Cosmology of Holographic and New Agegraphic Models
We consider the theory, where is the scalar curvature and is the trace of energy-momentum tensor, as an effective description for the holographic and new agegraphic dark energy and reconstruct the corresponding functions. In this study, we concentrate on two particular models of gravity namely, and . We conclude that the derived models can represent phantom or quintessence regimes of the universe which are compatible with the current observational data. In addition, the conditions to preserve the generalized second law of thermodynamics are established.
Keywords: Modified Gravity; Dark Energy; Thermodynamics.
PACS: 04.50.Kd; 95.36.+x; 97.60.Lf.
Supernovae type Ia (SNeIa) observations revealed the expanding behavior of the universe. This fact has further been affirmed by the observations of anisotropies in cosmic microwave background (CMB), large scale structure, baryon acoustic oscillations and weak lensing. A strange type of energy component with prominent negative pressure identified as dark energy (DE) is used to explain the current cosmic acceleration. The source and characteristics of DE are still a complicated story as several models have been suggested in the context of general relativity (GR) (for review see).
The most likely campaigner of DE is the cosmological constant or the vacuum energy whose equation of state (EoS) parameter is fixed, . The cosmological model that consists of cosmological constant plus cold dark matter is entitled as model, which appears to fit the observational data. However, despite of its success, this model experiences two notable cosmological problems namely, the “fine tuning” problem and the “cosmic coincidence” problem. Such issue primarily originates because the vacuum energy is counted in the setting of quantum field theory in Minkowski background. Nevertheless, it is considerably accepted that at cosmological measures where the quantum effects of gravity may be reported, the preceding sketch of vacuum energy would not sustain.
The accurate measurement of the vacuum energy may be indicated by comprehensive quantum theory of gravity. Though, we are lacking such a profound theory, it is possible to investigate the nature of DE corresponding to some principles of quantum gravity. In particular, the holographic principle is a significant characteristic that may play role to deal with cosmological and DE issues. Cohen et al. suggested a relation between the infrared (IR) and ultraviolet (UV) cutoffs because of the limit made by the formation of black hole, which adjusts up an upper bound for the vacuum energy , where is the vacuum energy associated with the UV cutoff, is the IR cutoff and is the reduced Planck mass. Li proposed the form of DE and suggested that the future event horizon is the appropriate choice for IR cutoff which seems to agree with recent measurements.
Introducing new ingredients of DE to the entire cosmic energy is the one approach to explain the mystery of cosmic acceleration. Another approach is based on modification of the Einstein-Hilbert action to get alternative theories of gravity such as , , where is the the torsion and theory etc. Harko et al. introduced theory by generalizing gravity and is established on the coupling between matter and geometry. Recently, this theory has gained attention and some worth mentioning results have been explored.
Many authors have discussed the cosmological reconstruction of modified theories of gravity according to holographic DE. Karami and Khaledian reconstructed models according to holographic and new agegraphic DE. Daouda et al. develped model using holographic DE which can imply unified scenario of dark matter with DE. Houndjo and Piattella numerically reconstructed the models which can represent the characteristics of holographic DE models. In this work, we consider the holographic and new agegraphic DE models, and reconstruct the corresponding gravity as an equivalent picture without utilizing any additional DE component. We also investigate the generalized second law of thermodynamics (GSLT) on the future event horizon and find out the necessary condition for its validity.
The paper is arranged as follows. In the next section, we introduce the general formulation of the field equations in gravity. Sections 3 and 4 provide the reconstruction of gravity according to holographic and new agegraphic DE respectively. In section 5, the validity of GSLT is investigated and the last section concludes our results.
2 Gravity: General Formalism
The gravity is an appealing modification to the Einstein-Hilbert action by setting an arbitrary function of scalar curvature and trace of the energy-momentum tensor . The action for this theory is defined as
where and . The energy-momentum tensor of matter component is determined as
The correspong field equations are found through the variation of (1) with respect to the metric tensor
where is the covariant derivative linked with the Levi-Civita connection symbol and is defined by
The matter content is assumed to be perfect fluid so that
where is the four velocity which satisfies , and are the energy density and pressure of the fluid, respectively. The matter Lagrangian can be assumed as , so that becomes
We assume the model as , where and are arbitrary functions of and , respectively. Thus the field equation (3) becomes
which can be reproduced as an effective Einstein field equation, i.e.,
Now, we formulate the field equations of models for particular choices of and .
We propose a particular case with and . Such model appears to be interesting and has been widely studied in literature. Accordingly, the field equations are obtained as follows
The line element of spatially flat FRW spacetime is given by
where is the scale factor and comprises the spatial part of the metric. In this background, the above field equations can be represented as
where is the Hubble parameter and dot represents differentiation with respect to time. The energy density () and pressure () of dark energy components are obtained as
The corresponding EoS parameter is
Let us consider a more complicated case choosing and , can be considered as correction term to gravity. For this model, the field equation (7) can be represented as
Using Eqs.(15) and (16), we can develop the evolution equation for as
This represents a third order differential equation in . In sections 3 and 4, we reconstruct the models for holographic DE (HDE) and new agegraphic DE (NADE) as follows.
3 Reconstruction from Holographic Dark Energy
According to holographic principle, the HDE density is given by
where is a constant. The IR cutoff (future event horizon) is defined as
For the homogeneous and isotropic universe with spatially flat geometry, comprising matter component and HDE, the Friedmann equation reads
where from the energy conservation equation of matter. By introducing critical energy density and dimensionless DE , we obtain
The HDE satisfies the conservation law
It can be seen that when in the future (i.e., the HDE dominates the contents of the universe), for , we have which depicts quintessence era such that the universe escapes from entering the de Sitter and Big Rip phases. For , it represents the de Sitter universe and if , it may end up with phantom phase and behaves as quintom era because EoS parameter intersects the cosmological constant boundary (the phantom divide) throughout evolution. Hence, the parameter plays a significant character in determining the evolutionary paradigm of HDE as well as ultimate fate of the universe. The HDE has been constrained from observations of SNeIa, CMB and galaxy clusters, the best fit favors , although is also compatible with the data in one-sigma error range.
Now we reconstruct the HDE models by considering two particular actions of Lagrangian.
Comparing EoS parameter of dark energy components for the above model with that of HDE, one obtains
This is the first order differential equation. For constant , its solution is of the form
We are interested to determine the model coming from HDE. Also, for a given , the gravity can be reconstructed corresponding to any DE model. The Hubble parameter is assumed to be
where and are positive constants and is the probable time when finite-time future singularity may appear. given by (26) specifies two type of singularities, type I (“Big rip singularity”) and type III which can occur for and respectively. One can find details of the classification of finite-time singularities in literature.
We look at the elementary case by choosing so that representing the phantom phase of the universe which may result in Big rip singularity within finite time . For this model, the future event horizon and are obtained as
Consequently, the solution of Eq.(25) yields
and the corresponding HDE model is
where is a constant depending on and is the integration constant. To find the constant , we need to develop initial condition on . The Friedmann equation (9) evaluated at yields
Applying initial condition (31), the constant is determined as
Hence, the explicit function of is given by
In this representation, we normalize and to and set and . The function is plotted against and in Figure 1. The difference among the values of is apparent for earlier times of the universe which vanishes in late times. Figure 1(b) shows the evolution in terms of redshift and here variation in curves is evident in future evolution for different values of . The function satisfies the EoS parameter which depicts the phantom era of DE. Figure 2 clearly shows that for this model, null energy condition (NEC) is violated and hence accelerated expansion of the universe is achievable. Here, NEC would violate even if one increases the value of which is in agreement with EoS parameter for this model.
Here, we reconstruct the function in the setting of HDE. For the choice of Hubble parameter , the future event horizon and matter energy density can be rewritten in terms of the Ricci scalar as
and are constants.
Now, we define necessary initial conditions to determine the values of constants. For this purpose, we make the same assumption as in ref.. In particular, we choose the initial conditions and which can be translated as
Applying the above initial conditions to the solution (36), it follows that
Consequently, the model corresponding to HDE turns out to be
We plot the function against for different choices of parameters and . In Figure 3(a), we fix (i.e., purely gravity), which represents the variation of for different values of parameter . It is obvious that curves become distinct for large and show increasing behavior. The effect of coupling parameter is shown in Figure 3(b) for . We can see that non-zero values of modify the evolutionary nature of curves. We have also represented these results in terms of redshift in Figure 4. These curves exhibit the future evolution of for different values of parameters and .
We also explore the behavior of NEC for the reconstructed in HDE and display the graphs for different values of parameters and . Figure 5 shows that NEC is violated i.e., which necessitates . To make sure the phantom regime of the DE, we also plot the evolution of against and shown in Figure 6. The plots clearly favors the accelerated expansion except for particular range of . Thus the model corresponding to HDE is consistent with present day observations.
4 Reconstruction from New Agegraphic Dark Energy
In this section, we discuss the reconstruction of gravity in the setting of NADE. The energy density of NADE is proposed as
where the numerical component is inserted to parameterize some uncertainties namely, the specific forms of cosmic quantum fields and the role of curvature of spacetime etc., is the conformal time in FRW background defined as
Wei and Cai developed the cosmological constraints on NADE and found that the resolution of coincidence problem may become more definite in the NADE model with specific value of nearly unity. They constrained the NADE by using the observational data of SNeIa, CMB and LSS and found the best fit parameter (with uncertainty) . The new agegraphic DE has been under consideration in both GR and modified theories scenario. The time derivative of is obtained as
We are concerned to demonstrate the possible correspondence between models and NADE. In the following, we discuss the two cases individually.
Comparing Eqs.(43) and (13), we obtain
For , its solution is , where is constant of integration and . Now, we develop initial constraint on for NADE model and find out the constant . Evaluating Eq.(44) at present day and manipulating with Eq.(30), we obtain the following initial condition on
Making use of Eq.(45) and relation , the model is constructed as
In case of NADE, we set and plot in terms of and redshift as shown in Figure 7. One can see that the difference in evolutionary curves of depending on the value of is not obvious in both graphs. These plots represent the future era where is increasing rapidly. The EoS of DE components for the model (46) is found as which represents the quintessence era of DE. We also plot the NEC for this model by varying the values of parameter shown in Figure 8. The NEC is found to be satisfied i.e., which confirms the regime with . Hence, the reconstructed for NADE represents the quintessence era of the universe.
The conformal time of FRW universe can be represented in terms of Ricci scalar as
Likewise and for NADE are determined as
Solving the differential equation (17) for NADE, it follows that
where and are given by
For the NADE, the function is plotted against for different values of parameter and as shown in Figure 9. In Figure 9(a), we fix (corresponds to gravity) and represent the behavior of for different values of . It shows that the curves for reconstructed in NADE are same. If one introduces the coupling parameter with , the variation in results is evident from Figure 9(b). We also plot these results in plane and represent the future evolution of as shown in Figure 10.
Now we check the validity of NEC for the in NADE shown in Figure 11. It is clear that NEC is satisfied i.e., except for the negative values of coupling parameter . Consequently, these models should imply , the quintessence EoS parameter. We show the evolution of for different values of parameters and . The plots in Figure 12 make it more definite that the reconstructed function favors the quintessence regime of the universe.
5 Generalized Second Law of Thermodynamics
Here, we discuss the validity of GSLT in this modified gravity on the future event horizon. The GSLT states that entropy of a black hole horizon summed to the entropy of matter and fluids inside the horizon is non-decreasing with time. The validity of GSLT has been discussed in the setting of modified theories of gravity . In, a non-equilibrium picture of thermodynamics is discussed on the apparent horizon of FRW spacetime in gravity. It is remarked that usual laws of thermodynamics do not hold in this modified theory and additional entropy production term is required. We consider a flat FRW universe consisting of ordinary matter plus the DE component. The modified first law of thermodynamics is stated as
where and represent temperature and entropy of entire contents within the horizon. We have to show that
where is the horizon entropy. For , Eq.(51) yields
We assume that temperature is proportional to Gibbson-Hawking temperature
where is a real constant. In the following, we study GSLT for two forms of function.
In GR, the Bekenstein-Hawking entropy is given by the relation , where represents the area of the event horizon. It was proposed that the horizon entropy is associated with Noether charge in the context of modified gravity theories. Brustein et al. interpreted that Wald entropy is equivalent to one-fourth of the horizon area with gravitational coupling being the effective one. Hence, the entropy in this modified gravity is defined as
its time rate is
Using the FRW equations for this model, Eq.(53) leads to
Thus, the total entropy for GSLT becomes
In GR, the above condition reduces to . The effective gravitational coupling constant for this model needs to be positive so that . To illustrate our result, let us consider the model given by Eq.(33). In this model, , and . By the direct replacement of these results, we obtain that GSLT is valid if , and . For , the condition holds if