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 X-ray 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;
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 cut-off at the Planck (or at the electro-weak 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 so-called Cosmological Constant problem, also known as fine tuning problem.
Moreover, it is also well-known 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 time-varying 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 k-essence (Armendariz-Picon et al.,, 2000; Chiba et al.,, 2000; Armendariz-Picon 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 (Arkani-Hamed 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 Gauss-Bonnet 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 Dvali-Gabadadze-Porrati (DGP) model, the modified gravity model, the Dirac-Born-Infeld (DBI) model and the Brans-Dicke 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; Arkani-Hamed 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 well-known 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 cut-off, which is indicated with , is related to the IR cut-off, 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 cut-off 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 system-size black hole, i.e. we must have that:
which implies that:
where represents the reduced Planck mass and represents the Newton’s gravitational constant. If the largest possible cut-off 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:
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 entropy-area 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 Bekenstein-Hawking 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 self-gravitation 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 Bekenstein-Hawking 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 fine-tuning 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 cut-off. 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 low-redshift 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 cut-offs, 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 scalar-tensor gravity, , DGP model, braneworld and Brans-Dicke 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 entropy-area relation can be modified considering quantum effects which are motivated from the Loop Quantum Gravity (LQG). The relation entropy-area has an interesting modification (correction), i.e. the power-law correction (Das et al.,, 2008; Radicella & Pavon,, 2010) which arises in dealing with the entanglement of quantum fields in and out the horizon.
The power-law corrected entropy-area relation has the following specific form (Das et al.,, 2008; Radicella & Pavon,, 2010):
where the term is given by the following power-law relation:
with and indicating two constant parameters, being the UV cut-off 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 (Bekenstein-Hawking) entropy.
Another useful way to write the expression of the entropy area relation for the power-law corrected entropy is given by the following relation:
with representing a dimensionless constant parameter and the term is a constant which is defined as follows:
where the term indicates the cross-over 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 power-law correction to the entropy-area law. In order the entropy is a well-defined 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 Power-Law Entropy-Corrected HDE (PLECHDE) model) was recently introduced as follows:
with being a positive dimensionless parameter and begin a positive exponent.
In the limiting case of (or, equivalently, for ), Eq. (8) reduces to the well-known 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 early-time inflation and late-time 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 non-vanishing 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 Hořava-Lifshitz gravity. The Hořava-Lifshitz 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, Hořava-Lifshitz theory is not Lorentz invariant (with the exception of the IR limit), test particles do not follow geodesics, it is non-relativistic and we also have that the speed of light diverges in the UV limit. We have four different versions of Hořava-Lifshitz 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), Hořava-Lifshitz gravity behaves like General Relativity plus DM. For some relevant works on the scenario where the cosmological evolution is ruled by Hořava-Lifshitz 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 Hořava-Lifshitz gravity as a cosmological context produces the Hořava-Lifshitz 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). Hořava-Lifhsitz 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 Hořava-Lifshitz cosmology.
Jamil et al. (Jamil et al.,, 2010) studied the behavior of the Generalized Second Law of Thermodynamics (GSLT) in the context of Hořava-Lifshitz cosmology using the dynamical apparent horizon as infrared (IR) cut-off of the system.
Karami et al. (Karami et al.,, 2012) studied the Logarithmic Entropy Corrected New Agegraphic DE (LECNADE) model in the context of Hořava-Lifshitz cosmology.
Jamil et al. (Jamil & Saridakis,, 2010) considered the NADE model in the context of the Hořava-Lifshitz cosmology.
Karami et al. (Karami et al.,, 2012) studied the Power-Law Entropy Corrected NADE (PLECNADE) model in the framework of Hořava-Lifshitz cosmology.
Pasqua et al. (Pasqua et al., 2015) studied the Power Law and the Logarithmic Ricci Dark Energy Models in the framework of Hořava-Lifshitz Cosmology.
Jawad et al. (Jawad et al., 2014) studied the power-law solution of the new agegraphic modified Hořava-Lifshitz gravity.
Chattopadhyay and Pasqua (Chattopadhyay & Pasqua, 2014) studied the modified holographic Ricci DE (RDE) model in the framework of modified Hořava-Lifshitz gravity.
Jawad et al. (Jawad et al., 2013) obtained a holographic reconstruction of the modified Hořava-Lifshitz gravity with the scale factor given in the in power-law form.
Anyway, even if this extended research is available, a lot of ambiguities are still presents about the fact that Hořava-Lifshitz 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 cut-off, known as Granda-Oliveros cut-off, based on purely dimensional grounds which Granda Oliveros recently proposed. We must also underline here that DE models with Granda-Oliveros cut-off belong to generalized Nojiri-Odintsov HDEs classes (Nojiri & Odintsov,, 2006). It also differs from the work of Pasqua Chattopadhyay (Pasqua & Chattopadhyay,, 2013a) since the Authors considered the Logarithmic Entropy-Corrected Holographic Dark Energy (LECHDE) model in the framework of Hořava-Lifshitz cosmology with Granda-Oliveros cut-off while we are considering here the power law correction to the entropy. This new cut-off 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):
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 cut-off for a non-flat 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 cut-off for some specific reasons. If the IR cut-off chosen is given by the particle horizon, the HDE model cannot produce an accelerated expansion of the Universe (Hsu,, 2008). If we consider as cut-off 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:
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 cut-off we are studying.
This paper is organized in the following way. In Section 2, we describe the most important features of Hořava-Lifshitz cosmology. In Section 3, we study the PLECHDE model with Granda-Oliveros cut-off in the context of Hořava-Lifshitz 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.