Research ArticleCosmological Implications of the Generalized Entropy BasedHolographic Dark Energy Models in Dynamical Chern-SimonsModified Gravity

M Younas1 Abdul Jawad 1 Saba Qummer1 H Moradpour 2 and Shamaila Rani 1

1Department of Mathematics COMSATS University Islamabad Lahore Campus-54000 Pakistan2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) PO Box 55134-441 Maragha Iran

Correspondence should be addressed to Abdul Jawad abduljawadcuilahoreedupk

Received 28 August 2018 Revised 8 November 2018 Accepted 5 December 2018 Published 8 January 2019

Academic Editor Chao-Qiang Geng

Copyright copy 2019 M Younas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Recently Tsallis Renyi and Sharma-Mittal entropies have widely been used to study the gravitational and cosmological setups Weconsider a flat FRWuniverse with linear interactionbetween dark energy and darkmatterWe discuss the dark energymodels usingTsallis Renyi and Sharma-Mittal entropies in the framework of Chern-Simons modified gravity We explore various cosmologicalparameters (equation of state parameter squared sound of speed ) and cosmological plane (120596119889 minus 1205961015840119889 where 1205961015840119889 is the evolutionaryequation of state parameter) It is observed that the equation of state parameter gives quintessence-like nature of the universe inmost of the cases Also the squared speed of sound shows stability of Tsallis and Renyi dark energy model but unstable behaviorfor Sharma-Mittal dark energy model The 120596119889 minus 1205961015840119889 plane represents the thawing region for all dark energy models

1 Introduction

In the last few years remarkable progress has been achievedin the understanding of the universe expansion It has beenapproved by current observational data that the universeundergoes an accelerated expansionTheobservations of typeIa Supernovae (SNeIa)[1ndash4] large scale structure (LSS) [5ndash8]and Cosmic Microwave Background Radiation (CMBR) [910] determined that the expansion of the universe is currentlyaccelerating There is also consensus that this accelerationis generally believed to be caused by a mysterious form ofenergy or exotic matter with negative pressure so called darkenergy (DE) [11ndash21]

The discovery of accelerating expansion of the universeis a milestone for cosmology It is considered that 95 ofour universe is composed of two components that is DEand dark matter [16] The dark matter constitutes about 25of the total energy density of the universe The existence ofthe universe is proved by astrophysical observation but thenature of dark matter is still unknown Mainly the DE is

also a curious component of our universe It is responsiblefor current accelerating universe and DE is entirely differentfrom baryonic matter DE constitutes almost 70 of the totalenergy density of our universe

In order to describe the accelerated expansion phe-nomenon two different approaches have been adopted Oneis the proposal of various dynamical DEmodels such as fam-ily of Chaplygin gas holographic dark energy quintessenceK-essence and ghost [16] A second approach for under-standing this strange component of the universe is modifyingthe standard theories of gravity namely general relativity(GR) Several modified theories of gravity are119891(119877) 119891(119879) [17]119891(119877T) [18] and 119891(119866) [19] where 119877 is the curvature scalar119879 denotes the torsion scalar T is the trace of the energy-momentum tensor and 119866 is the invariant of Gauss-Bonnet

Holographic DE (HDE) model is favorable techniqueto solve DE mystery which has attracted much attentionand is based upon the holographic principle that states thenumber of degrees of freedom of a system scales with itsarea instead of its volume In fact HDE relates the energy

HindawiAdvances in High Energy PhysicsVolume 2019 Article ID 1287932 9 pageshttpsdoiorg10115520191287932

2 Advances in High Energy Physics

density of quantum fields in vacuum (as the DE candidate)to the infrared and ultraviolet cut-offs In addition HDE isan interesting effort in exploring the nature of DE in theframework of quantum gravity Cohen et al [22] reportedthat the construction of HDE density is based on the relationwith the vacuum energy of the system whose maximumamount should not exceed the black hole mass Cosmologicalconsequences of some HDEmodels in the dynamical Chern-Simons framework as amodified gravity theory can be foundin [23]

By considering the long term gravity with the natureof spacetime different entropy formalism has been usedto observe the gravitational and cosmological effects [24ndash29] The HDE models such as Tsallis HDE (THDE) [27]Renyi HDE model (RHDE) [28] and Sharma-Mittal HDE(SMHDE) [29] have been recently proposed In the stan-dard cosmology framework and from the classical stabilityview of point while THDE is not stable [27] RHDE isstable during the cosmic evolution [28] and SMHDE isstable only whenever it becomes dominant in the world[29] In the present work we use the Tsallis Sharma-Mittal and Renyi entropies in the framework of dynamicalChern-Simons modified gravity and consider an interactionterm We investigate the different cosmological parameterssuch as equation of state parameter and the cosmological120596119889 minus 1205961015840119889 plane where 1205961015840119889 shows the evaluation with respectto ln 119886 We also investigate the squared sound speed ofthe HDE model to check the stability and the graphicalapproach

This paper is organized as follows In Section 2 weprovide the basics of Chern-Simons modified gravity InSection 3 we observe the equation of state parameter (EoS)cosmological plane and squared sound speed for THDEmodel Sections 4 and 5 are devoted to finding the cosmo-logical parameter cosmological plane and squared of soundspeed for RHDE and SMHDEmodels respectively In the lastsection we conclude the results

2 Dynamical Chern-Simons Modified Gravity

In this section we give a review of dynamical Chern-Simonsmodified gravity The action which describes the Chern-Simons modified gravity is given as

119878 = 116120587119866 int]1198894119909 [radicminus119892119877 + 1198974120579lowast119877120588120590120583]119877120588120590120583]

minus 12119892120583]nabla120583120579nabla]120579 + 119881 (120579)] + 119878119898119886119905(1)

where 119877 represents the Ricci scalar lowast119877120588120590120583]119877120588120590120583] is a topo-logical invariant called the Pontryagin term 119897 is a couplingconstant 120579 shows the dynamical variable 119878119898119886119905 represents theaction of matter and 119881(120579) is the potential term In the case ofstring theory we use119881(120579) = 0 By varying the action equation

with respect to 119892120583] and the scalar field 120579 we get the followingfield equations

119866120583] + 119897119862120583] = 8120587119866119879120583]119892120583]nabla120583nabla]120579 = minus 11989764120587lowast119877120588120590120583]119877120588120590120583] (2)

Here 119866120583] and 119862120583] are Einstein tensor and Cotton tensorrespectively The Cotton tensor 119862120583] is defined as

119862120583] = minus 12radicminus119892 ((nabla120588120579) 120576120588120573120591(120583nabla120591119877])120573 )

+ (nabla120590nabla120588120579) lowast119877120588(120583])120590 (3)

The energy-momentum tensor is given by

120579120583] = nabla120583120579nabla]120579 minus 12119892120583]nabla120588120579nabla120588120579119879120583] = (120588 + 119901) 119906120583119906] + 119901119892120583] (4)

where 119879120583] shows the matter contribution and 120579120583] representsthe scalar field contribution while 119875 and 120588 represent thepressure and energy density respectively Furthermore 119906120583 =(1 0 0 0) is the four velocity In the framework of Chern-Simons gravity we get the following Friedmann equation

1198672 = 13 (120588119898 + 120588119889) + 16 1205792 (5)

where 119867 = 119886119886 is the Hubble parameter and the dotrepresents the derivative of 119886 with respect to 119905 and 8120587119866 =1 For FRW spacetime the pony trying term lowast119877119877 vanishesidentically therefore the scalar field in (2) takes the followingform

119892120583]nabla120583nabla]120579 = 119892120583] [120597]120597120583120579] = 0 (6)

We set 120579 = 120579(119905) and get the following equation120579 + 3119867 120579 = 0 (7)

which implies that 120579 = 119887119886minus3 119887 is a constant of integrationUsing this result in (5) we have

1198672 = 13 (120588119898 + 120588119889) + 161198872119886minus6 (8)

We consider the interacting scenario between DE anddark matter and thus equation of continuity turns to thefollowing equations

120588119898 + 3119867120588119898 = 119876 (9)

120588119889 + 3119867(120588119889 + 119901119889) = minus119876 (10)

Here 120588119889 is the energy density of the DE 120588119898 is the energydensity of the pressureless matter and 119876 is the interactionterm Basically 119876 represents the rate of energy exchangebetween DE and dark matter If 119876 gt 0 it shows that energy

Advances in High Energy Physics 3

is being transferred from DE to the dark matter For 119876 lt 0the energy is being transferred from dark matter to the DEWe consider a specific form of interaction which is definedas119876 = 31198671198892120588119898 and 1198892 is interacting parameter which showsthe energy transfers between CDM and DE If we take 119889 = 0then it shows that each component that is the nonrelativisticmatter and DE is self-conserved Using the value of 119876 in (9)we have 120588119898 = 1205881198980119886minus3(1minus1198892) (11)

where 1205881198980 is an integration constant Hence (10) finally leadsto the expression for pressure as follows

119901119889 = minus(1198892120588119898 + 120588119889 + 1205881198893119867) (12)

The EoS parameter is used to categorize the deceleratedand accelerated phases of the universe This parameter isdefined as 120596 = 119901120588 (13)

If we take 120596 = 0 it corresponds to nonrelativistic matter andthe decelerated phase of the universe involves radiation era0 lt 120596 lt 13 120596 = minus1 minus1 lt 120596 lt minus13 and 120596 lt minus1 correspondto the cosmological constant quintessence and phantomeras respectively To analyze the dynamical properties of theDE models we use 120596 minus 1205961015840 plane [30] This plane describesthe evolutionary universe with two different cases freezingregion and thawing region In the freezing region the valuesof EoS parameter and evolutionary parameter are negative(120596 lt 0 and 1205961015840 lt 0) while for the thawing region the valueof EoS parameter is negative and evolutionary parameter ispositive (120596 lt 0 and 1205961015840 gt 0) In order to check the stability oftheDEmodels we need to evaluate the squared sound speedwhich is given by

V2119904 = 119889119901119889120588 = 119889119901119889119905119889120588119889119905 (14)

The sign of V2119904 decides its stability of DEmodels when V2119904 gt 0the model is stable otherwise it is unstable

3 Tsallis Holographic Dark Energy

The definition and derivation of standard HDE density aregiven by 120588119889 = 3119888211989811990121198712 where 1198981199012 represents reducedPlank mass and 119871 denotes the infrared cut-off It dependsupon the entropy area relationship of black holes ie 119878 sim119860 sim 1198712 where 119860 = 41205871198712 represents the area of the horizonTsallis and Cirto [31] showed that the horizon entropy of theblack hole can be modified as119878120575 = 120574119860120575 (15)

where 120575 is the nonadditivity parameter and 120574 is an unknownconstant [31] Cohen at al [22] proposed the mutual relation-ship between IR (L) cut-off system entropy (S) and UV (Λ)cut-off as 1198713Λ3 le (119878)34 (16)

Tsallis HDE

minus10

minus08

minus06

minus04

minus02

d

00 05minus05z

Figure 1 Plot of 120596119889 versus 119911 for THDE model where 120575 = 11 1205881198980 =1 1198892 = 0001 119861 = minus13 119887 = 05After combining (15) and (16) we get the following relation

Λ4 le 120574 (4120587)120575 1198712120575minus4 (17)

where Λ4 is vacuum energy density and 120588119889 sim Λ4 So theTsallis HDE density [29] is given as

120588119889 = 1198611198712120575minus4 (18)

Here 119861 is an unknown parameter and IR cut-off is taken asHubble radius which leads to 119871 = 1119867 where 119867 is Hubbleparameter The density of Tsallis HDE model along with itsderivative by using (18) becomes

120588119889 = 1198611198674minus2120575120588119889 = 119861 (4 minus 2120575)1198673minus2120575 (19)

Here is the derivative of Hubble parameter wrt 119905 Thevalue of is calculated in terms of 119911 using 119886 = 1(1 + 119911)which is given as follows

119889119867119889119911= (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)(1 minus (13) 119861 (4 minus 2120575)1198673minus2120575)119867 (1 + 119911)

(20)

Inserting these values in (12) yields

119901119889 = 13 (minus311988921205881198980119886minus3(1minus1198892)minus 1198611198672minus2120575 (31198672 + (4 minus 2120575) )) (21)

The EoS is obtained from (13)

120596119889 = 119901119889120588119889 = minus1 minus 11988921205881198980119886minus3(1minus1198892)1198672120575minus4119861 + (2120575 minus 4) 31198672 (22)

The plot of 120596119889 versus 119911 is shown in Figure 1 In this parameterand further results the function 119867(119911) is being utilizednumerically The other constant parameters are mentioned

4 Advances in High Energy Physics

Tsallis HDE

00

01

02

03

00 05 10minus05z

s2

Figure 2 Plot of V2119904 versus 119911 for THDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119861 = minus13 119887 = 05in Figure 1 The trajectory of EoS parameter remains inquintessence region at early present and latter epoch

The square of the sound speed is given by

V2119904 = 16119861 (120575 minus 2) 11988641198673 (91198892 (1198892 minus 1) 1205881198980119886311988921198672120575 119886minus 2119861 (120575 minus 2) 1198864119867times (31198672 minus 2 (120575 minus 1) 2 + 119867))

(23)

The plot of squared sound speed versus 119911 is shown inFigure 2 for different parametric values This graph is usedto analyze the stability of this model We can see that V2119904 gt 0for minus06 lt 119911 lt 1 which corresponds to the stability of THDEmodel However the model shows instability for 119911 lt minus06

Taking the derivative of the EoS parameter with respectto ln 119886 we get 1205961015840119889 as follows

1205961015840119889 = 1311986111988641198676 (minus311988921205883119889211989801198672120575 (3 (1198892 minus 1)119867 119886+ (2120575 minus 4) ) + 2119861 (120575 minus 2) times 11988641198672 (minus22+ 119867))

(24)

The graph of 120596119889 versus 1205961015840119889 is shown in Figure 3 for which1205961015840119889 depicts positive behavior Hence for 120596119889 lt 0 the evolutionparameter shows1205961015840119889 gt 0 which represents the thawing regionof evolving universe

4 Reacutenyi Holographic Dark Energy Model

Weconsider a systemwith119882 stateswith probability of getting119894th state 119875119894 and satisfying the condition Σ119882119894=1119875119894 = 1 Renyi andTsallis entropies are defined as

S = 1120575 lnΣ119882119894=11198751minus120575119894 119878119879 = 1120575Σ119882119894=1 (1198751minus120575119894 minus 119875119894) (25)

Tsallis HDE

minus08 minus06 minus04 minus02minus10d

0

1

2

3

4

(d)

Figure 3 Plot of 120596119889 versus 1205961015840119889 for THDE model where 120575 = 111205881198980 = 1 1198892 = 0001 119861 = minus13 119887 = 05where 120575 equiv 1minus119880 where119880 is a real parameter Now combiningthe above equations we find their mutual relation given as

S = 1120575 ln (1 + 120575119878119879) (26)

This equation shows that S belongs to the class of mostgeneral entropy functions of homogenous system Recentlyit has been observed that Bekenstein entropy 119878 = 1198604 is infact Tsallis entropy which gives the expression

119878 = 1120575 ln(1 + 1205751198604 ) (27)

which is the Renyi entropy of the system Now for theRHDE we focus on WMAP data for flat universe Using theassumption 120588119889119889V prop 119879119889119904 we can get RHDE density

120588119889 = 3119862211986728120587 (1 + 1205751205871198672) (28)

Considering the term 8120587 = 1 and substituting in (28) we getthe expression for density as

120588119889 = 3119862211986721 + 1205751205871198672 (29)

Now 119889119867119889119911 is given by the following

119889119867119889119911= (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)(1 minus (211988821198672 (1199112 + 120575120587) minus 11988821198674) (1198672 + 120575120587)2)119867 (1 + 119911)

(30)

The pressure for this case is obtained as

119901119889 = minus11988921205881198980119886minus3(1minus1198892)+ 11988821198672 (minus31198672 (120587120575 + 1198672) minus 2 (2120587120575 + 1198672) )(120587120575 + 1198672)2 (31)

Advances in High Energy Physics 5

Renyi HDE

00 05minus05z

minus10

minus08

minus06

minus04

minus02

00

d

Figure 4 Plot of 120596119889 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 005The expressions for EoS parameter 120596119889 can be evaluated from(12) as follows

120596119889 = (120587120575 + 1198672)(minus11988921205881198980119886minus3(1minus1198892)311988821198674minus (31198672 (120587120575 + 1198672) + 2 (2120587120575 + 1198672) )31198672 (120587120575 + 1198672)2 )

(32)

Figure 4 shows the plot of 120596119889 versus 119911 The trajectoryof EoS parameter evolutes the universe from quintessenceregion towards the ΛCDM limit The squared sound speedof this RHDEmodel is given by using (13) as

V1199042 = 3119867(1 minus 1198892) 11988921205881198980119886minus3(1minus1198892) (120587120575 + 1198672)2611988821198673 (2120587120575 + 1198672) minus 131198672 (2120587120575 + 1198672) (120587120575 + 1198672)times (6120587212057521198672 + 91205871205751198674 + 31198676 + 412058721205752)+ 119867 (120587120575 + 1198672) times (2120587120575 + 1198672)

(33)

The graph of squared speed of sound is shown in Figure 5versus 119911 In this case we have V2119904 gt 0 for all ranges of 119911 whichshows the stability of RHDE model at the early present andlatter epoch of the universe

The expression for 1205961015840119889 is evaluated as

1205961015840119889 = 13119888211988641198676 (120587120575 + 1198672)2 minus1198892120588119898011988631198892 (120587120575 + 1198672)2sdot (3119867 119886 (minus1 + 1198892)times (120587120575 + 1198672) minus 2119886 (2120587120575 + 1198672))+ 2119888211988641198672 (412058721205752 + 81205871205751198672 + 21198674) 2minus 2119867(120587120575+ 1198672) (2120587120575 + 1198672)

(34)

In Figure 6 we plot the EoS parameter with its evolutionparameter to discuss 120596119889 minus 1205961015840119889 plane for RHDE model The

Renyi HDE

00 05 10minus05z

0

50

100

150

200

s2

Figure 5 Plot of V2119904 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 15Renyi HDE

00minus06 minus04 minus02minus08minus10d

00

05

10

15

20

25

30

35

(d)

Figure 6 Plot of 120596119889 versus 1205961015840119889 for RHDE model where 120575 = 111205881198980 = 08 1198892 = 0001 119888 = 01 119887 = 005

graph shows that for 120596119889 lt 0 the evolutionary parameterremains positive at the early present and latter epoch Thistype of behavior depicts the thawing region of the evolvinguniverse

5 Sharma-Mittal Holographic DarkEnergy Model

From the Renyi entropy we have the generalized entropycontent of the system Using (26) Sharma-Mittal introduceda two-parametric entropy which is defined as

119878119878119872 = 11 minus 119903 ((Σ119882119894=11198751minus120575119894 )1minus119903120575 minus 1) (35)

where 119903 is a new free parameter We can observe that Renyiand Tsallis entropies can be recovered at the proper limitsusing (25) in (35) we have

119878119878119872 = 1119877 ((1 + 120575119878119879)119877120575 minus 1) (36)

6 Advances in High Energy Physics

where 119877 equiv 1minus119903 Using the argument that Bekenstein entropyis the proper candidate for Tsallis entropy by using 119878 = 1198604where 119860 is horizon entropy we get the following expression

119878119878119872 = 1119877 ((1 + 1205751198604 )119877120575 minus 1) (37)

and the relation of UV (Λ) cut-off IR (L) cut-off and systemhorizon (S) is given as follows

Λ4 prop 1198781198714 (38)

Now taking 119871 equiv 1119867 = radic1198604120587 then the energydensity of DE given by Sharma-Mittal [29] is consideredas

120588119889 = 3119888211986748120587119877 [(1 + 1205751205871198672)119877120575 minus 1] (39)

where 1198882 is an unknown free parameter Using 8120587 = 1 inabove equation we get the following expression for energydensity

120588119889 = 311988821198674119877 [(1 + 1205751205871198672)119877120575 minus 1] (40)

The differential equation of119867 is given by the following

119889119867119889119911 = (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)1 + 1198882120587 (1 + (1205751205871198672))119877120575minus1 minus (211988821198672119877) ((1 + 1205751205871198672)119877120575 minus 1)119867 (1 + 119911) (41)

The pressure can be evaluated by energy conservation (11) asfollows

119901119889 = minus11988921205881198980119886minus3(1minus1198892)minus 1198882(3((1 + 1205871205751198672)119877120575 minus 1)1198674119877minus 2120587(1 + 1205871205751198672)119877120575minus1

+ 4 ((1 + 1205871205751198672)119877120575 minus 1)1198672119877 )

(42)

The EoS parameter for this model is given by

120596119889 = 21198882 (120587(1 + 1205871205751198672)119877120575minus1minus 21198672119877 ((1 + 1205871205751198672 minus 1)119877120575))minus 11988921198771205881198980119886minus3(1minus1198892)311988821198674 ((1 + 1205751205871198672)119877120575 minus 1) minus 1

(43)

The plot of 120596119889 versus 119911 is shown in Figure 7 The EoSparameter represents the quintessence nature of the universeThe square of the sound speed is evaluated as

V2119904 = 161198882119867(minus120587 (1 + 1205871205751198672)119877120575minus1 + (21198672119877) ((1 + 1205871205751198672)119877120575 minus 1)) times minus31198892119867(minus1 + 1198892) 1205881198980119886minus3(1minus1198892)+ 21198882119867119877 (61198672 + 42 + 2119867 minus 1(120587120575 + 1198672)2 (1 + 1205871205751198672)119877120575sdot (31198672 (120587120575 + 1198672) (minus120587119877 + 2120587120575 + 21198672) + 2 (1205872 (119877 minus 2120575) (119877 minus 120575) minus 22120587 (119877 minus 2120575)1198672 + 21198674))+ 119867 (120587120575 + 1198672) times (minus120587119877 + 2120587120575 + 21198672) )

(44)

Advances in High Energy Physics 7

SharmandashMittal HDE

minus06minus05minus04minus03minus02minus01

00

d

00 05 10minus05z

Figure 7 Plot of 120596119889 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 0011198892 = 0001 119888 = 001 119887 = 04 119877 = 7In Figure 8 we draw V2119904 versus 119911which shows the unstable

behavior of the SMHDE model as V2119904 lt 0 at early presentand latter epoch

1205961015840119889 = minus 13 ((1 + 1205871205751198672)119877120575 minus 1)21198676 ( 1(120587120575 + 1198672)2sdot 21198672 (2(minus2 (120587120575 + 1198672)2 + (1 + 1205871205751198672)2119877120575sdot (1205872 (119877 minus 2120575) 120575 + 2120587 (119877 minus 2120575)1198672 minus 21198674)+ (1 + 1205871205751198672)119877120575times (minus1205872 (1198772 + 119877120575 minus 41205752) minus 2120587 (119877 minus 4120575)1198672 + 41198674))sdot 2 + (120587120575 + 1198672) times ((1 + 1205871205751198672)119877120575 minus 1)sdot 119867(minus2 (120587120575 + 1198672) + (1 + 1205871205751198672)119877120575times (minus120587119877 + 2120587120575 + 21198672)) ) + 31198892 (minus1 + 1198892)1198882sdot 1205881198980119877119886minus3(1minus1198892)1198672 times ((1 + 1205871205751198672)119877120575 minus 1)+ 211988921205881198980119877119886minus3(1minus1198892)1198882 (120587120575 + 1198672) ((120587 (119877 minus 2120575) minus 21198672) times (1+ 1205871205751198672)119877120575 + 2 (120587120575 + 1198672)) )

(45)

Figure 9 shows the plot of 120596119889-1205961015840119889 plane to classify thedynamical region for the given model We can see that 1205961015840119889 gt0 for 120596119889 lt 0 which indicates the thawing region of theuniverse

SharmandashMittal HDE

0200 04 06minus04minus06 minus02minus08z

minus6

minus5

minus4

minus3

minus2

s2

Figure 8 Plot of V2119904 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 081198892 = 0001 119888 = 08 119887 = 005 119877 = 7SharmandashMittal HDE

0

500

1000

1500

(d)

00minus04 minus03 minus02 minus01minus05minus06d

Figure 9 Plot of 120596119889 versus 1205961015840119889 for different values of 120575 for SMHDEwhere 120575 = 11 1205881198980 = 001 1198892 = 0001 119888 = 001 119887 = 04 119877 = 7

Table 1 Summary of the cosmological parameters and plane

DEmodels 120596119889 V2119904 120596119889 minus 1205961015840119889THDE quintessence-to-

vacuum partially stability thawing region

RHDE quintessence-to-vacuum stability thawing region

SMHDE quintessence un-stable thawing region

6 Conclusion

In this paper we have discussed the THDE RHDE andSMHDE models in the framework of Chern-Simons modi-fied theory of gravity We have taken the flat FRW universeand linear interaction term is chosen for the interactingscenario between DE and dark matter We have evaluatedthe different cosmological parameters (equation of stateparameter and squared sound speed) 120596119889 minus 1205961015840119889 cosmologicalplane The trajectories of all these models have been plottedwith different constant parametric values

We have summarized our results in Table 1Jawad et al [32] have explored various cosmological

parameters (equation of state squared speed of sound Om-diagnostic) and cosmological planes in the framework ofdynamical Chern-Simons modified gravity with the newholographic dark energy model They observed that the

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

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Advances in High Energy Physics 3

is being transferred from DE to the dark matter For 119876 lt 0the energy is being transferred from dark matter to the DEWe consider a specific form of interaction which is definedas119876 = 31198671198892120588119898 and 1198892 is interacting parameter which showsthe energy transfers between CDM and DE If we take 119889 = 0then it shows that each component that is the nonrelativisticmatter and DE is self-conserved Using the value of 119876 in (9)we have 120588119898 = 1205881198980119886minus3(1minus1198892) (11)

where 1205881198980 is an integration constant Hence (10) finally leadsto the expression for pressure as follows

119901119889 = minus(1198892120588119898 + 120588119889 + 1205881198893119867) (12)

The EoS parameter is used to categorize the deceleratedand accelerated phases of the universe This parameter isdefined as 120596 = 119901120588 (13)

If we take 120596 = 0 it corresponds to nonrelativistic matter andthe decelerated phase of the universe involves radiation era0 lt 120596 lt 13 120596 = minus1 minus1 lt 120596 lt minus13 and 120596 lt minus1 correspondto the cosmological constant quintessence and phantomeras respectively To analyze the dynamical properties of theDE models we use 120596 minus 1205961015840 plane [30] This plane describesthe evolutionary universe with two different cases freezingregion and thawing region In the freezing region the valuesof EoS parameter and evolutionary parameter are negative(120596 lt 0 and 1205961015840 lt 0) while for the thawing region the valueof EoS parameter is negative and evolutionary parameter ispositive (120596 lt 0 and 1205961015840 gt 0) In order to check the stability oftheDEmodels we need to evaluate the squared sound speedwhich is given by

V2119904 = 119889119901119889120588 = 119889119901119889119905119889120588119889119905 (14)

The sign of V2119904 decides its stability of DEmodels when V2119904 gt 0the model is stable otherwise it is unstable

3 Tsallis Holographic Dark Energy

The definition and derivation of standard HDE density aregiven by 120588119889 = 3119888211989811990121198712 where 1198981199012 represents reducedPlank mass and 119871 denotes the infrared cut-off It dependsupon the entropy area relationship of black holes ie 119878 sim119860 sim 1198712 where 119860 = 41205871198712 represents the area of the horizonTsallis and Cirto [31] showed that the horizon entropy of theblack hole can be modified as119878120575 = 120574119860120575 (15)

where 120575 is the nonadditivity parameter and 120574 is an unknownconstant [31] Cohen at al [22] proposed the mutual relation-ship between IR (L) cut-off system entropy (S) and UV (Λ)cut-off as 1198713Λ3 le (119878)34 (16)

Tsallis HDE

minus10

minus08

minus06

minus04

minus02

d

00 05minus05z

Figure 1 Plot of 120596119889 versus 119911 for THDE model where 120575 = 11 1205881198980 =1 1198892 = 0001 119861 = minus13 119887 = 05After combining (15) and (16) we get the following relation

Λ4 le 120574 (4120587)120575 1198712120575minus4 (17)

where Λ4 is vacuum energy density and 120588119889 sim Λ4 So theTsallis HDE density [29] is given as

120588119889 = 1198611198712120575minus4 (18)

Here 119861 is an unknown parameter and IR cut-off is taken asHubble radius which leads to 119871 = 1119867 where 119867 is Hubbleparameter The density of Tsallis HDE model along with itsderivative by using (18) becomes

120588119889 = 1198611198674minus2120575120588119889 = 119861 (4 minus 2120575)1198673minus2120575 (19)

Here is the derivative of Hubble parameter wrt 119905 Thevalue of is calculated in terms of 119911 using 119886 = 1(1 + 119911)which is given as follows

119889119867119889119911= (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)(1 minus (13) 119861 (4 minus 2120575)1198673minus2120575)119867 (1 + 119911)

(20)

Inserting these values in (12) yields

119901119889 = 13 (minus311988921205881198980119886minus3(1minus1198892)minus 1198611198672minus2120575 (31198672 + (4 minus 2120575) )) (21)

The EoS is obtained from (13)

120596119889 = 119901119889120588119889 = minus1 minus 11988921205881198980119886minus3(1minus1198892)1198672120575minus4119861 + (2120575 minus 4) 31198672 (22)

The plot of 120596119889 versus 119911 is shown in Figure 1 In this parameterand further results the function 119867(119911) is being utilizednumerically The other constant parameters are mentioned

4 Advances in High Energy Physics

Tsallis HDE

00

01

02

03

00 05 10minus05z

s2

Figure 2 Plot of V2119904 versus 119911 for THDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119861 = minus13 119887 = 05in Figure 1 The trajectory of EoS parameter remains inquintessence region at early present and latter epoch

The square of the sound speed is given by

V2119904 = 16119861 (120575 minus 2) 11988641198673 (91198892 (1198892 minus 1) 1205881198980119886311988921198672120575 119886minus 2119861 (120575 minus 2) 1198864119867times (31198672 minus 2 (120575 minus 1) 2 + 119867))

(23)

The plot of squared sound speed versus 119911 is shown inFigure 2 for different parametric values This graph is usedto analyze the stability of this model We can see that V2119904 gt 0for minus06 lt 119911 lt 1 which corresponds to the stability of THDEmodel However the model shows instability for 119911 lt minus06

Taking the derivative of the EoS parameter with respectto ln 119886 we get 1205961015840119889 as follows

1205961015840119889 = 1311986111988641198676 (minus311988921205883119889211989801198672120575 (3 (1198892 minus 1)119867 119886+ (2120575 minus 4) ) + 2119861 (120575 minus 2) times 11988641198672 (minus22+ 119867))

(24)

The graph of 120596119889 versus 1205961015840119889 is shown in Figure 3 for which1205961015840119889 depicts positive behavior Hence for 120596119889 lt 0 the evolutionparameter shows1205961015840119889 gt 0 which represents the thawing regionof evolving universe

4 Reacutenyi Holographic Dark Energy Model

Weconsider a systemwith119882 stateswith probability of getting119894th state 119875119894 and satisfying the condition Σ119882119894=1119875119894 = 1 Renyi andTsallis entropies are defined as

S = 1120575 lnΣ119882119894=11198751minus120575119894 119878119879 = 1120575Σ119882119894=1 (1198751minus120575119894 minus 119875119894) (25)

Tsallis HDE

minus08 minus06 minus04 minus02minus10d

0

1

2

3

4

(d)

Figure 3 Plot of 120596119889 versus 1205961015840119889 for THDE model where 120575 = 111205881198980 = 1 1198892 = 0001 119861 = minus13 119887 = 05where 120575 equiv 1minus119880 where119880 is a real parameter Now combiningthe above equations we find their mutual relation given as

S = 1120575 ln (1 + 120575119878119879) (26)

This equation shows that S belongs to the class of mostgeneral entropy functions of homogenous system Recentlyit has been observed that Bekenstein entropy 119878 = 1198604 is infact Tsallis entropy which gives the expression

119878 = 1120575 ln(1 + 1205751198604 ) (27)

which is the Renyi entropy of the system Now for theRHDE we focus on WMAP data for flat universe Using theassumption 120588119889119889V prop 119879119889119904 we can get RHDE density

120588119889 = 3119862211986728120587 (1 + 1205751205871198672) (28)

Considering the term 8120587 = 1 and substituting in (28) we getthe expression for density as

120588119889 = 3119862211986721 + 1205751205871198672 (29)

Now 119889119867119889119911 is given by the following

119889119867119889119911= (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)(1 minus (211988821198672 (1199112 + 120575120587) minus 11988821198674) (1198672 + 120575120587)2)119867 (1 + 119911)

(30)

The pressure for this case is obtained as

119901119889 = minus11988921205881198980119886minus3(1minus1198892)+ 11988821198672 (minus31198672 (120587120575 + 1198672) minus 2 (2120587120575 + 1198672) )(120587120575 + 1198672)2 (31)

Advances in High Energy Physics 5

Renyi HDE

00 05minus05z

minus10

minus08

minus06

minus04

minus02

00

d

Figure 4 Plot of 120596119889 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 005The expressions for EoS parameter 120596119889 can be evaluated from(12) as follows

120596119889 = (120587120575 + 1198672)(minus11988921205881198980119886minus3(1minus1198892)311988821198674minus (31198672 (120587120575 + 1198672) + 2 (2120587120575 + 1198672) )31198672 (120587120575 + 1198672)2 )

(32)

Figure 4 shows the plot of 120596119889 versus 119911 The trajectoryof EoS parameter evolutes the universe from quintessenceregion towards the ΛCDM limit The squared sound speedof this RHDEmodel is given by using (13) as

V1199042 = 3119867(1 minus 1198892) 11988921205881198980119886minus3(1minus1198892) (120587120575 + 1198672)2611988821198673 (2120587120575 + 1198672) minus 131198672 (2120587120575 + 1198672) (120587120575 + 1198672)times (6120587212057521198672 + 91205871205751198674 + 31198676 + 412058721205752)+ 119867 (120587120575 + 1198672) times (2120587120575 + 1198672)

(33)

The graph of squared speed of sound is shown in Figure 5versus 119911 In this case we have V2119904 gt 0 for all ranges of 119911 whichshows the stability of RHDE model at the early present andlatter epoch of the universe

The expression for 1205961015840119889 is evaluated as

1205961015840119889 = 13119888211988641198676 (120587120575 + 1198672)2 minus1198892120588119898011988631198892 (120587120575 + 1198672)2sdot (3119867 119886 (minus1 + 1198892)times (120587120575 + 1198672) minus 2119886 (2120587120575 + 1198672))+ 2119888211988641198672 (412058721205752 + 81205871205751198672 + 21198674) 2minus 2119867(120587120575+ 1198672) (2120587120575 + 1198672)

(34)

In Figure 6 we plot the EoS parameter with its evolutionparameter to discuss 120596119889 minus 1205961015840119889 plane for RHDE model The

Renyi HDE

00 05 10minus05z

0

50

100

150

200

s2

Figure 5 Plot of V2119904 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 15Renyi HDE

00minus06 minus04 minus02minus08minus10d

00

05

10

15

20

25

30

35

(d)

Figure 6 Plot of 120596119889 versus 1205961015840119889 for RHDE model where 120575 = 111205881198980 = 08 1198892 = 0001 119888 = 01 119887 = 005

graph shows that for 120596119889 lt 0 the evolutionary parameterremains positive at the early present and latter epoch Thistype of behavior depicts the thawing region of the evolvinguniverse

5 Sharma-Mittal Holographic DarkEnergy Model

From the Renyi entropy we have the generalized entropycontent of the system Using (26) Sharma-Mittal introduceda two-parametric entropy which is defined as

119878119878119872 = 11 minus 119903 ((Σ119882119894=11198751minus120575119894 )1minus119903120575 minus 1) (35)

where 119903 is a new free parameter We can observe that Renyiand Tsallis entropies can be recovered at the proper limitsusing (25) in (35) we have

119878119878119872 = 1119877 ((1 + 120575119878119879)119877120575 minus 1) (36)

6 Advances in High Energy Physics

where 119877 equiv 1minus119903 Using the argument that Bekenstein entropyis the proper candidate for Tsallis entropy by using 119878 = 1198604where 119860 is horizon entropy we get the following expression

119878119878119872 = 1119877 ((1 + 1205751198604 )119877120575 minus 1) (37)

and the relation of UV (Λ) cut-off IR (L) cut-off and systemhorizon (S) is given as follows

Λ4 prop 1198781198714 (38)

Now taking 119871 equiv 1119867 = radic1198604120587 then the energydensity of DE given by Sharma-Mittal [29] is consideredas

120588119889 = 3119888211986748120587119877 [(1 + 1205751205871198672)119877120575 minus 1] (39)

where 1198882 is an unknown free parameter Using 8120587 = 1 inabove equation we get the following expression for energydensity

120588119889 = 311988821198674119877 [(1 + 1205751205871198672)119877120575 minus 1] (40)

The differential equation of119867 is given by the following

119889119867119889119911 = (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)1 + 1198882120587 (1 + (1205751205871198672))119877120575minus1 minus (211988821198672119877) ((1 + 1205751205871198672)119877120575 minus 1)119867 (1 + 119911) (41)

The pressure can be evaluated by energy conservation (11) asfollows

119901119889 = minus11988921205881198980119886minus3(1minus1198892)minus 1198882(3((1 + 1205871205751198672)119877120575 minus 1)1198674119877minus 2120587(1 + 1205871205751198672)119877120575minus1

+ 4 ((1 + 1205871205751198672)119877120575 minus 1)1198672119877 )

(42)

The EoS parameter for this model is given by

120596119889 = 21198882 (120587(1 + 1205871205751198672)119877120575minus1minus 21198672119877 ((1 + 1205871205751198672 minus 1)119877120575))minus 11988921198771205881198980119886minus3(1minus1198892)311988821198674 ((1 + 1205751205871198672)119877120575 minus 1) minus 1

(43)

The plot of 120596119889 versus 119911 is shown in Figure 7 The EoSparameter represents the quintessence nature of the universeThe square of the sound speed is evaluated as

V2119904 = 161198882119867(minus120587 (1 + 1205871205751198672)119877120575minus1 + (21198672119877) ((1 + 1205871205751198672)119877120575 minus 1)) times minus31198892119867(minus1 + 1198892) 1205881198980119886minus3(1minus1198892)+ 21198882119867119877 (61198672 + 42 + 2119867 minus 1(120587120575 + 1198672)2 (1 + 1205871205751198672)119877120575sdot (31198672 (120587120575 + 1198672) (minus120587119877 + 2120587120575 + 21198672) + 2 (1205872 (119877 minus 2120575) (119877 minus 120575) minus 22120587 (119877 minus 2120575)1198672 + 21198674))+ 119867 (120587120575 + 1198672) times (minus120587119877 + 2120587120575 + 21198672) )

(44)

Advances in High Energy Physics 7

SharmandashMittal HDE

minus06minus05minus04minus03minus02minus01

00

d

00 05 10minus05z

Figure 7 Plot of 120596119889 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 0011198892 = 0001 119888 = 001 119887 = 04 119877 = 7In Figure 8 we draw V2119904 versus 119911which shows the unstable

behavior of the SMHDE model as V2119904 lt 0 at early presentand latter epoch

1205961015840119889 = minus 13 ((1 + 1205871205751198672)119877120575 minus 1)21198676 ( 1(120587120575 + 1198672)2sdot 21198672 (2(minus2 (120587120575 + 1198672)2 + (1 + 1205871205751198672)2119877120575sdot (1205872 (119877 minus 2120575) 120575 + 2120587 (119877 minus 2120575)1198672 minus 21198674)+ (1 + 1205871205751198672)119877120575times (minus1205872 (1198772 + 119877120575 minus 41205752) minus 2120587 (119877 minus 4120575)1198672 + 41198674))sdot 2 + (120587120575 + 1198672) times ((1 + 1205871205751198672)119877120575 minus 1)sdot 119867(minus2 (120587120575 + 1198672) + (1 + 1205871205751198672)119877120575times (minus120587119877 + 2120587120575 + 21198672)) ) + 31198892 (minus1 + 1198892)1198882sdot 1205881198980119877119886minus3(1minus1198892)1198672 times ((1 + 1205871205751198672)119877120575 minus 1)+ 211988921205881198980119877119886minus3(1minus1198892)1198882 (120587120575 + 1198672) ((120587 (119877 minus 2120575) minus 21198672) times (1+ 1205871205751198672)119877120575 + 2 (120587120575 + 1198672)) )

(45)

Figure 9 shows the plot of 120596119889-1205961015840119889 plane to classify thedynamical region for the given model We can see that 1205961015840119889 gt0 for 120596119889 lt 0 which indicates the thawing region of theuniverse

SharmandashMittal HDE

0200 04 06minus04minus06 minus02minus08z

minus6

minus5

minus4

minus3

minus2

s2

Figure 8 Plot of V2119904 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 081198892 = 0001 119888 = 08 119887 = 005 119877 = 7SharmandashMittal HDE

0

500

1000

1500

(d)

00minus04 minus03 minus02 minus01minus05minus06d

Figure 9 Plot of 120596119889 versus 1205961015840119889 for different values of 120575 for SMHDEwhere 120575 = 11 1205881198980 = 001 1198892 = 0001 119888 = 001 119887 = 04 119877 = 7

Table 1 Summary of the cosmological parameters and plane

DEmodels 120596119889 V2119904 120596119889 minus 1205961015840119889THDE quintessence-to-

vacuum partially stability thawing region

RHDE quintessence-to-vacuum stability thawing region

SMHDE quintessence un-stable thawing region

6 Conclusion

In this paper we have discussed the THDE RHDE andSMHDE models in the framework of Chern-Simons modi-fied theory of gravity We have taken the flat FRW universeand linear interaction term is chosen for the interactingscenario between DE and dark matter We have evaluatedthe different cosmological parameters (equation of stateparameter and squared sound speed) 120596119889 minus 1205961015840119889 cosmologicalplane The trajectories of all these models have been plottedwith different constant parametric values

We have summarized our results in Table 1Jawad et al [32] have explored various cosmological

parameters (equation of state squared speed of sound Om-diagnostic) and cosmological planes in the framework ofdynamical Chern-Simons modified gravity with the newholographic dark energy model They observed that the

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

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4 Advances in High Energy Physics

Tsallis HDE

00

01

02

03

00 05 10minus05z

s2

Figure 2 Plot of V2119904 versus 119911 for THDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119861 = minus13 119887 = 05in Figure 1 The trajectory of EoS parameter remains inquintessence region at early present and latter epoch

The square of the sound speed is given by

V2119904 = 16119861 (120575 minus 2) 11988641198673 (91198892 (1198892 minus 1) 1205881198980119886311988921198672120575 119886minus 2119861 (120575 minus 2) 1198864119867times (31198672 minus 2 (120575 minus 1) 2 + 119867))

(23)

The plot of squared sound speed versus 119911 is shown inFigure 2 for different parametric values This graph is usedto analyze the stability of this model We can see that V2119904 gt 0for minus06 lt 119911 lt 1 which corresponds to the stability of THDEmodel However the model shows instability for 119911 lt minus06

Taking the derivative of the EoS parameter with respectto ln 119886 we get 1205961015840119889 as follows

1205961015840119889 = 1311986111988641198676 (minus311988921205883119889211989801198672120575 (3 (1198892 minus 1)119867 119886+ (2120575 minus 4) ) + 2119861 (120575 minus 2) times 11988641198672 (minus22+ 119867))

(24)

The graph of 120596119889 versus 1205961015840119889 is shown in Figure 3 for which1205961015840119889 depicts positive behavior Hence for 120596119889 lt 0 the evolutionparameter shows1205961015840119889 gt 0 which represents the thawing regionof evolving universe

4 Reacutenyi Holographic Dark Energy Model

Weconsider a systemwith119882 stateswith probability of getting119894th state 119875119894 and satisfying the condition Σ119882119894=1119875119894 = 1 Renyi andTsallis entropies are defined as

S = 1120575 lnΣ119882119894=11198751minus120575119894 119878119879 = 1120575Σ119882119894=1 (1198751minus120575119894 minus 119875119894) (25)

Tsallis HDE

minus08 minus06 minus04 minus02minus10d

0

1

2

3

4

(d)

Figure 3 Plot of 120596119889 versus 1205961015840119889 for THDE model where 120575 = 111205881198980 = 1 1198892 = 0001 119861 = minus13 119887 = 05where 120575 equiv 1minus119880 where119880 is a real parameter Now combiningthe above equations we find their mutual relation given as

S = 1120575 ln (1 + 120575119878119879) (26)

This equation shows that S belongs to the class of mostgeneral entropy functions of homogenous system Recentlyit has been observed that Bekenstein entropy 119878 = 1198604 is infact Tsallis entropy which gives the expression

119878 = 1120575 ln(1 + 1205751198604 ) (27)

which is the Renyi entropy of the system Now for theRHDE we focus on WMAP data for flat universe Using theassumption 120588119889119889V prop 119879119889119904 we can get RHDE density

120588119889 = 3119862211986728120587 (1 + 1205751205871198672) (28)

Considering the term 8120587 = 1 and substituting in (28) we getthe expression for density as

120588119889 = 3119862211986721 + 1205751205871198672 (29)

Now 119889119867119889119911 is given by the following

119889119867119889119911= (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)(1 minus (211988821198672 (1199112 + 120575120587) minus 11988821198674) (1198672 + 120575120587)2)119867 (1 + 119911)

(30)

The pressure for this case is obtained as

119901119889 = minus11988921205881198980119886minus3(1minus1198892)+ 11988821198672 (minus31198672 (120587120575 + 1198672) minus 2 (2120587120575 + 1198672) )(120587120575 + 1198672)2 (31)

Advances in High Energy Physics 5

Renyi HDE

00 05minus05z

minus10

minus08

minus06

minus04

minus02

00

d

Figure 4 Plot of 120596119889 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 005The expressions for EoS parameter 120596119889 can be evaluated from(12) as follows

120596119889 = (120587120575 + 1198672)(minus11988921205881198980119886minus3(1minus1198892)311988821198674minus (31198672 (120587120575 + 1198672) + 2 (2120587120575 + 1198672) )31198672 (120587120575 + 1198672)2 )

(32)

Figure 4 shows the plot of 120596119889 versus 119911 The trajectoryof EoS parameter evolutes the universe from quintessenceregion towards the ΛCDM limit The squared sound speedof this RHDEmodel is given by using (13) as

V1199042 = 3119867(1 minus 1198892) 11988921205881198980119886minus3(1minus1198892) (120587120575 + 1198672)2611988821198673 (2120587120575 + 1198672) minus 131198672 (2120587120575 + 1198672) (120587120575 + 1198672)times (6120587212057521198672 + 91205871205751198674 + 31198676 + 412058721205752)+ 119867 (120587120575 + 1198672) times (2120587120575 + 1198672)

(33)

The graph of squared speed of sound is shown in Figure 5versus 119911 In this case we have V2119904 gt 0 for all ranges of 119911 whichshows the stability of RHDE model at the early present andlatter epoch of the universe

The expression for 1205961015840119889 is evaluated as

1205961015840119889 = 13119888211988641198676 (120587120575 + 1198672)2 minus1198892120588119898011988631198892 (120587120575 + 1198672)2sdot (3119867 119886 (minus1 + 1198892)times (120587120575 + 1198672) minus 2119886 (2120587120575 + 1198672))+ 2119888211988641198672 (412058721205752 + 81205871205751198672 + 21198674) 2minus 2119867(120587120575+ 1198672) (2120587120575 + 1198672)

(34)

In Figure 6 we plot the EoS parameter with its evolutionparameter to discuss 120596119889 minus 1205961015840119889 plane for RHDE model The

Renyi HDE

00 05 10minus05z

0

50

100

150

200

s2

Figure 5 Plot of V2119904 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 15Renyi HDE

00minus06 minus04 minus02minus08minus10d

00

05

10

15

20

25

30

35

(d)

Figure 6 Plot of 120596119889 versus 1205961015840119889 for RHDE model where 120575 = 111205881198980 = 08 1198892 = 0001 119888 = 01 119887 = 005

graph shows that for 120596119889 lt 0 the evolutionary parameterremains positive at the early present and latter epoch Thistype of behavior depicts the thawing region of the evolvinguniverse

5 Sharma-Mittal Holographic DarkEnergy Model

From the Renyi entropy we have the generalized entropycontent of the system Using (26) Sharma-Mittal introduceda two-parametric entropy which is defined as

119878119878119872 = 11 minus 119903 ((Σ119882119894=11198751minus120575119894 )1minus119903120575 minus 1) (35)

where 119903 is a new free parameter We can observe that Renyiand Tsallis entropies can be recovered at the proper limitsusing (25) in (35) we have

119878119878119872 = 1119877 ((1 + 120575119878119879)119877120575 minus 1) (36)

6 Advances in High Energy Physics

where 119877 equiv 1minus119903 Using the argument that Bekenstein entropyis the proper candidate for Tsallis entropy by using 119878 = 1198604where 119860 is horizon entropy we get the following expression

119878119878119872 = 1119877 ((1 + 1205751198604 )119877120575 minus 1) (37)

and the relation of UV (Λ) cut-off IR (L) cut-off and systemhorizon (S) is given as follows

Λ4 prop 1198781198714 (38)

Now taking 119871 equiv 1119867 = radic1198604120587 then the energydensity of DE given by Sharma-Mittal [29] is consideredas

120588119889 = 3119888211986748120587119877 [(1 + 1205751205871198672)119877120575 minus 1] (39)

where 1198882 is an unknown free parameter Using 8120587 = 1 inabove equation we get the following expression for energydensity

120588119889 = 311988821198674119877 [(1 + 1205751205871198672)119877120575 minus 1] (40)

The differential equation of119867 is given by the following

119889119867119889119911 = (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)1 + 1198882120587 (1 + (1205751205871198672))119877120575minus1 minus (211988821198672119877) ((1 + 1205751205871198672)119877120575 minus 1)119867 (1 + 119911) (41)

The pressure can be evaluated by energy conservation (11) asfollows

119901119889 = minus11988921205881198980119886minus3(1minus1198892)minus 1198882(3((1 + 1205871205751198672)119877120575 minus 1)1198674119877minus 2120587(1 + 1205871205751198672)119877120575minus1

+ 4 ((1 + 1205871205751198672)119877120575 minus 1)1198672119877 )

(42)

The EoS parameter for this model is given by

120596119889 = 21198882 (120587(1 + 1205871205751198672)119877120575minus1minus 21198672119877 ((1 + 1205871205751198672 minus 1)119877120575))minus 11988921198771205881198980119886minus3(1minus1198892)311988821198674 ((1 + 1205751205871198672)119877120575 minus 1) minus 1

(43)

The plot of 120596119889 versus 119911 is shown in Figure 7 The EoSparameter represents the quintessence nature of the universeThe square of the sound speed is evaluated as

V2119904 = 161198882119867(minus120587 (1 + 1205871205751198672)119877120575minus1 + (21198672119877) ((1 + 1205871205751198672)119877120575 minus 1)) times minus31198892119867(minus1 + 1198892) 1205881198980119886minus3(1minus1198892)+ 21198882119867119877 (61198672 + 42 + 2119867 minus 1(120587120575 + 1198672)2 (1 + 1205871205751198672)119877120575sdot (31198672 (120587120575 + 1198672) (minus120587119877 + 2120587120575 + 21198672) + 2 (1205872 (119877 minus 2120575) (119877 minus 120575) minus 22120587 (119877 minus 2120575)1198672 + 21198674))+ 119867 (120587120575 + 1198672) times (minus120587119877 + 2120587120575 + 21198672) )

(44)

Advances in High Energy Physics 7

SharmandashMittal HDE

minus06minus05minus04minus03minus02minus01

00

d

00 05 10minus05z

Figure 7 Plot of 120596119889 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 0011198892 = 0001 119888 = 001 119887 = 04 119877 = 7In Figure 8 we draw V2119904 versus 119911which shows the unstable

behavior of the SMHDE model as V2119904 lt 0 at early presentand latter epoch

1205961015840119889 = minus 13 ((1 + 1205871205751198672)119877120575 minus 1)21198676 ( 1(120587120575 + 1198672)2sdot 21198672 (2(minus2 (120587120575 + 1198672)2 + (1 + 1205871205751198672)2119877120575sdot (1205872 (119877 minus 2120575) 120575 + 2120587 (119877 minus 2120575)1198672 minus 21198674)+ (1 + 1205871205751198672)119877120575times (minus1205872 (1198772 + 119877120575 minus 41205752) minus 2120587 (119877 minus 4120575)1198672 + 41198674))sdot 2 + (120587120575 + 1198672) times ((1 + 1205871205751198672)119877120575 minus 1)sdot 119867(minus2 (120587120575 + 1198672) + (1 + 1205871205751198672)119877120575times (minus120587119877 + 2120587120575 + 21198672)) ) + 31198892 (minus1 + 1198892)1198882sdot 1205881198980119877119886minus3(1minus1198892)1198672 times ((1 + 1205871205751198672)119877120575 minus 1)+ 211988921205881198980119877119886minus3(1minus1198892)1198882 (120587120575 + 1198672) ((120587 (119877 minus 2120575) minus 21198672) times (1+ 1205871205751198672)119877120575 + 2 (120587120575 + 1198672)) )

(45)

Figure 9 shows the plot of 120596119889-1205961015840119889 plane to classify thedynamical region for the given model We can see that 1205961015840119889 gt0 for 120596119889 lt 0 which indicates the thawing region of theuniverse

SharmandashMittal HDE

0200 04 06minus04minus06 minus02minus08z

minus6

minus5

minus4

minus3

minus2

s2

Figure 8 Plot of V2119904 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 081198892 = 0001 119888 = 08 119887 = 005 119877 = 7SharmandashMittal HDE

0

500

1000

1500

(d)

00minus04 minus03 minus02 minus01minus05minus06d

Figure 9 Plot of 120596119889 versus 1205961015840119889 for different values of 120575 for SMHDEwhere 120575 = 11 1205881198980 = 001 1198892 = 0001 119888 = 001 119887 = 04 119877 = 7

Table 1 Summary of the cosmological parameters and plane

DEmodels 120596119889 V2119904 120596119889 minus 1205961015840119889THDE quintessence-to-

vacuum partially stability thawing region

RHDE quintessence-to-vacuum stability thawing region

SMHDE quintessence un-stable thawing region

6 Conclusion

In this paper we have discussed the THDE RHDE andSMHDE models in the framework of Chern-Simons modi-fied theory of gravity We have taken the flat FRW universeand linear interaction term is chosen for the interactingscenario between DE and dark matter We have evaluatedthe different cosmological parameters (equation of stateparameter and squared sound speed) 120596119889 minus 1205961015840119889 cosmologicalplane The trajectories of all these models have been plottedwith different constant parametric values

We have summarized our results in Table 1Jawad et al [32] have explored various cosmological

parameters (equation of state squared speed of sound Om-diagnostic) and cosmological planes in the framework ofdynamical Chern-Simons modified gravity with the newholographic dark energy model They observed that the

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 5

Renyi HDE

00 05minus05z

minus10

minus08

minus06

minus04

minus02

00

d

Figure 4 Plot of 120596119889 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 005The expressions for EoS parameter 120596119889 can be evaluated from(12) as follows

120596119889 = (120587120575 + 1198672)(minus11988921205881198980119886minus3(1minus1198892)311988821198674minus (31198672 (120587120575 + 1198672) + 2 (2120587120575 + 1198672) )31198672 (120587120575 + 1198672)2 )

(32)

Figure 4 shows the plot of 120596119889 versus 119911 The trajectoryof EoS parameter evolutes the universe from quintessenceregion towards the ΛCDM limit The squared sound speedof this RHDEmodel is given by using (13) as

V1199042 = 3119867(1 minus 1198892) 11988921205881198980119886minus3(1minus1198892) (120587120575 + 1198672)2611988821198673 (2120587120575 + 1198672) minus 131198672 (2120587120575 + 1198672) (120587120575 + 1198672)times (6120587212057521198672 + 91205871205751198674 + 31198676 + 412058721205752)+ 119867 (120587120575 + 1198672) times (2120587120575 + 1198672)

(33)

The graph of squared speed of sound is shown in Figure 5versus 119911 In this case we have V2119904 gt 0 for all ranges of 119911 whichshows the stability of RHDE model at the early present andlatter epoch of the universe

The expression for 1205961015840119889 is evaluated as

1205961015840119889 = 13119888211988641198676 (120587120575 + 1198672)2 minus1198892120588119898011988631198892 (120587120575 + 1198672)2sdot (3119867 119886 (minus1 + 1198892)times (120587120575 + 1198672) minus 2119886 (2120587120575 + 1198672))+ 2119888211988641198672 (412058721205752 + 81205871205751198672 + 21198674) 2minus 2119867(120587120575+ 1198672) (2120587120575 + 1198672)

(34)

In Figure 6 we plot the EoS parameter with its evolutionparameter to discuss 120596119889 minus 1205961015840119889 plane for RHDE model The

Renyi HDE

00 05 10minus05z

0

50

100

150

200

s2

Figure 5 Plot of V2119904 versus 119911 for RHDE model where 120575 = 11 1205881198980 =08 1198892 = 0001 119888 = 01 119887 = 15Renyi HDE

00minus06 minus04 minus02minus08minus10d

00

05

10

15

20

25

30

35

(d)

Figure 6 Plot of 120596119889 versus 1205961015840119889 for RHDE model where 120575 = 111205881198980 = 08 1198892 = 0001 119888 = 01 119887 = 005

graph shows that for 120596119889 lt 0 the evolutionary parameterremains positive at the early present and latter epoch Thistype of behavior depicts the thawing region of the evolvinguniverse

5 Sharma-Mittal Holographic DarkEnergy Model

From the Renyi entropy we have the generalized entropycontent of the system Using (26) Sharma-Mittal introduceda two-parametric entropy which is defined as

119878119878119872 = 11 minus 119903 ((Σ119882119894=11198751minus120575119894 )1minus119903120575 minus 1) (35)

where 119903 is a new free parameter We can observe that Renyiand Tsallis entropies can be recovered at the proper limitsusing (25) in (35) we have

119878119878119872 = 1119877 ((1 + 120575119878119879)119877120575 minus 1) (36)

6 Advances in High Energy Physics

where 119877 equiv 1minus119903 Using the argument that Bekenstein entropyis the proper candidate for Tsallis entropy by using 119878 = 1198604where 119860 is horizon entropy we get the following expression

119878119878119872 = 1119877 ((1 + 1205751198604 )119877120575 minus 1) (37)

and the relation of UV (Λ) cut-off IR (L) cut-off and systemhorizon (S) is given as follows

Λ4 prop 1198781198714 (38)

Now taking 119871 equiv 1119867 = radic1198604120587 then the energydensity of DE given by Sharma-Mittal [29] is consideredas

120588119889 = 3119888211986748120587119877 [(1 + 1205751205871198672)119877120575 minus 1] (39)

where 1198882 is an unknown free parameter Using 8120587 = 1 inabove equation we get the following expression for energydensity

120588119889 = 311988821198674119877 [(1 + 1205751205871198672)119877120575 minus 1] (40)

The differential equation of119867 is given by the following

119889119867119889119911 = (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)1 + 1198882120587 (1 + (1205751205871198672))119877120575minus1 minus (211988821198672119877) ((1 + 1205751205871198672)119877120575 minus 1)119867 (1 + 119911) (41)

The pressure can be evaluated by energy conservation (11) asfollows

119901119889 = minus11988921205881198980119886minus3(1minus1198892)minus 1198882(3((1 + 1205871205751198672)119877120575 minus 1)1198674119877minus 2120587(1 + 1205871205751198672)119877120575minus1

+ 4 ((1 + 1205871205751198672)119877120575 minus 1)1198672119877 )

(42)

The EoS parameter for this model is given by

120596119889 = 21198882 (120587(1 + 1205871205751198672)119877120575minus1minus 21198672119877 ((1 + 1205871205751198672 minus 1)119877120575))minus 11988921198771205881198980119886minus3(1minus1198892)311988821198674 ((1 + 1205751205871198672)119877120575 minus 1) minus 1

(43)

The plot of 120596119889 versus 119911 is shown in Figure 7 The EoSparameter represents the quintessence nature of the universeThe square of the sound speed is evaluated as

V2119904 = 161198882119867(minus120587 (1 + 1205871205751198672)119877120575minus1 + (21198672119877) ((1 + 1205871205751198672)119877120575 minus 1)) times minus31198892119867(minus1 + 1198892) 1205881198980119886minus3(1minus1198892)+ 21198882119867119877 (61198672 + 42 + 2119867 minus 1(120587120575 + 1198672)2 (1 + 1205871205751198672)119877120575sdot (31198672 (120587120575 + 1198672) (minus120587119877 + 2120587120575 + 21198672) + 2 (1205872 (119877 minus 2120575) (119877 minus 120575) minus 22120587 (119877 minus 2120575)1198672 + 21198674))+ 119867 (120587120575 + 1198672) times (minus120587119877 + 2120587120575 + 21198672) )

(44)

Advances in High Energy Physics 7

SharmandashMittal HDE

minus06minus05minus04minus03minus02minus01

00

d

00 05 10minus05z

Figure 7 Plot of 120596119889 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 0011198892 = 0001 119888 = 001 119887 = 04 119877 = 7In Figure 8 we draw V2119904 versus 119911which shows the unstable

behavior of the SMHDE model as V2119904 lt 0 at early presentand latter epoch

1205961015840119889 = minus 13 ((1 + 1205871205751198672)119877120575 minus 1)21198676 ( 1(120587120575 + 1198672)2sdot 21198672 (2(minus2 (120587120575 + 1198672)2 + (1 + 1205871205751198672)2119877120575sdot (1205872 (119877 minus 2120575) 120575 + 2120587 (119877 minus 2120575)1198672 minus 21198674)+ (1 + 1205871205751198672)119877120575times (minus1205872 (1198772 + 119877120575 minus 41205752) minus 2120587 (119877 minus 4120575)1198672 + 41198674))sdot 2 + (120587120575 + 1198672) times ((1 + 1205871205751198672)119877120575 minus 1)sdot 119867(minus2 (120587120575 + 1198672) + (1 + 1205871205751198672)119877120575times (minus120587119877 + 2120587120575 + 21198672)) ) + 31198892 (minus1 + 1198892)1198882sdot 1205881198980119877119886minus3(1minus1198892)1198672 times ((1 + 1205871205751198672)119877120575 minus 1)+ 211988921205881198980119877119886minus3(1minus1198892)1198882 (120587120575 + 1198672) ((120587 (119877 minus 2120575) minus 21198672) times (1+ 1205871205751198672)119877120575 + 2 (120587120575 + 1198672)) )

(45)

Figure 9 shows the plot of 120596119889-1205961015840119889 plane to classify thedynamical region for the given model We can see that 1205961015840119889 gt0 for 120596119889 lt 0 which indicates the thawing region of theuniverse

SharmandashMittal HDE

0200 04 06minus04minus06 minus02minus08z

minus6

minus5

minus4

minus3

minus2

s2

Figure 8 Plot of V2119904 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 081198892 = 0001 119888 = 08 119887 = 005 119877 = 7SharmandashMittal HDE

0

500

1000

1500

(d)

00minus04 minus03 minus02 minus01minus05minus06d

Figure 9 Plot of 120596119889 versus 1205961015840119889 for different values of 120575 for SMHDEwhere 120575 = 11 1205881198980 = 001 1198892 = 0001 119888 = 001 119887 = 04 119877 = 7

Table 1 Summary of the cosmological parameters and plane

DEmodels 120596119889 V2119904 120596119889 minus 1205961015840119889THDE quintessence-to-

vacuum partially stability thawing region

RHDE quintessence-to-vacuum stability thawing region

SMHDE quintessence un-stable thawing region

6 Conclusion

In this paper we have discussed the THDE RHDE andSMHDE models in the framework of Chern-Simons modi-fied theory of gravity We have taken the flat FRW universeand linear interaction term is chosen for the interactingscenario between DE and dark matter We have evaluatedthe different cosmological parameters (equation of stateparameter and squared sound speed) 120596119889 minus 1205961015840119889 cosmologicalplane The trajectories of all these models have been plottedwith different constant parametric values

We have summarized our results in Table 1Jawad et al [32] have explored various cosmological

parameters (equation of state squared speed of sound Om-diagnostic) and cosmological planes in the framework ofdynamical Chern-Simons modified gravity with the newholographic dark energy model They observed that the

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

6 Advances in High Energy Physics

where 119877 equiv 1minus119903 Using the argument that Bekenstein entropyis the proper candidate for Tsallis entropy by using 119878 = 1198604where 119860 is horizon entropy we get the following expression

119878119878119872 = 1119877 ((1 + 1205751198604 )119877120575 minus 1) (37)

and the relation of UV (Λ) cut-off IR (L) cut-off and systemhorizon (S) is given as follows

Λ4 prop 1198781198714 (38)

Now taking 119871 equiv 1119867 = radic1198604120587 then the energydensity of DE given by Sharma-Mittal [29] is consideredas

120588119889 = 3119888211986748120587119877 [(1 + 1205751205871198672)119877120575 minus 1] (39)

where 1198882 is an unknown free parameter Using 8120587 = 1 inabove equation we get the following expression for energydensity

120588119889 = 311988821198674119877 [(1 + 1205751205871198672)119877120575 minus 1] (40)

The differential equation of119867 is given by the following

119889119867119889119911 = (12) (1205881198980 (1 minus 1198892) (1 + 119911)3(1minus1198892) + 1198872 (1 + 119911)6)1 + 1198882120587 (1 + (1205751205871198672))119877120575minus1 minus (211988821198672119877) ((1 + 1205751205871198672)119877120575 minus 1)119867 (1 + 119911) (41)

The pressure can be evaluated by energy conservation (11) asfollows

119901119889 = minus11988921205881198980119886minus3(1minus1198892)minus 1198882(3((1 + 1205871205751198672)119877120575 minus 1)1198674119877minus 2120587(1 + 1205871205751198672)119877120575minus1

+ 4 ((1 + 1205871205751198672)119877120575 minus 1)1198672119877 )

(42)

The EoS parameter for this model is given by

120596119889 = 21198882 (120587(1 + 1205871205751198672)119877120575minus1minus 21198672119877 ((1 + 1205871205751198672 minus 1)119877120575))minus 11988921198771205881198980119886minus3(1minus1198892)311988821198674 ((1 + 1205751205871198672)119877120575 minus 1) minus 1

(43)

The plot of 120596119889 versus 119911 is shown in Figure 7 The EoSparameter represents the quintessence nature of the universeThe square of the sound speed is evaluated as

V2119904 = 161198882119867(minus120587 (1 + 1205871205751198672)119877120575minus1 + (21198672119877) ((1 + 1205871205751198672)119877120575 minus 1)) times minus31198892119867(minus1 + 1198892) 1205881198980119886minus3(1minus1198892)+ 21198882119867119877 (61198672 + 42 + 2119867 minus 1(120587120575 + 1198672)2 (1 + 1205871205751198672)119877120575sdot (31198672 (120587120575 + 1198672) (minus120587119877 + 2120587120575 + 21198672) + 2 (1205872 (119877 minus 2120575) (119877 minus 120575) minus 22120587 (119877 minus 2120575)1198672 + 21198674))+ 119867 (120587120575 + 1198672) times (minus120587119877 + 2120587120575 + 21198672) )

(44)

Advances in High Energy Physics 7

SharmandashMittal HDE

minus06minus05minus04minus03minus02minus01

00

d

00 05 10minus05z

Figure 7 Plot of 120596119889 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 0011198892 = 0001 119888 = 001 119887 = 04 119877 = 7In Figure 8 we draw V2119904 versus 119911which shows the unstable

behavior of the SMHDE model as V2119904 lt 0 at early presentand latter epoch

1205961015840119889 = minus 13 ((1 + 1205871205751198672)119877120575 minus 1)21198676 ( 1(120587120575 + 1198672)2sdot 21198672 (2(minus2 (120587120575 + 1198672)2 + (1 + 1205871205751198672)2119877120575sdot (1205872 (119877 minus 2120575) 120575 + 2120587 (119877 minus 2120575)1198672 minus 21198674)+ (1 + 1205871205751198672)119877120575times (minus1205872 (1198772 + 119877120575 minus 41205752) minus 2120587 (119877 minus 4120575)1198672 + 41198674))sdot 2 + (120587120575 + 1198672) times ((1 + 1205871205751198672)119877120575 minus 1)sdot 119867(minus2 (120587120575 + 1198672) + (1 + 1205871205751198672)119877120575times (minus120587119877 + 2120587120575 + 21198672)) ) + 31198892 (minus1 + 1198892)1198882sdot 1205881198980119877119886minus3(1minus1198892)1198672 times ((1 + 1205871205751198672)119877120575 minus 1)+ 211988921205881198980119877119886minus3(1minus1198892)1198882 (120587120575 + 1198672) ((120587 (119877 minus 2120575) minus 21198672) times (1+ 1205871205751198672)119877120575 + 2 (120587120575 + 1198672)) )

(45)

Figure 9 shows the plot of 120596119889-1205961015840119889 plane to classify thedynamical region for the given model We can see that 1205961015840119889 gt0 for 120596119889 lt 0 which indicates the thawing region of theuniverse

SharmandashMittal HDE

0200 04 06minus04minus06 minus02minus08z

minus6

minus5

minus4

minus3

minus2

s2

Figure 8 Plot of V2119904 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 081198892 = 0001 119888 = 08 119887 = 005 119877 = 7SharmandashMittal HDE

0

500

1000

1500

(d)

00minus04 minus03 minus02 minus01minus05minus06d

Figure 9 Plot of 120596119889 versus 1205961015840119889 for different values of 120575 for SMHDEwhere 120575 = 11 1205881198980 = 001 1198892 = 0001 119888 = 001 119887 = 04 119877 = 7

Table 1 Summary of the cosmological parameters and plane

DEmodels 120596119889 V2119904 120596119889 minus 1205961015840119889THDE quintessence-to-

vacuum partially stability thawing region

RHDE quintessence-to-vacuum stability thawing region

SMHDE quintessence un-stable thawing region

6 Conclusion

In this paper we have discussed the THDE RHDE andSMHDE models in the framework of Chern-Simons modi-fied theory of gravity We have taken the flat FRW universeand linear interaction term is chosen for the interactingscenario between DE and dark matter We have evaluatedthe different cosmological parameters (equation of stateparameter and squared sound speed) 120596119889 minus 1205961015840119889 cosmologicalplane The trajectories of all these models have been plottedwith different constant parametric values

We have summarized our results in Table 1Jawad et al [32] have explored various cosmological

parameters (equation of state squared speed of sound Om-diagnostic) and cosmological planes in the framework ofdynamical Chern-Simons modified gravity with the newholographic dark energy model They observed that the

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 7

SharmandashMittal HDE

minus06minus05minus04minus03minus02minus01

00

d

00 05 10minus05z

Figure 7 Plot of 120596119889 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 0011198892 = 0001 119888 = 001 119887 = 04 119877 = 7In Figure 8 we draw V2119904 versus 119911which shows the unstable

behavior of the SMHDE model as V2119904 lt 0 at early presentand latter epoch

1205961015840119889 = minus 13 ((1 + 1205871205751198672)119877120575 minus 1)21198676 ( 1(120587120575 + 1198672)2sdot 21198672 (2(minus2 (120587120575 + 1198672)2 + (1 + 1205871205751198672)2119877120575sdot (1205872 (119877 minus 2120575) 120575 + 2120587 (119877 minus 2120575)1198672 minus 21198674)+ (1 + 1205871205751198672)119877120575times (minus1205872 (1198772 + 119877120575 minus 41205752) minus 2120587 (119877 minus 4120575)1198672 + 41198674))sdot 2 + (120587120575 + 1198672) times ((1 + 1205871205751198672)119877120575 minus 1)sdot 119867(minus2 (120587120575 + 1198672) + (1 + 1205871205751198672)119877120575times (minus120587119877 + 2120587120575 + 21198672)) ) + 31198892 (minus1 + 1198892)1198882sdot 1205881198980119877119886minus3(1minus1198892)1198672 times ((1 + 1205871205751198672)119877120575 minus 1)+ 211988921205881198980119877119886minus3(1minus1198892)1198882 (120587120575 + 1198672) ((120587 (119877 minus 2120575) minus 21198672) times (1+ 1205871205751198672)119877120575 + 2 (120587120575 + 1198672)) )

(45)

Figure 9 shows the plot of 120596119889-1205961015840119889 plane to classify thedynamical region for the given model We can see that 1205961015840119889 gt0 for 120596119889 lt 0 which indicates the thawing region of theuniverse

SharmandashMittal HDE

0200 04 06minus04minus06 minus02minus08z

minus6

minus5

minus4

minus3

minus2

s2

Figure 8 Plot of V2119904 versus 119911 for SMHDE where 120575 = 11 1205881198980 = 081198892 = 0001 119888 = 08 119887 = 005 119877 = 7SharmandashMittal HDE

0

500

1000

1500

(d)

00minus04 minus03 minus02 minus01minus05minus06d

Figure 9 Plot of 120596119889 versus 1205961015840119889 for different values of 120575 for SMHDEwhere 120575 = 11 1205881198980 = 001 1198892 = 0001 119888 = 001 119887 = 04 119877 = 7

Table 1 Summary of the cosmological parameters and plane

DEmodels 120596119889 V2119904 120596119889 minus 1205961015840119889THDE quintessence-to-

vacuum partially stability thawing region

RHDE quintessence-to-vacuum stability thawing region

SMHDE quintessence un-stable thawing region

6 Conclusion

In this paper we have discussed the THDE RHDE andSMHDE models in the framework of Chern-Simons modi-fied theory of gravity We have taken the flat FRW universeand linear interaction term is chosen for the interactingscenario between DE and dark matter We have evaluatedthe different cosmological parameters (equation of stateparameter and squared sound speed) 120596119889 minus 1205961015840119889 cosmologicalplane The trajectories of all these models have been plottedwith different constant parametric values

We have summarized our results in Table 1Jawad et al [32] have explored various cosmological

parameters (equation of state squared speed of sound Om-diagnostic) and cosmological planes in the framework ofdynamical Chern-Simons modified gravity with the newholographic dark energy model They observed that the

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

8 Advances in High Energy Physics

equation of state parameter gives consistent ranges by usingdifferent observational schemes They also found that thesquared speed of sound shows a stable solution They sug-gested that the results of cosmological parameters showconsistency with recent observational data Jawad et al [33]have also considered the power law and the entropy correctedHDE models with Hubble horizon in the dynamical Chern-Simons modified gravity They have also explored variouscosmological parameters and planes and found consistentresults with observational data Nadeem et al [34] have alsoinvestigated the interacting modified QCD ghost DE andgeneralized ghost pilgrim DE with cold dark matter in theframework of dynamical Chern-Simons modified gravity Itis found that the results of cosmological parameters as well asplanes explain the accelerated expansion of the universe andare compatible with observational data

However the present work is different from the above-mentioned works in which we have recently proposed DEmodels along with nonlinear interaction term and foundinteresting and compatible results regarding current acceler-ated expansion of the universe

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework of HMoradpour has been supported financially byResearch Institute for Astronomy ampAstrophysics of Maragha(RIAAM) under research project No 15237 minus 8References

[1] A G Riess ldquoObservational evidence from supernovae foran accelerating universe and a cosmological constantrdquo eAstronomical Journal vol 116 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from 42 High-Redshift Supernovaerdquoe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[3] P de Bernardis P A R Ade and J J Bock ldquoA flatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[4] S Perlmutter et al ldquoNew constraints on ΩM ΩΛ and w froman independent set of 11 high-redshift supernovae observedwith the hubble space telescoperdquoe Astrophysical Journal vol598 2003

[5] M Colless et al ldquoThe 2dF galaxy redshift survey luminositydependence of galaxy clusteringrdquoMonthly Notices of the RoyalAstronomical Society vol 328 2001

[6] M Tegmark et al ldquoCosmological parameters from SDSS andWMAPrdquo Physical Review D vol 69 Article ID 103501 2004

[7] S Cole ldquoThe 2dF galaxy redshift survey power-spectrumanalysis of the final data set and cosmological implicationsrdquoMonthly Notices of the Royal Astronomical Society vol 3622005

[8] V Springel C S Frenk and S D M White ldquoThe large-scalestructure of the Universerdquo Nature vol 440 no 7088 pp 1137ndash1144 2006

[9] C B Nettereld P A R Ade and J J Bock ldquoA measurement byBOOMERANG of multiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquoe AstrophysicalJournal vol 571 no 2 pp 604ndash614 2002

[10] D N Spergel et al ldquoFirst-year wilkinson microwave anisotropyprobe (WMAP)lowast observations determination of cosmologicalparametersrdquo e Astrophysical Journal Supplement Series vol148 2003

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 62 Article ID 023511 2000

[12] T M Aliev M Savcı and B B Sirvanlı ldquoDouble-leptonpolarization asymmetries in Λ 119887 997888rarr Λℓ+ℓminus- decay in universalextra dimension modelrdquo e European Physical Journal C vol52 no 2 pp 375ndash382 2007

[13] G Perez-Nadal ldquoStability of de Sitter spacetime under isotropicperturbations in semiclassical gravityrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 77 Article ID124033 2008

[14] S D H Hsu ldquoEntropy bounds and dark energyrdquo Physics LettersB vol 594 no 1-2 pp 13ndash16 2004

[15] S P de Alwis ldquoBrane worlds in 5D and warped compactifica-tions in IIBrdquo Physics Letters B Particle Physics Nuclear Physicsand Cosmology vol 603 no 3-4 pp 230ndash238 2004

[16] K Bamba S Capozziello S Nojiri S D Odintsov and KBamba ldquoDark energy cosmology the equivalent descriptionvia different theoretical models and cosmography testsrdquo Astro-physics and Space Science vol 342 no 1 pp 155ndash228 2012

[17] WA Ponce J B Florez andL A Sanchez ldquoAnalysis of SU(3)119888timesSU(3)119871 times U(1)119883 local Gauge theoryrdquo International Journal ofModern Physics A vol 17 p 643 2002

[18] E H Baffou M J S Houndjo and J Tossa ldquoExploring stablemodels in 119891(119877 119879 119877120583] 119879120583]) gravityrdquo Astrophysics and SpaceScience vol 361 article 376 2016

[19] S Nojiri and S D Odintsov ldquoModified gauss-bonnet theory asgravitational alternative for dark energyrdquo Physics Letters B p 12005

[20] M Roos Introduction to Cosmology John Wiley and Sons UK2003

[21] S Nojiri and S D Odintsov ldquoThe new form of the equation ofstate for dark energy fluid and accelerating universerdquo PhysicsLetters B vol 639 no 3-4 pp 144ndash150 2006

[22] A G Cohen et al ldquoEffective field theory black holes andthe cosmological constantrdquo Physical Review Letters vol 82 pp4971ndash4974 1999

[23] A Pasqua R da Rocha and S Chattopadhyay ldquoHolographicdark energymodels andhigher order generalizations in dynam-ical Chern-Simons modified gravityrdquo e European PhysicalJournal C vol 75 article 44 2015

[24] HMoradpour A Sheykhi C Corda and I G Salako ldquoImplica-tions of the generalized entropy formalisms on the Newtoniangravity and dynamicsrdquo Physics Letters B vol 783 pp 82ndash852018

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 9

[25] H Moradpour A Bonilla E M C Abreu and J A NetoldquoAccelerated cosmos in a nonextensive setuprdquo Physical ReviewD vol 96 Article ID 123504 2017

[26] H Moradpour ldquoImplications consequences and interpreta-tions of generalized entropy in the cosmological setupsrdquo Inter-national Journal of eoretical Physics vol 55 no 9 pp 4176ndash4184 2016

[27] M Tavayef A Sheykhi K Bamba and H Moradpour ldquoTsallisholographic dark energyrdquoPhysics Letters B vol 781 pp 195ndash2002018

[28] HMoradpour et al ldquoThermodynamic approach to holographicdark energy and the Renyi entropyrdquo General Physics 2018

[29] A Sayahian Jahromi S A Moosavi H Moradpour et alldquoGeneralized entropy formalism and a new holographic darkenergy modelrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 780 pp 21ndash24 2018

[30] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[31] C Tsallis and L J L Cirto ldquoBlack hole thermodynamicalentropyrdquoe European Physical Journal C vol 73 no 7 p 24872013

[32] A Jawad S Rani and T Nawaz ldquoInteracting new holographicdark energy in dynamical Chern-Simons modified gravityrdquoeEuropean Physical Journal Plus vol 131 p 282 2016

[33] A Jawad S Rani andNAzhar ldquoEntropy corrected holographicdark energy models in modified gravityrdquo International Journalof Modern Physics D vol 26 Article ID 1750040 2016

[34] N Azhar ldquoCosmological implications of dark energymodels inmodified gravityrdquo International Journal of Geometric Methods inModern Physics vol 15 Article ID 1850034 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom