Research Article Israel-Stewart Approach to Viscous ...downloads.hindawi.com/journals/ahep/2016/8515967.pdf · lled with a bulk viscous cosmological uid, the energy-momentum tensor

Research ArticleIsrael-Stewart Approach to Viscous Dissipative ExtendedHolographic Ricci Dark Energy Dominated Universe

Surajit Chattopadhyay

Pailan College of Management and Technology (MCA Division) Bengal Pailan Park Kolkata 700 104 India

Correspondence should be addressed to Surajit Chattopadhyay surajitchattooutlookcom

Received 16 April 2016 Revised 24 May 2016 Accepted 27 June 2016

Academic Editor Elias C Vagenas

Copyright copy 2016 Surajit Chattopadhyay This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

This paper reports a study on the truncated Israel-Stewart formalism for bulk viscosity using the extended holographic Ricci darkenergy (EHRDE) Under the consideration that the universe is dominated by EHRDE the evolution equation for the bulk viscouspressure Π in the framework of the truncated Israel-Stewart theory has been taken as 120591Π + Π = minus3120585119867 where 120591 is the relaxationtime and 120585 is the bulk viscosity coefficient Considering effective pressure as a sum of thermodynamic pressure of EHRDE and bulkviscous pressure it has been observed that under the influence of bulk viscosity the EoS parameter 119908DE is behaving like phantomthat is 119908DE le minus1 It has been observed that the magnitude of the effective pressure 119901eff = 119901 + Π is decaying with time Wealso investigated the case for a specific choice of scale factor namely 119886(119905) = (119905 minus 119905

0)120573(1minus120572) For this choice we have observed that a

transition from quintessence to phantom is possible for the equation of state parameter However theΛCDMphase is not attainableby the state-finder trajectories for this choice Finally it has been observed that in both of the cases the generalized second law ofthermodynamics is valid for the viscous EHRDE dominated universe enveloped by the apparent horizon

1 Introduction

Accelerated expansion of the current universe was reportedby Riess et al [1] of high-redshift supernovae search team andPerlmutter et al [2] of supernovae cosmology project teamthrough accumulation of observational data from distanttype Ia Supernovae Discovery of [1 2] was truly groundbreaking and subsequently this has been further confirmedby other observational studies including more detailed stud-ies of supernovae and independent evidence from clustersof galaxies large-scale structure (LSS) and the cosmicmicrowave background (CMB) [3] The reason behind thisexpansion is referred to as ldquodark energyrdquo (DE) which isregarded as an exotic matter characterized by negative pres-sure and having equation of state parameter119908 = 119901120588 lt minus13required for accelerated expansion of the universe Nature ofthis DE is not yet clear and different DE candidates have beenproposed till date Some remarkable reviews on DE include[4ndash7]While proposing a phantom cosmology based unifyingapproach to early and late-time universe Nojiri andOdintsov[8] suggested generalized holographic dark energy (HDE)

involving infrared cutoff combined with FRW parametersand also discussed the entropy bound in phantom era HDEis based on ldquoholographic principlerdquo and the density of HDEis [9] 120588

Λ= 311988821198722

119901119871minus2 where 119871 is the infrared cutoff Some

notable works on HDE include [10ndash15] Furthermore thereexist plethora of literatures on HDE in theoretical aspects aswell as observational constraints for example [16ndash18]

In the present work we consider a special form of HDE[19] dubbed as ldquoextended holographic Ricci dark energyrdquo(EHRDE) [20 21] whose density has the folowing form

120588DE = 31198722

119901(1205721198672+ 120573) (1)

where the upper dot represents derivative with respect tocosmic time 119905 1198722

119901is the reduced Planck mass and 120572 and

120573 are constants to be determined In this context we shouldmention that the role of a distance proportional to the Ricciscale as a causal connection scale for perturbations wasnoticed in [22 23] and used for the first time as a DE cutoffscale Detailed cosmology of Ricci DE was discussed in [24]and some further literatures on Ricci DE include [25ndash28]

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 8515967 14 pageshttpdxdoiorg10115520168515967

2 Advances in High Energy Physics

Wang and Xu [29] found the best-fit values in order to makethe cutoff of [20] to be consistent with observational dataas 120572 = 08502+00984+01299

minus00875minus01064and 120573 = 04817+00842+01176

minus00773minus00955 In

the current work we will take 120572 = 098 and 120573 = 037In the early evolution of the universe dissipative effectsincluding both bulk and shear viscosity are supposed toplay a very important role [30] Chimento et al [31] hadshown that accelerated expansion can be derived by thecombination of a cosmic fluid with bulk dissipative pressureand quintessence matter and it can also solve the coincidenceproblem Evolution of the universe involves a sequence ofimportant dissipative processes that includes GUT phasetransition at 119905 asymp 10minus34 s and a temperature of about 119879 asymp1027 K [30] Eckart [32] and Landau and Lifshitz [33] were

the first ones to attempt to create a theory of relativisticdissipative fluids A relativistic second-order theory wasdeveloped by Israel and Stewart [34]

A time dependent viscosity consideration was made toDE by Nojiri and Odintsov [35] while considering EoS withinhomogeneous Hubble parameter dependent term and itwas demonstrated that the thermodynamic entropy may bepositive even in phantom era as a result of crossing the phan-tom boundary Pun et al [30] considered a generalization ofthe Chaplygin gas model and by assuming the presence ofa bulk viscous type dissipative term in the effective thermo-dynamic pressure of the gas and by considering dissipativeeffects described by truncated Israel-Stewart model they[30] had shown that viscous Chaplygin gas model offers aneffective dynamical possibility for replacing the cosmologicalconstant Effects of viscosity in Chaplygin gas with differentmodifications have been studied in [36ndash38] Cataldo et al[39] investigated dissipative processes in the universe in thefull causal Israel-Stewart-Hiscock theory and showed thatthe negative pressure generated by the bulk viscosity cannotavoid the idea that the dark energy of the universe is to bephantom energy Brevik and Gorbunova [40] assumed thebulk viscosity to be proportional to the scalar expansion ina spatially flat FRW universe and had shown that it couldlead the universe to phantom phase even if the universelies in the quintessence phase in the nonviscous scenarioRen and Meng [41] considered a generally parameterizedEoS in the cosmological evolution with bulk viscosity mediamodelled as dark fluid Setare and Sheykhi [42] studied thevalidity of generalized second law of thermodynamics in thepresence of viscous dark energy in a nonflat universe andconcluded with validity of the generalized second law FengandLi [43] investigated viscousRicci dark energy and showedthat once viscosity is taken into account the problem onage of the universe gets alleviated Amirhashchi and Pradhan[44] considered viscous and nonviscous dark energy EoSparameter in anisotropic Bianchi type I space-time and couldget a transition from quintessence to phantom Brevik et al[45] investigated interacting dark energy and dark matterin flat FRW universe taking bulk viscosity as a function ofHubble parameter 119867 and cosmic time 119905 and studied subse-quent corrections of thermodynamic parameters Velten et al[46] derived conditions under which either viscous matter orradiation cosmologies can be mapped into the phantom darkenergy scenario with constraints frommultiple observational

data sets Jamil and Farooq [47] presented a generalizationof interacting HDE using the viscous generalized Chaplygingas and reconstructed the potential and the dynamics of thescalar field Setare and Kamali [48] study warm-viscous infla-tionary universe model on the brane in a tachyon field theoryand obtained the general conditions which are required forthis model to be realizable In a very recent work Bamba andOdintsov [49] investigated a fluid model in which EoS for afluid includes bulk viscosity and found that the spectral indexof the curvature perturbations the tensor-to-scalar ratio ofthe density perturbations and the running of the spectralindex can be consistent with the recent Planck results It maybe noted that in a remarkablework Brevik et al [50] discussedentropy of DE filled FRW universe in the framework ofholographic Cardy-Verlinde formula and expressed entropyin terms of energy and Casimir energy depending on theEoS and in a relatively recent work Brevik et al [51] deriveda formula for the entropy for a multicomponent coupledfluid that under certain conditions may reduce to the Cardy-Verlinde form to relate the entropy of a closed FRW universeto the energy contained in it together with its Casimir energyIn some recent studies bulk viscosity has been incorporatedin the studies of modified gravity too and studies in thisdirection include [52ndash54]

Plan of the present work is as follows In Section 2 wewill apply Israel-Stewart theory to study the behavior ofviscous extended holographic Ricci dark energy (EHRDE)without any specific choice of scale factor as well as the choice119886(119905) = (119905 minus 119905

0)120573(1minus120572) In Section 3 we will study the state-

finder parameters and investigate whether ΛCDM phase isattainable for both the cases In Section 4 we will examinevalidity of the generalized second law of thermodynamics forboth the cases and in Section 5 we will conclude the paper

2 Israel-Stewart Approach

21 Israel-Stewart Approach without Any Specific Choice ofScale Factor In the cosmological framework bulk viscositycan be thought of as an internal friction due to the differentcooling rates in an expanding gas [30] As the dissipationdue to bulk viscosity converts kinetic energy of the particlesinto heat the effective pressure is expected to be reducedin an expanding fluid For flat homogeneous Friedmann-Robertson-Walker (FRW) with a line element

1198891199042= 1198891199052minus 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112) (2)

filled with a bulk viscous cosmological fluid the energy-momentum tensor is given by [30]

119879119896

119894= (120588 + 119901 + Π) 119906

119894119906119896minus (119901 + Π) 120575

119896

119894 (3)

where 120588 119901 and Π are energy density thermodynamicpressure and the bulk viscous pressure respectively 119906

119894is for

velocity that satisfies the condition 119906119894119906119894= 1 Here 119873119894 =

119899119906119894 and 119878119894 = 120590119873119894 minus (120591Π22120585119879)119906119894 are particle and entropy

fluxes respectively where 119899 120590 and 119879 ge 0 imply the numberdensity the specific entropy and temperature respectivelyAlso bulk viscosity coefficient and relaxation time are 120585 and

Advances in High Energy Physics 3

120591 ge 0 respectively If the Hubble parameter is119867 = 119886 thenthe gravitational field equations together with the continuityequation are [30]

31198672= 120588 (4)

2 + 31198672= minus119901 minus Π (5)

+ 3119867 (120588 + 119901) = minus3119867Π (6)

The effect of the bulk viscosity can be considered by addingthermodynamic pressure 119901 to the bulk viscous pressure Πthat is

119901eff = 119901 + Π (7)

terms in the energy-momentum tensor Taking 119909 = ln 119886 in(1) we have (1198722

119901= 1)

120588DE = 3(1205721198672+120573

2

1198891198672

119889119909) (8)

Considering 120588 = 120588DE in (4) we have the Hubble parameter

119867(119909) = 1198670exp(119909 (1 minus 120572)

120573) (9)

Putting 119909 = ln 119886 in (9) we can write

119867 = 1198670119886(1minus120572)120573

(10)

which can be written in the form of a differential equation asfollows

(119905) = 1198670119886 (119905)1+(1minus120572)120573

(11)

where the upper dot implies time derivative with respect tocosmic time 119905 Since for the current time 119905 = 119905

0we have 119886 = 1

we have the following particular solution for 119886(119905)

119886 (119905) = (1198670(119905 minus 1199050) (minus1 + 120572) + 120573

120573)

120573(minus1+120572)

(12)

Hence Hubble parameter as expressed in (9) can be writtenas a function of 119905

119867 =1198670120573

1198670(119905 minus 1199050) (minus1 + 120572) + 120573

(13)

Using (13) in (1) we have the reconstructed RDE as

120588DE = 31198672=

31198672

01205732

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2 (14)

Using (14) in (6) we have the EoS parameter

119908DE = minus1

+1

3(minus2

120573+2120572

120573minusΠ (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2

1198672

01205732

)

(15)

Hence for present acceleration that is119908 lt minus13 at 119905 = 1199050we

need

120572 lt 1 +120573

2(6 +

Π (119905 = 1199050)

1198672

0

) (16)

If the current universe is in phantom phase that is 119908 lt minus1then we will require

120572 lt 1 +Π (119905 = 119905

0)

21198672

0

120573 (17)

For the bulk viscosity coefficient 120585 and for the relaxation time120591 of the viscous extended holographic Ricci dark energy weassume the following phenomenological laws [30]

120585 = 120578120588]

120591 = 120585120588minus1= 120578120588

]minus1

(18)

where 120578 ge 0 and ] ge 0 are constants At this juncture itmay be stated that the bulk viscosity coefficient 120585 is beingconsidered as a function of 120588(119905) Hence possibility is open toa variety of 120585(120588(119905)) The case ] = 12 yields 120585 = 12057812058812 thatcorresponds to a power-law expansion for the scale factorTo obtain solution with Big Rip no restriction is imposed on] and similar approach was adopted in [39] Another workthat is noteworthy in this context is done by Colistete Jr etal [55] where for viscous generalized Chaplygin gas a varietyof solutions were presented for different ranges of ] Taking] = 2 and subsequently solving the evolution equation for Πin the framework of the truncated Israel-Stewart theory givenby

120591Π + Π = minus3119867120585 (19)

we obtain the evolution equation of the bulk viscous pressureas

Π = 119890minus(1205931119905+1205932)3912059321198672

01205781198621

minus 119890minus(1205931119905+1205932)3912059321198672

01205781198673

0(minus9119890

(1205931119905+1205932)3912059321198672

0120578

+ 323(minus(1205931119905 + 1205932)3

12059321198672

0120578)

23

Γ[1

3 minus(1205931119905 + 1205932)3

912059321198672

0120578])

sdot (21205932(1205931119905 + 1205932)2)minus1

(20)

where

1205931=1198670

120573(120572 minus 1)

1205932= 1 +

11986701199050

120573(1 minus 120572)

(21)


And 1198621is the constant of integration Hence EoS parameter

when expressed as a function of 119909 (= ln 119886) takes the form

119908DE = minus1 +1

3(2 (120572 minus 1)

120573minus1

21198672

0

1198902119909(120572minus1)120573minus119883

(2

minus1198673

0120573(minus9119862

1119890119883+ 323(minus119883)23Γ [1

3 minus119883])

sdot ((1198670(1199050 (1 minus 120572) + 120573))

sdot (

119883 (91198672

0(11986701199050(1 minus 120572) + 120573) 120578)

120573)

23

)

minus1

))

(22)

where

119883 = 120573(1198670(120572 minus 1)

120573+ (1 minus

11986701199050(120572 minus 1)

120573)

sdot (1199050+

(minus1 + 119890119909(120572minus1)120573

) 120573

1198670(120572 minus 1)

))

3

sdot (1198672

0(11986701199050(1 minus 120572) + 120573) 120578)

minus1

(23)

Equation (22) imposes one more constraint on the relation-ship between 120572 and 120573 as

120572 = 1 +120573

11986701199050

(24)

We would like to add a note at this juncture Equation(18) comes as a phenomenological law for bulk viscositycoefficient and relaxation time with 120578 ge 0 that is possibilityof 0 is not excluded Clearly for 120578 = 0 the bulk viscouspressure will vanish and we will get back the nonviscousscenario However it is clear that the evolution equation forΠ in the framework of Israel-Stewart theory is written forviscous scenario and for nonviscous scenario it is irrelevantEquation (19) is a linear differential equation on Π whoseintegrating factor is expint(119889119905120591) and obviously the integratingfactor will not exist if 120578 = 0 and hence there will exist nosolution and (23) will be of no existence

From (22) we observe that for 120572 = 098 and 120573 = 037 inEHRDE as mentioned in Section 1 we have for very late stageof the universe

119909 997888rarr infin 997904rArr

119908DE 997888rarr minus1 +1

3(minus0108 minus

1

1198672

0

minus(10minus51198673

0(minus9 + 19577259(

(0137 + 0014811986701199050+ 4 times 10

minus41198672

0(minus1 + 119905

2

0))3

1198675

0(minus037 minus 002119867

01199050) 120578

)

23

sdot Γ [

[

1

3

101452804 (01369 + 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))3

1198675

0(minus037 minus 002119867

01199050) 120578

]

]

))((minus037 minus 00211986701199050) (01369

+ 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))2

)

minus1

)

(25)

and for early stage

119909 997888rarr minus014 997904rArr

119908DE 997888rarr minus1 +1

3

sdot (minus1

1198672

0

05119890minus004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578

sdot (2 minus (0371198673

0(minus9119890004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578


+ 107(minus

(minus014 + 09911986701199050+ 1198672

0(minus0054 + 0054119905

2

0))3

1198675

0(037 + 002119867

01199050) 120578

)

23

sdot Γ [033 minus004 (minus0054119867

0+ (minus014119867

0+ 1199050) (1 + 0054119867

01199050))3

1198672

0(037 + 002119867

01199050) 120578

]))

sdot ((037 + 00211986701199050) (minus0054119867

0+ (minus

014

1198670

+ 1199050) (1 + 0054119867

01199050))

2

)

minus1

))

(26)

where Γ(119886 119911) = intinfin119911119890minus119905119905119886minus1119889119905

From the above limits one can see that the EoS parameteris determined by the viscosity of the EHRDE and does notblow up in the past or future This observation is consistentwith [43] where the RDE was considered with barotropicfluid and the EoS parameter was seen not to blow up inthe past of the future In Figure 1 we have plotted theEoS parameter and observed that 119908DE lt minus1 that is thephantom phase is attained It is noted that the phantombarrier 119908DE = minus1 is never being crossed and in the verylate stage 119908DE ≪ minus1 In Figure 2 the effect of bulk viscosityon the thermodynamic pressure is visualized We observethat 119901 + Π ≪ 0 during the evolution Figure 3 shows thatΠ gt 0 and from this we can understand that the effect ofbulk viscosity on the thermodynamic pressure is increasingwith evolution of the universe However |119901eff | is graduallydecreasing with evolution of the universe Figure 4 showsthat the bulk viscosity coefficient 120585 is a monotone increasingfunction of cosmic time

22 Israel-Stewart Approach for 119886(119905) = (119905 minus 1199050)120573(1minus120572) In

Section 21 instead ofmaking any assumption on scale factorwe have derived solution for scale factor in (12) In the presentsection the scale factor is chosen as

119886 (119905) = 1198860(119905 minus 1199050)120573(1minus120572)

(27)

This leads to

119867 =120573

(119905 minus 1199050) (1 minus 120572)

= minus120573

(119905 minus 1199050)2

(1 minus 120572)

(28)

Based on (28) the reconstructed density of EHRDE is

120588DE =31205732(2120572 minus 1)

(119905 minus 1199050)2

(1 minus 120572)2 (29)

For 120588 gt 0 one needs 120572 gt 12 Accordingly the bulk viscositycoefficient and relaxation times (18) get reconstructed and

hence evolution equation for Π in the framework of thetruncated Israel-Stewart theory given by (19) is solved to get

Π = 1198622119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

+ 119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(minus9119890

(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

+ 323(minus(119905 minus 1199050)3

(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578]) (2 (119905 minus 119905

0)2

sdot (minus1 + 120572)3)minus1

(30)

and hence

Π =119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

6 (119905 minus 1199050)3

(minus1 + 120572)3(minus1 + 2120572) 120573

21205782(120578 (minus2119862

2(119905

minus 1199050)5

(minus1 + 120572)5+ 9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(minus1

+ 2120572) 1205733((119905 minus 119905

0)3

(minus1 + 120572)2+ 6 (minus1 + 2120572) 120573

2120578))

+ (119905 minus 1199050)6

(minus1 + 120572)412057311986423 [119879])

(31)

where 119879 = minus(119905 minus 1199050)3(minus1 + 120572)

29(minus1 + 2120572)120573

2120578 Subsequently

119901DE becomes

119901DE = 1198622119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)


sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

minus 323(minus(119905 minus 1199050)3

(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


(32)

and hence EoS parameter 119908DE is

119908DE

=(119905 minus 1199050)2

(minus1 + 120572)2

3 (minus1 + 2120572) 1205732(minus1198622119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


)

(33)

In the above equations 119864119899[119911] = int

infin

1(119890minus119911119905119905119899)119889119905

In Figures 5 6 and 7 the red green and blue linescorrespond to 120578 = 000009 00002 and 000015 respectivelyIn all the cases in this section 120572 = 09180 and120573 = 03701In Figure 5 we plot 119908DE and observe that the EoS parametershows a clear transition from 119908DE gt minus1 to 119908DE lt minus1 thatis from quintessence to phantom Hence the model behaveslike ldquoQuintomrdquoThis is in contradiction with what happenedin themodel without any specific choice of scale factor where119908DE lt minus1 that implies ldquophantomrdquo behavior of the EoSparameter Similar to the earlier case the effective pressure(Figure 6) is decaying with the evolution of the universeHowever time derivative of the bulk viscous pressure Π staysat positive level (Figure 7) This indicates that the effect ofbulk viscous pressure increases with timeThismeans that thenegative bulk viscous pressure gives a significant contributionto the total negative pressure of the EHRDEThepositive timederivative of bulk viscous pressure and gradually decaying

minus30

minus25

minus20

minus15

minus10

minus5

0

wDE

02 04 06 08 1000t

Figure 1 EoS parameter based on (22) Red green and blue linescorrespond to 120572 120573 combination of 09735 03701 with 120578 = 35 times10minus4 30 times 10minus4 and 25 times 10minus4 respectively

minus2500

minus2000

minus1500

minus1000

minus500

0

peff

08 10 12 14 16 18 2006t

Figure 2 Plot of 119901eff = 119901+Π Red green and blue lines correspondto 120572 120573 combination of 09735 3701 and 09735 3701 with 120578 =35 times 10

minus4 30 times 10minus4 and 25 times 10minus4 respectively

effective pressure indicates that nonequilibrium bulk viscouspressure is small compared to the local equilibrium pressure

3 State-Finder Parameters

31Model withoutAny SpecificChoice of Scale Factor Sahni etal [56] and Alam et al [57] introduced a pair of cosmological


02 04 06 08 1000t

0

10000

20000

30000

40000

Π

Figure 3 Plot of Π based on (20) Red green and blue linescorrespond to 120572 120573 combination of 09735 3701 with 120578 = 35 times10minus4 30 times 10minus4 and 25 times 10minus4 respectively

00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000

Figure 4 Plot of the bulk viscosity coefficient 120585 Red green andblue lines correspond to 120572 120573 combination of 09735 3701 with120578 = 35 times 10


parameters 119903 119904 (the so-called ldquostate-finder parametersrdquo)that seem to be promising candidates for the purpose of dis-crimination between the various contenders of dark energyIf the 119903 minus 119904 trajectory meets the point 119903 = 1 119904 = 0 thenthe model is said to attain ΛCDM phase of the universe Inthe literature quite a good number of works are available

wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t

Figure 5 Plot of 119908DE for scale factor in (33)

minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t

Figure 6 Effective pressure 119901eff = 119901DE + Π based on (30) and (32)

where dark energy models have been explored through state-finder trajectories for example [58ndash60] The state-finderparameters are given by

119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)

In (34) deceleration parameter 119902 is given by

119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5

Figure 7 Time evolution of bulk viscous pressure Π based on (31)

where 119886 is the scale factor as available in (12) Hence in thecurrent framework (34) take the form

119903 =1198672

0(120572 minus 1)

2(minus2 + 120573 (120572 minus 1)) (minus1 + 120573 (120572 minus 1))

(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672

0(120572 minus 1 minus 120573) (minus2 + 2120572 minus 120573)

(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)

sdot (3(minus3 minus2

120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM

Figure 8 The 119903 minus 119904 trajectory based on (36) Red green and bluelines correspond to 120572 120573 combination of 09735 3701 with 120578 =35 times 10


sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)

In Figure 8 we observe that the 119903 minus 119904 trajectory can attainthe ΛCDM point that is 119903 = 1 119904 = 0 Again for finite119903 we observe that 119904 rarr minusinfin that corresponds to dust phaseThus we may conclude that the viscous EHRDE interpolatesbetween dust and ΛCDM phases of the universe Howeverthe model deviates significantly from ΛCDM

32 Model with Scale Factor 119886(119905) = (119905 minus 1199050)120573(1minus120572) For the

choice of scale factor 119886(119905) = (119905 minus 1199050)120573(1minus120572) we get from (34)

the state-finder parameters as

119903 =1

2(1 + 119890

minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578 + 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573

sdot (minus4 (minus1 + 120572)2minus 6 (minus1 + 120572) 120573 + 9 (minus1 + 2120572) 120573

2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1

6 (1 + 120572 (minus3 + 2120572)) 12057351205782119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)2

sdot (minus1 + 120572)3(1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)

sdot ((minus1 + 2120572) 120573)minus1minus 12119862

2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])

minus (119905 minus 1199050) (minus2119862

2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578

sdot (minus1 + 119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

minus119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785

sFigure 9 State-finder trajectories based on (37)

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)

+ (1 + 119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)

In Figure 9 we have plotted the state-finder trajectories forthemodel with scale factor 119886(119905) = (119905minus119905

0)120573(1minus120572) It is observed

that the fixed ΛCDM point 119903 = 1 119904 = 0 is not attained bythe trajectories However for finite 119903 we observe 119904 rarr minusinfin

that corresponds to the dust phase of the universe

4 Generalized Second Law ofThermodynamics

Discovery of black hole thermodynamics in the 70s [61ndash63]prompted physicists to study the thermodynamics of cosmo-logical models of the universe Semiclassical description inblack hole physics shows that a black hole behaves like a black


body that is emitting thermal radiation with temperatureand entropy This temperature and the entropy are known asHawking temperature and Bekenstein entropy respectively[64ndash66] This entropy is proportional to surface area 119860 ofblack hole which according to Hawkings area theoremcannot decrease Based on the conjectured proportionalitybetween entropy and horizon area of black hole a generalizedversion of the second law of thermodynamics was proposedby Bekenstein According to the proposal of Bekensteinthe sum of black hole entropy and the entropy of matterand radiation in the region exterior to black hole cannotdecrease The GSL provides a relation between gravitationthermodynamics and quantum theory In a very recentwork [67] conjectured a novel GSL that can be applied incosmology irrespective of the presence of event horizon

Reference [42] examined the validity of the generalizedGSL in a nonflat universe in the presence of viscous darkenergy Setare [68] investigated the validity of the generalizedsecond law of thermodynamics for the Quintom model ofdark energy Considering the universe as a closed boundedsystem filled with 119899 component fluids Bamba et al [69]studied the generalized second law in 119891(119879) cosmologyReference [70] investigated the validity of the generalizedsecond law in the context of interacting 119891(119877) gravity Weconsider the universe To check the generalized second lawof thermodynamics we have to examine the evolution ofthe total entropy 119878

119860+ 119878DE where 119878119860 denotes the entropy

of the apparent horizon and 119878DE denotes the entropy of thefluid inside the horizon For the FRW universe the apparenthorizon radius reads [42]

119860=

1

radic1198672 + 1198961198862 (39)

In a flat universe 119896 = 0 and (39) becomes

119860=1

119867 (40)

Temperature on the apparent horizon is defined as [42]

119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)

The entropy associated with the apparent horizon is [42]

119878119860=119860

4119866=1205872

119860

119866 (42)

where 119860 = 41205872119860is the area of the apparent horizon It has

been shown by some calculations in [42] that for viscous darkenergy dominated flat universe enveloped by the apparenthorizon

119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)

In the present case the above can be rewritten as

119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)

The entropy of the viscous dark energy inside the apparenthorizon 119878DE can be related to its energy 119864D = 120588D119881 and itspressure as

119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)

where 119879DE is the temperature of the viscous dark energy and119881 = (43)120587

3

119860is the volume enveloped by the apparent hori-

zon Under the assumption that the thermal system boundedby the apparent horizon remains in equilibrium that istemperature of the system is uniform and the temperatureon the horizon is equal to temperature of the fluid inside thehorizon we have 119879

119860= 119879DE = 119879 It has been shown in [42] by

some simple calculation that

119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)

Using (14) (15) and (20) in (46) we get

119860+ DE =

2120587

11986701205733(minus1 + 120572) (1198670 (119905 minus 1199050) (minus1 + 120572) + 120573)

sdot (120573

119866minus

8120587

120572 minus 2120573 minus 1(minus2120573

2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2

sdot (1198621119890minus120594minus

119890minus1205941198673

01205733(9119890120594minus 323(minus120594)23Γ [13 minus120594])

2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]

(11986701199050(minus1 + 120572) minus 120573) 120578 (minus120594)

13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673

0120573 (9119890120594minus 323(minus120594)23Γ [13 minus120594])

(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890

minus120594(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +

120572) minus 120573)1205732120578 In Figure 10 we have plotted 119879(

119860+ DE) based

on (47) with cosmic time 119905 and 120578 ranging from 00002 to00005 We have observed that 119879(

119860+ DE) is staying at

positive level Since 119879 gt 0 Figure 10 indicates that 119860+

DE ge 0 This indicates validity of the generalized secondlaw of thermodynamics in a universe dominated by viscousEHRDE Using (29) (30) and (33) in (46) we get the timeevolution of the total entropy for the case 119886(119905) = (119905minus 119905

0)120573(1minus120572)

as

119860+ DE =

1

1205732 (minus1 + 120572 + 2120573)81205872(119905 minus 1199050) (minus1 + 120572)

2(2

+ (minus119905 + 1199050) (minus1 + 120572) minus 2120572


29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)

+119890minus(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)

Figure 11 shows that like the previous case total = 119860+

DE gt 0 and hence the GSL of thermodynamics is validatedin a universe dominated by viscous EHRDE and expandingaccording to 119886(119905) = (119905 minus 119905

0)120573(1minus120572) However contrary to

Figure 10 the time derivative of total entropy is increasingwith evolution of the universe

5 Conclusions

Motivated by [30 43] we have presented a study on viscousextended holographic Ricci dark energy (EHRDE) in flatFRW universe based on Israel-Stewart approach The workhas been carried out in two phases In one phase instead ofchoosing any specific form of scale factor we have recon-structed Hubble parameter119867 based on the field equation (4)with120588 = 120588DE = 3119872

2

119901(1205721198672+120573) and subsequently solving the

truncated Israel-Stewart theory given by 120591Π + Π = minus3119867120585 forΠwe studied the behavior of EoS parameter119908DE bulk viscouspressure Π state-finder parameters and the thermodynamicconsequences in terms of generalized second law (GSL) ofthermodynamics In this phase of study we have observed thefollowing

02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578

Figure 10 Plot of time evolution of total entropy (119860+ DE) based

on (47) The 120572 120573 combination is taken as 09735 03701 and 120578ranges from 00002 to 00005

1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578


on (47) for 119886(119905) = (119905 minus 1199050)120573(1minus120572) The 120572 120573 combination is taken as

09180 03701 and 120578 ranges from 000009 to 000015

Under the consideration that the universe is dominatedby EHRDE we have taken evolution equation for the bulkviscous pressure Π in the framework of the truncated Israel-Stewart theory as 120591Π + Π = minus3120585119867 where 120591 is the relaxationtime and 120585 is the bulk viscosity coefficient Consideringeffective pressure as a sum of thermodynamic pressure ofEHRDE and bulk viscous pressure we have observed thatunder the influence of bulk viscosity the EoS parameter 119908DEis behaving like phantom that is 119908DE le minus1 (see Figure 1)Furthermore it has been observed that the effect of bulkviscosity is not blowing up in the very late stage or very early


stage of the universeThis observation is consistent with [43]Some constraints have been derived for themodel parameters120572 and 120573 and other constants Obeying the observationalstudies that show that for EHRDE 120572 = 08502+00984+01299

minus00875minus01064

and 120573 = 04817+00842+01176

minus00773minus00955we have taken 120572 = 09735

120573 = 03701 It has been already mentioned that like [39] toobtain solution with Big Rip no restriction is imposed on ]Figure 1 shows that 119908DE is decaying with evolution of theuniverse It is observable that alongwith120588+119901 lt 0we also have120588 + 3119901 lt 0 with passage of cosmic time Also with passageof cosmic time minus(120588 + 3119901) increases and so all gravitationallybound systems will be dissociated [71] and the universe willend up in Big Rip We have observed that the magnitude ofthe effective pressure 119901eff = 119901 + Π is decaying with time(see Figure 2) Moreover the nonnegative derivative of Π(see Figure 3) has indicated that the effect of bulk viscosityis increasing with time The bulk viscosity coefficient 120585 hasbeen found to be an increasing function of time (see Figure 4)The state-finder parameters 119903 119904 have also been studied Ithas been observed that under the effect of bulk viscosity thestate-finder trajectory 119903minus119904 for EHRDE is capable of reachingthe ΛCDM point that is 119903 = 1 119904 = 0 We have furtherobserved that for finite 119903 119904 rarr minusinfin which correspondsto dust phase This shows that the viscous EHRDE interpo-lates between dust and ΛCDM phases of the universe (seeFigure 8) Finally we have studied the thermodynamics ofthe viscous EHRDE under the assumption that the universeis enveloped by apparent horizon We have derived theexpression of the time derivative of the total entropy andwe have observed that (see Figure 10) the time derivative ispositive throughout the evolution of the universeThis showsthat the generalized second law of thermodynamics is validfor the viscous EHRDE

In the next phase of the study we have chosen the scalefactor as 119886(119905) = (119905 minus 119905

0)120573(1minus120572) and subsequently studied the

cosmological parameters similar to that in the previous phaseThe effective pressure and time derivative of bulk viscouspressure Π appear to be of similar pattern to those in theearlier phase However the significant difference is observedin the case of EoS parameter119908DE (see Figure 5) and the state-finder trajectories 119903 minus 119904 (see Figure 9) The EoS parameter119908DE is found to cross the phantom barrier that is transitingfrom quintessence (119908DE gt minus1) to phantom (119908DE lt minus1)Thus the model is found to behave like ldquoQuintomrdquo for thesaid form of scale factor In the state-finder trajectory unlikethe previous case theΛCDMfixed point is not attainable andthe model is deviated significantly from ΛCDM Howeverthe dust phase is attainable like the previous case Like theprevious case the GSL is found to be valid that is timederivative of the total entropy stays in the nonnegative levelHowever contrary to what happened in the previous casethe time derivative of the total entropy is increasing withevolution of the universe

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

Financial support from DST Government of India underProject Grant no SRFTPPS-1672011 is thankfully acknowl-edged Warm hospitality provided by the Inter-UniversityCentre for Astronomy and Astrophysics (IUCCA) PuneIndia during a scientific visit in May 2016 is duly acknowl-edged by the author

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 pp 1009ndash1038 1998

[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 article 565 1999

[3] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 no 1 pp 385ndash432 2008

[4] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 pp 1753ndash1935 2006

[5] K Bamba S Capozziello S Nojiri and S D Odintsov ldquoDarkenergy cosmology the equivalent description via differenttheoretical models and cosmography testsrdquo Astrophysics andSpace Science vol 342 no 1 pp 155ndash228 2012

[6] R R Caldwell and M Kamionkowski ldquoThe physics of cosmicaccelerationrdquoAnnual Review ofNuclear andParticle Science vol59 no 1 pp 397ndash429 2009

[7] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from 119865(119877) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[8] S Nojiri and S D Odintsov ldquoUnifying phantom inflation withlate-time acceleration scalar phantom-non-phantom transitionmodel and generalized holographic dark energyrdquo General Rela-tivity and Gravitation vol 38 no 8 pp 1285ndash1304 2006

[9] M Li ldquoA model of holographic dark energyrdquo Physics Letters Bvol 603 no 1-2 pp 1ndash5 2004

[10] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[11] S Nojiri and S D Odintsov ldquoUnifying phantom inflation withlate-time acceleration scalar phantomndashnon-phantom transi-tion model and generalized holographic dark energyrdquo GeneralRelativity and Gravitation vol 38 no 8 pp 1285ndash1304 2006

[12] S del Campo J C Fabris R Herrera and W ZimdahlldquoHolographic dark-energy modelsrdquo Physical Review D vol 83Article ID 123006 2011

[13] J-L Cui and J-F Zhang ldquoComparing holographic dark energymodels with statefinderrdquo The European Physical Journal C vol74 article 2849 2014

[14] S Chattopadhyay A Jawad and S Rani ldquoHolographic poly-tropic f (T) gravity modelsrdquo Advances in High Energy Physicsvol 2015 Article ID 798902 15 pages 2015

[15] S Chattopadhyay andA Pasqua ldquoConsequences of holographicscalar field dark energy models in chameleon Brans-Dickecosmologyrdquo inXXIDAE-BRNSHighEnergy Physics SymposiumB Bhuyan Ed vol 174 of Springer Proceedings in Physics pp487ndash492 2016


[16] Q-G Huang and Y Gong ldquoSupernova constraints on a holo-graphic dark energymodelrdquo Journal of Cosmology andAstropar-ticle Physics vol 2004 no 8 p 6 2004

[17] Q-G Huang and M Li ldquoAnthropic principle favours the holo-graphic dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2005 no 3 article 001 2005

[18] X Zhang and F-Q Wu ldquoConstraints on holographic darkenergy from the latest supernovae galaxy clustering and cos-mic microwave background anisotropy observationsrdquo PhysicalReview D vol 76 Article ID 023502 2007

[19] L N Granda and A Oliveros ldquoInfrared cut-off proposal for theholographic densityrdquo Physics Letters B vol 669 no 5 pp 275ndash277 2008

[20] Y U Fei and Z Jing-Fei ldquoStatefinder diagnosis for the extendedholographic Ricci dark energy model without and with interac-tionrdquo Communications in Theoretical Physics vol 59 no 2 pp243ndash248 2013

[21] S Chattopadhyay ldquoExtended holographic Ricci dark energy inchameleon Brans-Dicke cosmologyrdquo ISRN High Energy Physicsvol 2013 Article ID 414615 7 pages 2013

[22] R Brustein and G Veneziano ldquoCausal entropy bound for aspacelike regionrdquo Physical Review Letters vol 84 no 25 pp5695ndash5698 2000

[23] C Gao F Wu X Chen and Y-G Shen ldquoHolographic darkenergy model from Ricci scalar curvaturerdquo Physical Review Dvol 79 no 4 Article ID 043511 7 pages 2009

[24] S del Campo J C Fabris R Herrera and W ZimdahlldquoCosmology with Ricci dark energyrdquo Physical Review D vol 87no 12 Article ID 123002 2013

[25] S Chattopadhyay ldquoOn the various aspects of interacting Riccidark energy and tachyonic fieldrdquoAstrophysics and Space Sciencevol 331 no 2 pp 651ndash655 2011

[26] P George and T K Mathew ldquoHolographic Ricci dark energy asrunning vacuumrdquoModern Physics Letters A Particles and FieldsGravitation Cosmology Nuclear Physics vol 31 no 13 ArticleID 1650075 12 pages 2016

[27] C-J Feng ldquoStatefinder diagnosis for Ricci dark energyrdquo PhysicsLetters B vol 670 no 3 pp 231ndash234 2008

[28] T K Mathew J Suresh and D Divakaran ldquoModified holo-graphic Ricci dark energy model and statefinder diagnosis inflat universerdquo International Journal ofModern Physics D vol 22no 9 Article ID 1350056 17 pages 2013

[29] Y Wang and L Xu ldquoCurrent observational constraints to theholographic dark energy model with a new infrared cutoff viatheMarkov chainMonte Carlomethodrdquo Physical ReviewD vol81 no 8 11 pages 2010

[30] C S J Pun L A Gergely M K Mak Z Kovacs G M Szaboand T Harko ldquoViscous dissipative Chaplygin gas dominatedhomogenous and isotropic cosmological modelsrdquo PhysicalReview D vol 77 no 6 Article ID 063528 2008

[31] L P Chimento A S Jakubi and D Pavon ldquoEnlarged quintes-sence cosmologyrdquo Physical Review D vol 62 no 6 Article ID063508 9 pages 2000

[32] C Eckart ldquoThe thermodynamics of irreversible processes IIIRelativistic theory of the simple fluidrdquo Physical Review vol 58no 10 pp 919ndash924 1940

[33] L D Landau and E M Lifshitz Fluid Mechanics ButterworthHeinemann Oxford UK 1987

[34] W Israel and J M Stewart ldquoThermodynamics of nonstationaryand transient effects in a relativistic gasrdquo Physics Letters A vol58 no 4 pp 213ndash215 1976

[35] S Nojiri and S D Odintsov ldquoInhomogeneous equation of stateof the universe phantom era future singularity and crossingthe phantom barrierrdquo Physical Review D vol 72 no 2 ArticleID 023003 12 pages 2005

[36] H B Benaoum ldquoModified Chaplygin gas cosmology with bulkviscosityrdquo International Journal of Modern Physics D vol 23 no10 Article ID 1450082 2014

[37] J Naji S Heydari and R Darabi ldquoNew version of viscousChaplygin gas cosmology with varying gravitational constantrdquoCanadian Journal of Physics vol 92 no 12 pp 1556ndash1561 2014

[38] K Karimiyan and J Naji ldquoInteracting viscous modified Chap-lygin gas cosmology in presence of cosmological constantrdquoInternational Journal of Theoretical Physics vol 53 no 7 pp2396ndash2403 2014

[39] M Cataldo N Cruz and S Lepe ldquoViscous dark energy andphantom evolutionrdquo Physics Letters B vol 619 no 1-2 pp 5ndash102005

[40] I Brevik and O Gorbunova ldquoDark energy and viscous cosmol-ogyrdquoGeneral Relativity andGravitation vol 37 no 12 pp 2039ndash2045 2005

[41] J Ren and X-H Meng ldquoCosmological model with viscositymedia (dark fluid) described by an effective equation of staterdquoPhysics Letters B vol 633 no 1 pp 1ndash8 2006

[42] M R Setare and A Sheykhi ldquoViscous dark energy and gener-alized second law of thermodynamicsrdquo International Journal ofModern Physics D vol 19 no 7 pp 1205ndash1215 2010

[43] C-J Feng and X-Z Li ldquoViscous Ricci dark energyrdquo PhysicsLetters B vol 680 no 4 pp 355ndash358 2009

[44] H Amirhashchi and A Pradhan ldquoViscous dark energy andphantom field in an anisotropic universerdquo Astrophysics andSpace Science vol 351 no 1 pp 59ndash65 2014

[45] I Brevik V V Obukhov and A V Timoshkin ldquoDark energycoupled with dark matter in viscous fluid cosmologyrdquo Astro-physics and Space Science vol 355 no 2 pp 399ndash403 2015

[46] H Velten J Wang and X Meng ldquoPhantom dark energy as aneffect of bulk viscosityrdquo Physical Review D vol 88 Article ID123504 2013

[47] M Jamil and M U Farooq ldquoInteracting holographic viscousdark energy modelrdquo International Journal ofTheoretical Physicsvol 49 article 42 2010

[48] M R Setare and V Kamali ldquoCosmological perturbationsin warm-tachyon inflationary universe model with viscouspressure on the branerdquo Journal of High Energy Physics vol 2013article 66 2013

[49] K Bamba and S D Odintsov ldquoInflation in a viscous fluidmodelrdquoThe European Physical Journal C vol 76 p 18 2016

[50] I Brevik S Nojiri S D Odintsov and L Vanzo ldquoEntropy anduniversality of the Cardy-Verlinde formula in a dark energyuniverserdquo Physical Review D vol 70 no 4 Article ID 0435202004

[51] I Brevik S Nojiri S D Odintsov andD Saez-Gomez ldquoCardyndashVerlinde formula in FRW Universe with inhomogeneous gen-eralized fluid and dynamical entropy bounds near the futuresingularityrdquo The European Physical Journal C vol 69 no 3-4pp 563ndash574 2010

[52] M Sharif and S Rani ldquoViscous dark energy in 119891(119879) gravityrdquoModern Physics Letters A vol 28 no 27 Article ID 1350118 13pages 2013

[53] P Kumar and C P Singh ldquoViscous cosmology with mattercreation in modified f (RT) gravityrdquo Astrophysics and SpaceScience vol 357 no 2 article 120 2015


[54] V R Chirde and S H Shekh ldquoBarotropic bulk viscous FRWcosmological model in teleparallel gravityrdquo Bulgarian Journal ofPhysics vol 41 no 4 pp 258ndash273 2014

[55] R Colistete Jr J C Fabris J Tossa and W Zimdahl ldquoBulkviscous cosmologyrdquo Physical Review D vol 76 no 10 ArticleID 103516 2007

[56] V Sahni T D Saini A A Starobinsky and U Alam ldquoState-findermdasha new geometrical diagnostic of dark energyrdquo JETPLetters vol 77 no 5 pp 201ndash206 2003

[57] U Alam V Sahni T D Saini and A A Starobinsky ldquoExploringthe expanding Universe and dark energy using the statefinderdiagnosticrdquo Monthly Notices of the Royal Astronomical Societyvol 344 no 4 pp 1057ndash1074 2003

[58] M Jamil DMomeni RMyrzakulov and P Rudra ldquoStatefinderanalysis of f (T) cosmologyrdquo Journal of the Physical Society ofJapan vol 81 Article ID 114004 2012

[59] W Zimdahl and D Pavon ldquoStatefinder parameters for interact-ing dark energyrdquoGeneral Relativity and Gravitation vol 36 no6 pp 1483ndash1491 2004

[60] C-J Feng ldquoStatefinder diagnosis for Ricci dark energyrdquo PhysicsLetters B vol 670 pp 231ndash234 2008

[61] J D Bekenstein ldquoBlack holes and the second lawrdquo Lettere AlNuovo Cimento vol 4 no 15 pp 737ndash740 1972

[62] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review Dvol 7 pp 2333ndash2346 1973

[63] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquoPhysical ReviewD vol 9 no 12 pp 3292ndash3300 1974

[64] S Chakraborty ldquoGeneralizedBekensteinndashHawking system log-arithmic correctionrdquo The European Physical Journal C vol 74article 2876 2014

[65] S Hod ldquoBekensteinrsquos generalized second law of thermodynam-ics the role of the hoop conjecturerdquo Physics Letters B vol 751pp 241ndash245 2015

[66] H Moradpour and R Dehghani ldquoThermodynamical study ofFRW universe in quasi-topological theoryrdquo Advances in HighEnergy Physics vol 2016 Article ID 7248520 10 pages 2016

[67] R Bousso and N Engelhardt ldquoGeneralized second law for cos-mologyrdquo Physical Review D vol 93 no 2 Article ID 0240252016

[68] M R Setare ldquoGeneralized second law of thermodynamics inquintom dominated universerdquo Physics Letters B vol 641 no 2pp 130ndash133 2006

[69] K Bamba M Jamil D Momeni and R Myrzakulov ldquoGen-eralized second law of thermodynamics in 119891(119879) gravity withentropy correctionsrdquo Astrophysics and Space Science vol 344no 1 pp 259ndash267 2013

[70] R Herrera and N Videla ldquoThe generalized second law ofthermodynamics for interacting f(R) gravityrdquo InternationalJournal of Modern Physics D vol 23 no 8 Article ID 145007116 pages 2014

[71] R R Caldwell M Kamionkowski and N NWeinberg ldquoPhan-tom energy dark energy with 119908 lt minus1 causes a cosmic dooms-dayrdquo Physical Review Letters vol 91 no 7 Article ID 071301 4pages 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


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Statistical MechanicsInternational Journal of


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Computational Methods in Physics

Journal of



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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


Wang and Xu [29] found the best-fit values in order to makethe cutoff of [20] to be consistent with observational dataas 120572 = 08502+00984+01299

minus00875minus01064and 120573 = 04817+00842+01176

minus00773minus00955 In

the current work we will take 120572 = 098 and 120573 = 037In the early evolution of the universe dissipative effectsincluding both bulk and shear viscosity are supposed toplay a very important role [30] Chimento et al [31] hadshown that accelerated expansion can be derived by thecombination of a cosmic fluid with bulk dissipative pressureand quintessence matter and it can also solve the coincidenceproblem Evolution of the universe involves a sequence ofimportant dissipative processes that includes GUT phasetransition at 119905 asymp 10minus34 s and a temperature of about 119879 asymp1027 K [30] Eckart [32] and Landau and Lifshitz [33] were

the first ones to attempt to create a theory of relativisticdissipative fluids A relativistic second-order theory wasdeveloped by Israel and Stewart [34]

A time dependent viscosity consideration was made toDE by Nojiri and Odintsov [35] while considering EoS withinhomogeneous Hubble parameter dependent term and itwas demonstrated that the thermodynamic entropy may bepositive even in phantom era as a result of crossing the phan-tom boundary Pun et al [30] considered a generalization ofthe Chaplygin gas model and by assuming the presence ofa bulk viscous type dissipative term in the effective thermo-dynamic pressure of the gas and by considering dissipativeeffects described by truncated Israel-Stewart model they[30] had shown that viscous Chaplygin gas model offers aneffective dynamical possibility for replacing the cosmologicalconstant Effects of viscosity in Chaplygin gas with differentmodifications have been studied in [36ndash38] Cataldo et al[39] investigated dissipative processes in the universe in thefull causal Israel-Stewart-Hiscock theory and showed thatthe negative pressure generated by the bulk viscosity cannotavoid the idea that the dark energy of the universe is to bephantom energy Brevik and Gorbunova [40] assumed thebulk viscosity to be proportional to the scalar expansion ina spatially flat FRW universe and had shown that it couldlead the universe to phantom phase even if the universelies in the quintessence phase in the nonviscous scenarioRen and Meng [41] considered a generally parameterizedEoS in the cosmological evolution with bulk viscosity mediamodelled as dark fluid Setare and Sheykhi [42] studied thevalidity of generalized second law of thermodynamics in thepresence of viscous dark energy in a nonflat universe andconcluded with validity of the generalized second law FengandLi [43] investigated viscousRicci dark energy and showedthat once viscosity is taken into account the problem onage of the universe gets alleviated Amirhashchi and Pradhan[44] considered viscous and nonviscous dark energy EoSparameter in anisotropic Bianchi type I space-time and couldget a transition from quintessence to phantom Brevik et al[45] investigated interacting dark energy and dark matterin flat FRW universe taking bulk viscosity as a function ofHubble parameter 119867 and cosmic time 119905 and studied subse-quent corrections of thermodynamic parameters Velten et al[46] derived conditions under which either viscous matter orradiation cosmologies can be mapped into the phantom darkenergy scenario with constraints frommultiple observational

data sets Jamil and Farooq [47] presented a generalizationof interacting HDE using the viscous generalized Chaplygingas and reconstructed the potential and the dynamics of thescalar field Setare and Kamali [48] study warm-viscous infla-tionary universe model on the brane in a tachyon field theoryand obtained the general conditions which are required forthis model to be realizable In a very recent work Bamba andOdintsov [49] investigated a fluid model in which EoS for afluid includes bulk viscosity and found that the spectral indexof the curvature perturbations the tensor-to-scalar ratio ofthe density perturbations and the running of the spectralindex can be consistent with the recent Planck results It maybe noted that in a remarkablework Brevik et al [50] discussedentropy of DE filled FRW universe in the framework ofholographic Cardy-Verlinde formula and expressed entropyin terms of energy and Casimir energy depending on theEoS and in a relatively recent work Brevik et al [51] deriveda formula for the entropy for a multicomponent coupledfluid that under certain conditions may reduce to the Cardy-Verlinde form to relate the entropy of a closed FRW universeto the energy contained in it together with its Casimir energyIn some recent studies bulk viscosity has been incorporatedin the studies of modified gravity too and studies in thisdirection include [52ndash54]

Plan of the present work is as follows In Section 2 wewill apply Israel-Stewart theory to study the behavior ofviscous extended holographic Ricci dark energy (EHRDE)without any specific choice of scale factor as well as the choice119886(119905) = (119905 minus 119905

0)120573(1minus120572) In Section 3 we will study the state-

finder parameters and investigate whether ΛCDM phase isattainable for both the cases In Section 4 we will examinevalidity of the generalized second law of thermodynamics forboth the cases and in Section 5 we will conclude the paper

2 Israel-Stewart Approach

21 Israel-Stewart Approach without Any Specific Choice ofScale Factor In the cosmological framework bulk viscositycan be thought of as an internal friction due to the differentcooling rates in an expanding gas [30] As the dissipationdue to bulk viscosity converts kinetic energy of the particlesinto heat the effective pressure is expected to be reducedin an expanding fluid For flat homogeneous Friedmann-Robertson-Walker (FRW) with a line element

1198891199042= 1198891199052minus 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112) (2)

filled with a bulk viscous cosmological fluid the energy-momentum tensor is given by [30]

119879119896

119894= (120588 + 119901 + Π) 119906

119894119906119896minus (119901 + Π) 120575

119896

119894 (3)

where 120588 119901 and Π are energy density thermodynamicpressure and the bulk viscous pressure respectively 119906

119894is for

velocity that satisfies the condition 119906119894119906119894= 1 Here 119873119894 =

119899119906119894 and 119878119894 = 120590119873119894 minus (120591Π22120585119879)119906119894 are particle and entropy

fluxes respectively where 119899 120590 and 119879 ge 0 imply the numberdensity the specific entropy and temperature respectivelyAlso bulk viscosity coefficient and relaxation time are 120585 and



31198672= 120588 (4)

2 + 31198672= minus119901 minus Π (5)

+ 3119867 (120588 + 119901) = minus3119867Π (6)


119901eff = 119901 + Π (7)


119901= 1)

120588DE = 3(1205721198672+120573

2

1198891198672

119889119909) (8)


119867(119909) = 1198670exp(119909 (1 minus 120572)

120573) (9)


119867 = 1198670119886(1minus120572)120573

(10)


(119905) = 1198670119886 (119905)1+(1minus120572)120573

(11)


0we have 119886 = 1


119886 (119905) = (1198670(119905 minus 1199050) (minus1 + 120572) + 120573

120573)

120573(minus1+120572)

(12)


119867 =1198670120573

1198670(119905 minus 1199050) (minus1 + 120572) + 120573

(13)


120588DE = 31198672=

31198672

01205732

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2 (14)


119908DE = minus1

+1

3(minus2

120573+2120572


2

1198672

01205732

)

(15)


need

120572 lt 1 +120573

2(6 +

Π (119905 = 1199050)

1198672

0

) (16)


120572 lt 1 +Π (119905 = 119905

0)

21198672

0

120573 (17)


120585 = 120578120588]

120591 = 120585120588minus1= 120578120588

]minus1

(18)


120591Π + Π = minus3119867120585 (19)


Π = 119890minus(1205931119905+1205932)3912059321198672

01205781198621

minus 119890minus(1205931119905+1205932)3912059321198672

01205781198673

0(minus9119890

(1205931119905+1205932)3912059321198672

0120578

+ 323(minus(1205931119905 + 1205932)3

12059321198672

0120578)

23

Γ[1

3 minus(1205931119905 + 1205932)3

912059321198672

0120578])

sdot (21205932(1205931119905 + 1205932)2)minus1

(20)

where

1205931=1198670

120573(120572 minus 1)

1205932= 1 +

11986701199050

120573(1 minus 120572)

(21)




119908DE = minus1 +1

3(2 (120572 minus 1)

120573minus1

21198672

0

1198902119909(120572minus1)120573minus119883

(2

minus1198673

0120573(minus9119862

1119890119883+ 323(minus119883)23Γ [1

3 minus119883])

sdot ((1198670(1199050 (1 minus 120572) + 120573))

sdot (

119883 (91198672

0(11986701199050(1 minus 120572) + 120573) 120578)

120573)

23

)

minus1

))

(22)

where

119883 = 120573(1198670(120572 minus 1)

120573+ (1 minus

11986701199050(120572 minus 1)

120573)

sdot (1199050+

(minus1 + 119890119909(120572minus1)120573

) 120573

1198670(120572 minus 1)

))

3

sdot (1198672

0(11986701199050(1 minus 120572) + 120573) 120578)

minus1

(23)


120572 = 1 +120573

11986701199050

(24)



119909 997888rarr infin 997904rArr

119908DE 997888rarr minus1 +1

3(minus0108 minus

1

1198672

0


0(minus9 + 19577259(

(0137 + 0014811986701199050+ 4 times 10

minus41198672

0(minus1 + 119905

2

0))3

1198675

0(minus037 minus 002119867

01199050) 120578

)

23

sdot Γ [

[

1

3

101452804 (01369 + 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))3

1198675

0(minus037 minus 002119867

01199050) 120578

]

]

))((minus037 minus 00211986701199050) (01369

+ 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))2

)

minus1

)

(25)

and for early stage

119909 997888rarr minus014 997904rArr

119908DE 997888rarr minus1 +1

3

sdot (minus1

1198672

0

05119890minus004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578

sdot (2 minus (0371198673

0(minus9119890004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578


+ 107(minus

(minus014 + 09911986701199050+ 1198672

0(minus0054 + 0054119905

2

0))3

1198675

0(037 + 002119867

01199050) 120578

)

23


0+ (minus014119867

0+ 1199050) (1 + 0054119867

01199050))3

1198672

0(037 + 002119867

01199050) 120578

]))

sdot ((037 + 00211986701199050) (minus0054119867

0+ (minus

014

1198670

+ 1199050) (1 + 0054119867

01199050))

2

)

minus1

))

(26)





119886 (119905) = 1198860(119905 minus 1199050)120573(1minus120572)

(27)

This leads to

119867 =120573

(119905 minus 1199050) (1 minus 120572)

= minus120573

(119905 minus 1199050)2

(1 minus 120572)

(28)


120588DE =31205732(2120572 minus 1)

(119905 minus 1199050)2

(1 minus 120572)2 (29)




29(minus1+2120572)120573

2120578


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(minus9119890

(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

+ 323(minus(119905 minus 1199050)3

(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578]) (2 (119905 minus 119905

0)2


(30)

and hence


29(minus1+2120572)120573

2120578

6 (119905 minus 1199050)3

(minus1 + 120572)3(minus1 + 2120572) 120573

21205782(120578 (minus2119862

2(119905

minus 1199050)5


29(minus1+2120572)120573

2120578(minus1

+ 2120572) 1205733((119905 minus 119905

0)3

(minus1 + 120572)2+ 6 (minus1 + 2120572) 120573

2120578))

+ (119905 minus 1199050)6

(minus1 + 120572)412057311986423 [119879])

(31)


29(minus1 + 2120572)120573


119901DE becomes


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)


sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


(32)


119908DE

=(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


)

(33)


infin

1(119890minus119911119905119905119899)119889119905


minus30

minus25

minus20

minus15

minus10

minus5

0

wDE

02 04 06 08 1000t


minus2500

minus2000

minus1500

minus1000

minus500

0

peff

08 10 12 14 16 18 2006t







02 04 06 08 1000t

0

10000

20000

30000

40000

Π


00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000




wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t


minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t



119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)


119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





31198672= 120588 (4)

2 + 31198672= minus119901 minus Π (5)

+ 3119867 (120588 + 119901) = minus3119867Π (6)


119901eff = 119901 + Π (7)


119901= 1)

120588DE = 3(1205721198672+120573

2

1198891198672

119889119909) (8)


119867(119909) = 1198670exp(119909 (1 minus 120572)

120573) (9)


119867 = 1198670119886(1minus120572)120573

(10)


(119905) = 1198670119886 (119905)1+(1minus120572)120573

(11)


0we have 119886 = 1


119886 (119905) = (1198670(119905 minus 1199050) (minus1 + 120572) + 120573

120573)

120573(minus1+120572)

(12)


119867 =1198670120573

1198670(119905 minus 1199050) (minus1 + 120572) + 120573

(13)


120588DE = 31198672=

31198672

01205732

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2 (14)


119908DE = minus1

+1

3(minus2

120573+2120572


2

1198672

01205732

)

(15)


need

120572 lt 1 +120573

2(6 +

Π (119905 = 1199050)

1198672

0

) (16)


120572 lt 1 +Π (119905 = 119905

0)

21198672

0

120573 (17)


120585 = 120578120588]

120591 = 120585120588minus1= 120578120588

]minus1

(18)


120591Π + Π = minus3119867120585 (19)


Π = 119890minus(1205931119905+1205932)3912059321198672

01205781198621

minus 119890minus(1205931119905+1205932)3912059321198672

01205781198673

0(minus9119890

(1205931119905+1205932)3912059321198672

0120578

+ 323(minus(1205931119905 + 1205932)3

12059321198672

0120578)

23

Γ[1

3 minus(1205931119905 + 1205932)3

912059321198672

0120578])

sdot (21205932(1205931119905 + 1205932)2)minus1

(20)

where

1205931=1198670

120573(120572 minus 1)

1205932= 1 +

11986701199050

120573(1 minus 120572)

(21)




119908DE = minus1 +1

3(2 (120572 minus 1)

120573minus1

21198672

0

1198902119909(120572minus1)120573minus119883

(2

minus1198673

0120573(minus9119862

1119890119883+ 323(minus119883)23Γ [1

3 minus119883])

sdot ((1198670(1199050 (1 minus 120572) + 120573))

sdot (

119883 (91198672

0(11986701199050(1 minus 120572) + 120573) 120578)

120573)

23

)

minus1

))

(22)

where

119883 = 120573(1198670(120572 minus 1)

120573+ (1 minus

11986701199050(120572 minus 1)

120573)

sdot (1199050+

(minus1 + 119890119909(120572minus1)120573

) 120573

1198670(120572 minus 1)

))

3

sdot (1198672

0(11986701199050(1 minus 120572) + 120573) 120578)

minus1

(23)


120572 = 1 +120573

11986701199050

(24)



119909 997888rarr infin 997904rArr

119908DE 997888rarr minus1 +1

3(minus0108 minus

1

1198672

0


0(minus9 + 19577259(

(0137 + 0014811986701199050+ 4 times 10

minus41198672

0(minus1 + 119905

2

0))3

1198675

0(minus037 minus 002119867

01199050) 120578

)

23

sdot Γ [

[

1

3

101452804 (01369 + 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))3

1198675

0(minus037 minus 002119867

01199050) 120578

]

]

))((minus037 minus 00211986701199050) (01369

+ 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))2

)

minus1

)

(25)

and for early stage

119909 997888rarr minus014 997904rArr

119908DE 997888rarr minus1 +1

3

sdot (minus1

1198672

0

05119890minus004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578

sdot (2 minus (0371198673

0(minus9119890004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578


+ 107(minus

(minus014 + 09911986701199050+ 1198672

0(minus0054 + 0054119905

2

0))3

1198675

0(037 + 002119867

01199050) 120578

)

23


0+ (minus014119867

0+ 1199050) (1 + 0054119867

01199050))3

1198672

0(037 + 002119867

01199050) 120578

]))

sdot ((037 + 00211986701199050) (minus0054119867

0+ (minus

014

1198670

+ 1199050) (1 + 0054119867

01199050))

2

)

minus1

))

(26)





119886 (119905) = 1198860(119905 minus 1199050)120573(1minus120572)

(27)

This leads to

119867 =120573

(119905 minus 1199050) (1 minus 120572)

= minus120573

(119905 minus 1199050)2

(1 minus 120572)

(28)


120588DE =31205732(2120572 minus 1)

(119905 minus 1199050)2

(1 minus 120572)2 (29)




29(minus1+2120572)120573

2120578


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(minus9119890

(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

+ 323(minus(119905 minus 1199050)3

(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578]) (2 (119905 minus 119905

0)2


(30)

and hence


29(minus1+2120572)120573

2120578

6 (119905 minus 1199050)3

(minus1 + 120572)3(minus1 + 2120572) 120573

21205782(120578 (minus2119862

2(119905

minus 1199050)5


29(minus1+2120572)120573

2120578(minus1

+ 2120572) 1205733((119905 minus 119905

0)3

(minus1 + 120572)2+ 6 (minus1 + 2120572) 120573

2120578))

+ (119905 minus 1199050)6

(minus1 + 120572)412057311986423 [119879])

(31)


29(minus1 + 2120572)120573


119901DE becomes


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)


sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


(32)


119908DE

=(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


)

(33)


infin

1(119890minus119911119905119905119899)119889119905


minus30

minus25

minus20

minus15

minus10

minus5

0

wDE

02 04 06 08 1000t


minus2500

minus2000

minus1500

minus1000

minus500

0

peff

08 10 12 14 16 18 2006t







02 04 06 08 1000t

0

10000

20000

30000

40000

Π


00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000




wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t


minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t



119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)


119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119908DE = minus1 +1

3(2 (120572 minus 1)

120573minus1

21198672

0

1198902119909(120572minus1)120573minus119883

(2

minus1198673

0120573(minus9119862

1119890119883+ 323(minus119883)23Γ [1

3 minus119883])

sdot ((1198670(1199050 (1 minus 120572) + 120573))

sdot (

119883 (91198672

0(11986701199050(1 minus 120572) + 120573) 120578)

120573)

23

)

minus1

))

(22)

where

119883 = 120573(1198670(120572 minus 1)

120573+ (1 minus

11986701199050(120572 minus 1)

120573)

sdot (1199050+

(minus1 + 119890119909(120572minus1)120573

) 120573

1198670(120572 minus 1)

))

3

sdot (1198672

0(11986701199050(1 minus 120572) + 120573) 120578)

minus1

(23)


120572 = 1 +120573

11986701199050

(24)



119909 997888rarr infin 997904rArr

119908DE 997888rarr minus1 +1

3(minus0108 minus

1

1198672

0


0(minus9 + 19577259(

(0137 + 0014811986701199050+ 4 times 10

minus41198672

0(minus1 + 119905

2

0))3

1198675

0(minus037 minus 002119867

01199050) 120578

)

23

sdot Γ [

[

1

3

101452804 (01369 + 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))3

1198675

0(minus037 minus 002119867

01199050) 120578

]

]

))((minus037 minus 00211986701199050) (01369

+ 0014811986701199050+ 4 times 10

minus41198672

0(1199052

0minus 1))2

)

minus1

)

(25)

and for early stage

119909 997888rarr minus014 997904rArr

119908DE 997888rarr minus1 +1

3

sdot (minus1

1198672

0

05119890minus004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578

sdot (2 minus (0371198673

0(minus9119890004(minus014+099119867

01199050+1198672

0(minus0054+0054119905

2

0))31198675

0(037+002119867

01199050)120578


+ 107(minus

(minus014 + 09911986701199050+ 1198672

0(minus0054 + 0054119905

2

0))3

1198675

0(037 + 002119867

01199050) 120578

)

23


0+ (minus014119867

0+ 1199050) (1 + 0054119867

01199050))3

1198672

0(037 + 002119867

01199050) 120578

]))

sdot ((037 + 00211986701199050) (minus0054119867

0+ (minus

014

1198670

+ 1199050) (1 + 0054119867

01199050))

2

)

minus1

))

(26)





119886 (119905) = 1198860(119905 minus 1199050)120573(1minus120572)

(27)

This leads to

119867 =120573

(119905 minus 1199050) (1 minus 120572)

= minus120573

(119905 minus 1199050)2

(1 minus 120572)

(28)


120588DE =31205732(2120572 minus 1)

(119905 minus 1199050)2

(1 minus 120572)2 (29)




29(minus1+2120572)120573

2120578


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(minus9119890

(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

+ 323(minus(119905 minus 1199050)3

(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578]) (2 (119905 minus 119905

0)2


(30)

and hence


29(minus1+2120572)120573

2120578

6 (119905 minus 1199050)3

(minus1 + 120572)3(minus1 + 2120572) 120573

21205782(120578 (minus2119862

2(119905

minus 1199050)5


29(minus1+2120572)120573

2120578(minus1

+ 2120572) 1205733((119905 minus 119905

0)3

(minus1 + 120572)2+ 6 (minus1 + 2120572) 120573

2120578))

+ (119905 minus 1199050)6

(minus1 + 120572)412057311986423 [119879])

(31)


29(minus1 + 2120572)120573


119901DE becomes


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)


sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


(32)


119908DE

=(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


)

(33)


infin

1(119890minus119911119905119905119899)119889119905


minus30

minus25

minus20

minus15

minus10

minus5

0

wDE

02 04 06 08 1000t


minus2500

minus2000

minus1500

minus1000

minus500

0

peff

08 10 12 14 16 18 2006t







02 04 06 08 1000t

0

10000

20000

30000

40000

Π


00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000




wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t


minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t



119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)


119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




+ 107(minus

(minus014 + 09911986701199050+ 1198672

0(minus0054 + 0054119905

2

0))3

1198675

0(037 + 002119867

01199050) 120578

)

23


0+ (minus014119867

0+ 1199050) (1 + 0054119867

01199050))3

1198672

0(037 + 002119867

01199050) 120578

]))

sdot ((037 + 00211986701199050) (minus0054119867

0+ (minus

014

1198670

+ 1199050) (1 + 0054119867

01199050))

2

)

minus1

))

(26)





119886 (119905) = 1198860(119905 minus 1199050)120573(1minus120572)

(27)

This leads to

119867 =120573

(119905 minus 1199050) (1 minus 120572)

= minus120573

(119905 minus 1199050)2

(1 minus 120572)

(28)


120588DE =31205732(2120572 minus 1)

(119905 minus 1199050)2

(1 minus 120572)2 (29)




29(minus1+2120572)120573

2120578


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(minus9119890

(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578

+ 323(minus(119905 minus 1199050)3

(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578]) (2 (119905 minus 119905

0)2


(30)

and hence


29(minus1+2120572)120573

2120578

6 (119905 minus 1199050)3

(minus1 + 120572)3(minus1 + 2120572) 120573

21205782(120578 (minus2119862

2(119905

minus 1199050)5


29(minus1+2120572)120573

2120578(minus1

+ 2120572) 1205733((119905 minus 119905

0)3

(minus1 + 120572)2+ 6 (minus1 + 2120572) 120573

2120578))

+ (119905 minus 1199050)6

(minus1 + 120572)412057311986423 [119879])

(31)


29(minus1 + 2120572)120573


119901DE becomes


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)


sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


(32)


119908DE

=(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


)

(33)


infin

1(119890minus119911119905119905119899)119889119905


minus30

minus25

minus20

minus15

minus10

minus5

0

wDE

02 04 06 08 1000t


minus2500

minus2000

minus1500

minus1000

minus500

0

peff

08 10 12 14 16 18 2006t







02 04 06 08 1000t

0

10000

20000

30000

40000

Π


00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000




wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t


minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t



119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)


119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


(32)


119908DE

=(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578

minus2120573

(119905 minus 1199050)2

(minus1 + 120572)

minus31205732

(119905 minus 1199050)2

(minus1 + 120572)2


29(minus1+2120572)120573

2120578(minus1 + 2120572)

sdot 1205733(9119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578


(minus1 + 120572)2

(minus1 + 2120572) 1205732120578

)

23

sdot Γ [1

3 minus(119905 minus 1199050)3

(minus1 + 120572)2

9 (minus1 + 2120572) 1205732120578])(2 (119905 minus 119905

0)2


)

(33)


infin

1(119890minus119911119905119905119899)119889119905


minus30

minus25

minus20

minus15

minus10

minus5

0

wDE

02 04 06 08 1000t


minus2500

minus2000

minus1500

minus1000

minus500

0

peff

08 10 12 14 16 18 2006t







02 04 06 08 1000t

0

10000

20000

30000

40000

Π


00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000




wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t


minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t



119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)


119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




02 04 06 08 1000t

0

10000

20000

30000

40000

Π


00 02 04 06 08 10t

120585

0

5000

10000

15000

20000

25000

30000




wDE

minus103

minus102

minus101

minus100

minus099

minus098

1 2 3 40t


minus2000

minus1500

minus1000

minus500

0

peff

02 04 06 08 1000t



119903 = 119902 + 21199022+

119867

119904 =119903 minus 1

3 (119902 minus 12)

(34)


119902 = minus119886

2 (35)


Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Π

00105 00110 00115 00120 0012500100t

times1011

0

1

2

3

4

5



119903 =1198672

0(120572 minus 1)


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

119904 = 2(minus1 +1198672


(1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2)


120573+2120572

120573

+ (1198621119890minus(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

2120578

sdot (1198670(119905 minus 1199050) (120572 minus 1) + 120573)

2

sdot (21198672

0(1 minus 4119905119905

0+ 311990521199052

0) (120572 minus 1)

2120573

minus 21198670119905 (minus2 + 3119905119905

0) (120572 minus 1) 120573

2+ 211990521205733

+ 1198673

0(minus21199050(120572 minus 1)

3+ 41199051199052

0(120572 minus 1)

3

minus 211990521199053

0(120572 minus 1)

3+ 9

sdot 119890(1198670(minus1+119905119905

0)(120572minus1)minus119905120573)

391198672

0(11986701199050(120572minus1)minus120573)120573

21205781205733)

minus 3231198673

01205733(minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

1198672

0(11986701199050 (120572 minus 1) minus 120573) 120573

2120578)

23

Dust

s

200 400 600 8000r

minus4

minus3

minus2

minus1

0ΛCDM



sdot Γ [1

3 minus(1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

3

91198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2120578]))

sdot (21198672

0(11986701199050(120572 minus 1) minus 120573) 120573

2

sdot (1198670(minus1 + 119905119905

0) (120572 minus 1) minus 119905120573)

2)minus1

))

minus1

(36)





119903 =1

2(1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (2 (1 + 120572 (minus3 + 2120572)) 120573

2120578)minus1

minus1


29(minus1+2120572)120573

2120578(119905 minus 1199050)2


29(minus1+2120572)120573

2120578(119905 minus 1199050)



2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2)


2 (minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (minus1 + 2120572)minus1) + (1 + 119890


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

)

(37)

119904 = (2119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(1 + 120572 (minus3 + 2120572)) 120573

2120578


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1


29(minus1+2120572)120573

2120578

6 (1 + 120572 (minus3 + 2120572)) 12057351205782(119905 minus 1199050)2

(minus1 + 120572)3

sdot (1205732120578(119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578(119905 minus 1199050)


2)


2(minus1 + 120572) 120578

+(119905 minus 1199050)4

(minus1 + 120572)2119864minus13 [119879]

(minus1 + 2120572) 120573120578+ 9 (119905 minus 119905

0) 12057311986423 [119879])


2(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

23875 23880 23885 23890 23895 2390023870r

minus6789

minus6788

minus6787

minus6786

minus6785


+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]) (minus1 + 2120572)

minus1)


29(minus1+2120572)120573

2120578

sdot (minus21198622(119905 minus 1199050)2


29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879])

sdot (2 (1 + 120572 (minus3 + 2120572)) 1205732120578)minus1

)

2

))

sdot (3 (minus21198622(119905 minus 1199050)2

(minus1 + 120572)3120578

+ 119890(119905minus1199050)3(minus1+120572)

29(minus1+2120572)120573

2120578120573


2) 120578

+ (119905 minus 1199050)3

(minus1 + 120572)212057311986423 [119879]))

minus1

(38)












119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119860=

1

radic1198672 + 1198961198862 (39)


119860=1

119867 (40)


119879119860=

1

2120587119860

(1 minus

119903119860

2119867119860

) (41)


119878119860=119860

4119866=1205872

119860

119866 (42)



119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) minus 3119867120585)(1 minus

119903119860

2119867119860

)

(43)


119879119860119860

= 41205871198673

119860(120588DE (1 + 119908DE) + 120591Π + Π)(1 minus

119903119860

2119867119860

)

(44)


119879DD = 119881DE + (120588DE (1 + 119908DE) minus 3119867120585)

= 119881DE + (120588DE (1 + 119908DE) + 120591Π + Π) (45)


3





119879 (119860+ DE) =

119860

2(120588DE (1 + 119908DE) + 120591Π + Π)

119903119860 (46)


119860+ DE =

2120587


sdot (120573

119866minus

8120587


2+1

1198672

0

(1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2


119890minus1205941198673


2 (11986701199050(minus1 + 120572) minus 120573) (119867

0(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 119890minus1205941205732120578(minus

21198621(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

1205732120578

minus63231198673

0120573Γ [13 minus120594]


13

+

61198675


(1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

3

+

1198673


(11986701199050(minus1 + 120572) minus 120573) 120578

)

sdot (2 (1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

+ 1198672

0(2120573 (120572 minus 1) minus 119890


2

sdot (minus91198901205941198673

01205733+ 21198621(11986701199050 (minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2

+ 3231198673

01205733(minus120594)23Γ [1

3 minus120594])


sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




sdot (21198672

0(11986701199050(minus1 + 120572) minus 120573)

sdot (1198670(minus1 + 119905119905

0) (minus1 + 120572) minus 119905120573)

2)minus1

)

sdot ((1198670(119905 minus 1199050) (minus1 + 120572) + 120573)

2)minus1

)))

(47)

where 120594 = (1198670(minus1 + 119905119905

0)(minus1 + 120572) minus 119905120573)

391198672

0(11986701199050(minus1 +


119860+ DE) based





0)120573(1minus120572)

as

119860+ DE =

1


2(2



29(minus1+2120572)120573

2120578(119905 minus 1199050)2

(minus1 + 120572)2

120573minus 3120573

+ 3 (minus1 + 2120572) 120573 +9 (minus1 + 2120572) 120573

2

2 (minus1 + 120572)

+ (minus1 + 120572)(3 (minus1 + 2120572) 120573

2120578

(119905 minus 1199050)2

(minus1 + 120572)2)


29(minus1+2120572)120573

2120578(119905 minus 1199050)3

(minus1 + 120572) 11986423 [119879]

2120578)

(48)





5 Conclusions


2



02

03

04

05

t

0030

0035

0040

0045

0050

0

5

10

15

dd

t(S A

+S D

E)

120578



1

2

3

4

t000010

000012

000014

0

2

4

6

dd

t(S A

+S D

E)

120578








and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






and 120573 = 04817+00842+01176






Competing Interests


Acknowledgments


References















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Israel-Stewart Approach to Viscous ...downloads.hindawi.com/journals/ahep/2016/8515967.pdf · lled with a bulk viscous cosmological uid, the energy-momentum tensor