DARK ENERGY MODEL WITH GENERALIZED COS MO LOGICAL HORIZON
M. Sharifa* A. Jawada'h**
" Department of Mathematics, University of the Punjab Lahore-54590, Pakistan
h Department of Mathematics. Lahore Leads University Lahore, Pakistan
Received March 14, 2014
We discuss the evolution of the newly proposed dark-energy model with a generalized event horizon (a generalized form of the holographic dark-energy model with a future event horizon) in the flat and nonflat universes. We consider the interacting scenario of this model with cold dark matter. We use the well-known logarithmic approach to evaluate the equation of state parameter and explore its present values. It is found that this parameter shows phantom crossing in some cases of the generalized event horizon parameters. The w-w' plane is also developed for three different cases of the generalized event horizon parameters. The corresponding phase plane provides thawing and freezing regions. Finally, the validity of a generalized second law of thermodynamics is explored which holds in certain ranges of constant parameters.
DOI: 10.7868/S0044451014100113
1. INTRODUCTION
The prediction about the accelerated expansion of the universe is a revolutionary change in modern cosmology. The debate 011 this topic has been extensive in the last decade in both observational and 110110b-servational terms. The main focus of this discussion remained 011 the unknown type of matter, which is assumed to be the major factor of the accelerating universe. A consensus has been developed 011 dark energy (DE), but its nature is still unclear. I11 order to resolve this problem, a plethora of work has been done within two main approaches: modification of the gravitational part and of the matter part of the Einstein field equations.
The modification approach in the matter part has led to different dynamical DE models such as the Cliap-lygin gas fl], holographic [2, 3], agegraphic [4], new agegraphic [5], and scalar field DE models [6 13]. The holographic DE (HDE) model is one of the famous models developed in the framework of quantum gravity. The main motivation behind this model is to achieve
E-mail: msharif.math'fflpu.edu.pk
** E-mail: jawadablSl'fflyahoo.com
consensus about the ambiguous nature of DE. The holographic principle is the origin of this model, according to which the number of degrees of freedom of a physical system should scale with its bounding area rather than its volume [14].
Later 011, Cohen et al. [15] developed a relation between ultraviolet (UV) and infrared (IR) cutoffs using the idea of black hole formation in quantum field theory. They argued that the total energy of a system of size L should not exceed the black hole IIlclSS of the same size. Using this argument, Hsu [2] developed a model for the density of HDE in the form
o\2 2 r —2
Pi) = 3A m L
where A is an arbitrary constant and mp is the reduced Planck rilctSS. Different expressions for the IR cutoff L have been proposed such as Hubble, event, particle horizons [3], Ricci scalar [16] and its generalized form [17]. However, the HDE model with an event horizon has been discussed extensively in the absence [3, 18, 19] and presence [20 22] of interaction with dark matter (DM). These models have also been tested in the framework of different observational schemes and used to develop reliable constraints 011 different cosnio-logical parameters such as the equation-of-state (EoS)
paraniotor, Hubble parameter, fractional energy densities, etc [23, 24].
Li [3] explored HDE with a future event horizon using the logarithmic approach and found the present value of the EoS parameter = ^0.90. Huang and Li [18] used this approach to examine the evolution of the universe by checking all possible values of the HDE parameter A and also found that a generalized second law of thermodynamics (GSLT) is preserved for HDE with a future event horizon in a flat as well as closed universe for A < 1. They also revealed that HDE with this horizon can cross the phantom region. Jamil et al. [19] investigated the HDE scenario with a varying gravitational constant (G) in both flat and nonflat universes by using the logarithmic approach. They found corrections to the evolution of the EoS parameter in [3] due to variation of G. Lu et al. [24] checked these results within observational schemes and argued that the scenario of HDE with a varying G is compatible with the present observations. They also found the present values of different cosmological parameters in this scenario within a 1(7 error range.
Recently, the holographic, agegraphic, and new age-graphic DE models (with event and particle horizons) have been extended to the most general class characterized by dimensionless constant parameters (m,n). The behavior of these models in terms of the EoS parameter, noninteracting and interacting with DM in a flat universe, was investigated in [25]. Cosmological behavior of the universe for a general class of HDE with a particle horizon was explored in [26] within observational schemes in a flat universe. In this paper, we choose an (in. ii} type DE model with a generalized cosmological horizon (GCH) (a generalized form of the HDE model with a future event horizon) in flat and nonflat universes. We use the logarithmic approach to evaluate the EoS parameter in the context of interaction with cold DM (CDM). We also discuss the ui^ ui'^ plane and the validity of the GSLT.
The rest of the paper is arranged as follows. In Sec. 2, we investigate the EoS parameter, ui^ and the GSLT in a flat universe. Section 3 explores the EoS parameter, ui^ and the GSLT in a nonflat universe. In the Sec. 4, we summarize our results.
2. FLAT UNIVERSE
In this section, we elaborate a basic cosmological scenario in a flat Friedman Robertson Walker (FRW) universe for DE with a GCH. The generalized form of the cosmological horizon is defined as [25, 26]
oo
L = RGCH = -^ f uw(t)dí (1)
where u(t) is the cosmic scale factor. We can recover the original HDE with a future event horizon for m = n = The time derivative of the above relation yields
ROCH = -UHRCCH - am~", m <0, (2)
where H is the Hubble parameter. The first FRW equation leads to
H'2 = +Pwh ny + n'« = (3)
where and pm are the respective DE and CDM densities, while
O — q _ Pm
""3mjH^ ~ 3nipH2
are the corresponding fractional energy densities. The continuity equations in the interacting case become
Pm + 3 Hpm = 'Ait2 H pi), (4)
Pi) + 2>H(p.i) + pi)) = —3-u2Hp#, (5)
where u2 is an interaction parameter.
Currently, there are no prior conditions imposed on the possible interactions between DM and DE because neither DE nor DM is understood fundamentally. However,
without violating the observational constraints, DE can interact with DM in various fashions by means of energy transfer between each other. The interaction between DE and DM yields a richer cosmological dynamics as compared to noninteracting models and it is possible to solve the cosmic coincidence problem within this framework. However, we cannot describe interaction between these vague nature components from first principles. Therefore, we have to take a specific interaction or set it from phenomenological requirements. The DE density with a GCH is defined as
Pi) = 3A 2m2,R^2CH, (6)
and its evolutionary form is given by
p'a =2 po ('»- + -—^— J. en
where the prime denotes differentiation with respect to x = lna. By taking the derivative of Eq. (3) with respect to the cosmic time, we obtain
= ^3 + (3(1 + «2) + 2n)n, + "" . (8)
Differentiating ÍI0 with respect to x and using Eqs. (7) and (8) yields
dil i)
dx
= íly(l — íly) X
x 3 + 2 n ■
2a"
~Ä 1 -2.1. Cosmological implications
• 0)
We now evaluate the EoS parameter within the logarithmic approach. The DE density is obtained from Eq. (5) in the form
Pi) = Pmu
-Sd+Wri + li2
(10)
where p.¿>0 serves as the current value of the DE density. We uso a Taylor series expansion for about the present value of u0 = 1 as follows:
In p, = In p% + In a + ± (In a)2
1 (P In pi)
illna
2 ii(lna)2
1 d?' In pi)
(Ilia)3
(ID
6 il(lnu)3
The series is terminated at the second-order derivative because of the small-redshift approximation, i.e., Ilia = — ln(l + z) « —z, and it follows from (10) and (11) that
LO.i) = U>i)Q + U>i)l Î
where
ui.yo = — 1
1 din pi) 3 dlnu
Mm =
1 d2 In
Pi)
(12)
(13)
6 ii(lna)2
Here, the derivatives are taken at the present value of uo- Expressing in terms of fractional densities as Pd = Qi)pm/Qm, after some calculations, we obtain
(111! Pi)
d In u d2 In p.j) ii(lllu)2
= 2 n
24r,-n
A
m-n /q
l0
t»0
A
x í 3+2n+2«¿"-"A-1 v/Sîyo-
2(m-n)+(l-iîOT) x (14)
'¿U2üi)0
1-íí
t»0
Using Eqs. (12) (14), we obtain the EoS parameter as follows:
LOtf = — 1 — W
6A
3 + 2n + 2am-n\-1
2(m - n) + (1 - flyo) x
'¿U2üi)0
1-íí
(15)
O-'i?
-0.7 -0.8 -0.9 -1.0
a- = 0 u2 = 0.058
-1.0
-0.5
0.5
1.0
-1.40 -1.45 -1.50 -1.55 -1.60
N N v N \ N \ S « b -
a2 = 0 N \ —
--a2 = 0.058 N \
N S ■
-1.0
-0.5
0.5
1.0
a;,}
— 2.12 c ~
— 2.16 "V.
— 2.20 u2 - 0 --il2 = 0.058 ■v. •v.
— 2.24 -.......... ......... i-
-1.0
-0.5
0.5
1.0
Fig. 1. Plots of uis versus z with u2 = 0, 0.058 in the flat case for n = —1, in = —2 (a), n = 0, in = —1 (b), and n = 1, in =0 (c). We use the present value of the fractional DE density i2,jo « 0.73 and choose A = 0.91
Using the observational dataset from WMAP^ +SNIa^BAO^iio. the best-fit values for the coupling parameter u2 were presented in [27]. It was also commented there that positive values of this parameter alleviate the cosmological coincidence problem. Here, we take u2 = 0.058 [27] for the interacting case and plot the EoS parameter versus i in the noninteracting case as well (Fig. 1). We choose three different well-settled pairs of the values of m and n by using well-known ob-
sorvational data [26]. It is found that for a given n, the models with n — m = 1 are most suitable for discussing the
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