořavaLifshitz theory of gravity:
(i) with projectability condition, (ii) without projectability condition, (iii) with detailed balance and (iv) without detailed balance. Having a first look, it seems that this model of Quantum Gravity (QG) has a well defined IR limit and it also reduces to General Relativity, but as it was obtained by Mukohyama (Mukohyama,, 2010, 2009), HořavaLifshitz gravity behaves like General Relativity plus DM. For some relevant works on the scenario where the cosmological evolution is ruled by HořavaLifshitz gravity see (Neupane,, 2009; Saridakis,, 2010; Majhi & Samanta,, 2010; Wang,, 2011).
Because of these characteristics, a great effort in extending, examining and improving the physical features and properties of the theory itself have been done (Volovik,, 2009; Nishioka,, 2009; Orlando & Reffert,, 2009; Visser,, 2009). Furthermore, applications of HořavaLifshitz gravity as a cosmological context produces the HořavaLifshitz cosmology, which has some interesting features. For example, it is possible to examine the perturbation spectrum (Wang & Maartens,, 2010), some particular solution subclasses (Minamitsuji,, 2010), the matter bounce (Brandenberger,, 2009), the production of gravitational waves (Takahashi,, 2009), the phenomenology of DE (Appignani,, 2010), the properties of black hole (Danielsoon,, 2009; Mann,, 2009; Bertoldi et al.,, 2009) and the astrophysical phenomenology (Lin et al.,, 2012). HořavaLifhsitz cosmology has been recently investigated raking into account and choosing different infrared cut (IR)offs and different approaches.
Setare Jamil (Setare & Jamil,, 2010a) considered the HDE model with a varying Newton’s gravitational constant in the framework of HořavaLifshitz cosmology.
Jamil et al. (Jamil et al.,, 2010) studied the behavior of the Generalized Second Law of Thermodynamics (GSLT) in the context of HořavaLifshitz cosmology using the dynamical apparent horizon as infrared (IR) cutoff of the system.
Karami et al. (Karami et al.,, 2012) studied the Logarithmic Entropy Corrected New Agegraphic DE (LECNADE) model in the context of HořavaLifshitz cosmology.
Jamil et al. (Jamil & Saridakis,, 2010) considered the NADE model in the context of the HořavaLifshitz cosmology.
Karami et al. (Karami et al.,, 2012) studied the PowerLaw Entropy Corrected NADE (PLECNADE) model in the framework of HořavaLifshitz cosmology.
Pasqua et al. (Pasqua et al., 2015) studied the Power Law and the Logarithmic Ricci Dark Energy Models in the framework of HořavaLifshitz Cosmology.
Jawad et al. (Jawad et al., 2014) studied the powerlaw solution of the new agegraphic modified HořavaLifshitz gravity.
Chattopadhyay and Pasqua (Chattopadhyay & Pasqua, 2014) studied the modified holographic Ricci DE (RDE) model in the framework of modified HořavaLifshitz gravity.
Jawad et al. (Jawad et al., 2013) obtained a holographic reconstruction of the modified HořavaLifshitz gravity with the scale factor given in the in powerlaw form.
Anyway, even if this extended research is available, a lot of ambiguities are still presents about the fact that HořavaLifshitz gravity can be considered a reliable theory and it is able to accurately describe the cosmological behavior of our Universe.
This work differs from the ones cited above and other available in literature since we are considering an IR cutoff, known as GrandaOliveros cutoff, based on purely dimensional grounds which Granda Oliveros recently proposed. We must also underline here that DE models with GrandaOliveros cutoff belong to generalized NojiriOdintsov HDEs classes (Nojiri & Odintsov,, 2006). It also differs from the work of Pasqua Chattopadhyay (Pasqua & Chattopadhyay,, 2013a) since the Authors considered the Logarithmic EntropyCorrected Holographic Dark Energy (LECHDE) model in the framework of HořavaLifshitz cosmology with GrandaOliveros cutoff while we are considering here the power law correction to the entropy. This new cutoff contains a term proportional to the time derivative of the Hubble parameter and one term proportional to the squared Hubble parameter and it is indicated with . The final expression of is given by (Granda & Oliveros,, 2009, 2008):
(9) 
and indicate two constant dimensionless parameters. In the limiting case corresponding to and , we obtain that the expression of defined in Eq. (9) becomes proportional to the average radius of the Ricci scalar curvature when the curvature parameter assumes the values of zero. In a recent paper, Wang Xu (Wang & Xu,, 2010) have constrained the HDE model with GO cutoff for a nonflat Universe using observational data. The best fit values of the pair with their confidence level they found are given by and for non flat Universe (i.e. for ), while for a flat Universe (i.e. for ) they found that are and .
We decided to consider the GO scale defined in Eq. (9) as IR cutoff for some specific reasons. If the IR cutoff chosen is given by the particle horizon, the HDE model cannot produce an accelerated expansion of the Universe (Hsu,, 2008). If we consider as cutoff of the system the future event horizon, the HDE model has a causality problem. The DE models which consider the GO scale depend only on local quantities, then it is possible to avoid the causality problem, moreover it is also possible to obtain the accelerated phase of the Universe.
Replacing with in the expression of the energy density of DE given in Eq. (8), we get the energy density of the PLECHDE model as follows:
(10) 
In the following Sections we will study the main properties and features of the cosmological parameters obtained using the energy density given in Eq. (10) of the PELCHDE model with GO cutoff we are studying.
This paper is organized in the following way. In Section 2, we describe the most important features of HořavaLifshitz cosmology. In Section 3, we study the PLECHDE model with GrandaOliveros cutoff in the context of HořavaLifshitz cosmology. Moreover, we derive the evolutionary form of the energy density of DE, the Equation of State (EoS) parameter, the evolutionary form of the fractional energy density and the deceleration parameter for both non interacting and interacting Dark Sectors. In Section 4, we study the low redshift limit of the EoS parameter, which is parametrized as , obtaining the expressions of and for both cases corresponding to non interacting and interacting Dark Sectors. In Section 5, we study the statefinder pair for the model we are studying. In Section 6, we derive and study the expressions of the snap and of the lerk cosmographic parameters for the model taken into account in this paper. In Section 7, we study the squared speed of the sound for the model considered in order to check its stability. Finally, in Section 8, we write the Conclusions of this work.
2 HořavaLifshitz Gravity
In this Section, we introduce the main physical and cosmological characteristics of HořavaLifshitz gravity. These information will be useful in order to obtain the cosmological information we want to derive for the model we are studying.
Considering the projectability condition, we have that the metric in the (3+1)dimensional ArnowittDeserMisner formalism can be written as follows (Arnowitt,, 2004):
(11) 
where indicates the cosmic time while the dynamical variables , and indicate, respectively, the 3dimensional metric tensor, the lapse function and the shift vector. The projectability condition leads to the fact that the lapse function is spaceindependent, instead the 3dimensional metric and the shift vector still depend on both space and time. Moreover, we have that the indices are raised and lowered thanks to the metric tensor . The scaling transformations of the coordinates and are given by the following relations:
(12)  
(13) 
where the quantities , , and represent, respectively, the temporal coordinate, the dynamical critical exponent, the scaling factor and the spatial coordinates.
In this paper, we have that , then the scaling transformation of the temporal coordinate defined in Eq. (13) can be rewritten as follows:
(14) 
The gravitational action of HořavaLifshitz cosmology, indicated with , can be decomposed into two different parts, i.e. a kinetic part, given by , and a potential part, given by , and it is given by the following relation:
(15) 
with indicating the determinant of the metric tensor .
The assumption of detailed balance (Hořava,, 2009a) allows to reduce the number of possible terms in
the expression of the gravitational action . Moreover, it also permit a quantum inheritance principle, because the dimensional
theory takes the renormalization properties of the dimensional theory. Considering the detailed balance condition, the gravitational action of the HořavaLifshitz gravity is given by the following expression (Hořava,, 2009a):
(16)  
where the terms and represent, respectively, the Cotton tensor and the extrinsic curvature which are defined in the following way:
(17)  
(18) 
Furthermore, the quantity represents a positive dimensionless constant which is related to the cosmological constant in the infrared (IR) limit, the quantity indicates the totally antisymmetric unit tensor and represents a dimensionless constant (also known as running parameter). More information about the running parameter will be given later on.
The three parameters , and represents three constants which has mass dimension, respectively, of 1, 1 and 0.
If we want to include the matter component in a Universe ruled by HořavaLifshitz gravity, we have that there are two options which can be taken into account. In the first option, we include a scalar field with action which is given by the following relation (Calcagni,, 2009):
(19)  
where the three quantities , and represent three constant parameters while the term indicated with represents the potential term. Furthermore, we have that the equation of motion for the field can be written as follows:
(20) 
with the condition , i.e. in order to avoid singularities. Moreover, an overdot indicates a derivative with respect to the cosmic time .
The second option we have in order to insert the matter component is obtained taking into account a hydrodynamical approximation adding a cosmological stressenergy tensor to the gravitational field equations; we must also consider the condition that the formalism of the General Relativity must be obtained when the lowenergy limit is considered (Chaichian et al.,, 2010). In this case, the energy density and the pressure of DM satisfy the following continuity equation:
(21) 
In this paper, we have decided to consider the hydrodynamical approximation.
Eq. (16), as it is wellknown, has several problems, like strong coupling problems, instability and inconsistency (Mukohyama,, 2010). It is possible overcome these problems invoking the Vainshtein mechanism, as it was already done in the paper of Mukohyama in the case of spherical spacetimes (Mukohyama,, 2010) and in the paper of Wang Wu in the cosmological setting (Wang & Wu,, 2011). These considerations were also carried out by considering the gradient expansion method (Izumi & Mukohyama,, 2011). Another possible approach which can be taken into account is given by the introduction of an extra symmetry: this kind of approach was considered for the first time by Hořava MelbyThompson (Hořava & MelbyThompson,, 2010) in the limiting case corresponding to , and subsequently generalized to the case with any possible value of in the paper of da Silva (da Silva,, 2011). These works were also extended to the case with absence of the projectability condition (Zhu et al.,, 2011). In both cases, i.e. with and without the projectability condition, the spin0 gravitons are eliminated because of the symmetry, therefore all the problems related to them are then solved.
In the cosmological framework, we consider a FLRW metric which is recovered for the following values of , and :
(22)  
(23)  
(24) 
where is given be the following relation:
(25) 
with the term representing the angular part of the metric.
Taking the variation of the action obtained in Eq. (16) with respect to the metric components and , we obtain the modified Friedmann equations in the framework of HořavaLifshitz cosmology as follows:
(26)  
(27)  
In the limiting case of a flat Universe, i.e. for , the higher order derivative terms do not produce contributions to the action. Instead, for a non flat universe, i.e. for , the higher derivative terms give a relevant contribution for small volumes, i.e. for small values of ,
while this contribution becomes practically negligible when assumes large values (in this case we recover a good agreement with the results of General Relativity).
Considering the Friedmann equations given in Eqs. (26) and (27), we define the energy density and the pressure of DE as follows:
(28)  
(29) 
The first term of the right hand side of both Eqs. (28) and (29) (which scales as ) indicates effectively the dark radiation term which is present in HořavaLifshitz cosmology, instead the second term (which is constant) has a cosmological constant termlike behavior. Furthermore, Eqs. (28) and (29) obey the following continuity equation:
(30) 
We also have that Eqs. (26) and (27) lead to the standard Friedmann equations if we take into account the following considerations:
(31)  
(32) 
The term indicates the Newton’s cosmological constant. We must underline that, in gravitational theories which lead to a violation of the Lorentz invariance (which happens in theories like HořavaLifshitz gravity), the Newton’s gravitational constant (which is one of the terms present in the gravitational action) is different from the Newton’s cosmological constant (which is one of the terms present in Friedmann equations). We have that and are equal if Lorentz invariance is recovered.
For completeness, we give the definition of , which can we written as follows:
(33) 
as we easily derive using the results of Eq. (16). Moreover, we observe that, in the IR limit (corresponding to the limiting case of ), which also implies that the Lorentz invariance is restored, and assume an equivalent form. Then, we can also state that the running parameter gives information about
possibility of breaking the Lorentz invariance. In fact, a value of indicates validity of
Lorentz invariance, while indicates that Lorentz invariance has been broken.
In a recent work of Dutta Saridakis (Dutta & Saridakis,, 2013), authors concluded that with confidence level while its best fit value is .
Moreover, using Eqs. (28), (29), (31) and (32), it is possible to rewrite the modified Friedmann equations given in Eqs. (26) and (27) in the usual forms as follows:
(34)  
(35) 
In the following Section, we will derive some important cosmological quantities for the model considered, in particular the Equation of State (EoS) parameter , the evolutionary form of the energy density of DE and the deceleration parameter .
3 PLECHDE Model with GO cutoff IN HořavaLifshitz Cosmology
We now discuss the main features and properties of the PLECHDE model with GrandaOliveros cutoff in the context of HořavaLifhsitz cosmology. We must underline that we consider a spatially nonflat FLRW Universe which is filled by both Dark Sectors, i.e. DE and DM.
We start remembering that the DE energy density with GO cutoff can be written as follows:
(36) 
Since we have that the Planck mass can be expressed as , we can rewrite the expression of the energy density defined in Eq. (36) as follows:
(37) 
We now introduce the expressions of the dimensionless fractional energy densities for DM, DE and also for the curvature parameter which are defined, respectively, in the following way:
(38)  
(39)  
(40) 
where indicates the critical energy density (i.e. the energy density necessary to obtain the flatness of the Universe) which is defined as follows:
(41) 
and the term is given by the following relation:
(42) 
Using the expressions of the fractional energy densities defined in Eqs. (38), (39) and (40), the first Friedmann equation defined in Eq. (34) can be written in an equivalent way as follows:
(43) 
Eq. (43) has the property that it relates all the fractional energy densities considered in this work.
1 Non Interacting Case
We start considering the case corresponding to a FLRW Universe containing DE and pressureless DM (i.e., we have ) and in absence of interaction between DE and DM. Moreover, we consider that the Dark Sectors evolve according to conservation laws which expressions are given by the following continuity equations:
(44)  
(45) 
which are equivalent to the expressions:
(46)  
(47) 
We must also underline that the prime and the overdot are related by the following relation:
(48) 
where we have that the parameter is defined as .
As described in the Introduction, we decided to choose as IR cutoff the GO scale , which has been previously defined in Eq. (9).
The first time derivative of the expression of given in Eq. (9) is given by the following expression:
(49) 
Instead, the first derivative with respect to the cosmic time of the energy density of DE given in Eq. (8) is given by:
(50) 
where we have used the result of Eq. (49).
Differentiating the Friedmann equation given in Eq. (34) with respect to the cosmic time and using Eq. (50), we obtain that the term can be rewritten as follows:
(51) 
where the dimensionless parameter is defined as follows:
(52) 
In Eq. (52), we used the result of Eq. (43) in order to find a result for .
Using the definition of given in Eq. (9) and the expression of the fractional energy density of DE given in Eq. (35), after some algebraic calculations, we obtain that the term can be written as follows:
(53) 
Inserting the result of Eq. (53) in Eq. (51) and later on Eq. (51) into Eq. (50), we obtain the following expression for :
(54) 
which leads to the following expression of the evolutionary form of the DE energy density :
(55) 
Using the definition of the fractional energy density of DE given in Eq. (39), we can rewrite the result of Eq. (55) as follows:
(56) 
From the continuity equations for DE given in Eqs. (44) and (46), we can derive the following expression for the EoS parameter of DE :
(57)  
Using in Eq. (57) the expression of obtained in Eq. (54) or equivalently the expression of obtained in Eq. (55), we can write as follows:
(58) 
We now derive an expression for the evolutionary form of the fractional energy density of DE .
Differentiating the expression of given in Eq. (39) with respect to the variable , we obtain the following expression:
(59) 
Using in Eq. (59) the expressions of and given, respectively, in Eqs. (53) and (54), we obtain the evolutionary form of the fractional energy density of DE as follows:
(60) 
2 Interacting Case
We now extend the calculations we have made in the previous subsection obtaining the same cosmological quantities but in the case of presence of a kind of interaction between the two Dark Sectors, i.e. DM and DE.
The presence of an interaction between the two Dark Sectors implies that the energy conservation laws are not held separately, therefore we have the following expressions:
(61)  
(62) 
which are equivalent to the following relations:
(63)  
(64) 
The quantity in the continuity equations given in Eqs. (61), (62), (63) and (64) indicates the interaction term which is in general a function of other cosmological parameters, like for example energy densities of DE and DM and , the Hubble parameter and the deceleration parameter . Many candidates have been suggested in order to describe the behavior of the interaction term , we decided to consider in this paper one interaction term which is proportional to the energy density of DE and to the Hubble parameter as follows (Perivolaropoulos,, 2005):
(65) 
where the quantity indicates a coupling parameter (also known with the name of transfer strength) between the Dark Sectors (Amendola & TocchiniValentini,, 2001; Setare & Jamil,, 2010b; Farooq et al.,, 2010; Zimdhal,, 2001; Jamil & Farooq,, 2010b). The limiting case corresponding to leads to the noninteracting FLRW Universe, which has been studied in the previous subsection.
The presence of interaction between DE and DM can be detected during the formation of the Large Scale Structures (LSS). It is suggested that the dynamical equilibrium of some collapsed structures like for example the clusters of galaxies (the cluster Abell A586 is one good example) results to be modified because of the coupling between the two Dark Sectors (Abdalla et al.,, 2009; Bertolami et al.,, 2009, 2007). The main concept is that the virial theorem is affected by the exchange of energy between the two Dark Sectors which leads to a bias in the estimation made for the virial masses of clusters when the usual virial conditions are taken into account. This fact gives a probe in the near Universe of the coupling of the Dark components. Some other observational signatures on the dark sectors mutual interaction can be also found in the probes of the cosmic expansion history by using the results we know about Supernovae Ia (SNe Ia), Baryonic Acoustic Oscillations (BAO) and CMBR shift data (Guo,, 2007; He et al.,, 2009). Thanks to observational data of Gold SNe Ia samples, CMBR anisotropies and the Baryonic Acoustic Oscillations (BAO), it was possible to estimate that the coupling parameter between DM and DE must assume a small positive value. This condition satisfies the requirement for solving the cosmic coincidence problem and also constraints given by the second law of thermodynamics (Feng et al.,, 2008).
Observations of the CMBR and of clusters of galaxies indicate a value of the coupling parameter which is . (Ichiki et al.,, 2008). Negative values of are not taken into account since they would lead to the violation of laws of thermodynamics. We also need to emphasize that other interaction terms can be also taken into account (Jamil & Rashid,, 2008).
Following the same procedure made for the non interacting case, we obtain the following expression for :
(66) 
which leads to the following evolutionary form of the DE energy density:
(67) 
Using in Eq. (67) the expression of we have considered in Eq. (65), we can write as follows:
(68) 
Using the definition of fractional energy density of DE given in Eq. (39), we can rewrite Eq. (68) as follows:
(69) 
From the continuity equations for DE given in Eqs. (61) and (63), we can derive the following expression for the EoS parameter :
(70)  
Using in Eq. (70) the expression of obtained in Eq. (66) or equivalently the expression of obtained in Eq. (67) along with the definition of given in Eq. (65), we can write as follows:
(71) 
which is the same result of the non interacting case.
We now want to find the expression for for the interacting case.
Using the general expression of given in Eq. (59) along with the expressions of and given, respectively, in Eqs. (53) and (67), we obtain the evolutionary form of the fractional energy density of DE as follows:
(72) 
For completeness, we here obtain also the expression of the deceleration parameter , which is generally defined as follows:
(73)  
The deceleration parameter can be used in order to quantify the status of the acceleration of the Universe (Da̧browski,, 2005). In particular, a negative value of the present day value of indicates an accelerating Universe, whereas a positive value of the present day value of indicates a Universe which is either decelerating or expanding at the coasting (Alam et al.,, 2003). In order to have a negative value of the deceleration parameter, we must have .
Using the expression of given in Eq. (53), we can write the deceleration parameter as follows:
(74)  
We can easily derive that the expression of given in Eq. (71) and the expression of given in Eq. (74) are related through the following relation:
(75) 
We must also underline that, in the limiting case of (i.e. in absence of interaction), we recover the same results of the non interacting case obtained in the previous subsection.
4 Low Redshift Expansion
In the previous Section, we have derived the general expression of the EoS parameter of DE as function of the other cosmological parameters.
We now consider a particular parametrization of the EoS parameter of DE as function of the redshift , which is given by , and we will calculate the expressions of the two parameters and for both non interacting and interacting Dark Sectors.
1 Non Interacting Case
We start studying the case corresponding to absence of interaction between the two Dark Sectors.
As previously stated, the EoS parameter of DE written in a parameterized way as function of the redshift is given by the following relation (Huterer & Turner,, 1999):
(76) 
therefore we obtain an expression of the EoS parameter of DE which is function of the redshift too while the dependence on the other cosmological parameters will appear in the final expressions of the two parameters and .
We must also remember here that the relation between the scale factor and the redshift is given by the following expression:
(77) 
which leads to the following expression for the redshift :
(78) 
Using the continuity equation for DE obtained in Eq. (44) in the definition of given in (76), we derive that the energy density of DE evolves according to the following relation (Weller & Albrecht,, 2001):
(79) 
where indicates the present day value of the energy density of DE .
The Taylor expansion of the energy density of DE around the point yields: