research article will there be future deceleration? a study...

Research ArticleWill There Be Future Deceleration A Study of Particle CreationMechanism in Nonequilibrium Thermodynamics

Supriya Pan and Subenoy Chakraborty

Department of Mathematics Jadavpur University Kolkata West Bengal 700032 India

Correspondence should be addressed to Supriya Pan spanresearchjdvuacin

Received 15 December 2014 Revised 20 March 2015 Accepted 26 March 2015

Academic Editor Juan Jose Sanz-Cillero

Copyright copy 2015 S Pan and S Chakraborty 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

The paper deals with nonequilibrium thermodynamics based on adiabatic particle creation mechanism with the motivation ofconsidering it as an alternative choice to explain the recent observed accelerating phase of the universe Using Friedmannrsquosequations it is shown that the deceleration parameter (119902) can be obtained from the knowledge of the particle production rate (Γ)Motivated by thermodynamical point of view cosmological solutions are evaluated for the particle creation rates in three cosmicphases namely inflation matter dominated era and present late time acceleration The deceleration parameter (119902) is expressedas a function of the redshift parameter (119911) and its variation is presented graphically Also statefinder analysis has been presentedgraphically in three different phases of the universe Finally two noninteracting fluids with different particle creation rates areconsidered as cosmic substratum and deceleration parameter (119902) is evaluated Whether more than one transition of 119902 is possibleor not is examined by graphical representations

1 Introduction

There was a dramatic change in our knowledge of the evolu-tion history of the universe based on the standard cosmologyat the end of the last century due to some observationalpredictions from type Ia Supernova [1 2] and others [3 4]Riess et al [1] and Perlmutter et al [2] observed that distantSupernovae at redshift 119911 sim 05 and Δ119898 sim 025mag are foundto be about 25 fainter than the prediction from standardcosmology and hence they concluded that the universe atpresent is undergoing an accelerated expansion rather thandeceleration (as predicted by the standard cosmology) Thispresent accelerating phase was also supported by the cosmicmicrowave background (CMB) [3] and the baryon acousticoscillations (BAO) [4] The explanation of this unexpectedaccelerating phase is a great challenge to the theoreticalphysics Since the discovery of this accelerating universepeople are trying to explain this observational fact in twodifferent waysmdasheither bymodifying the Einstein gravity itselfor by introducing someunknownkind ofmatter in the frame-work of Einstein gravity In the second option cosmological

constant is the common choice for this unknown matter Butit suffers from two serious problemsmdashthe measured valueof the cosmological constant being far below the predictionfrom quantum field theory and secondly the coincidenceproblem [5] So people choose this unknownmatter as somekind of dynamical fluid with negative and time dependentequation of state which is termed dark energy (DE) Thougha lot of works have been done with several models of DE(see references [6ndash9] for reviews) still its origin is totallymysterious to us

Among other possibilities to explain the present acceler-ating stage inclusion of back reaction in the Einstein fieldequations through (minusve) effective pressure is much relevantin the context of cosmology and the gravitational productionof particles (radiation or cold dark matter (CDM)) provides amechanism for cosmic acceleration [10ndash16] In particular incomparison with dark energy models the particle creationscenario has a strong physical basis the nonequilibriumthermodynamics Also the particle creation mechanism notonly unifies the dark sectors (DE + DM) [15 16] but alsocontains only one free parameter as we need only a single

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 654025 10 pageshttpdxdoiorg1011552015654025

2 Advances in High Energy Physics

dark component (DM) Further statistical Bayesian analysiswith one free parameter should be preferred along with thehierarchy of cosmologicalmodels [17] So the present particlecreation model which simultaneously fits the observationaldata and alleviates the coincidence and fine-tuning problemsis better compared to the known (one-parameter) modelsnamely (i) the concordance ΛCDM which however suffersfrom the coincidence and fine-tuning problems [18ndash23] and(ii) the brane world cosmology [24] which does not fitthe SNIa + BAO + CMB (shift-parameter) data [25 26]Furthermore it should be mentioned that the thermody-namics of dark energy has been studied in equilibrium andnonequilibrium situations in the literature [27ndash32]

The homogeneous and isotropic flat FLRW model of theuniverse is chosen as an open thermodynamics systemwhichis adiabatic in nature Although the entropy per particle isconstant for this system still there is entropy production dueto expansion of the universe (ie enlargement of the phasespace) [33 34] As a result the dissipative pressure (ie bulkviscous pressure) is linearly related to the particle creationrate Γ [33ndash35] Further using the Friedmann equationsone can relate Γ to the evolution of the universe (see (8)below) Choosing Γ as a function of the Hubble parameterfrom thermodynamical view point it is possible to describedifferent phases of the evolution of the universe and 119902 can beobtained as a function of 119911 the redshift parameter Finallywe consider two components of matter which have differentparticle creation rate [36] and 119902 has been evaluated andplotted to examine whether more than one transition of119902 is possible or not In formulation of the general theoryof relativity in terms of the spin connection coefficientsit has been shown [37] that the cosmological evolution ofthe metrics is induced by the dilaton without the inflationhypothesis and the Λ term Further it is found that thedilaton evolution yields the vacuum creation of matter andthe dilaton vacuum energy plays a role of the dark energyOn the other hand in the Hamiltonian approach to thegravitational model with the aid of Dirac-ADM foliationPervushin et al [38] showed a natural separation of thedilatonic and gravitational dynamics in terms of the Maurer-Carton forms As a result the dominance of the Casimir vac-uum energy of physical fields provides a good description ofthe type Ia Supernovae luminosity distance-redshift relationFurthermore introducing the uncertainty principle at thePlanckrsquos epoch it is found that the hierarchy of the universersquosenergy scales is supported by the observational data Alsothis Hamiltonian dynamics of the model describe the effectof an intensive vacuum creation of gravitons and theminimalcoupling scalar (Higgs) bosons in the early universe

Moreover the motivation of the present work in theframework of the particle creation mechanism comes fromsome recent related works It has been shown in [39 40]that the entire cosmic evolution from inflationary stage canbe described by particle creation mechanism with somespecific choices of the particle creation rates As these worksshow late-time acceleration without any concept of darkenergy so it is very interesting to think of the particlecreation mechanism as an alternative way of explaining theidea of dark energy The present work is an extension of

these works by considering two fluid systems as the cosmicfluid The paper is organized as follows Section 2 dealswith nonequilibrium thermodynamics in the background ofparticle creation mechanics while several choices of Γ as afunction of the Hubble parameter are shown in Section 3and the deceleration parameter 119902 has been presented bothanalytically and graphically A field theoretic analysis ofthe particle creation mechanism is presented in Section 4Section 5 is related to interacting two dark fluids havingdifferent particle creation rates and it is examined whethertwo transitions for 119902 are possible or not Finally there is asummary of the work in Section 6

2 Nonequilibrium ThermodynamicsMechanism of Particle Creation

Suppose the homogeneous and isotropic flat FLRWmodel ofthe universe is chosen as an open thermodynamical systemThe metric ansatz takes the form

1198891199042= minus119889119905

2+ 1198862(119905) [119889119903

2+ 1199032(1198891205792+ sin21205791198891206012)] (1)

Then the Friedmann equations are

31198672= 120581120588 2 = minus120581 (120588 + 119901 + Π) (2)

where 120581 = 8120587119866 119867 = 119886119886 is the Hubble rate 119886 = 119886(119905) isthe scale factor of the universe 120588 and 119901 are the total energydensity and the thermodynamical pressure of the cosmicfluid Π is related to some dissipative phenomena (bulkviscous pressure) and the overdot denotes the derivative withrespect to the cosmic time 119905 It should be noted that thereare several choices of Π and corresponding solutions in theliterature [41 42] The energy conservation relation reads

120588 + 3119867 (120588 + 119901 + Π) = 0 (3)

As the particle number is not conserved (ie 119873120583120583

= 0)so themodified particle number conservation equation takesthe form [36]

119899 + Θ119899 = 119899Γ (4)

where 119899 = 119873119881 is the particle number density119873 is the totalnumber of particles in a comoving volume 119881 119873120583 = 119899119906

120583

is the particle flow vector 119906120583 is the particle velocity Θ =

119906120583

120583= 3119867 stands for the fluid expansion Γ represents the

particle creation rate and notationally 120578 = 120578120583119906120583 The sign

of Γ indicates creation (for Γ gt 0) or annihilation (for Γ lt 0)of particles and due to Π which adds some dissipative effectto the cosmic fluid nonequilibrium thermodynamics comesinto picture

Now from the Gibbrsquos equation using Clausius relation wehave [36]

119879119889119904 = 119889 (120588

119899) + 119901119889(

1

119899) (5)

where ldquo119904rdquo represents entropy per particle and 119879 is the fluidtemperature Using the conservation relations (3) and (4) thevariation of entropy can be expressed as [33ndash35]

119899119879 119904 = minusΠΘ minus Γ (120588 + 119901) (6)

Advances in High Energy Physics 3

Further for simplicity if we assume the thermal processto be adiabatic (ie 119904 = 0) then from (6) we have [33ndash35]

Π = minusΓ

Θ(120588 + 119901) (7)

Thus the dissipative pressure is completely characterizedby the particle creation rate for the above simple (isentropic)thermodynamical system In other words the cosmic sub-stratum may be considered as a perfect fluid with barotropicequation of state 119901 = (120574 minus 1)120588 (23 lt 120574 le 2) together withdissipative phenomena which comes into picture throughthe particle creation mechanism Furthermore althoughthe ldquoentropy per particlerdquo is constant still there is entropygeneration due to particle creation that is enlargement ofthe phase space through expansion of the universe So insome sense the nonequilibrium configuration is not theconventional one due to the effective bulk pressure rathera state with equilibrium properties as well (but not theequilibrium era with Γ = 0) Now eliminating 120588 119901 and Π

from the Friedmann equations (2) the isentropic condition(7) and using barotropic equation of state 120574 = 1 + 119901120588 weobtain

Γ

Θ= 1 +

2

3120574

1198672 (8)

The above equation shows that in case of adiabaticprocess the particle creation rate is related to the evolutionof the universe

3 Particle Creation Rate asa Function of the Hubble Parameter andEvolution of the Universe

Introducing the deceleration parameter

119902 equiv minus(1 +

1198672) (9)

and using (8) we have

119902 = minus1 +3120574

2(1 minus

Γ

Θ) (10)

In the following subsections we shall choose Γ as differentfunctions of the Hubble parameter to describe differentstages of evolution of the universe and examine whether119902 obtained from (10) has any transition (from decelerationto acceleration or vice versa) or not Furthermore it wouldbe worthwhile to see the statefinder analysis for the par-ticle creation rates in different stages of our universe Thestatefinder parameters were introduced by Sahni et al [43]as a geometrical concept to filter several observationallysupported dark energymodels from other phenomenologicaldark energymodels existing in the literatureThey introducedtwo new geometrical variables 119903 119904 as follows [43]

119903 =1

1198861198673

119886 119904 =

119903 minus 1

3 (119902 minus 12) (11)

31 Early Epochs In the very early universe (starting from aregular vacuum)most of the particle creation effectively takesplace and from thermodynamic point of view we have thefollowing [44]

(i) At the beginning of the expansion there shouldbe maximal entropy production rate (ie maximalparticle creation rate) so that universe evolves fromnonequilibrium thermodynamical state to equilib-rium era with the expansion of the universe

(ii) A regular (true) vacuum for radiation initially that is120588 rarr 0 as 119886 rarr 0

(iii) Γ gt 119867 in the very early universe so that the cre-ated radiation behaves as thermalized heat bath andsubsequently the creation rate should fall slowerthan expansion rate and particle creation becomesdynamically insignificant

Now according to Gunzig et al [45] the simplest choicesatisfying the above requirements is that particle creation rateis proportional to the energy density that is Γ = Γ

01198672 where

Γ0is a proportionality constant For this choice of Γ119867 can be

solved from (8) as

119867 =119867119890

120573 + (1 minus 120573) (119886119886119890)31205742

(12)

where 120573 is related to Γ0as Γ0

= 3120573119867119890 119867119890and 119886

119890are

chosen to be the values of the Hubble parameter and thescale factor respectively at some instant We note that as119886 rarr 0 119867 rarr 120573

minus1119867119890= constant indicating an exponential

expansion ( 119886 gt 0) in the inflationary era while for 119886 ≫ 119886119890

119867 prop 119886minus31205742 indicates the standard FLRW cosmology ( 119886 lt 0)

So if we identify ldquo119886119890rdquo at some intermediate value of ldquo119886rdquo where

119886 = 0 that is a transition from de Sitter accelerating phaseto the standard decelerating radiation phase then we have119890= minus119867

2

119890 and (8) gives 120573 = 1 minus 23120574

Now using (10) we obtain

119902 (119911) = minus1 +3120574

2[1 minus

120573

120573 + (1 minus 120573) (1 + 119911)minus31205742

] (13)

where the redshift parameter is defined as 119886119890119886 = 1 + 119911

Figure 1 shows the transition of the deceleration param-eter 119902(119911) from early inflationary era to the deceleratedradiation era Figures 2 and 3 display the graphical behavior ofthe statefinder parameters showing that in early stage of theuniverse 119903 changes its sign from +ve (equiv inflationary era) tominusve (equiv decelerating phase) while 119904 stays positive throughoutthe transition

32 IntermediateDecelerating Phase Here the simple naturalchoice is Γ prop 119867 It should be noted that this choice ofΓ does not satisfy the third thermodynamical requirement(mentioned above) at the early universe Also the solutionwillnot satisfy the above condition (ii) of Section 31 In this case119902 does not depend on the expansion rate it only depends on120574 For radiation (ie 120574 = 43) era 119902 = 1 minus 2Γ

13 while for


08

06

04

02

0

minus02

minus04

minus06

minus08

1 2 3 4 5 6 7 8 9

z

q

120574 = 19

120574 = 17

120574 = 15

Figure 1 The figure shows the transition from early inflationaryphase rarr decelerating stage (see (13))

z

r

120574 = 19

120574 = 17

120574 = 15

minus05

minus1

1

2

3

0 05 1 15 2

Figure 2The figures show the variation of 119903 against 119911 in early stageof the universe for three different choices of 120574

matter dominated era (ie 120574 = 1) 119902 = 12minusΓ12 So if Γ

1lt 1

then we have deceleration in both the epochs as in standardcosmology while there will be acceleration if Γ

1gt (3 minus 2120574)

The solution for theHubble parameter and the scale factorare given by

119867minus1

=3120574

2(1 minus

Γ1

3) 119905 119886 = 119886

0119905119897 (14)

where 119897 = 23120574(1 minus Γ13) which represents the usual power

law expansion of the universe in standard cosmology withparticle production rate decreasing as 119905minus1

120574 = 19

120574 = 17

120574 = 15

z

s

0

05

1

15

2

25

3

05 1 15 2

Figure 3 The second statefinder parameter 119904 is shown over 119911 forthree different choices of 120574

The statefinder parameters 119903 and 119904 in this case are constantwith the values

119903 = (1 minus1

119897) (1 minus

2

119897)

119904 =2

3(1 minus

1

119897) (1 minus

2

119897) (

2

119897minus 3)

minus1

(15)

which indicates that for 119897 rarr infin 119903 and 119904 change their sign 119903becomes +ve while 119904 becomes minusve

33 Late Time Evolution Accelerated Expansion In thiscase the thermodynamical requirements of Section 31 aremodified as follows [45]

(i) There should be minimum entropy production rateat the beginning of the late time accelerated expan-sion and the universe again becomes nonequilibriumthermodynamically

(ii) The late time false vacuum should have 120588 rarr 0 as119886 rarr infin

(iii) The creation rate should be faster than the expansionrate

We shall show that another simple choice of Γ namelyΓ prop 1119867 that is Γ = Γ

3119867 where Γ

3is a proportionality

constant will satisfy these requirements For this choice of Γthe Hubble parameter is related to the scale factor as

1198672=Γ3

3+ (

119886

119886119891

)

minus3120574

(16)


02 04 06 08 10 12 14 16 18 20

minus02

q

z

minus01

0

01

02

03

120574 = 099

120574 = 090

120574 = 085

Figure 4 It describes the cosmic evolution from decelerating phaserarr recent accelerating stage (see (15))

where 119886119891is some intermediate value of ldquo119886rdquo such that

119867 sim 119886minus31205742

for 119886 ≪ 119886119891

119867 simΓ3

3 for 119886 ≫ 119886

119891

(17)

So we have a transition from the standard cosmologicalera ( 119886 lt 0) to late time acceleration ( 119886 gt 0) and 119886

119891rsquos can

be identified as the value of the scale factor at the instant oftransition ( 119886 = 0) The deceleration parameter now has theexpression

119902 = minus1 +3120574

2[

1

1 + (Γ33) (1 + 119911)

minus31205742] (18)


Figure 4 displays the transition of the universe from matterdominated era to the present late time acceleration Furtherwe have presented the statefinder analysis in Figures 5 and 6for 119903 and 119904 parameters respectively

4 Field Theoretic Analysis andParticle Creation

This section deals with particle creation from vacuum usingquantum field theory [46] In particular the quantum effectof particle creation is considered in the context of thermo-dynamics of open systems and is interpreted as an additionalnegative pressure

The energy-momentum tensor corresponding to thequantum vacuum energy is

119879119876

120583] equiv ⟨119879119876

120583]⟩ = Λ (119905) 119892120583] (19)

03

0 1 2

z

3

r

04

05

06

07

120574 = 1

120574 = 09

120574 = 08

Figure 5 In late time this is the variation of 119903 over 119911

01

02

03

04

05

06

07

s

0 1 2 3 4 5 6 7

z

120574 = 1

120574 = 09

120574 = 08

Figure 6Thefigure shows the behavior of 119904 for three different valuesfor 120574

So the Friedmann equations and the correspondingenergy conservation equation of a perfect fluid are nowmodified as

31198672= 8120587119866 (120588 + Λ) (20)

2 = minus8120587119866 (119901 + 120588) (21)

120588 + 3119867 (119901 + 120588) = minusΛ (22)

which shows energy transfer from the decaying vacuum tomatter This modified energy conservation equation can be


considered as the energy balance equation for an imperfectfluid with bulk viscous pressure

Π =Λ

3119867 (23)

Now if the perfect fluid is considered as a scalar field withpotential 119881(120601) that is

120588120601=1

2

1206012+ 119881 (120601)

119901120601=1

2

1206012minus 119881 (120601)

(24)

then the Einstein field equations (20) and (21) becomerespectively (taking 8120587119866 = 1)

31198672=1

2

1206012+ 119881 (120601) + Λ (119905) 2 = minus 120601

2 (25)

and the evolution equation (22) of the scalar field is given by

120601 120601 + 120601119889119881

119889120601+ 3119867( 120601

2+

Λ

3119867) = 0 (26)

So we have

120601 = intradicminus21198671015840

119886119867 119881 = minusΛ + 3119867

2[1 +

1198861198671015840

3119867] (27)

where ldquo 1015840rdquo stands for the differentiation with respect to thescale factor ldquo119886rdquo Hence for adiabatic process the particlecreation rate can be written as

Γ =119867

2 (1 + 119902)[(4 minus 119903) +

1

1198673

119889119881

119889120601

radicminus2] (28)

where 119903 =119886119886119867

3 is the state finder parameter [43] and 119902 =

minus(1 + (1198672)) is the usual deceleration parameter

It may be noted that if we have only quantum energyand there is no other matter then the energy conservationequation (22) demands ldquoΛrdquo should be a constant Also from(8) we have constant particle creation rateradic3Λ

5 Two Noninteracting Fluids as CosmicSubstratum and Particle Creations

In this section we suppose that the present open thermo-dynamical system contains two noninteracting dark fluidswhich have different particle creation rates Let (120588

1 1199011) and

(1205882 1199012) be the energy density and thermodynamic pressure

of the fluids respectively Suppose that (1198991 1198992) denote the

number density of the two fluids having balance equations[36]

1198991+ 3119867119899

1= Γ11198991 119899

2+ 3119867119899

2= minusΓ21198992 (29)

where Γ1gt 0 and Γ

2gt 0 The above equations imply that

there is creation of particles of fluid-1 while particles of fluid-2 decay Now combining equations in (29) the total numberof particles 119899 = 119899

1+ 1198992 will have the balance equation

119899 + 3119867119899 = (Γ11198991minus Γ21198992

119899) 119899 = Γ119899 (30)

So the total number of particles will remain conserve ifΓ = 0 that is Γ

11198991= Γ21198992

Again from the isentropic condition in (7) the dissipative(bulk) pressures of the matter components are given by

Π1= minus

Γ1

3119867(1205881+ 1199011) Π

2= minus

Γ2

3119867(1205882+ 1199012) (31)

As a consequence the energy conservation relations are

1205881+ 3119867 (120588

1+ 1199011) = Γ1(1205881+ 1199011)

1205881+ 3119867 (120588

1+ 1199011) = minusΓ

2(1205882+ 1199012)

(32)

which imply an exchange of energy between the two fluidsIn this connection it should be mentioned that Barrowand Clifton [47] found cosmological solutions with energyexchange Now if 120596

1= 11990111205881and 120596

2= 11990121205882are the

equations of state of the two fluid components respectivelythen from the above two conservation relations the effectiveequations of state parameters are

119908eff1

= 1205961minus

Γ1

3119867(1 + 120596

1)

119908eff2

= 1205962+

Γ2

3119867(1 + 120596

2)

(33)

Thus from the Einstein equations we have

31198672= 1205881+ 1205882

2 = minus [(1205881+ 1199011+ Π1) + (120588

2+ 1199012+ Π2)]

(34)

Then the deceleration parameter can be written as [48]

119902 =1

2+3

2[minus

Γ1

3119867Ω1(1 + 120596

1)

+Γ2

3119867Ω2(1 + 120596

2) + (Ω

11205961+ Ω21205962)]

(35)

In particular if Γ1= Γ2= Γ (say) then the above form of

the deceleration parameter (119902) reads

119902 =1

2+3

2[ (Ω11205961+ Ω21205962)

+Γ

3119867((Ω21205962minus Ω11205961) + (Ω

2minus Ω1))]

(36)

Figures 7ndash12 describe how the deceleration parameterbehaves with the Hubble parameter for different choices ofthe particle creation rate (equal or unequal) and it alsomatches with the present day observations It should bementioned that the choice for Γ

1and Γ

2in Figure 9 is not

fixed rather the cosmic evolution can be obtained for anyΓ1

= 1198601198672 and Γ

2= (119861119867) + 119862 (where 119860 119861 and 119862 are

constants) Similarly for any PCR of the form (for Figure 12)Γ = 119863119867

2+ (119864119867) + 119865 (where 119863 119864 and 119865 are constants) we

can have the same evolution of the universe


minus02

minus04

0

02

04

q

H

1 2 3 4 5

Γ1 = H2 Γ2 = minus1

Γ1 = H2 Γ2 = minusH

minus1

Γ1 = Hminus1 Γ2 = minusH

2

Figure 7 It is a comparative study of the deceleration parameter (119902)with the Hubble parameter (119867) for different particle creation rates(Γ) for the set of values Ω

1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08

Figure 8The figure shows the variation of the deceleration param-eter (119902) with theHubble parameter (119867) in different equation of statesfor the following unequal particle creation rates Γ

1= 1198672 and Γ

2=

minus1119867 forΩ1= 04 Ω

2= 06

6 Discussions and Future Prospects

The present work deals with nonequilibrium thermodynam-ics based on particle creation formalism In the context ofuniversal thermodynamics flat FLRW model of the uni-verse is considered as the open thermodynamical systemAlthough cosmic fluid is chosen in the formof a perfect fluiddissipative effect in the form of bulk viscous pressure arises

12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01

Figure 9 The variation of 119902 against 119867 for unequal PCR hasbeen shown in this way inflation rarr deceleration rarr late timeacceleration rarr future deceleration Here Γ

1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03

Figure 10 The evolution from inflation rarr deceleration rarr latetime acceleration for a single PCR Γ = 2119867

2+3119867 has been shown

Here Ω1= 06 and Ω

2= 04

due to the particle production mechanism For simplicityof calculations we are restricted to the adiabatic processwhere the dissipative pressure is linearly related to the particleproduction rate (Γ) From thermodynamic point of view Γis chosen as a function of the Hubble parameter (119867) andthe deceleration parameter is shown to be a function ofthe redshift parameter In particular by proper choices ofΓ cosmological solutions are evaluated and the decelera-tion parameter is presented graphically in Figures 1 and 4


1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06

Figure 11 It describes our universe as follows inflation rarr decel-eration rarr late time acceleration for the particle creation rate Γ =

21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01

Figure 12 The figure shows that for equal PCR the completescenario from inflation to present late time acceleration can bedescribed and not only that but it also predicts a possible futuredeceleration of the universe Here Γ asymp 034119867

2minus 1818119867 + 1869

The graphs show a transition from early inflationary stage tothe radiation dominated era (Figure 1) and also the transitionfrom matter dominated era to the present late time acceler-ation (Figure 4) Further we have presented the statefinderanalysis in early (Figures 2 and 3) intermediate (119903 and 119904 areconstant in this stage) and late phases (Figures 5 and 6)respectively Then a field theoretic analysis has been shownfor the particle creation mechanism in Section 4 Finallyin Section 5 a combination of noninteracting two perfectfluids having different particle creation rate is considered

as a cosmic substratum and the deceleration parameter isevaluated The behavior of the deceleration parameter isexamined graphically in Figures 7ndash12 Figures 7 8 10 and11 show two transitions of 119902mdashone in the early epoch fromacceleration to deceleration and the other one correspondingto the transition in the recent past from deceleration topresent accelerating stage Figures 7 and 8 correspond to twodifferent choices of unequal particle creation parameterswhile for two distinct equal particle creation rates thevariations of 119902 are presented in Figures 10 and 11 There arethree distinct transitions of 119902 for unequal and equal particlecreation parameters in Figures 9 and 12 respectively Boththe figures show that there is a chance of our universeto decelerate again in future from the present acceleratingstage Thus theoretically considering noninteracting two-fluid system as cosmic substratum it is possible to have againa decelerating phase of the universe in future Therefore wemay conclude that the present observed accelerating phase isdue to nonequilibrium thermodynamics having particle cre-ation processes or in otherwords in addition to the presentlyknown two possibilities (namely modification of Einsteingravity or introduction of some unknown exotic fluid ieDE) for explaining the recent observations nonequilibriumthermodynamics through particle creation mechanism maynot only explain the late time acceleration but also exhibitearly inflationary scenario and predict future transition todecelerating era again and this transient phenomenon issupported by the works in [39 49 50] Finally we remarkthat only future evolution of the universe can test whetherour prediction from particle creation mechanism is corrector wrong

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Supriya Pan acknowledges CSIR Government of India forfinancial support through SRF scheme (File no 09096(0749)2012-EMR-I) Subenoy Chakraborty thanks UGC-DRS programme at the Department of MathematicsJadavpur University Both the authors thank Inter UniversityCentre for Astronomy and Astrophysics (IUCAA) PuneIndia for their warm hospitality as a part of the work wasdone during a visit there The authors thank Prof JDBarrow for introducing some references which were usefulfor the present work and their other works Finally theyare thankful to the anonymous referees for their valuablecomments on the earlier version of the paper which helpedus to improve the paper considerably

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 pp1009ndash1038 1998


[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 pp 565ndash586 1999

[3] E Komatsu K M Smith J Dunkley et al ldquoSeven-yearWilkinson Microwave Anisotropy Probe (WMAP) observa-tions cosmological interpretationrdquo The Astrophysical JournalSupplement Series vol 192 no 2 article 18 2011

[4] A G Sanchez C G Scoccola A J Ross et al ldquoThe clusteringof galaxies in the SDSS-III Baryon Oscillation SpectroscopicSurvey cosmological implications of the large-scale two-pointcorrelation functionrdquoMonthly Notices of the Royal AstronomicalSociety vol 425 pp 415ndash437 2012

[5] S M Carroll ldquoThe Cosmological Constantrdquo Living Reviews inRelativity vol 4 no 1 2001

[6] T Padmanabhan ldquoCosmological constantmdashthe weight of thevacuumrdquo Physics Reports vol 380 no 5-6 pp 235ndash320 2003

[7] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

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

[9] J Yoo and Y Watanabe ldquoTheoretical models of dark energyrdquoInternational Journal of Modern Physics D vol 21 Article ID1230002 53 pages 2012

[10] I Prigogine J Geheniau E Gunzig and P Nardone ldquoThermo-dynamics and cosmologyrdquo General Relativity and Gravitationvol 21 no 8 pp 767ndash776 1989

[11] M O Calvao J A S Lima and I Waga ldquoOn the thermody-namics of matter creation in cosmologyrdquo Physics Letters A vol162 no 3 pp 223ndash226 1992

[12] J A S Lima M O Calvao and I Waga Frontier Physics Essaysin Honor of Jayme Tiomno World Scientific Singapore 1990

[13] J A S Lima andA SM Germano ldquoOn the equivalence of bulkviscosity and matter creationrdquo Physics Letters A vol 170 no 5pp 373ndash378 1992

[14] S Basilakos and J A S Lima ldquoConstraints on cold dark mat-ter accelerating cosmologies and cluster formationrdquo PhysicalReview D vol 82 Article ID 023504 2010

[15] J A S Lima F E Silva andR C Santos ldquoAccelerating cold darkmatter cosmology (Ω

Λequiv 0)rdquo Classical and Quantum Gravity

vol 25 Article ID 205006 2008[16] J A S Lima J F Jesus and F A Oliveira ldquoCDM accelerating

cosmology as an alternative to ΛCDM modelrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 027 2010

[17] A C C Guimaraes J V Cunha and J A S Lima ldquoBayesiananalysis and constraints on kinematicmodels fromunion SNIardquoJournal of Cosmology and Astroparticle Physics vol 2009 no 10article 10 2009

[18] I Zlatev L Wang and P J Steinhardt ldquoQuintessence cosmiccoincidence and the cosmological constantrdquo Physical ReviewLetters vol 82 no 5 pp 896ndash899 1999

[19] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 Article ID 083513 2003

[20] S Nojiri and S DOdintsov ldquoThe oscillating dark energy futuresingularity and coincidence problemrdquo Physics Letters B vol 637no 3 pp 139ndash148 2006

[21] S del Campo R Herrera and D Pavon ldquoToward a solution ofthe coincidence problemrdquo Physical Review D vol 78 Article ID021302 2008

[22] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 no 2 Article ID023508 6 pages 2010

[23] J P Ostriker and P J Steinhardt ldquoCosmic concordancerdquohttparxivorgabsastro-ph9505066

[24] C Deffayet G Dvali and G Gabadadze ldquoAccelerated universefrom gravity leaking to extra dimensionsrdquo Physical Review DThird Series vol 65 Article ID 044023 2002

[25] S Basilakos M Plionis and J A S Lima ldquoConfronting darkenergy models using galaxy cluster number countsrdquo PhysicalReview D vol 82 Article ID 083517 2010

[26] S Basilakos M Plionis M E S Alves and J A S LimaldquoDynamics and constraints of the massive graviton dark matterflat cosmologiesrdquo Physical Review D vol 83 Article ID 1035062011

[27] M Jamil E N Saridakis and M R Setare ldquoThermodynamicsof dark energy interacting with dark matter and radiationrdquoPhysical ReviewD vol 81 no 2 Article ID 023007 6 pages 2010

[28] M Jamil E N Saridakis and M R Setare ldquoThe generalizedsecond law of thermodynamics in Horava-Lifshitz cosmologyrdquoJournal of Cosmology and Astroparticle Physics no 11 article 3220 pages 2010

[29] H M Sadjadi and M Jamil ldquoGeneralized second law of ther-modynamics for FRW cosmology with logarithmic correctionrdquoEurophysics Letters vol 92 no 6 Article ID 69001 2010

[30] K Karami M Jamil and N Sahraei ldquoIrreversible thermo-dynamics of dark energy on the entropy-corrected apparenthorizonrdquo Physica Scripta vol 82 Article ID 045901 2010

[31] H M Sadjadi and M Jamil ldquoCosmic accelerated expansionand the entropy-corrected holographic dark energyrdquo GeneralRelativity and Gravitation vol 43 no 6 pp 1759ndash1775 2011

[32] M U Farooq and M Jamil ldquoNonequilibrium thermodynamicsof dark energy on the power-law entropy-corrected apparenthorizonrdquo Canadian Journal of Physics vol 89 no 12 pp 1251ndash1254 2011

[33] W Zimdahl ldquoBulk viscous cosmologyrdquo Physical Review D vol53 p 5483 1996

[34] W Zimdahl ldquoCosmological particle production causal ther-modynamics and inflationary expansionrdquo Physical Review Dvol 61 Article ID 083511 2000

[35] S Chakraborty ldquoIs emergent universe a consequence of particlecreation processrdquo Physics Letters B vol 732 pp 81ndash84 2014

[36] T Harko and F S N Lobo ldquoIrreversible thermodynamicdescription of interacting dark energy-dark matter cosmolog-ical modelsrdquo Physical Review D vol 87 Article ID 044018 2013

[37] A B Arbuzov BM Barbashov R G Nazmitdinov et al ldquoCon-formal Hamiltonian dynamics of general relativityrdquo PhysicsLetters B vol 691 no 5 pp 230ndash233 2010

[38] V N Pervushin A B Arbuzov B M Barbashov et al ldquoCon-formal and affine Hamiltonian dynamics of general relativityrdquoGeneral Relativity andGravitation vol 44 no 11 pp 2745ndash27832012

[39] S Chakraborty S Pan and S Saha ldquoA third alternative toexplain recent observations future decelerationrdquo Physics LettersB vol 738 pp 424ndash427 2014

[40] S Chakraborty and S Saha ldquoComplete cosmic scenario frominflation to late time acceleration nonequilibrium thermody-namics in the context of particle creationrdquo Physical Review Dvol 90 Article ID 123505 2014


[41] J D Barrow ldquoString-driven inflationary and deflationary cos-mological modelsrdquo Nuclear Physics B vol 310 no 3-4 pp 743ndash763 1988

[42] J A S Lima R Portugal and I Waga ldquoBulk-viscosity-drivenasymmetric inflationary universerdquo Physical Review D vol 37no 10 pp 2755ndash2760 1988

[43] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics vol 77 no 5pp 201ndash206 2003

[44] J A S Lima S Basilakos and F E M Costa ldquoNew cosmicaccelerating scenario without dark energyrdquo Physical Review Dvol 86 Article ID 103534 2012

[45] E Gunzig R Maartens and A V Nesteruk ldquoInflationary cos-mology and thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 4 pp 923ndash932 1998

[46] L L Graef F E M Costa and J A S Lima ldquoOn the equiv-alence of Λ(119905) and gravitationally induced particle productioncosmologiesrdquo Physics Letters B vol 728 pp 400ndash406 2014

[47] J D Barrow and T Clifton ldquoCosmologies with energyexchangerdquo Physical Review D vol 73 no 10 Article ID 1035202006

[48] S Pan and S Chakraborty ldquoWill there be again a transitionfrom acceleration to deceleration in course of the dark energyevolution of the universerdquoTheEuropean Physical Journal C vol73 article 2575 2013

[49] F C Carvalho J S Alcaniz J A S Lima and R Silva ldquoScalar-field-dominated cosmologywith a transient accelerating phaserdquoPhysical Review Letters vol 97 Article ID 081301 2006

[50] A C C Guimaraes and J A S Lima ldquoCould the cosmicacceleration be transient A cosmographic evaluationrdquoClassicaland Quantum Gravity vol 28 Article ID 125026 2011

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


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


dark component (DM) Further statistical Bayesian analysiswith one free parameter should be preferred along with thehierarchy of cosmologicalmodels [17] So the present particlecreation model which simultaneously fits the observationaldata and alleviates the coincidence and fine-tuning problemsis better compared to the known (one-parameter) modelsnamely (i) the concordance ΛCDM which however suffersfrom the coincidence and fine-tuning problems [18ndash23] and(ii) the brane world cosmology [24] which does not fitthe SNIa + BAO + CMB (shift-parameter) data [25 26]Furthermore it should be mentioned that the thermody-namics of dark energy has been studied in equilibrium andnonequilibrium situations in the literature [27ndash32]

The homogeneous and isotropic flat FLRW model of theuniverse is chosen as an open thermodynamics systemwhichis adiabatic in nature Although the entropy per particle isconstant for this system still there is entropy production dueto expansion of the universe (ie enlargement of the phasespace) [33 34] As a result the dissipative pressure (ie bulkviscous pressure) is linearly related to the particle creationrate Γ [33ndash35] Further using the Friedmann equationsone can relate Γ to the evolution of the universe (see (8)below) Choosing Γ as a function of the Hubble parameterfrom thermodynamical view point it is possible to describedifferent phases of the evolution of the universe and 119902 can beobtained as a function of 119911 the redshift parameter Finallywe consider two components of matter which have differentparticle creation rate [36] and 119902 has been evaluated andplotted to examine whether more than one transition of119902 is possible or not In formulation of the general theoryof relativity in terms of the spin connection coefficientsit has been shown [37] that the cosmological evolution ofthe metrics is induced by the dilaton without the inflationhypothesis and the Λ term Further it is found that thedilaton evolution yields the vacuum creation of matter andthe dilaton vacuum energy plays a role of the dark energyOn the other hand in the Hamiltonian approach to thegravitational model with the aid of Dirac-ADM foliationPervushin et al [38] showed a natural separation of thedilatonic and gravitational dynamics in terms of the Maurer-Carton forms As a result the dominance of the Casimir vac-uum energy of physical fields provides a good description ofthe type Ia Supernovae luminosity distance-redshift relationFurthermore introducing the uncertainty principle at thePlanckrsquos epoch it is found that the hierarchy of the universersquosenergy scales is supported by the observational data Alsothis Hamiltonian dynamics of the model describe the effectof an intensive vacuum creation of gravitons and theminimalcoupling scalar (Higgs) bosons in the early universe

Moreover the motivation of the present work in theframework of the particle creation mechanism comes fromsome recent related works It has been shown in [39 40]that the entire cosmic evolution from inflationary stage canbe described by particle creation mechanism with somespecific choices of the particle creation rates As these worksshow late-time acceleration without any concept of darkenergy so it is very interesting to think of the particlecreation mechanism as an alternative way of explaining theidea of dark energy The present work is an extension of

these works by considering two fluid systems as the cosmicfluid The paper is organized as follows Section 2 dealswith nonequilibrium thermodynamics in the background ofparticle creation mechanics while several choices of Γ as afunction of the Hubble parameter are shown in Section 3and the deceleration parameter 119902 has been presented bothanalytically and graphically A field theoretic analysis ofthe particle creation mechanism is presented in Section 4Section 5 is related to interacting two dark fluids havingdifferent particle creation rates and it is examined whethertwo transitions for 119902 are possible or not Finally there is asummary of the work in Section 6

2 Nonequilibrium ThermodynamicsMechanism of Particle Creation

Suppose the homogeneous and isotropic flat FLRWmodel ofthe universe is chosen as an open thermodynamical systemThe metric ansatz takes the form

1198891199042= minus119889119905

2+ 1198862(119905) [119889119903

2+ 1199032(1198891205792+ sin21205791198891206012)] (1)

Then the Friedmann equations are

31198672= 120581120588 2 = minus120581 (120588 + 119901 + Π) (2)

where 120581 = 8120587119866 119867 = 119886119886 is the Hubble rate 119886 = 119886(119905) isthe scale factor of the universe 120588 and 119901 are the total energydensity and the thermodynamical pressure of the cosmicfluid Π is related to some dissipative phenomena (bulkviscous pressure) and the overdot denotes the derivative withrespect to the cosmic time 119905 It should be noted that thereare several choices of Π and corresponding solutions in theliterature [41 42] The energy conservation relation reads

120588 + 3119867 (120588 + 119901 + Π) = 0 (3)

As the particle number is not conserved (ie 119873120583120583

= 0)so themodified particle number conservation equation takesthe form [36]

119899 + Θ119899 = 119899Γ (4)

where 119899 = 119873119881 is the particle number density119873 is the totalnumber of particles in a comoving volume 119881 119873120583 = 119899119906

120583

is the particle flow vector 119906120583 is the particle velocity Θ =

119906120583

120583= 3119867 stands for the fluid expansion Γ represents the

particle creation rate and notationally 120578 = 120578120583119906120583 The sign

of Γ indicates creation (for Γ gt 0) or annihilation (for Γ lt 0)of particles and due to Π which adds some dissipative effectto the cosmic fluid nonequilibrium thermodynamics comesinto picture

Now from the Gibbrsquos equation using Clausius relation wehave [36]

119879119889119904 = 119889 (120588

119899) + 119901119889(

1

119899) (5)

where ldquo119904rdquo represents entropy per particle and 119879 is the fluidtemperature Using the conservation relations (3) and (4) thevariation of entropy can be expressed as [33ndash35]

119899119879 119904 = minusΠΘ minus Γ (120588 + 119901) (6)



Π = minusΓ

Θ(120588 + 119901) (7)



Γ

Θ= 1 +

2

3120574

1198672 (8)





1198672) (9)


119902 = minus1 +3120574

2(1 minus

Γ

Θ) (10)


119903 =1

1198861198673

119886 119904 =

119903 minus 1

3 (119902 minus 12) (11)






01198672 where


solved from (8) as

119867 =119867119890

120573 + (1 minus 120573) (119886119886119890)31205742

(12)


= 3120573119867119890 119867119890and 119886

119890are







2

119890 and (8) gives 120573 = 1 minus 23120574


119902 (119911) = minus1 +3120574

2[1 minus

120573

120573 + (1 minus 120573) (1 + 119911)minus31205742

] (13)




13 while for


08

06

04

02

0

minus02

minus04

minus06

minus08

1 2 3 4 5 6 7 8 9

z

q

120574 = 19

120574 = 17

120574 = 15


z

r

120574 = 19

120574 = 17

120574 = 15

minus05

minus1

1

2

3

0 05 1 15 2



1lt 1


1gt (3 minus 2120574)


119867minus1

=3120574

2(1 minus

Γ1

3) 119905 119886 = 119886

0119905119897 (14)



120574 = 19

120574 = 17

120574 = 15

z

s

0

05

1

15

2

25

3

05 1 15 2



119903 = (1 minus1

119897) (1 minus

2

119897)

119904 =2

3(1 minus

1

119897) (1 minus

2

119897) (

2

119897minus 3)

minus1

(15)







3119867 where Γ



1198672=Γ3

3+ (

119886

119886119891

)

minus3120574

(16)


02 04 06 08 10 12 14 16 18 20

minus02

q

z

minus01

0

01

02

03

120574 = 099

120574 = 090

120574 = 085



119867 sim 119886minus31205742

for 119886 ≪ 119886119891

119867 simΓ3

3 for 119886 ≫ 119886

119891

(17)


119891rsquos can


119902 = minus1 +3120574

2[

1

1 + (Γ33) (1 + 119911)

minus31205742] (18)






119879119876

120583] equiv ⟨119879119876

120583]⟩ = Λ (119905) 119892120583] (19)

03

0 1 2

z

3

r

04

05

06

07

120574 = 1

120574 = 09

120574 = 08


01

02

03

04

05

06

07

s

0 1 2 3 4 5 6 7

z

120574 = 1

120574 = 09

120574 = 08



31198672= 8120587119866 (120588 + Λ) (20)

2 = minus8120587119866 (119901 + 120588) (21)

120588 + 3119867 (119901 + 120588) = minusΛ (22)




Π =Λ

3119867 (23)


120588120601=1

2

1206012+ 119881 (120601)

119901120601=1

2

1206012minus 119881 (120601)

(24)


31198672=1

2

1206012+ 119881 (120601) + Λ (119905) 2 = minus 120601

2 (25)


120601 120601 + 120601119889119881

119889120601+ 3119867( 120601

2+

Λ

3119867) = 0 (26)

So we have

120601 = intradicminus21198671015840

119886119867 119881 = minusΛ + 3119867

2[1 +

1198861198671015840

3119867] (27)


Γ =119867

2 (1 + 119902)[(4 minus 119903) +

1

1198673

119889119881

119889120601

radicminus2] (28)

where 119903 =119886119886119867






1 1199011) and




1198991+ 3119867119899

1= Γ11198991 119899

2+ 3119867119899

2= minusΓ21198992 (29)





119899 + 3119867119899 = (Γ11198991minus Γ21198992

119899) 119899 = Γ119899 (30)


11198991= Γ21198992


Π1= minus

Γ1

3119867(1205881+ 1199011) Π

2= minus

Γ2

3119867(1205882+ 1199012) (31)


1205881+ 3119867 (120588

1+ 1199011) = Γ1(1205881+ 1199011)

1205881+ 3119867 (120588

1+ 1199011) = minusΓ

2(1205882+ 1199012)

(32)


1= 11990111205881and 120596

2= 11990121205882are the


119908eff1

= 1205961minus

Γ1

3119867(1 + 120596

1)

119908eff2

= 1205962+

Γ2

3119867(1 + 120596

2)

(33)


31198672= 1205881+ 1205882

2 = minus [(1205881+ 1199011+ Π1) + (120588

2+ 1199012+ Π2)]

(34)


119902 =1

2+3

2[minus

Γ1

3119867Ω1(1 + 120596

1)

+Γ2

3119867Ω2(1 + 120596

2) + (Ω

11205961+ Ω21205962)]

(35)



119902 =1

2+3

2[ (Ω11205961+ Ω21205962)

+Γ

3119867((Ω21205962minus Ω11205961) + (Ω

2minus Ω1))]

(36)


1and Γ

2in Figure 9 is not


= 1198601198672 and Γ

2= (119861119867) + 119862 (where 119860 119861 and 119862 are





minus02

minus04

0

02

04

q

H

1 2 3 4 5



minus1


2


1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08


1= 1198672 and Γ

2=


2= 06



12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03



Here Ω1= 06 and Ω

2= 04



1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





Acknowledgments


References




























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Π = minusΓ

Θ(120588 + 119901) (7)



Γ

Θ= 1 +

2

3120574

1198672 (8)





1198672) (9)


119902 = minus1 +3120574

2(1 minus

Γ

Θ) (10)


119903 =1

1198861198673

119886 119904 =

119903 minus 1

3 (119902 minus 12) (11)






01198672 where


solved from (8) as

119867 =119867119890

120573 + (1 minus 120573) (119886119886119890)31205742

(12)


= 3120573119867119890 119867119890and 119886

119890are







2

119890 and (8) gives 120573 = 1 minus 23120574


119902 (119911) = minus1 +3120574

2[1 minus

120573

120573 + (1 minus 120573) (1 + 119911)minus31205742

] (13)




13 while for


08

06

04

02

0

minus02

minus04

minus06

minus08

1 2 3 4 5 6 7 8 9

z

q

120574 = 19

120574 = 17

120574 = 15


z

r

120574 = 19

120574 = 17

120574 = 15

minus05

minus1

1

2

3

0 05 1 15 2



1lt 1


1gt (3 minus 2120574)


119867minus1

=3120574

2(1 minus

Γ1

3) 119905 119886 = 119886

0119905119897 (14)



120574 = 19

120574 = 17

120574 = 15

z

s

0

05

1

15

2

25

3

05 1 15 2



119903 = (1 minus1

119897) (1 minus

2

119897)

119904 =2

3(1 minus

1

119897) (1 minus

2

119897) (

2

119897minus 3)

minus1

(15)







3119867 where Γ



1198672=Γ3

3+ (

119886

119886119891

)

minus3120574

(16)


02 04 06 08 10 12 14 16 18 20

minus02

q

z

minus01

0

01

02

03

120574 = 099

120574 = 090

120574 = 085



119867 sim 119886minus31205742

for 119886 ≪ 119886119891

119867 simΓ3

3 for 119886 ≫ 119886

119891

(17)


119891rsquos can


119902 = minus1 +3120574

2[

1

1 + (Γ33) (1 + 119911)

minus31205742] (18)






119879119876

120583] equiv ⟨119879119876

120583]⟩ = Λ (119905) 119892120583] (19)

03

0 1 2

z

3

r

04

05

06

07

120574 = 1

120574 = 09

120574 = 08


01

02

03

04

05

06

07

s

0 1 2 3 4 5 6 7

z

120574 = 1

120574 = 09

120574 = 08



31198672= 8120587119866 (120588 + Λ) (20)

2 = minus8120587119866 (119901 + 120588) (21)

120588 + 3119867 (119901 + 120588) = minusΛ (22)




Π =Λ

3119867 (23)


120588120601=1

2

1206012+ 119881 (120601)

119901120601=1

2

1206012minus 119881 (120601)

(24)


31198672=1

2

1206012+ 119881 (120601) + Λ (119905) 2 = minus 120601

2 (25)


120601 120601 + 120601119889119881

119889120601+ 3119867( 120601

2+

Λ

3119867) = 0 (26)

So we have

120601 = intradicminus21198671015840

119886119867 119881 = minusΛ + 3119867

2[1 +

1198861198671015840

3119867] (27)


Γ =119867

2 (1 + 119902)[(4 minus 119903) +

1

1198673

119889119881

119889120601

radicminus2] (28)

where 119903 =119886119886119867






1 1199011) and




1198991+ 3119867119899

1= Γ11198991 119899

2+ 3119867119899

2= minusΓ21198992 (29)





119899 + 3119867119899 = (Γ11198991minus Γ21198992

119899) 119899 = Γ119899 (30)


11198991= Γ21198992


Π1= minus

Γ1

3119867(1205881+ 1199011) Π

2= minus

Γ2

3119867(1205882+ 1199012) (31)


1205881+ 3119867 (120588

1+ 1199011) = Γ1(1205881+ 1199011)

1205881+ 3119867 (120588

1+ 1199011) = minusΓ

2(1205882+ 1199012)

(32)


1= 11990111205881and 120596

2= 11990121205882are the


119908eff1

= 1205961minus

Γ1

3119867(1 + 120596

1)

119908eff2

= 1205962+

Γ2

3119867(1 + 120596

2)

(33)


31198672= 1205881+ 1205882

2 = minus [(1205881+ 1199011+ Π1) + (120588

2+ 1199012+ Π2)]

(34)


119902 =1

2+3

2[minus

Γ1

3119867Ω1(1 + 120596

1)

+Γ2

3119867Ω2(1 + 120596

2) + (Ω

11205961+ Ω21205962)]

(35)



119902 =1

2+3

2[ (Ω11205961+ Ω21205962)

+Γ

3119867((Ω21205962minus Ω11205961) + (Ω

2minus Ω1))]

(36)


1and Γ

2in Figure 9 is not


= 1198601198672 and Γ

2= (119861119867) + 119862 (where 119860 119861 and 119862 are





minus02

minus04

0

02

04

q

H

1 2 3 4 5



minus1


2


1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08


1= 1198672 and Γ

2=


2= 06



12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03



Here Ω1= 06 and Ω

2= 04



1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





Acknowledgments


References




























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




08

06

04

02

0

minus02

minus04

minus06

minus08

1 2 3 4 5 6 7 8 9

z

q

120574 = 19

120574 = 17

120574 = 15


z

r

120574 = 19

120574 = 17

120574 = 15

minus05

minus1

1

2

3

0 05 1 15 2



1lt 1


1gt (3 minus 2120574)


119867minus1

=3120574

2(1 minus

Γ1

3) 119905 119886 = 119886

0119905119897 (14)



120574 = 19

120574 = 17

120574 = 15

z

s

0

05

1

15

2

25

3

05 1 15 2



119903 = (1 minus1

119897) (1 minus

2

119897)

119904 =2

3(1 minus

1

119897) (1 minus

2

119897) (

2

119897minus 3)

minus1

(15)







3119867 where Γ



1198672=Γ3

3+ (

119886

119886119891

)

minus3120574

(16)


02 04 06 08 10 12 14 16 18 20

minus02

q

z

minus01

0

01

02

03

120574 = 099

120574 = 090

120574 = 085



119867 sim 119886minus31205742

for 119886 ≪ 119886119891

119867 simΓ3

3 for 119886 ≫ 119886

119891

(17)


119891rsquos can


119902 = minus1 +3120574

2[

1

1 + (Γ33) (1 + 119911)

minus31205742] (18)






119879119876

120583] equiv ⟨119879119876

120583]⟩ = Λ (119905) 119892120583] (19)

03

0 1 2

z

3

r

04

05

06

07

120574 = 1

120574 = 09

120574 = 08


01

02

03

04

05

06

07

s

0 1 2 3 4 5 6 7

z

120574 = 1

120574 = 09

120574 = 08



31198672= 8120587119866 (120588 + Λ) (20)

2 = minus8120587119866 (119901 + 120588) (21)

120588 + 3119867 (119901 + 120588) = minusΛ (22)




Π =Λ

3119867 (23)


120588120601=1

2

1206012+ 119881 (120601)

119901120601=1

2

1206012minus 119881 (120601)

(24)


31198672=1

2

1206012+ 119881 (120601) + Λ (119905) 2 = minus 120601

2 (25)


120601 120601 + 120601119889119881

119889120601+ 3119867( 120601

2+

Λ

3119867) = 0 (26)

So we have

120601 = intradicminus21198671015840

119886119867 119881 = minusΛ + 3119867

2[1 +

1198861198671015840

3119867] (27)


Γ =119867

2 (1 + 119902)[(4 minus 119903) +

1

1198673

119889119881

119889120601

radicminus2] (28)

where 119903 =119886119886119867






1 1199011) and




1198991+ 3119867119899

1= Γ11198991 119899

2+ 3119867119899

2= minusΓ21198992 (29)





119899 + 3119867119899 = (Γ11198991minus Γ21198992

119899) 119899 = Γ119899 (30)


11198991= Γ21198992


Π1= minus

Γ1

3119867(1205881+ 1199011) Π

2= minus

Γ2

3119867(1205882+ 1199012) (31)


1205881+ 3119867 (120588

1+ 1199011) = Γ1(1205881+ 1199011)

1205881+ 3119867 (120588

1+ 1199011) = minusΓ

2(1205882+ 1199012)

(32)


1= 11990111205881and 120596

2= 11990121205882are the


119908eff1

= 1205961minus

Γ1

3119867(1 + 120596

1)

119908eff2

= 1205962+

Γ2

3119867(1 + 120596

2)

(33)


31198672= 1205881+ 1205882

2 = minus [(1205881+ 1199011+ Π1) + (120588

2+ 1199012+ Π2)]

(34)


119902 =1

2+3

2[minus

Γ1

3119867Ω1(1 + 120596

1)

+Γ2

3119867Ω2(1 + 120596

2) + (Ω

11205961+ Ω21205962)]

(35)



119902 =1

2+3

2[ (Ω11205961+ Ω21205962)

+Γ

3119867((Ω21205962minus Ω11205961) + (Ω

2minus Ω1))]

(36)


1and Γ

2in Figure 9 is not


= 1198601198672 and Γ

2= (119861119867) + 119862 (where 119860 119861 and 119862 are





minus02

minus04

0

02

04

q

H

1 2 3 4 5



minus1


2


1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08


1= 1198672 and Γ

2=


2= 06



12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03



Here Ω1= 06 and Ω

2= 04



1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





Acknowledgments


References




























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




02 04 06 08 10 12 14 16 18 20

minus02

q

z

minus01

0

01

02

03

120574 = 099

120574 = 090

120574 = 085



119867 sim 119886minus31205742

for 119886 ≪ 119886119891

119867 simΓ3

3 for 119886 ≫ 119886

119891

(17)


119891rsquos can


119902 = minus1 +3120574

2[

1

1 + (Γ33) (1 + 119911)

minus31205742] (18)






119879119876

120583] equiv ⟨119879119876

120583]⟩ = Λ (119905) 119892120583] (19)

03

0 1 2

z

3

r

04

05

06

07

120574 = 1

120574 = 09

120574 = 08


01

02

03

04

05

06

07

s

0 1 2 3 4 5 6 7

z

120574 = 1

120574 = 09

120574 = 08



31198672= 8120587119866 (120588 + Λ) (20)

2 = minus8120587119866 (119901 + 120588) (21)

120588 + 3119867 (119901 + 120588) = minusΛ (22)




Π =Λ

3119867 (23)


120588120601=1

2

1206012+ 119881 (120601)

119901120601=1

2

1206012minus 119881 (120601)

(24)


31198672=1

2

1206012+ 119881 (120601) + Λ (119905) 2 = minus 120601

2 (25)


120601 120601 + 120601119889119881

119889120601+ 3119867( 120601

2+

Λ

3119867) = 0 (26)

So we have

120601 = intradicminus21198671015840

119886119867 119881 = minusΛ + 3119867

2[1 +

1198861198671015840

3119867] (27)


Γ =119867

2 (1 + 119902)[(4 minus 119903) +

1

1198673

119889119881

119889120601

radicminus2] (28)

where 119903 =119886119886119867






1 1199011) and




1198991+ 3119867119899

1= Γ11198991 119899

2+ 3119867119899

2= minusΓ21198992 (29)





119899 + 3119867119899 = (Γ11198991minus Γ21198992

119899) 119899 = Γ119899 (30)


11198991= Γ21198992


Π1= minus

Γ1

3119867(1205881+ 1199011) Π

2= minus

Γ2

3119867(1205882+ 1199012) (31)


1205881+ 3119867 (120588

1+ 1199011) = Γ1(1205881+ 1199011)

1205881+ 3119867 (120588

1+ 1199011) = minusΓ

2(1205882+ 1199012)

(32)


1= 11990111205881and 120596

2= 11990121205882are the


119908eff1

= 1205961minus

Γ1

3119867(1 + 120596

1)

119908eff2

= 1205962+

Γ2

3119867(1 + 120596

2)

(33)


31198672= 1205881+ 1205882

2 = minus [(1205881+ 1199011+ Π1) + (120588

2+ 1199012+ Π2)]

(34)


119902 =1

2+3

2[minus

Γ1

3119867Ω1(1 + 120596

1)

+Γ2

3119867Ω2(1 + 120596

2) + (Ω

11205961+ Ω21205962)]

(35)



119902 =1

2+3

2[ (Ω11205961+ Ω21205962)

+Γ

3119867((Ω21205962minus Ω11205961) + (Ω

2minus Ω1))]

(36)


1and Γ

2in Figure 9 is not


= 1198601198672 and Γ

2= (119861119867) + 119862 (where 119860 119861 and 119862 are





minus02

minus04

0

02

04

q

H

1 2 3 4 5



minus1


2


1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08


1= 1198672 and Γ

2=


2= 06



12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03



Here Ω1= 06 and Ω

2= 04



1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





Acknowledgments


References




























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Π =Λ

3119867 (23)


120588120601=1

2

1206012+ 119881 (120601)

119901120601=1

2

1206012minus 119881 (120601)

(24)


31198672=1

2

1206012+ 119881 (120601) + Λ (119905) 2 = minus 120601

2 (25)


120601 120601 + 120601119889119881

119889120601+ 3119867( 120601

2+

Λ

3119867) = 0 (26)

So we have

120601 = intradicminus21198671015840

119886119867 119881 = minusΛ + 3119867

2[1 +

1198861198671015840

3119867] (27)


Γ =119867

2 (1 + 119902)[(4 minus 119903) +

1

1198673

119889119881

119889120601

radicminus2] (28)

where 119903 =119886119886119867






1 1199011) and




1198991+ 3119867119899

1= Γ11198991 119899

2+ 3119867119899

2= minusΓ21198992 (29)





119899 + 3119867119899 = (Γ11198991minus Γ21198992

119899) 119899 = Γ119899 (30)


11198991= Γ21198992


Π1= minus

Γ1

3119867(1205881+ 1199011) Π

2= minus

Γ2

3119867(1205882+ 1199012) (31)


1205881+ 3119867 (120588

1+ 1199011) = Γ1(1205881+ 1199011)

1205881+ 3119867 (120588

1+ 1199011) = minusΓ

2(1205882+ 1199012)

(32)


1= 11990111205881and 120596

2= 11990121205882are the


119908eff1

= 1205961minus

Γ1

3119867(1 + 120596

1)

119908eff2

= 1205962+

Γ2

3119867(1 + 120596

2)

(33)


31198672= 1205881+ 1205882

2 = minus [(1205881+ 1199011+ Π1) + (120588

2+ 1199012+ Π2)]

(34)


119902 =1

2+3

2[minus

Γ1

3119867Ω1(1 + 120596

1)

+Γ2

3119867Ω2(1 + 120596

2) + (Ω

11205961+ Ω21205962)]

(35)



119902 =1

2+3

2[ (Ω11205961+ Ω21205962)

+Γ

3119867((Ω21205962minus Ω11205961) + (Ω

2minus Ω1))]

(36)


1and Γ

2in Figure 9 is not


= 1198601198672 and Γ

2= (119861119867) + 119862 (where 119860 119861 and 119862 are





minus02

minus04

0

02

04

q

H

1 2 3 4 5



minus1


2


1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08


1= 1198672 and Γ

2=


2= 06



12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03



Here Ω1= 06 and Ω

2= 04



1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





Acknowledgments


References




























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus02

minus04

0

02

04

q

H

1 2 3 4 5



minus1


2


1= 04 Ω

2= 06 120596

1= 01 and 120596

2= 04

minus02

0

02

04

06

08

q

H

1 2 3 4 5 6

1205961 = 001 1205962 = 04

1205961 = 01 1205962 = 06

1205961 = 03 1205962 = 08


1= 1198672 and Γ

2=


2= 06



12

1

08

06

04

02

0

q

2 4 6 8 10 12 14 16 18

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


1asymp 012119867

2 and Γ2asymp

2071119867 minus 18

1205961 = 03 1205962 = 01

1205961 = 04 1205962 = 02

H

1 15 2 25 3

q

minus05

minus04

minus03

minus02

minus01

0

01

02

03



Here Ω1= 06 and Ω

2= 04



1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





Acknowledgments


References




























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1205961 = 04 1205962 = 01

1205961 = 03 1205962 = 02

H

105 15 2 25 3

q

minus06

minus04

minus02

0

02

04

06


21198672+ 1 withΩ

1= 06 Ω

2= 04

minus01

0

01

02

03

q

2 4 6 8 10 12

H

Ω1 = 06 1205961 = 03 Ω2 = 04 1205962 = 01


2minus 1818119867 + 1869





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



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