irreversible thermodynamic description of dark matter and

Research ArticleIrreversible Thermodynamic Description of Dark Matter andRadiation Creation during Inflationary Reheating

Juntong Su12 Tiberiu Harko34 and Shi-Dong Liang15

1School of Physics Sun Yat-Sen University Guangzhou 510275 China2Yat Sen School Sun Yat-Sen University Guangzhou 510275 China3Department of Physics Babes-Bolyai University Kogalniceanu Street 400084 Cluj-Napoca Romania4Department of Mathematics University College London Gower Street London WC1E 6BT UK5State Key Laboratory of Optoelectronic Material and Technology and Guangdong Province Key Laboratory ofDisplay Material and Technology Guangzhou China

Correspondence should be addressed to Tiberiu Harko tharkouclacuk

Received 10 July 2017 Accepted 23 August 2017 Published 18 October 2017

Academic Editor Orlando Luongo

Copyright copy 2017 Juntong Su et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We investigate matter creation processes during the reheating period of the early Universe by using the thermodynamic of opensystemsTheUniverse is assumed to consist of the inflationary scalar field which through its decay generates relativisticmatter andpressureless dark matterThe inflationary scalar field transfers its energy to the newly created matter particles with the field energydecreasing to near zero The equations governing the irreversible matter creation are obtained by combining the thermodynamicsdescription of the matter creation and the gravitational field equations The role of the different inflationary scalar field potentialsis analyzed by using analytical and numerical methods The values of the energy densities of relativistic matter and dark matterreach their maximum when the Universe is reheated up to the reheating temperature which is obtained as a function of the scalarfield decay width the scalar field particle mass and the cosmological parameters Particle production leads to the acceleration ofthe Universe during the reheating phase with the deceleration parameter showing complex dynamics Once the energy density ofthe scalar field becomes negligible with respect to the matter densities the expansion of the Universe decelerates and inflation hasa graceful exit

1 Introduction

The inflationary paradigm is one of the corner stones ofpresent day cosmology Inflation was first proposed in [1]to solve the spatial flatness the horizon and the monopoleproblems of the early Universe The main theoretical ingre-dients in inflation are a scalar field 120601 with self-interactionpotential 119881(120601) its energy density 120588120601 = 12060122 + 119881(120601) andpressure 119901120601 = 12060122 minus 119881(120601) respectively [2ndash5] In the initialapproach the inflation potential arrives at a local minimumat 120601 = 0 through supercooling from a phase transitionand then the Universe expands exponentially However inthis scenario called ldquoold inflationrdquo there is no graceful exitto the inflationary stage To solve this problem the ldquonewinflationrdquo model was proposed in [6ndash9] In this model the

inflationary scalar field is initially at its local maximum 120601 = 0The potential is required to be very flat near the minimumat 120601 = 0 so that the field rolls slowly down withoutchanging much before the Universe expands exponentiallyThe ldquoflatnessrdquo requirement is not easy to obtain and anotherproblem of this scenario is that the inflation has to beginvery late and last a long time or it may never happen insome cases [10] By extending the new inflation model thechaotic inflation theory was proposed [11] which solvedthe problems above The model does not require a thermalequilibrium state of the early Universe and various formsof the scalar field potentials are acceptable so that a largenumber potential forms can be used in this scenario Latera subtler theory of hybrid inflation with two scalar fields inthe model was proposed [12] The potential takes the form of

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 7650238 24 pageshttpsdoiorg10115520177650238

2 Advances in High Energy Physics

119881(120590 120601) = (1198722minus1205821205902)(4120582)+119898212060122+1198922120601212059022 with119872119898120582and 120590 constants One scalar field accounts for the exponentialgrowth of the Universe and the other one is responsible forthe graceful exit of inflation

Presently very precise observations of the CosmicMicrowave Background (CMB) radiation give us the oppor-tunity to test the fundamental predictions of inflation onprimordial fluctuations such as scale independence andGaussianity [13ndash17] One can calculate cosmological param-eters from CMB fluctuations and use them as constraints ofthe inflationary modelsThe slow-roll parameters 120598 and 120578 canbe calculated directly from the potentials of the inflationaryscalar field so that these parameters can be observed andthe corresponding inflationary models can be tested [18]The slow-roll parameters and the inflationary parameters likescalar spectral index 119899119904 and tensor-to-scalar ratio 119903 of severalforms of the inflationary potentials were investigated in [19]Further investigations of the parameters 119899119904 and 119903 can be foundin [20ndash22]

The prediction of the statistical isotropy claiming thatinflation removes the classical anisotropy of the Universehas been a central prediction of inflationary scenarios alsostrongly supported by the cosmic no-hair conjecture How-ever recently several observations of the large scale structureof the Universe have raised the possibility that the principlesof homogeneity and isotropy may not be valid at all scalesand that the presence of an intrinsic large scale anisotropyin the Universe cannot be ruled out a priori [23] Althoughgenerally the recent Planck observations tend to confirmthe fundamental principles of the inflationary paradigmthe CMB data seem to show the existence of some tensionbetween the models and the observations For example thecombined Planck andWMAPpolarization data show that theindex of the power spectrum is 119899119904 = 09603 plusmn 00073 [14 15]at the pivot scale 1198960 = 005Mpcminus1 a result which definitelyrules out the exact scale-invariance (119899119904 = 1) as predicted bysome inflationarymodels [24 25] atmore than 5120590Moreoverthis case can also be ruled out based on some fundamentaltheoretical considerations since it is connected with a deSitter phase and more importantly it does not lead to aconsistent inflationary model mainly due to the gracefulexit problem On the other hand the joint constraints on 119903(tensor-to-scalar ratio) and 119899119904 can put strong restrictions onthe inflationary scenarios For example inflationary modelswith power law potentials of the form 1206014 cannot provide anacceptable number of 119890-folds (of the order of 50ndash60) in therestricted space 119903 minus 119899119904 at a 2120590 level [15] In actuality thetheories of inflation have not yet reached a general consensusbecause not only the overall inflationary picture but also thedetails of the theoretical models are based on physics beyondthe Standard Model of particle physics

During inflation the exponential growth of the Universeled to a homogeneous isotropic and empty Universe Ele-mentary particles are believed to be created in the period ofreheating at the end of inflation defrosting the Universe andtransferring energy from the inflationary scalar field to mat-ter Reheating was suggested along with the new inflationaryscenario [6] and further developed in [26ndash28] When the

inflationary scalar field reaches its minimum after the expan-sion of the Universe it oscillates around the minimum ofthe potential and decays into Standard Model particles Thenewly created particles interact with each other and reach athermal equilibrium state of temperature 119879 However quan-tum field theoretical investigations indicate that the reheat-ing period is characterised by complicated nonequilibriumprocesses the main characteristic of which are initial violentparticle production via parametric resonance and inflatondecay (preheating) with a highly nonequilibrium distribu-tion of the produced particles subsequently relaxing to anequilibrium state Such explosive particle production is dueto parametric amplification of quantum fluctuations for thenonbroken symmetry case (appropriate for chaotic inflation)or spinodal instabilities in the broken symmetry phase [29ndash32]

The reheating temperature 119879reh is thought to be the max-imum temperature of the radiation dominated Universe afterreheating However it was argued that the reheating tem-perature is not necessarily the maximum temperature of theUniverse after reheating [33] In other words reheatingmightbe just one of the stages of the matter creation and othersubstance with larger freeze-out temperature may be createdafter inflation through other process [34]

In the study of reheating the phenomenological approachintroduced in [26] has been widely investigated Thisapproach is based on the introduction of a suitable loss termin the scalar field equation which also appears as a sourceterm for the energy density of the newly created matterfluid It is this source term that if chosen appropriately isresponsible for the reheating process that follows adiabaticsupercooling during the de Sitter phase In this simple modelthe corresponding dynamics is considered within a twocomponent model The first component is a scalar field withenergy density and pressure 120588120601 and 119901120601 respectively and withenergy-momentum tensor (120601)119879]120583 = (120588120601 + 119901120601)119906120583119906] minus 119901120601120575]120583The second component is represented by normal matterwith energy density and pressure 120588119898 and 119901119898 and withenergy-momentum tensor (119898)119879]120583 = (120588119898 + 119901119898)119906120583119906] minus 119901119898120575]120583In the above equations for simplicity we assume that bothcosmological fluid components share the same four velocities119906120583 normalized as 119906120583119906120583 = 1 Einsteinrsquos field equations implythe relation nabla120583((120601)119879]120583+(119898)119879]120583) = 0 which in the case of a flatgeometry reduces to

120601 120601 + 3119867 1206012 + 1198811015840 (120601) 120601 + 120588119898 + 3119867 (120588119898 + 119901119898) = 0 (1)

where 119867 = 119886119886 is the Hubble function and 119886 is the scalefactor of the Universe In order to describe the transitionprocess between the scalar field and the matter componentit is convenient to introduce phenomenologically a frictionterm Γ describing the decay of the scalar field componentand acting as a source term for the matter fluid Hence (1)is assumed to decompose into two separate equations

120601 + 3119867 120601 + 1198811015840 (120601) + Γ 120601 = 0 (2)

120588119898 + 3119867 (120588119898 + 119901119898) minus Γ 1206012 = 0 (3)

Advances in High Energy Physics 3

In the framework of the braneworld models in whichour Universe is a 3-brane embedded in a five-dimensionalbulk the reheating era after the inflationary period wasanalyzed [35] by assuming the possibility of brane-bulkenergy exchangeThe inflaton fieldwas assumed to decay intonormal matter only while the dark matter is injected intothe brane from the bulk The observational constraint of anapproximately constant ratio of the dark and the baryonicmatter requires that the dark matter must be nonrelativistic(cold) The model predicts a reheating temperature of theorder of 3 times 106 GeV and brane tension of the order of1025 GeV4 The composition of the Universe thus obtainedis consistent with the observational data The problem ofperturbative reheating and its effects on the evolution ofthe curvature perturbations in tachyonic inflationary modelswas investigated in [36] It was found that reheating doesnot affect the amplitude of the curvature perturbationsand after the transition the relative nonadiabatic pressureperturbation decays extremely rapidly during the early stagesof the radiation dominated epoch These behaviors ensurethat the amplitude of the curvature perturbations remainsunaffected during reheating It was pointed out in [37]that among primordial magnetogenesis models inflationis a prime candidate to explain the current existence ofcosmological magnetic fields Their energy density decreasesas radiation during the decelerating eras of the Universeand in particular during reheating Avoiding magnetic fieldbackreaction is always complementary to CMB and can givestronger limits on reheating for all high energy models ofinflation The reheating dynamics after the inflation inducedby 1198772-corrected 119891(119877) models were considered [38] Theinflationary and reheating dynamics was analyzed in theEinstein frame Observational constraints on the model werealso discussed

The processes of particle production from the inflatontheir subsequent thermalization and evolution of infla-tonplasma system by taking dissipation of the inflaton in ahot plasma into account were investigated in [39] and it wasshown that the reheating temperature is significantly affectedby these effects

In 119891(119877)modified gravity models the reheating dynamicsafter the inflation is significantly modified and affects theshape of the gravitational wave background spectrum [40] In[41] it was shown that reheating considerations may provideadditional constraints to some classes of inflationary modelsThe current Planck satellite measurements of the CosmicMicrowave Background anisotropies constrain the kinematicproperties of the reheating era for most of the inflationarymodels [42]This result is obtained by deriving the marginal-ized posterior distributions of the reheating parameter forabout 200models It turns out that the precision of the currentCMB data is such that estimating the observational perfor-mance of an inflationary model now requires incorporatinginformation about its reheating history It was pointed out in[43] that the inflaton coupling to the StandardModel particlesis generated radiatively even if absent at tree level Hencethe dynamics of the Higgs field during inflation can changedramatically The inflaton decay and reheating period after

the end of inflation in the nonminimal derivative couplinggravity model with chaotic potential was investigated in [44]This model provides an enhanced slow-roll inflation causedby gravitationally enhanced friction A scale-invariant modelof quadratic gravity with a nonminimally coupled scalar fieldwas studied in [45] At the end of inflation the Hubbleparameter and the scalar field converge to a stable fixed pointthrough damped oscillations that can reheat the Universe invarious ways For reviews on the inflationary reheating see[46] and [47] respectively

If the importance of the nonequilibrium and dissipa-tive aspects of particle creation during reheating has beenalready pointed out a long time ago [48] its irreversibleand open character has not received a similar attentionThermodynamical systems in which matter creation occursbelong to the class of open thermodynamical systems inwhich the usual adiabatic conservation laws are modifiedby explicitly including irreversible matter creation [49] Thethermodynamics of open systems has been also applied firstto cosmology in [49] The explicit inclusion of the mattercreation in the matter energy stress tensor in the Einsteinfield equations leads to a three-stage cosmological modelbeginning with an instability of the vacuum During the firststage the Universe is driven from an initial fluctuation of thevacuum to a de Sitter space and particle creation occursThede Sitter space exists during the decay time of its constituents(second stage) and ends after a phase transition in the usualFRWphaseThe phenomenological approach of [49] was fur-ther discussed and generalized in [50 51] through a covariantformulation allowing specific entropy variations as usuallyexpected for nonequilibrium processes The cosmologicalimplications of the irreversible matter creation processes inthe framework of the thermodynamics of open or quantumsystems have been intensively investigated in [52ndash75]

From the analysis of some classes of scalar field potentialsincluding the 1206011 potential which provide a good fit to theCMB measurements in [76] it was found that the Planck2015 68 confidence limit upper bound on the spectral index119899119904 implies an upper bound on the reheating temperature of119879reh le 6 times 1010 GeV The observations also exclude instan-taneous reheating Hence the low reheating temperaturesallowed by some scalar field models open the possibility thatdark matter could have been produced during the reheatingperiod instead of the radiation dominated era in the historyof the Universe Such a possibility could lead to a majorchange in our understanding of the early evolution stages oftheUniverse leading to different predictions for the relic den-sity andmomentumdistribution ofWIMPs sterile neutrinosand axions

It is the purpose of the present paper to apply thethermodynamics of open systems as introduced in [49 50]to a cosmological fluid mixture in which particle decay andproduction occur consisting of two basic components scalarfield and matter respectively From a cosmological pointof view this physical situation is specific to the preheatingreheating period of the inflationary cosmological modelsThe thermodynamics of irreversible processes as applied toinflationary cosmological models leads to a self-consistent


description of the matter creation processes which deter-mines the whole dynamics and future evolution of theUniverse By combining the basic principles of the thermo-dynamics of open systems with the cosmological Einsteinequations for a flat homogeneous and isotropic Universewe obtain a set of first-order ordinary differential equationsdescribing matter creation due to the decay of the scalarfield As for the matter components we specifically considera model in which ordinary matter is a mixture of two com-ponents a relativistic one (radiation) and a pressureless fluidassumed to correspond to dark matter In order to simplifythe analysis of the reheating equations we reformulate themin a dimensionless form by introducing a set of appropriatedimensionless variables As a cosmological application ofthe general formalism we investigate matter creation froma scalar field by adopting different mathematical forms ofthe self-interaction potential 119881 of the field More exactlywe will consider power law and exponential and Higgs typepotentials For each case the evolution of the postinflationaryUniverse with matter creation is investigated in detail bysolving numerically the set of the cosmological reheatingevolution equations The results display the process of infla-tionary scalar field decaying to matter and dark matter andthe effects of the expansion of the Universe on the decay Weconcentrate on the evolution of the darkmatter and radiationcomponents which increase from zero to a maximum valueas well as on the scalar field decay A simple model whichcan be fully solved analytically is also presented As a generalresult we find that the scalar field potential also plays animportant role in the reheating process and in the decay ofthe scalar field Some of the cosmological parameters of eachmodel are also presented and analyzed in detail The modelsare constrained by the observational parameters called theinflationary parameters including the scalar spectral index119899119904 the tensor-to-scalar ratio 119903 the number of 119890-folds 119873and the reheating temperature 119879reh These parameters can becalculated directly from the potentials used in the modelsand they can be used together with the observational datato constrain the free parameters in the potentials Thestudy of the reheating models and the development of newformalisms and constraints to the theories are undoubtedlyan essential part of the completion to the theory Analysisof the current experimental data including the calculationof the inflationary parameters is able to test the reheatingmodels

The present paper is organized as follows The basicresults in the thermodynamics of open systems are brieflyreviewed in Section 2 The full set of equations describingmatter creation due to the decay of a scalar field are obtainedin Section 3 where the observational constraints on theinflationary and reheating models are also presented Anexact solution of the system of the reheating equationscorresponding to simple form of the scalar field (coherentwave) is obtained in Section 4 Several reheating modelscorresponding to different choices of the scalar field self-interaction potential are considered in Section 5 by numer-ically solving the set of reheating evolution equations Thetime dynamics of the cosmological scalar field and of the

matter components is obtained and analyzed in detail Weconclude and discuss our results in Section 6

2 Irreversible Thermodynamics ofMatter Creation

In this Section we briefly review the thermodynamics ofmatter creation in open systems which we will use for thestudy of the postinflationary reheating processes In ourpresentation we start from the fundamental laws of thermo-dynamics we proceed to the general covariant formulationof the theory and then we apply our results to the particularcase of homogeneous and isotropic cosmological models

21 The First Law of Thermodynamics The main novelelement that appears in the thermodynamics of open systemswith energymatter exchange is the creation pressure relatedto the variation of the particle number The expression ofthe creation pressure can be derived from the fundamentallaws of thermodynamics [49]We consider a thermodynamicsystem consisting of N particles inside a volume V Fora closed system N is a constant and the first law ofthermodynamics the energy conservation equation is

dE = dQ minus 119901dV (4)

where E is the internal energy Q is the heat received by thesystem and 119901 is the thermodynamic pressure We can alsowrite the energy conservation as

d(120588119899) = d119902 minus 119901d(1119899) (5)

where the energy density 120588 = EV the particle numberdensity 119899 = NV and d119902 = dQN respectively Adiabatictransformations (dQ = 0) are described by

d (120588V) + 119901dV = 0 (6)

For an open system the particle number can vary in timeso thatN = N(119905) By including the thermodynamic param-eters of the newly created particles the energy conservationequation is modified to [49]

d (120588V) = dQ minus 119901dV + ℎ119899d (119899V) (7)

where ℎ = 120588 + 119901 is the enthalpy per unit volume Adiabatictransformations are then expressed as

d (120588V) + 119901dV minus ℎ119899d (119899V) = 0 (8)

The matter creation process is a source of the internal energyof the system From (8) we obtain the relation

120588 = (120588 + 119901) 119899119899 (9)

describing the variation of the matter density due to thecreation processes On the other hand the above equation


can also describe physical systems in the absence of mattercreation like for example the photon gas in which for aparticular temperature the particle number 119873 varies withthe volume in a fixed manner adjusting itself to have aconstant density of photons Hence for a photon gas theparticle number119873 is a variable and not a constant quantitylike in an ordinary gas The physical mechanism by whichthermodynamic equilibrium can be established consists inthe absorption and emission of photons by matter For aradiation like equation of state with 119901 = 1205883 from (9) weobtain 120588 = 1205880(1198991198990)43 with 1205880 and 1198990 constants giving thewell-known energy density-particle number relation for theultrarelativistic photon gas

Alternatively we can reformulate the law of the energyconservation by introducing the matter creation pressure sothat the energy conservation equation for open systems takesa form similar to (6) with the pressure written as 119901 = 119901 + 119901119888

d (120588V) = minus (119901 + 119901119888) dV (10)

where the creation pressure associated describing the creationof the matter is

119901119888 = minus120588 + 119901119899 d (119899V)dV

(11)

22 The Second Law of Thermodynamics In order to formu-late the second law of thermodynamics for open systems wedefine first the entropy flow d119890S and the entropy creationd119894S ge 0 so that the total entropy variation of the system iswritten as [49]

dS = d119890S + d119894S ge 0 (12)

For a closed and adiabatic system d119894S equiv 0 For an opensystem with matter creation in the following we assumethat matter is created in a thermal equilibrium state d119890S =0 Therefore the entropy increases only due to the particlecreation processes

The total differential of the entropy is

119879d (119904V) = d (120588V) minus ℎ119899d (119899V) + 119879d (119904V)= d (120588V) minus (ℎ minus 119879119904) dV= d (120588V) minus 120583d (119899V)

(13)

where the entropy density 119904 = SV ge 0 and the chemicalpotential 120583 = (ℎ minus 119879119904)119899 ge 0 Hence the entropy productionfor an open system is

119879d119894S = 119879dS = (ℎ119899) d (119899V) minus 120583d (119899V)= 119879 ( 119904119899) d (119899V) ge 0

(14)

The matter creation processes can also be formulated inthe framework of general relativity In order to apply the ther-modynamics of open systems to cosmology in the following

we consider the formulation of the thermodynamics of opensystems in a general relativistic covariant form [50] Then bystarting from this formulation we particularize it to the caseof a homogeneous and isotropic Universe by also derivingthe entropy of the system [72]

23 General Relativistic Description of the Thermodynamicof Open Systems In the following we denote the energy-momentum tensor of the system as T120583] and we introducethe entropy flux vector 119904120583 the particle flux vector N120583 andthe four-velocity 119906120583 of the relativistic fluid These variablesare defined as

T120583] = (120588 + 119901 + 119901119888) 119906120583119906] minus (119901 + 119901119888) 119892120583] (15)

119904120583 = 119899120590119906120583N120583 = 119899119906120583 (16)

where 119901119888 is the matter creation pressure 120590 is the specificentropy per particle and 119899 is the particle number density

The energy conservation law requires

nabla]T120583] = 0 (17)

while the second law of thermodynamics imposes the con-straint

nabla120583119904120583 ge 0 (18)

on the entropy flux four-vector The balance equation of theparticle flux vector is given by

nabla120583N120583 = Ψ (19)

where Ψ gt 0 represents a particle source or a particle sink ifΨ lt 0 For an open system the Gibbs equation is [50]

119899119879d120590 = d120588 minus 120588 + 119901119899 d119899 (20)

In order to derive the energy balance equation we multiplyboth sides of (17) by 119906120583 thus obtaining119906120583nabla]T120583] = 119906120583nabla] (120588 + 119901 + 119901119888) 119906120583119906] + 119906120583 (120588 + 119901 + 119901119888)

times nabla] (119906120583119906]) minus 119906120583nabla120583 (119901 + 119901119888)= 119906]nabla] (120588 + 119901 + 119901119888) + (120588 + 119901 + 119901119888)times (119906120583119906]nabla]119906120583 + 119906120583119906120583nabla]119906]) minus ( + 119901119888)

= 120588 + (120588 + 119901 + 119901119888) nabla]119906] = 0(21)

where we have used the relations 120588 = 119906120583nabla120583120588 = d120588d119904 119906120583119906120583 =1 and 119906120583119906]nabla]119906120583 = 0 respectively Hence we have obtainedthe energy conservation equation for open systems as

120588 + (120588 + 119901 + 119901119888) nabla120583119906120583 = 0 (22)


In order to obtain the entropy variation we substitute therelation (22) into theGibbs equation (20) andwe use (16) and(19) to obtain first

119899119879119906120583nabla120583120590 = 120588 minus 120588 + 119901119899 119906120583nabla120583119899119879nabla120583119904120583 minus 119879120590nabla120583N120583= minus (120588 + 119901 + 119901119888) nabla120583119906120583 minus 120588 + 119901119899times (nabla120583N120583 minus 119899nabla120583119906120583) 119879nabla120583119904120583

= minus119901119888nabla120583119906120583 minus (120588 + 119901119899 minus 119879120590)nabla120583N120583(23)

Thus we obtain the entropy balance equation as

nabla120583119904120583 = minus119901119888Θ119879 minus 120583Ψ119879 (24)

where 120583 = (120588 + 119901)119899 minus 119879120590 is the chemical potential and Θ =nabla120583119906120583 is the expansion of the fluid From (16) we obtain

nabla120583119904120583 = 120590119906120583nabla120583119899 + 119899119906120583nabla120583120590 = 120590Ψ + 119899 (25)

where we have constrained the specific entropy per particleto be a constant = 0 From (24) and (25) it follows that

minus119901119888Θ119879 minus 120583Ψ119879 = 120590Ψ (26)

and thus we obtain matter creation pressure as

119901119888 = minus120588 + 119901119899 ΨΘ (27)

One can see from the above expression that if there is nomatter creationΨ = 0 and then the matter creation pressure119901119888 vanishes and the entropy becomes a constant If matteris created with Ψ gt 0 and Θ gt 0 then the matter creationpressure will be negative 119901119888 lt 024 Matter Creation in Homogeneous and Isotropic UniversesIn the case of a homogeneous and isotropic flat Friedmann-Robertson-Walker (FRW) geometry the line element is givenby

d1199042 = d1199052 minus 1198862 (119905) (d1199092 + d1199102 + d1199112) (28)

where 119886(119905) is the scale factor In the comoving frame the four-velocity is given by 119906120583 = (1 0 0 0) For any thermodynamicfunction F its derivative is given by F = 119906120583nabla120583F = dFd119905For the covariant derivative of the four-velocity by taking intoaccount the fact thatV = 1198863 we findnabla120583119906120583 = 1radicminus119892 120597 (radicminus119892119906

120583)120597119909120583 = 11198863 120597 (1198863119906120583)120597119909120583 = V

V= 3119867 (29)

where119867 the expansion of the fluid (the Hubble function) isgiven bynabla120583119906120583 = VV = 119886119886The components of the energy-momentum tensor (15) in the presence of particle creation are

T00 = 120588

T11 = T

22 = T

33 = minus (119901 + 119901119888) (30)

From (19)we obtain the particle number balance equationas

Ψ = 119906120583nabla120583119899 + 119899nabla120583119906120583 = 119899 + 119899VV (31)

Hence by substituting the above equation into (14) we obtainthe variation of the entropy of the open system due to particlecreation processes as

S = S0 exp(int1199051199050

Ψ (119905)119899 d119905) (32)

where S0 = S(1199050) is an arbitrary integration constant

3 Reheating Dynamics Irreversible OpenSystems Thermodynamics Description

Reheating is the period at the end of inflation when matterparticles were created through the decay of the inflationaryscalar field The particle creation is essentially determinedby the energy density 120588120601 and pressure 119901120601 of the inflationaryscalar field 120601 For all inflationary models inflation begins atthe energy scale of order 1014 GeV or equivalently at an ageof the Universe of the order of 10minus34 s [3] Hence reheatingstarts at the end of inflation at a time of the order of 10minus32 sand it should end before the hot big bang at 10minus18 s whichmeans that the energy scale of reheating is from around10minus7 GeV up to 107 GeV The reheating temperature is animportant parameter of the model which we will discuss indetail

In this Section we derive first the general equationsdescribing the reheating process in the framework of thethermodynamics of open systems We also review the mainobservational constraints and parameters of the inflationarycosmological models prior and during the reheating

31 Irreversible Thermodynamical Description of ReheatingWe consider the Universe in the latest stages of inflation as anopen thermodynamic system with matter creationThe threeconstituents of the Universe are the inflationary scalar fieldthe ordinary matter with energy density 120588m and pressure 119901mand the dark matter with energy density 120588DM and pressure119901DM respectively Both the ordinary matter and the darkmatter are assumed to be perfect fluids The total energydensity 120588 of the system is

120588 = 120588120601 + 120588DM + 120588m (33)

Similarly the total pressure 119901 and the total particle numberdensity 119899 of the Universe at the end of inflation are given by

119901 = 119901120601 + 119901DM + 119901m119899 = 119899120601 + 119899DM + 119899m (34)

In the presence of particle creation processes the stress-energy tensorT120583] of the inflationary Universe is obtained as

T120583] = (120588 + 119901 + 119901(total)119888 ) 119906120583119906] minus (119901 + 119901(total)119888 ) 119892120583] (35)


where 119901(total)119888 is the total creation pressure describing mattercreation andwehave assumed that all three fluid componentsare comoving thus having the same four-velocity 119906120583 In thefollowing we will restrict our analysis to the case of the flatFriedmann-Robertson-Walker Universe with line elementgiven by (28) The basic equations describing the reheatingprocess in the framework of the thermodynamic of opensystems can be formulated as follows

(a) The Particle Number Balance Equation In the FRWgeometry the balance equation of the particle flux vector(19) can be written for each species of particle with particlenumbers 119899120601 119899m and 119899DM as

Ψ119886 = 119899119886 + 119899119886nabla120583119906120583 = 119899119886 + 3119867119899119886 119886 = 120601mDM (36)

We assume that the particle source function Ψ is propor-tional to the energy density of the inflationary scalar field120588120601 with the proportionality factor Γ representing a decay(creation) width Hence the balance equations of the particlenumber densities are given by

119899120601 + 3119867119899120601 = minus Γ120601119898120601 120588120601119899DM + 3119867119899DM = ΓDM119898DM

120588120601119899m + 3119867119899m = Γm119898m

120588120601(37)

where Γ120601 ΓDM and Γm are generally different nonzero func-tions and 119898120601 119898DM and 119898m denote the masses of the scalarfield dark matter and ordinary matter particles respectivelyFor the whole system the particle balance equation is

119899 + 3119867119899 = (Γm + ΓDM minus Γ120601) 120588120601 (38)

(b) The Energy Balance Equations From (9) we obtain theequations of the energy density balance of the cosmologicalfluids as

120588120601 + 3119867(120588120601 + 119901120601) = minusΓ120601 (120588120601 + 119901120601) 120588120601119898120601119899120601 (39)

120588DM + 3119867 (120588DM + 119901DM) = ΓDM (120588DM + 119901DM) 120588120601119898DM119899DM (40)

120588m + 3119867 (120588m + 119901m) = Γm (120588m + 119901m) 120588120601119898m119899m (41)

120588 + 3119867120588 = 120588120601 [Γm (120588m + 119901m)119898m119899m + ΓDM (120588DM + 119901DM)119898DM119899DMminus Γ120601 (120588120601 + 119901120601)119898120601119899120601 ]

(42)

respectively

(c) The Matter Creation Pressures From the expression of thecreation pressure associated with the newly created particlesEq (27) we find

119901(119886)119888 = minus120588(119886) + 119901(119886)119899(119886) Ψ(119886)Θ(119886) = minus(120588(119886) + 119901(119886))Ψ(119886)3119867119899(119886) (43)

where superscript (119886) denotes the different cosmologicalfluids During the reheating phase for each component thedecay and creation pressures are

119901(120601)119888 = Γ120601 (120588120601 + 119901120601) 1205881206013119867119898120601119899120601 119901(DM)119888 = minusΓDM (120588DM + 119901DM) 1205881206013119867119898DM119899DM 119901(m)119888 = minusΓm (120588m + 119901m) 1205881206013119867119898m119899m

(44)

The total creation pressure is

119901(total)119888 = 1205881206013119867 [ Γ120601119898120601119899120601 (120588120601 + 119901120601) minus ΓDM119898DM119899DMtimes (120588DM + 119901DM) minus Γm119898m119899m (120588m + 119901m)]

(45)

(d) The Gravitational Field Equations The Einstein gravita-tional field equations in a flat FRWUniverse with inflationaryscalar field ordinary matter and dark matter fluids in thepresence of matter creation can be written as follows

31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) 2 + 31198672 = minus 11198982Pl (119901120601 + 119901DM + 119901m + 119901(total)119888 ) (46)

In the present paper we use natural units with 119888 = ℏ = 119896119861 = 1in which 8120587119866 = 11198982Pl = 1687 times 10minus43MeVminus2 where 119898Pl isthe ldquoreducedrdquo Planck mass

(e) The Entropy Generation The entropy generated by thenewly created matter is

Sm (119905)Sm (1199050) = exp(int119905

1199050

Γm 120588120601119898m119899m d119905) (47)

32 The Models of Reheating Presently the acceptedparadigm for reheating is based on particle productionand parametric amplification as extensively discussed in[46 47] According to thismodel themain process of particleproduction is parametric amplification in combination witha slowing expansion This mechanism entails a descriptionin terms of Mathieu equations either for the scalar orfermion fields These equations feature unstable bands withthe concomitant parametric instabilities This mechanismcannot be described in terms of a simple Γ functions and


different bands feature different Floquet exponents (inMinkowski or near Minkowski spacetime) and correspondto different wave-vector bands In this picture matter iscreated in a highly nonthermal state (determined primarilyby the position and widths of the unstable parametric bands)and thermalizes (at least in principle although never quiteproven directly) via scattering This last step requires a clearknowledge of relaxation times which entail understandingthe microscopic processes and cross sections

In the following we will consider models of reheatinginvolving energy transfer between the inflationary scalar fieldand the newly created ordinary matter and dark matterrespectively We assume that particles are created through anirreversible thermodynamic process In this Subsection wepresent explicitly the general equations governing the matterand dark matter creation during inflationary reheating Inour reheatingmodels the energy density and pressure for theinflationary scalar field have the form

120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)

where119881(120601) is the self-interaction potential of the scalar fieldSubstituting (48) into (39) we obtain the generalized Klein-Gordon equation for the scalar field in the presence of mattercreation The equation of state of the dark matter is definedas 119908DM = 119901DM120588DM equiv 0 since we assume that the pressureof the newly created dark matter particles is 119901DM = 0 Theenergy density of dark matter is proportional to its particlenumber density 120588DM sim 119899DM so that 120588DM = 119898DM119899DM where119898DM is the mass of the dark matter particles For pressurelessmatter from (9) we obtain the general relation

119899 120588 = 120588 119899 (49)

The newly created matter is considered to be in form ofradiation (an ultrarelativistic fluid) with the equation of state119908m = 119901m120588m = 13 The relation between the energy densityand the particle number density of radiation is 120588m sim 11989943m

In summary the set of equations governing the mattercreation process during reheating is written below as

31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)

where Γ120601 ΓDM and Γm are considered as constants Eq (52)is the generalized Klein-Gordon equation describing theevolution of the scalar field The last term on the left handside of this equation represents an effective friction termdescribing the effects of irreversible matter creation (decay)on the scalar field dynamics If 120588120601119899120601 = constant we recover(2) introduced on a purely phenomenological basis in [26]

321 Dimensionless Form of the Reheating Equations Inorder to simplify the mathematical formalism we introduce aset of dimensionless variables 120591 ℎ 119903120601 119903DM 119903m119873120601119873m 120574DM120574m and V(120601) respectively defined as

119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)

where119872120601 is a constant with the physical dimension of amassIn the new variables (50)ndash(54) take the dimensionless

form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)

2dℎd120591 + 3ℎ2 = minus119875120601 minus 119875DM minus 119875(total)119888 (57)

d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)


Eqs (59) and (60) can be solved exactly and we obtainthe following exact representation of the scalar field particlenumber and matter densities

119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)

When 120588120601 rarr 0 the particles creation processes end and theUniverse enters in the standard expansionary phase with itschemical composition consisting of two matter componentsas well as a possible background of scalar field particles Afterreheating the energy density of the darkmatter and radiationvaries as

119903DM = 119903DM01198863 119903m = 119903m01198864

(62)

where 119903DM0 and 119903m0 are the dark matter and radiation energydensities at the end of reheating The evolution of the scalefactor is described by the equation

ℎ (119886) = 1119886 d119886d120591 = 1radic3radic119903DM01198863 + 119903m01198864 (63)

giving

120591 (119886) minus 1205910 = 2 (119886119903DM0 minus 2119903m0)radic119903DM0119886 + 119903m0radic31199032DM0 (64)

The deceleration parameter for the postreheating phase isgiven by

119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)

In the limit 119903DM0119886 ≫ 119903m0 the deceleration parameter tendsto the value 119902 = 12 indicating a decelerating expansion ofthe Universe

33 Observational Tests of Inflationary Models In order tofacilitate the comparison of the theoretical models of thereheating based on the thermodynamics of open systemswith the observational results we also present and calculatevarious cosmological parameters either directly from theinflationary models or from the potentials of the inflationaryscalar field

(a) Slow-Roll Parameters The slow-roll parameters wereintroduced in 1992 and are based on the slow-roll approxi-mation [18] Their calculations are presented in [4] Duringthe period before reheating the inflationary scalar field has

not yet decayed and the field equation and the equation ofthe scalar field can be written as

31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)

The slow-roll approximation requires that the terms 12060122 and120601 be sufficiently small so that the above equations can besimplified to

31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)

The slow-roll parameters are defined as 120598(120601) = (12)(1198811015840119881)2and 120578(120601) = 11988110158401015840119881 minus (12)(1198811015840119881)2 respectively [4 18]The slow-roll approximation only holds when the followingconditions are satisfied 120598(120601) ≪ 1 and 120578(120601) ≪ 1 respectively(b) Inflationary Parameters During inflation the Universeexpands exponentially The scale of inflation is representedby the scale factor 119886 The ratio of the scale factor at the endof inflation and at its beginning is always large so that thelogarithm of this quantity is of physical interest The numberof 119890-folds is defined as 119873(119905) = ln[119886(119905end)119886(119905)] where 119905end isthe time when inflation ends The number of 119890-folds can alsobe obtained from the comoving Hubble length [4] Under theslow-roll approximation it can be calculated as

119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)

where 120601end is defined as 120598(120601end) = 1 when the slow-rollapproximation is violated and inflation ends Typically 119873 ≃50 Sometimes the integral is multiplied by a factor of 1radic2

The scalar spectral index 119899119904 is defined by the slow-rollparameters 119899119904 = 1 minus 4120598(120601CMB) + 2120578(120601CMB) The index 119899119904 isusually a function of the parameters of the model The valueof 119899119904 can be obtained from the observations [16] and is used toconstrain the inflationary models The tensor-to-scalar ratio119903 is similarly defined as 119903 = 137 times 120598(120601CMB) One can seefrom their definition that the two parameters 119899119904 and 119903 are notindependent

(c) Cosmological Parameters The deceleration parameter isdefined as

119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)

It indicates the nature of the acceleration of the expansion ofthe Universe 119902 lt 0means that the expansion of the Universeis accelerating while a positive 119902 indicates deceleration

The density parameters of the inflationary scalar fieldmatter and dark matter are defined as Ω120601 = 120588120601120588 ΩDM =120588DM120588 andΩm = 120588m120588 The evolution of density parametersof eachmatter component in the earlyUniverse gives us a hinton the matter creation process during inflationary reheating


Table 1 Scalar field potentials

Number Potentials Expressions

(1) Large-field polynomial potentials 1198722120601 (120601120583)119901 119901 gt 1(2) Small-field polynomial potentials 1198722120601[1 minus (120601120583)119901] 120601 ≪ 120583 119901 gt 2(3) Small-field quadratic potentials 1198722120601 [1 minus (120601120583)2] 120601 ≪ 120583(4) Linear potentials 1198722120601(120601120583) and1198722120601 (1 minus 120601120583)(5) Exponential potentials 1198722120601 exp (plusmnradic21199011206012) 119901 gt 0(6) Higgs potentials 1198722120601 [1 minus (120601120583)2 + (120601])4]

Table 2 Slow-roll parameters for each model [19]

Number 1206012end 1206012CMB 120598(120601CMB) 120578(120601CMB)(1) 11990122 119901 (119901 + 200) 2 119901(119901 + 200) (119901 minus 2)(119901 + 200)(2) 1205832 (120583119901)2(119901minus1) 1205832 [120583250119901(119901 minus 2)]2(119901minus2) ≃0 minus(119901 minus 1)50(119901 minus 2)(3) 1206012endexp (minus2001205832) ≃0 minus21205832(4) 121205832 minus121205832(5) 1119901 1119901(6) 1206012end exp (minus2001205832) ≃0 minus21205832331 Model Parameters In studying the reheating modelswe consider different scalar field potentials that influence thedynamics of the early Universe During inflation the energydensity of the scalar field is potential energy dominated

(a) Scalar Field PotentialsWe picked six potentials of cosmo-logical interest that we are going to investigate and presentedthem in Table 1

Some of these potentials are simple and satisfy the slow-roll conditions like the large-field small-field and linearpotentials However since the slow-roll condition is a suf-ficient but not necessary condition for inflation [4] someother forms of the potentials are also considered We havealso selected the well-studied exponential potential which iseasy to analyze and constrain observationally Finally as theonly scalar field in the Standard Model of particle physicsthe Higgs type potentials are also worth investigating Theobservational constraints of the six potentials are not exactlythe same Each potential has its own slow-roll and inflation-ary parameters while they share similar constraints from thecosmological parameters and the reheating temperature Thevalues of the constants are discussed in the next paragraphs

(b) Slow-Roll Parameters The slow-roll parameters areobtained for each potentialTheir values are shown in Table 2

(c) Inflationary Parameters Similarly the inflationary param-eters scalar spectral index 119899119904 and tensor-to-scalar ratio 119903 arepresented in Table 3

(d) Mass Scale119872120601 In order to compare the different poten-tials we assume that in all of them the mass scale119872120601 is thesame We constrain 119872120601 in the potentials by estimating thevalues of 119881 at the beginning and ending time of reheating

Table 3 Inflationary parameters for each model

Number 119899119904 119903(1) 1 minus (2119901 + 4)(119901 + 200) 137(119901(119901 + 200))(2) 1 minus (119901 minus 1)25(119901 minus 2) ≃0(3) 1 minus 41205832 ≃0(4) 1 minus 31205832 13721205832(5) 1 minus 2119901 137119901(6) 1 minus 41205832 ≃0Reheating begins at the end of inflation at around 10minus32 s andends before 10minus16 s [3]Thus the value of the mass scale119872120601 is

1018 sminus1 le 119872120601 le 1032 sminus1658 times 10minus7 GeV le 119872120601 le 658 times 107 GeV (70)

In the set of reheating equations119872120601 is defined as

119872120601 = 120591119905 = 120591119905 (s) times 658 times 10minus25 GeV = 0658120591 keV (71)

When the dimensionless time variable is in the range 03 le120591 le 3 the value of119872120601 varies as01974 keV le 119872120601 le 1974 keV (72)

In our irreversible thermodynamic study of reheatingmodelswe adopt the value119872120601 asymp 1 keV


(e) Reheating Temperature The reheating temperature is usu-ally referred to as the maximum temperature during reheat-ing period However it was argued that the reheating temper-ature is not necessarily themaximum temperature of theUni-verse [33] In the radiation dominated Universe the relationbetween energy density and the temperature of the matter(radiation) is 120588m sim 1198794 With 120588m = 11987221206011198982Pl119903m we have 119879 sim11987212120601 11989812Pl 4radic119903m When the dimensionless variable is approx-imately 01 le 119903m le 05 we obtain

119879 sim 0511989812Pl 11987212120601 (73)

Hence for themaximum temperature of the reheating periodwe find

119879reh sim 329 times 107 GeV (74)

(f)DecayWidth Γ120601 of the Inflationary Scalar FieldThe relationbetween decay width of inflationary scalar field and 119872120601 isgiven by [77]

Γ120601 = 120572120601119872120601radic1 minus ( 119879119872120601)2 ≃ 120572120601119872120601 = 120572120601 times 1 keV (75)

where 119879 is the temperature of the decay product and 120572120601is a constant As we have previously seen the maximumtemperature during reheating is of the order of 107 GeV Themass scale of the inflationary scalar field is of the order of thePlanck Mass which leads to 119879 ≪ 119872120601 so the decay widthof inflationary scalar field is similarly proportional to 119872120601Hence for the decay width in the reheating model we adoptthe value Γ120601 asymp 120572120601 times 1 keV(g) Mass of the Dark Matter Particle The mass of darkmatter particles is investigated model independently in [78]Two kinds of dark matter particles and their decouplingtemperature from the dark matter galaxies were obtainedOne is possibly the sterile neutrino gravitino the light neu-tralino or majoron with mass about 1sim2 keV and decouplingtemperature above 100GeV The other one is the WeaklyInteracting Massive particles (WIMP) with mass sim100GeVand decoupling temperature sim5GeV From the analysis ofreheating temperature above we adopt the value of the massof the dark matter particles as119898DM sim 1 keV(h) Other Parameters For the reduced Planck mass weadopt the value 119898Pl = 2435 times 1018 GeV If not otherwisespecified the numerical value of the spectral index is takenas 119899119904 = 0968 In order to numerically solve the system of thereheating equations we use the initial conditions 119886(0) = 033119903DM(0) = 119903m(0) = 119899m(0) = 0 119873120601(0) = 119873(120601)0 120601(0) = 1206010 and(d120601d119905)|120591=0 = 1199060 respectively4 Coherent Scalar Field Model

As a first example of the analysis of the reheating process inthe framework of the thermodynamics of open systems weassume that the inflationary scalar field can be regarded as

a homogeneous oscillating coherent wave with the energydensity of the wave 120601(119905) given by [2 63]

120588120601 = 119898120601119899120601 (76)

where 119898120601 is the mass of the scalar field particle (inflaton)Then from relations (76) and (9) we obtain

120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)

In the following we assume that the energy density of thenewly created matter and dark matter particles is smallenough and they do not affect the evolution of the energydensity of the inflationary scalar field That is we assume theapproximations

120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)

Then (50)ndash(54) take the simplified form

2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)

Eqs (80) can be transformed into dimensionless form byintroducing the dimensionless variables 120591 ℎ 119873DM and 119873mdefined as

119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)

Hence we obtain the dimensionless equations

dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)


The solution for ℎ is obtained as

ℎ (120591) = 1119890120591 minus 1 (83)

while the time variations of the particle numbers are

119873119886 = (1198901205910 minus 1119890120591 minus 1 )211987311988601198902(120591minus1205910) + 1198902(120591minus1205910) minus 12 (119890120591 minus 1)2

119886 = DMm(84)

where for dark matter 1198731198860 = 119873DM0 = 119873DM(1205910) and 1198731198860 =119873m0 = 119873m(1205910) for the ordinary matterThe scale factor in this model is given by

119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)

where 1198860 is an integration constant while the decelerationparameter 119902 is found to be

119902 = 32119890120591 minus 1 (86)

Since 119902 gt 0 for all 120591 ge 0 it follows that the expansion rateof the Universe continues to decrease Inflation ends at thebeginning of the reheating period

Let us consider now the beginning of the inflationaryreheating when 120591 is small Then the solutions (83)ndash(85) canbe approximated as

ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)

When 120591 reaches the value 120591max = 21205910 minus 2119873012059120 the particlenumber reaches its maximum value

119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)

where 1198991198860 = 2Γ120601119898120601Γ1198863 For dark matter Γ119886 = ΓDM and formatter Γ119886 = Γm

The entropy production of matter can also be calculated[63] and for small times it shows a linear increase so that

Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)

5 Effect of the Scalar Field Potential on theReheating Dynamics

In the present section we will investigate the reheating andgeneration of thematter content of the Universe as describedby the thermodynamics of open systems In order to performthis study we will investigate the role played by the differentscalar field potentials on the reheating dynamics

51 Large-Field Polynomial and Linear Potentials We willbegin our analysis by assuming that the scalar field potentialis a large-field polynomial potential of the form

119881 (120601) = 1198722120601 (120601120583)119901 (90)

where 119901 ge 1 119872120601 is a parameter of the dimension of masswhile 120583 is a dimensionless constant This kind of large-fieldpolynomial potential is used in chaotic inflation models Aspecial case of the large-field polynomial potentials are thelinear potentials corresponding to 119901 = 1 These types ofpotentials are usually suggested by particle physics mod-els with dynamical symmetry breaking or nonperturbativeeffects [79 80] The inverse power law potentials appear ina globally supersymmetric 119878119880(119873119888) gauge theory with 119873119888colors and the condensation of 119873119891 flavors Positive powerlaw potentials can drive an inflationary expansion with thescalar field becoming progressively less important as thecosmological evolution proceeds Hence the use of powerlaw scalar field potentials can convincingly justify the neglectof the scalar field terms during the postinflationary and latetimes cosmological evolution in the Friedmann equations

The dimensionless energy density and pressure of theinflationary scalar field take then the form

119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)

The dimensionless equations (56)ndash(60) describing theevolution of theUniverse during the reheating phase can thenbe written as

d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1

minus radic3radic11990622 + (120601120583)119901 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906

d119873120601d120591 = minusradic3radic119906

2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]

d119903DMd120591 = minus 4radic3radic119906

2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)

Figure 1 Time variation of the dark matter energy density (a) and of the radiation (b) during the reheating period in the presence of a large-field polynomial potential for different values 120572120601 of the decay width of the scalar field 120572120601 = 001 (solid curve) 120572120601 = 003 (dotted curve)120572120601 = 005 (dash dotted curve) 120572120601 = 007 (dashed curve) and 120572120601 = 009 (dash double dotted curve) respectively For 119901 we have adopted thevalue 119901 = 18 while 120583 = 35 The initial conditions used to numerically integrate (92) are 120601(0) = 11 119906(0) = minus1119873120601(0) = 3 119903DM(0) = 0 and119903m(0) = 0 respectively

01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)

Figure 2 Time variation of the energy density of the scalar field (a) and of the deceleration parameter (b) during the reheating period in thepresence of a large-field polynomial potential for different values 120572120601 of the decay width of the scalar field 120572120601 = 001 (solid curve) 120572120601 = 003(dotted curve) 120572120601 = 005 (dash dotted curve) 120572120601 = 007 (dashed curve) and 120572120601 = 009 (dash double dotted curve) respectively For 119901 wehave adopted the value 119901 = 18 while 120583 = 35 The initial conditions used to numerically integrate (92) are 120601(0) = 11 119906(0) = minus1119873120601(0) = 3119903DM(0) = 0 and 119903m(0) = 0 respectively

d119903119898d120591 = minus 4radic3radic119906

2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)

The system of (92) must be integrated with the initialconditions 120601(0) = 1206010 119906(0) = 1199060 119873120601(0) = 1198731206010 119903DM(0) = 0and 119903m(0) = 0 respectively

The variations with respect to the dimensionless time 120591 ofthe energy density of the dark matter of the radiation of thescalar field particle number and of the Hubble function are

presented in Figures 1 and 2 respectively In order to numeri-cally integrate (92) we have fixed the parameters of the poten-tial as 119901 = 18 and 120583 = 35 respectively For the initial con-ditions of the scalar field and of its variation we have adoptedthe values 120601(0) = 11 and 119906(0) = minus10 The initial numberof the scalar field particles was fixed to 119873120601(0) = 3 whilefor the coefficients 120574DM and 120574m we have used the values 120574DM =(45)120572120601 and 120574m = (15)120572120601

As one can see from Figure 1 the evolution of thematter component (dark plus radiation) of the Universe canbe divided into two distinct phases In the first phase thematter energy densities increase from zero to a maximumvalue 119903(max)

119886 119886 = DMm which is reached at a time interval


120591max due to the energy transfer from the scalar field Fortime intervals 120591 gt 120591max the expansion of the Universebecomes the dominant force describing the matter dynamicsleading to a decrease in thematter energy densityThe energydensity of the scalar field shown in Figure 2(a) as well asthe associated particle number decreases rapidly during thereheating process and tends to zero in the large time limitTheenergy density of the scalar field is not affected significantlyby the variation of the numerical values of the parameter120572120601 The deceleration parameter of the Universe depicted inFigure 2(b) shows a complex evolution depending on thenature of the particle creation processes For the adoptedinitial values the particle creation phase in the inflationaryhistory of the Universe begins with values of the decelerationparameter of the order 119902 asymp minus020 During the particle cre-ation processes theUniverse continues to accelerate a processspecifically associated with the presence of the negative cre-ation pressure which induces a supplementary accelerationThis acceleration periods end once the number of the newlycreated particles declines significantly and at values of 119902 ofthe order of 119902 = minus025 the Universe starts decelerating withthe deceleration parameter tending to zero corresponding toa marginally expanding Universe When 119902 asymp 0 the energydensity of the scalar field also becomes extremely smalland the Universe enters in the standard matter dominatedcosmological phase with dark matter and radiation as themajor components plus a small remnant of the inflationaryscalar field which may play the role of the dark energyand become again dominant in the latest stages of expansion

We define the reheating temperature 119879reh as the temper-ature corresponding to the maximum value of the energydensity of the relativistic matter 119903(max)

m whose energy densityis defined as 120588m = (120587230)119892reh1198794 where 119892reh asymp 100 isthe number of relativistic degrees of freedomThe maximummatter energy density 119903(max)

m is obtained from the condition119889119903m119889120591 = 0 which gives the equation

4ℎ (120591max 119879reh) 120587230119892reh 1198794reh11987221206011198982Pl = 15120572120601119903120601 (120591max) (93)

or equivalently

119879reh = 156 times 109 times [ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]

14

times11987212120601 GeV(94)

Eqs (93) and (94) determine the reheating temperature as afunction of119872120601 the decay width of the scalar field the valueof the scalar field at maximum reheating and its potentialrespectively If the reheating temperature is known fromobservations from (93) one can constrain the parameters ofthe inflationary model and of the scalar field potential Byadopting the bound 119879reh le 6 times 1010 [76] we obtain therestriction on the parameter119872120601 of the potential as119872120601 le 16 times 103 times [ 241205872119892reh

120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)

The functions 119903120601 and ℎ are very little influenced by themodifications of the parameters of the potential or of 120572120601 (butthey depend essentially on the initial conditions) Hence forthis class of potentials we approximate 119903120601(120591max) asymp 020 andℎ(120591max) asymp 070 Hence we can approximate the coefficient inthe scalar field parameter as

[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]

minus14 asymp 346 times 120572minus14120601 (96)

Therefore the second important parameter restricting thereheating temperature or the potential parameters is thedecay width of the scalar field For 120572120601 = 10minus3 120572120601 = 1034 =562 while for120572120601 = 103120572120601 = 10minus34 = 017 Hence the uncer-tainty in the knowledge of 120572120601 can lead to variations over twoorders of magnitude in the potential parameters

52 Small-Field Power Law and Small-Field Quadratic Poten-tials The small-field polynomial potentials are of the form

119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)

where 120601 ≪ 120583 and 119901 ge 2 119872120601 is a parameter of thedimension of mass while 120583 is a dimensionless constantThese potentials appear as a result of a phase transition fromspontaneous symmetry breaking They are usually used inthe ldquonew inflationrdquo models A particular case of this class ofpotentials is small-field quadratic potentials correspondingto 119901 = 2 Natural inflation models usually use a cosinepotential and the small-field quadratic potentials are itsexpansion near 120601 = 0 for119901 = 2 For this class of potentials thedimensionless energy density and pressure of the scalar fieldtake the form

119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)

The dimensionless equations describing particle productionfrom inflationary scalar fields with small-field polynomialpotentials are

d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1

minus radic3radic11990622 + [1 minus (120601120583)119901] + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)

Figure 3 Time variation of the dark matter energy density (a) and of the radiation (b) during the reheating period in the presence of asmall-field power law potential for different values 120572120601 of the decay width of the scalar field 120572120601 = 015 (solid curve) 120572120601 = 030 (dotted curve)120572120601 = 045 (dash dotted curve) 120572120601 = 060 (dashed curve) and 120572120601 = 075 (dash double dotted curve) respectively For 119901 we have adopted thevalue 119901 = 3 while 120583 = 103 The initial conditions used to numerically integrate (99) are 120601(0) = 1 119906(0) = minus4 119873120601(0) = 5 119903DM(0) = 0 and119903m(0) = 0 respectively


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]

d119903DMd120591 = minusradic3radic11990622 + [1 minus (120601120583)

119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)

119901] d119903md120591 = minus 4radic3radic119906

2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]

(99)The system of (99) must be integrated with the initialconditions 120601(0) = 1206010 119906(0) = 1199060 119873120601(0) = 1198731206010 119903DM(0) = 0and 119903m(0) = 0 respectively

The variation of the dark matter energy density of theradiation energy density of the scalar field particle numberand of the Hubble function respectively are depicted inFigures 3 and 4

In order to numerically integrate the system of (99) wehave fixed the value of 119901 at 119901 = 3 and the value of 120583 as120583 = 103 We have slightly varied the numerical values ofthe parameter 120572120601 which resulted in significant variations inthe time evolution of the cosmological quantities The initialconditions used for the numerical integration of the evolutionequations are 119906(0) = minus4 120601(0) = 1 and 119873120601 = 5 whilethe initial values of the energy density of the dark matterand of radiation were taken as zero As one can see fromFigure 3(a) the energy density of the dark matter is initially

a monotonically increasing function of time reaching itsmaximum value 119903DM at a time 120591max For time intervals 120591 gt120591max the energy density of the dark matter decreases dueto the expansion of the Universe The time variation of theradiation presented in Figure 3(b) shows a similar behavioras the dark matter component The energy density of thescalar field decreases shown in Figure 4(a) tends to zeroin a finite time interval The deceleration parameter of theUniverse presented in Figure 4(b) shows an interestingbehavior For the chosen initial values the Universe is at 120591 = 0in a marginally accelerating state with 119902 asymp 0 Howeverduring the particle creation phase the Universe starts toaccelerate and reaches a de Sitter phase with 119902 = minus1 at atime interval around 120591 = 120591fin asymp 1 when the energy densityof the scalar field becomes negligibly small Since 120588120601 asymp 0 for120591 gt 120591fin the particle creation processes basically stop and theUniverse enters in the matter dominated decelerating phaseFor the chosen range of initial conditions the numericalvalues of 120588120601(120591max) and ℎ120591max

are roughly the same as in theprevious case thus leading to the same restrictions on thepotential parameter11987212060153The Exponential Potential Theexponential potentials areof the form

119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)

where 119872120601 is a parameter of the dimension mass and 119901 is aconstant Exponential potentials are sometimes used in extradimensionsmodels and superstring theories For example anexponential potential is generated from compactification ofthe higher-dimensional supergravity or superstring theoriesin four-dimensional Kaluza-Klein type effective theories[81] The moduli fields associated with the geometry ofthe extra dimensions can have in string or Kaluza-Kleintheories effective exponential type potentials which are due


0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)

Figure 4 Time variation of the energy density of the scalar field with a small power law potential (a) and of the deceleration parameter(b) during the reheating period in the presence of irreversible matter creation for different values 120572120601 of the decay width of the scalar field120572120601 = 015 (solid curve) 120572120601 = 030 (dotted curve) 120572120601 = 045 (dash dotted curve) 120572120601 = 060 (dashed curve) and 120572120601 = 075 (dash doubledotted curve) respectively For 119901 we have adopted the value 119901 = 3 while 120583 = 103 The initial conditions used to numerically integrate (99)are 120601(0) = 1 119906(0) = minus4119873120601(0) = 5 119903DM(0) = 0 and 119903m(0) = 0 respectively

to the curvature of the internal spaces or alternatively tothe interaction on the internal spaces of the moduli fieldswith form fields Nonperturbative effects such as gauginocondensation could also lead to the presence of exponentialpotentials [82]

The dimensionless energy density and pressure of theinflationary scalar field are

119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)

The dimensionless equations (56)ndash(60) describing theevolution of the Universe in the presence of irreversiblematter creation can then be formulated as

d120601d120591 = 119906d119906d120591 = ∓radic 2119901119890plusmnradic2119901120601

minus radic3radic11990622 + 119890plusmnradic2119901120601 + 119903DM + 119903m119906minus 120572120601119873120601 [119906

2

2 + 119890plusmnradic2119901120601] 119906d119873120601d120591 = minusradic3radic119906

2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 119890plusmnradic2119901120601] d119903DMd120591 = minusradic3radic11990622 + 119890plusmnradic2119901120601 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 119890plusmnradic2119901120601] d119903md120591 = minus 4radic3radic119906

2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119903m+ 120572m [11990622 + 119890plusmnradic2119901120601]

(102)

The systemof Eqs (102)must be integratedwith the initialconditions 120601(0) = 1206010 119906(0) = 1199060 119873120601(0) = 1198731206010 119903DM(0) = 0and 119903m(0) = 0 respectively

In order to numerically integrate the cosmological evolu-tion equations we have fixed the values of 119901 as 119901 = 5 and wehave adopted the negative sign for the scalar field exponentialpotential The initial conditions used for the numerical studyare 120601(0) = 15 119906(0) = minus55 and119873120601(0) = 20 together with thenull initial conditions for the matter energy densities As onecan see from Figure 5(a) the amount of darkmatter increasesvery rapidly during the first phase of the reheating For thechosen values of the parameters and initial conditions thereis a sharp peak in the dark matter distribution followed bya decreasing phase A similar dynamic is observed for thetime evolution of the radiation shown in Figure 6(b) Theenergy density of the scalar field presented in Figure 6(a)decreases rapidly and reaches the value 120588120601 asymp 0 at 120591 asymp 1The evolution of the scalar field energy density is basicallyindependent of the variation of the numerical values of thedecay width of the scalar field 120572120601The numerical values of the


00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)

Figure 5 Time variation of the dark matter energy density (a) and of the radiation (b) during the reheating period in the presence of anexponential scalar field potential for different values 120572120601 of the decay width of the scalar field 120572120601 = 115 (solid curve) 120572120601 = 135 (dottedcurve) 120572120601 = 155 (dash dotted curve) 120572120601 = 175 (dashed curve) and 120572120601 = 195 (dash double dotted curve) respectively For 119901 we haveadopted the value 119901 = 5 The initial conditions used to numerically integrate (102) are 120601(0) = 15 119906(0) = minus55119873120601(0) = 20 119903DM(0) = 0 and119903m(0) = 0 respectively

05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)

Figure 6 Time variation of the energy density of the scalar field (a) and of the deceleration parameter (b) during the reheating period in thepresence of an exponential scalar field potential for different values 120572120601 of the decay width of the scalar field 120572120601 = 115 (solid curve) 120572120601 = 135(dotted curve) 120572120601 = 155 (dash dotted curve) 120572120601 = 175 (dashed curve) and 120572120601 = 195 (dash double dotted curve) respectively For 119901 wehave adopted the value 119901 = 5 The initial conditions used to numerically integrate (102) are 120601(0) = 15 119906(0) = minus55119873120601(0) = 20 119903DM(0) = 0and 119903m(0) = 0 respectivelydeceleration parameter (Figure 6(b)) do depend significantlyon 120572120601 At the beginning of the particle creation epoch theUniverse is in amarginally inflating phase with 119902 asymp 0 Particlecreation induces a reacceleration of the Universe with thedeceleration parameter reaching values of the order of 119902 asympminus015 at the moment when the matter energy density reachesits peak value However the acceleration of the Universecontinues and at the moment the energy density of thescalar field becomes (approximately) zero the decelerationparameter has reached values of the order of 119902 asymp minus040 Oncethe energy density of the scalar field is zero the Universestarts its matter dominated expansion For the chosen initialconditions 120588120601(120591max) and ℎ120591max

have significantly greater valuesthan in the cases of the power law potentials However sincethe model parameters depend on their ratio no significant

variations in the reheating temperature or model parametersare expected

54 The Higgs Potential TheHiggs potentials are of the form

119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)

where 119872120601 is a parameter of the dimension mass and 120583 and] are constants The Higgs field is the only scalar field inthe Standard Model of particle physics The Higgs boson isa very special particle in physics and it plays a central role inthe Standard Model [83] It provides a physical mechanismfor the inclusion of the weakly interacting massive vectorbosons in the Standard Model and for giving masses to


the fundamental particle of nature like quarks and leptonsThe Higgs field may have also played an important role incosmology It could have been the basic field making theUniverse flat homogeneous and isotropic and it could havegenerated the small scale fluctuations that eventually led tocosmic structure formation The Higgs field could have alsoallowed the occurrence of the radiation dominated phase ofthe hot Big Bang [84]

It is an intriguing question whether the initial scalarfield driving the inflation is the same field as the Higgs field[84] The dimensionless energy density and pressure of theinflationary Higgs scalar field are given by

120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)

The equations describing irreversible matter creationprocesses from a decaying scalar field in the presence of theHiggs type potential are

d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)

minus radic3radic11990622 + 1 minus (120601120583)2 + (120601

])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906

d119873120601d120591= minusradic3times radic11990622 + 1 minus (120601120583)

2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]

d119903DMd120591= minusradic3times radic11990622 + 1 minus (120601120583)

2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]

d119903md120591= minus 4radic3times radic11990622 + 1 minus (120601120583)

2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)

In order to obtain the numerical evolution of theUniversein the presence of irreversible matter creation triggered bythe decay of a cosmological scalar field with Higgs typepotential we fix the parameters of the potential as 120583 = 2and ] = 3 respectively For the initial conditions of thescalar field and of the scalar field particle number we haveadopted the numerical values phi(0) = 105 and 119873120601(0) =100 respectively while for 119906(0) we have taken the value119906(0) = minus005 The time variations of the dark matter andradiation energy densities of the scalar field potential andof the Hubble function are presented in Figures 7 and 8respectively

For the adopted set of the parameters the dark matterenergy density depicted in Figure 7(b) increases from zeroto a maximum value and then begins to decrease due tothe combined effects of the expansion of the Universe andof the decay of the scalar field The radiation energy densitypresented in Figure 7(b) shows a similar behavior like darkmatter The evolution of both matter components is stronglydependent on the numerical values of the model parametersas well as on the initial conditions The energy density ofthe Higgs type scalar field shown in Figure 8(a) becomesnegligibly small at 120591 asymp 1 Its evolution also strongly dependson the model parameters The deceleration parameter 119902presented in Figure 8(b) indicates that in the case of scalarfield with Higgs potential the irreversible matter creationprocess starts when the Universe is in a de Sitter stage with119902 = minus1 During the first phase of the particle creation theUniverse decelerates but not significantly and it continues itsdecelerating trend up to themoment when the energy densityof the scalar field vanishes at around 120591 asymp 1 and standardmatter dominated evolution takes over the expansionarydynamics of the Universe However at this moment in timethe deceleration parameter still has a large negative value ofthe order of 119902 asymp minus07 For the chosen set of initial conditionsthe numerical values of 120588120601(120591max) and ℎ120591max

are similar to thecase of the exponential potential and hence no significant dif-ferences in the predicted reheating temperature or in modelparameter constraints can take place

6 Discussions and Final Remarks

In the present paper we have investigated the implicationsof the irreversible thermodynamic description of particlecreation during the reheating period of the early Universe Byinterpreting the postinflationary Universe as an open system


0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)

Figure 7 Time variation of the dark matter energy density (a) and of the radiation (b) during the reheating period in the presence of a Higgstype scalar field potential for different values 120572120601 of the decay width of the scalar field 120572120601 = 100 (solid curve) 120572120601 = 120 (dotted curve)120572120601 = 140 (dash dotted curve) 120572120601 = 160 (dashed curve) and 120572120601 = 250 (dash double dotted curve) respectively For the potential parameters120583 and ] we have adopted the values 120583 = 2 and ] = 3 respectively The initial conditions used to numerically integrate (105) are 120601(0) = 105119906(0) = minus005119873120601(0) = 100 119903DM(0) = 0 and 119903m(0) = 0 respectively

0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)

Figure 8 Time variation of the energy density of the Higgs type scalar field (a) and of the deceleration parameter (b) during the reheatingperiod with irreversible particle creation for different values of 120572120601 120572120601 = 100 (solid curve) 120572120601 = 120 (dotted curve) 120572120601 = 140 (dash dottedcurve) 120572120601 = 160 (dashed curve) and 120572120601 = 180 (dash double dotted curve) respectively For the potential parameters 120583 and ] we haveadopted the values 120583 = 2 and ] = 3 respectively The initial conditions used to numerically integrate (105) are 120601(0) = 105 119906(0) = minus005119873120601(0) = 100 119903DM(0) = 0 and 119903m(0) = 0 respectively

with energy-matter transfer between the fluid componentsa complete thermodynamic description of the basic physicalprocesses taking place at the end of inflation can be obtainedThe basic feature of our model is the decay of the inflationaryscalar field and the transfer of its energy to the newly createdparticles As for the matter content of the postinflationaryUniverse we have assumed that it consists of pressurelessdark matter and a relativistic radiation type component Byusing the fundamental principles of the thermodynamics ofopen systems and the Einstein field equations a systematicand consistent approach describing the time evolution ofthe matter components after inflation can be obtained Wehave investigated the effects on particle creation of variousforms of the scalar field potential which generally lead tosimilar qualitative results for the reheating scenario What

we have found from our numerical analysis is that thechoice of the functional form of the scalar field potential hasan essential influence on the cosmological dynamics Thisinfluence can be seen in the time interval for which themaximum energy densities of the dark matter and radiationare obtained in the decay rate of the energy density ofthe scalar field and its conversion to matter as well as inthe evolution of the deceleration parameter Moreover thecosmological dynamics also depends on the numerical valuesof the potential parameters

However the scalar field potentials have a unique quan-titative imprint on the matter creation during reheatingBy adopting a set of initial conditions for the physicalvariables the general equations describing reheating can besolved numerically During reheating the particle number


density of the scalar field drops to near zero while thenumber of the dark matter and matter particles increasesfrom zero to a maximum value thus heating up the Universeto a maximum reheating temperature 119879reh The reheatingtemperature depends on the physical parameters of theinflation (scalar field energy density field decay width andpotential parameters) and on the cosmological quantitieslike the value of the Hubble function at the moment whenthe matter density reached its maximum value By using thegeneral constraint on the reheating temperature as obtainedin [76] we have obtained some general constraints on themass parameter in the scalar field potential However in ourinvestigation these results are only of qualitative nature sincea significant increase in the accuracy of the observationaldata is still needed to obtain an accurate reconstructionof the physical conditions in the early Universe Large-field potential models lead to similar results as the kineticdominated ones since the observation of the scalar spectralindex 119899119904 constrains the value of 119901 to be relatively small withobservations preferring the potentials with 119901 = 2 [41] Theexponential potential models speed up the matter creationprocess specifically the preheating process to 120591 lt 02 andkeep creating new particles even after the matter energydensity has reached itsmaximumvalueThe process is similarto the one induced by the Higgs potentials model In ourpresent approach we have neglected the backreaction of thenewly created particles on the dynamics of the Universe

We have also carefully investigated the behavior ofthe deceleration parameter during the reheating periodA general result is that for all considered potentials theUniverse still accelerates during the reheating phase Thisis not an unexpected result from a physical point of viewsince in the formalism of the thermodynamics of opensystems particle creation is associatedwith a negative creationpressure that accelerates the system Irreversible particlecreation has been proposed as a possible explanation ofthe late time acceleration of the Universe [64 85ndash89] Asimilar process takes place in the irreversible thermodynamicdescription of the reheating with the Universe acquiringa supplementary acceleration Moreover this accelerationextends beyond the moment of the reheating (when themaximum temperature is reached) and indicates that theUniverse does not enter the standard matter and radiationdominated phase in a decelerating state Hence the use ofthe thermodynamic of open systems with irreversible mattercreation leads to an accelerating expansion of the Universeduring the particle production stage corresponding to thepostinflationary reheating This effect is strongly scalar fieldpotential dependent andwhile for somemodels the Universeacceleration is marginal with 119902 asymp 0 (coherent scalar fieldand large-field polynomial potential with 119881(120601) prop 120601119901) forother types of potentials (small-field power law exponentialHiggs) the deceleration parameter at the end of the reheatingperiod could have negative values in the range 119902 isin (minus1 minus04)These extremely accelerating models may be unrealisticsince they would indeed pose some serious problems tothe nucleosynthesis and structure formation and contradictsome well-established cosmological bounds on the chemical

composition of theUniverse and the time framewhen the firstchemical elements were produced

On the other hand in the presentmodel the cosmologicalbehavior and particle creation rates also strongly depend onthe scalar field potential parameters whose full range hasnot been systematically and completely investigated in thepresent study Therefore the use of the thermodynamic ofopen processes for the description of the postinflationaryreheating may also provide some strong constraints on theadmissible types of scalar field potentials In this context wewould like to note that model independent constraints on thecosmological parameters can be obtained within the frame-work of the so-called cosmography of the Universe Cosmog-raphy is essentially a Taylor expansion as applied to cosmol-ogy Model independent constraints on the evolution of thedeceleration parameter have been found by using the cosmo-graphic approach in [90ndash94]

When the temperature of the decay products 119879approaches the mass scale 119872120601 in the scalar field potentials119879 rarr 119872120601 then as one can see from (75) the decay width Γ120601of the scalar field tends to zero Γ120601 rarr 0 Moreover the energydensity of the scalar field becomes negligibly small meaningthat after the end of the reheating phase corresponding toΓ120601 asymp 0 and 120588120601 asymp 0 the number of the dark matter particlesand of the photons is conserved Hence in the postreheatingphase the cosmological evolution of the Universe is governedby the equations

31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)

where 120588119894 and 119901119894 119894 = 1 2 denote the thermodynamicdensities and pressures of the matter components (radiationdark matter baryonic matter etc) of the Universe Eq (106)represents a cosmological model in which the Universe iscomposed from matter and dark energy described by ascalar field with a self-interaction potential 119881(120601) In this waywe have recovered the quintessence dark energy models inwhich dark energy is described by a canonical scalar field120601 minimally coupled to gravity [95] As compared to otherscalar field models like phantom and 119896-essence dark energyscenarios the quintessence model is the simplest scalar fieldapproach not having major theoretical shortcomings suchas the presence of ghosts and of the Laplacian instabilitiesMoreover a slowly varying scalar field together with a poten-tial119881(120601) can lead to the late time acceleration of theUniverseThe quintessence scalar field has the important property ofbeing very weakly coupled tomatter (baryonic and dark) butit contributes a negative pressure to the global equation ofstate The magnitude of the energy scale of the quintessencepotential must be of the order of 120588DE asymp 10minus47 GeV4 atthe present time a value that is much smaller than that ofthe inflaton potential triggering the inflationary expansionIn order to achieve a late time cosmic acceleration the mass119898120601 of the quintessence field given by1198982120601 = d2119881(120601)d1206012must


be extremely small of the order of |119898120601| sim 1198670 asymp 1033 eV [95]where1198670 is the present day value of the Hubble functionTheevolution of the Universe as described by (106) starts fromthe initial conditions fixed at the end of reheating includingthe initial matter densities scalar field and Hubble functionvalues In the early stages of the evolution of the Universe(postreheating phase) the quintessence scalar field had a verysmall contribution to the energy balance of the Universe butduring the later cosmological evolution it decreases slowerwith the scale factor as compared to the matter and radiationdensities and consequently it is dominant at the presenttime triggering an exponential increase of the scale factorTherefore even the Universe enters in the conservative phasein an accelerating state in the first stages of the conservativeevolution in which the effect of the scalar field can beneglected the cosmological evolution will strongly deceler-ate By assuming that the energy density of the scalar field isnegligibly small and the Universe is matter dominated thenthe scale factor is given by 119886(119905) = 11990523 with the correspondingdeceleration parameter having the value 119902 = 12

An interesting and important question is the physicalnature of the dark matter particles that could have been cre-ated during reheating Presently the true physical nature ofthe dark matter particle is not known One possibility is thatdarkmatter may consist of ultralight particles with masses ofthe order of 119898 asymp 10minus24 eV (see [96] and references therein)Such an ultralight dark matter particle may represent froma physical point of view a pseudo Nambu-Goldstone bosonOther important ultralight dark matter particle candidatesare the axions with masses of the order of 119898 le 10minus22 eV[97] Very lowmass particles can be easily created even in veryweak gravitational fields and hence their production in largenumbers during reheating could have taken place Anotherhypothesis about the nature of dark matter is represented bythe so-called scalar field dark matter (SFDM)models [98] Inthese models dark matter is described as a real scalar fieldminimally coupled to gravity with the mass of the scalarfield particle having a very small value of the order of 119898 lt10minus21 eV An important property of scalar field dark mattermodels is that at zero temperature all particles condense tothe samequantumground state thus forming aBose-EinsteinCondensate (BEC) Hence from a physical point of view itturns out that SFDM models are equivalent to the BEC darkmatter models [99ndash104] On the other hand in the openirreversible thermodynamic model of reheating introducedin the present paper dark matter particle creation can takeplace also in the form of a scalar field one we assume that thescalar field does not fully decay into some ordinary matterHence scalar field dark matter could be a particular resultof the inflationary scalar field dynamics during the reheatingphase in the evolution of the Universe

Particle creation can also be related to what is called thearrow of time a physical process that provides a directionto time and distinguishes the future from the past Thereare two different arrows of time the thermodynamical arrowof time the direction in which entropy increases and thecosmological arrow of time the direction in which the Uni-verse is expanding Particle creation introduces asymmetry

in the evolution of the Universe and enables us to assigna thermodynamical arrow of time which agrees in thepresent model with the cosmological one This coincidenceis determined in a natural way by the physical nature ofinflation

The present day observations cannot accurately constrainthe main predictions of the present model However evenwithout a full knowledge of the true cosmological scenarioone can make some predictions about reheating In themodels we are considering that the energy density of theinflationary scalar field does not drop completely to zeroduring reheating This may suggest that the dark energy thatdrives the accelerating expansion of the Universe today mayhave also originated from the original inflationary scalar fieldNewly created dark matter particles may cluster with matterthrough gravitational interactions and form galaxies withbaryonic matter clustering in its centerThe initial conditionsof the reheating as well as the physics of the decay of the scalarfields are not fully known and theymay need physics beyondthe StandardModel of particle physics to account for the darkmatter creation process And certainlywe needmore accurateobservational data as well to constrain the models and findout what truly happens in the early Universe

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Tiberiu Harko would like to thank the Institute of AdvancedStudies of the Hong Kong University of Science and Technol-ogy for the kind hospitality offered during the preparationof this work Shi-Dong Liang would like to thank theNatural Science Funding of Guangdong Province for support(2016A030313313)

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A Linde Particle Physics and Inflationary Cosmology HarwoodAcademic Publishers London UK 1992

[3] EWKolb andM S TurnerTheEarlyUniverseWestviewPressNew York NY USA 1994

[4] A R Liddle and D H Lyth Cosmological Inflation and Large-Scale Structure Cambridge University Press Cambridge UK2000

[5] V Mukhanov Physical Foundations of Cosmology CambridgeUniversity Press New York NY USA 2005

[6] A D Linde ldquoA new inflationary universe scenario a possiblesolution of the horizon flatness homogeneity isotropy andprimordial monopole problemsrdquo Physics Letters B vol 108 no6 pp 389ndash393 1982

[7] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982


[8] A D Dolgov and A D Linde ldquoBaryon asymmetry in theinflationary universerdquo Physics Letters B vol 116 no 5 pp 329ndash334 1982

[9] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[10] A Linde Physica Scripta T85 168 2000[11] A Linde ldquoChaotic inflationrdquo Physics Letters B vol 129 no 3-4

pp 177ndash181 1983[12] A Linde ldquoHybrid inflationrdquo Physical Review D vol 49 no 2

pp 748ndash754 1994[13] P A R Ade et al Planck 2015 results I Overview of products

and scientific results arXiv150201582 2015[14] P A R Ade et al Planck 2015 results XIII Cosmological

parameters arXiv150201589 [astro-phCO] 2015[15] P A R Ade et al Planck 2015 results XVII Constraints on

primordial non-Gaussianity arXiv150201592 [astro-phCO]2015

[16] P A R Ade et al Planck 2015 results XVIII Backgroundgeometry ampamp topology arXiv150201593 [astro-phCO]2015

[17] P A R Ade and et al Planck 2015 results XX Constraints oninflation arXiv150202114 [astro-phCO] 2015

[18] A R Liddle and D H Lyth ldquoCOBE gravitational wavesinflation and extended inflationrdquo Physics Letters B vol 291 no4 pp 391ndash398 1992

[19] S DodelsonW H Kinney and EW Kolb ldquoCosmicmicrowavebackground measurements can discriminate among inflationmodelsrdquo Physical Review D vol 56 no 6 pp 3207ndash3215 1997

[20] W H Kinney ldquoConstraining inflation with cosmic microwavebackground polarizationrdquo Physical Review D - Particles FieldsGravitation and Cosmology vol 58 no 12 pp 123506E7ndash123506E7 1998

[21] W H Kinney A Melchiorri and A Riotto ldquoNew constraintson inflation from the cosmic microwave backgroundrdquo PhysicalReview D vol 63 no 2 Article ID 023505 2001

[22] W H Kinney E W Kolb A Melchiorri and A Riotto ldquoLatestinflation model constraints from cosmic microwave back-ground measurements Addendumrdquo Physical Review D - Par-ticles Fields Gravitation and Cosmology vol 78 no 8 ArticleID 087302 2008

[23] D Huterer C J Copi D J Schwarz and G D StarkmanldquoLarge-angle anomalies in the CMBrdquo Advances in Astronomyvol 2010 Article ID 847541 2010

[24] S W Hawking ldquoThe development of irregularities in a singlebubble inflationary universerdquo Physics Letters B vol 115 no 4pp 295ndash297 1982

[25] V F Mukhanov Journal of Experimental and TheoreticalPhysics Letters 41 493 1985

[26] A Albrecht P J Steinhardt M S Turner and F WilczekldquoReheating an inflationary universerdquo Physical Review Lettersvol 48 no 20 pp 1437ndash1440 1982

[27] L Kofman A Linde and A A Starobinsky ldquoReheating afterinflationrdquo Physical Review Letters vol 73 no 24 pp 3195ndash31981994

[28] L Kofman A Linde and A A Starobinsky ldquoTowards thetheory of reheating after inflationrdquo Physical Review D vol 56no 6 pp 3258ndash3295 1997

[29] J Repond and J Rubio ldquoCombined preheating on the latticewith applications to Higgs inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2016 no 7 article no 043 2016

[30] K D Lozanov and M A Amin ldquoThe charged inflaton and itsgauge fields Preheating and initial conditions for reheatingrdquoJournal of Cosmology and Astroparticle Physics vol 2016 no 6article no 032 2016

[31] S Antusch D Nolde and S Orani ldquoHill crossing duringpreheating after hilltop inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2015 no 6 article no 009 2015

[32] I Rudenok Y Shtanov and S Vilchinskii ldquoPost-inflationarypreheating with weak couplingrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 733pp 193ndash197 2014

[33] R J Scherrer and M S Turner ldquoDecaying particles do not heatup the Universerdquo Physical Review D vol 31 no 4 pp 681ndash6881985

[34] G F Giudice E W Kolb A Riotto D V Semikoz and II Tkachev ldquoStandard model neutrinos as warm dark matterrdquoPhysical Review D vol 64 no 4 2001

[35] T Harko W F Choi K C Wong and K S Cheng ldquoReheatingthe Universe in braneworld cosmological models with bulk-brane energy transferrdquo Journal of Cosmology and AstroparticlePhysics vol 2008 no 6 article no 002 2008

[36] R K Jain P Chingangbam and L Sriramkumar ldquoReheatingin tachyonic inflationary models Effects on the large scalecurvature perturbationsrdquo Nuclear Physics B vol 852 no 2 pp366ndash389 2011

[37] V Demozzi and C Ringeval ldquoReheating constraints in infla-tionary magnetogenesisrdquo Journal of Cosmology and Astroparti-cle Physics vol 2012 no 5 article no 009 2012

[38] H Motohashi and A Nishizawa ldquo Reheating after rdquo PhysicalReview D vol 86 no 8 2012

[39] K Mukaida and K Nakayama ldquoDissipative effects on reheatingafter inflationrdquo Journal of Cosmology and Astroparticle Physicsvol 2013 no 3 article no 002 2013

[40] A Nishizawa andHMotohashi ldquo Constraint on reheating afterrdquo Physical Review D vol 89 no 6 2014

[41] L Dai M Kamionkowski and J Wang ldquoReheating Constraintsto Inflationary Modelsrdquo Physical Review Letters vol 113 no 42014

[42] J Martin C Ringeval and V Vennin ldquoObserving InflationaryReheatingrdquo Physical Review Letters vol 114 no 8 2015

[43] C Gross O Lebedev and M Zatta ldquoHiggs-inflaton couplingfrom reheating and the metastable Universerdquo Physics LettersSection B Nuclear Elementary Particle andHigh-Energy Physicsvol 753 pp 178ndash181 2016

[44] Y S Myung and T Moon ldquoInflaton decay and reheating innonminimal derivative couplingrdquo Journal of Cosmology andAstroparticle Physics vol 2016 no 7 article no 014 2016

[45] M Rinaldi and L Vanzo ldquoInflation and reheating in theorieswith spontaneous scale invariance symmetry breakingrdquo Physi-cal Review D vol 94 no 2 024009 13 pages 2016

[46] R Allahverdi R Brandenberger F-Y Cyr-Racine and AMazumdar ldquoReheating in inflationary cosmology Theory andapplicationsrdquoAnnual Review ofNuclear andParticle Science vol60 pp 27ndash51 2010

[47] M A Amin M P Hertzberg D I Kaiser and J Karouby Int JMod Phys D 24 1530003 2015

[48] W Zimdahl J Triginer and D Pavon ldquoCollisional equilibriumparticle production and the inflationary universerdquo PhysicalReview D vol 54 no 10 pp 6101ndash6110 1996

[49] I Prigogine J Geheniau E Gunzig and P Nardone Proceed-ings Of The National Academy Of Sciences 7428 1988


[50] 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

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

[52] T Harko and M K Mak ldquoIrreversible matter creation ininflationary cosmologyrdquo Astrophysics and Space Science vol253 no 1 pp 161ndash175 1997

[53] T Harko and M K Mak ldquoParticle creation in varying speed oflight cosmologicalmodelsrdquoClassical andQuantumGravity vol16 no 8 pp 2741ndash2752 1999

[54] T Harko and M K Mak ldquoParticle creation in cosmologicalmodels with varying gravitational and cosmological ldquocon-stantsrdquordquo General Relativity and Gravitation vol 31 no 6 pp849ndash862 1999

[55] M K Mak and T Harko ldquoCosmological particle production infive-dimensional Kaluza-KLEin theoryrdquoClassical andQuantumGravity vol 16 no 12 pp 4085ndash4099 1999

[56] M K Mak and T Harko ldquoCosmological particle productionand causal thermodynamicsrdquo Australian Journal of Physics vol52 no 4 pp 659ndash679 1999

[57] V V Papoyan V N Pervushin and D V Proskurin ldquoCos-mological creation of matter in conformal cosmologyrdquo Astro-physics vol 46 no 1 pp 92ndash102 2003

[58] R Brandenberger and A Mazumdar ldquoDynamical relaxation ofthe cosmological constant and matter creation in the UniverserdquoJournal of Cosmology andAstroparticle Physics vol 2004 no 08pp 015-015 2004

[59] Y Qiang T J Zhang and Z L Yi Astrophys Space Sci 311 4072007

[60] S KModak andD Singleton ldquoInflationwith a graceful exit andentrance driven by Hawking radiationrdquo Physical Review D vol86 no 12 2012

[61] S K Modak and D Singleton ldquoHawking radiation as a mecha-nism for inflationrdquo International Journal of Modern Physics DGravitation Astrophysics Cosmology vol 21 no 11 1242020 6pages 2012

[62] J Chen FWu H Yu and Z LiThe European Physical JournalC 72 1861 2012

[63] T Harko F S Lobo M K Mak and S V Sushkov ldquoModified-gravity wormholes without exotic matterrdquo Physical Review Dvol 87 no 6 2013

[64] R O Ramos M Vargas dos Santos and I Waga ldquoMattercreation and cosmic accelerationrdquo Physical Review D vol 89no 8 2014

[65] S Chakraborty Phys Lett B 732 81 2014[66] J Quintin Y Cai and R H Brandenberger ldquoMatter creation in

a nonsingular bouncing cosmologyrdquo Physical Review D vol 90no 6 2014

[67] S Chakrabor and S Saha Phys Rev D 90 123505 2014[68] J Fabris J d Pacheco and O Piattella ldquoIs the continuous

matter creation cosmology an alternative toΛCDMrdquo Journal ofCosmology and Astroparticle Physics vol 2014 no 06 pp 038-038 2014

[69] J Lima and I Baranov ldquoGravitationally induced particle pro-duction Thermodynamics and kinetic theoryrdquo Physical ReviewD vol 90 no 4 2014

[70] T Harko ldquoThermodynamic interpretation of the general-ized gravity models with geometry-matter couplingrdquo PhysicalReview D vol 90 no 4 Article ID 044067 2014

[71] S Pan and S Chakraborty Adv High Energy Phys 6540252015

[72] T Harko F S N Lobo J P Mimoso and D Pavon ldquoGrav-itational induced particle production through a nonminimalcurvaturendashmatter couplingrdquo European Physical Journal C vol75 no 8 article no 386 2015

[73] R C Nunes and S Pan ldquoCosmological consequences of anadiabatic matter creation processrdquoMonthly Notices of the RoyalAstronomical Society vol 459 no 1 pp 673ndash682 2016

[74] V Singh and C P Singh ldquoFriedmann Cosmology with MatterCreation in Modified 119891(119877 119879) Gravityrdquo International Journal ofTheoretical Physics vol 55 no 2 pp 1257ndash1273 2016

[75] S Shandera N Agarwal and A Kamal arXiv170800493 2017[76] T Rehagen and G B Gelmini ldquoLow reheating temperatures

in monomial and binomial inflationary modelsrdquo Journal ofCosmology and Astroparticle Physics vol 2015 no 6 article no039 2015

[77] E W Kolb A Notari and A Riotto ldquoReheating stage afterinflationrdquo Physical Review D vol 68 no 12 Article ID 1235052003

[78] H De Vega and N Sanchez Monthly Notices Of The RoyalAstronomical SocietY 404 885 2010

[79] P J E Peebles and B Ratra Ap J Lett 325 L17 1988[80] B Ratra and P J E Peebles ldquoCosmological consequences of a

rolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[81] C G Callan D Friedan E J Martinec and M J PerryldquoStrings in background fieldsrdquo Nuclear Physics B Theoreti-cal Phenomenological and Experimental High Energy PhysicsQuantum Field Theory and Statistical Systems vol 262 no 4pp 593ndash609 1985

[82] B de Carlos J A Casas and C Munoz Nucl Phys B 399 6231993

[83] M Shaposhnikov Phil Trans R Soc A 373 20140038 2015[84] F Bezrukov and M Shaposhnikov ldquoThe Standard Model Higgs

boson as the inflatonrdquo Physics Letters B vol 659 no 3 pp 703ndash706 2008

[85] M P Freaza R S De Souza and I Waga ldquoCosmic accelerationand matter creationrdquo Physical Review D vol 66 no 10 ArticleID 103502 2002

[86] J A S Lima J F Jesus and F A Oliveira ldquoCDM acceleratingcosmology as an alternative to CDMmodelrdquo Journal of Cosmol-ogy and Astroparticle Physics vol 11 article 027 2010

[87] S Chakraborty and S Saha Phys Rev D 90 123505 2014[88] J A S Lima S Basilakos and J Sola ldquoThermodynamical

aspects of running vacuum modelsrdquo The European PhysicalJournal C vol 76 no 4 article 228 2016

[89] C Pigozzo S Carneiro J S Alcaniz H A Borges and J CFabris ldquoEvidence for cosmological particle creationrdquo Journal ofCosmology and Astroparticle Physics vol 2016 no 5 article no022 2016

[90] C Cattoen and M Visser ldquoThe Hubble series convergenceproperties and redshift variablesrdquo Classical and QuantumGrav-ity vol 24 no 23 pp 5985ndash5997 2007

[91] A Aviles C Gruber O Luongo and H Quevedo ldquoCosmog-raphy and constraints on the equation of state of the Universein various parametrizationsrdquo Physical Review D vol 86 no 12Article ID 123516 2012

[92] A de la P K S Cruz-Dombriz O Luongo and L ReverberiJournal of Cosmology and Astroparticle Physics 12 042 2016


[93] O Luongo G B Pisani and A Troisi ldquoCosmological degener-acy versus cosmography a cosmographic dark energy modelrdquoInternational Journal of Modern Physics D Gravitation Astro-physics Cosmology vol 26 no 3 Article ID 1750015 1750015 18pages 2017

[94] A A Mamon K Bamba and S Das ldquoConstraints on recon-structed dark energy model from SN Ia and BAOCMB obser-vationsrdquoThe European Physical Journal C vol 77 no 1 2017

[95] S Tsujikawa ldquoQuintessence a reviewrdquo Classical and QuantumGravity vol 30 no 21 Article ID 214003 2013

[96] J W Lee Phys Lett B 681 118 2009[97] C Park J Hwang and H Noh ldquoAxion as a cold dark matter

candidate Low-mass caserdquo Physical Review D vol 86 no 82012

[98] L Martinez-Medina V Robles and T Matos ldquoDwarf galaxiesin multistate scalar field dark matter halosrdquo Physical Review Dvol 91 no 2 2015

[99] C G Bohmer and T Harko ldquoCan dark matter be a Bose-Einstein condensaterdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 6 2007

[100] V H Robles and T Matos Mon Not Roy Astron Soc 422282 2012

[101] T Rindler-Daller and P R ShapiroMonNot Roy Astron Soc422 135 2012

[102] H Velten and E Wamba Phys Lett B709 1 2012[103] T Harko and M J Lake ldquoNull fluid collapse in brane world

modelsrdquo Physical Review DmdashParticles Fields Gravitation andCosmology vol 89 no 6 Article ID 064038 2014

[104] M-H Li and Z-B Li Phys Rev D 89 103512 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

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Advances in Condensed Matter Physics

OpticsInternational Journal of



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Statistical MechanicsInternational Journal of


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ThermodynamicsJournal of


119881(120590 120601) = (1198722minus1205821205902)(4120582)+119898212060122+1198922120601212059022 with119872119898120582and 120590 constants One scalar field accounts for the exponentialgrowth of the Universe and the other one is responsible forthe graceful exit of inflation

Presently very precise observations of the CosmicMicrowave Background (CMB) radiation give us the oppor-tunity to test the fundamental predictions of inflation onprimordial fluctuations such as scale independence andGaussianity [13ndash17] One can calculate cosmological param-eters from CMB fluctuations and use them as constraints ofthe inflationary modelsThe slow-roll parameters 120598 and 120578 canbe calculated directly from the potentials of the inflationaryscalar field so that these parameters can be observed andthe corresponding inflationary models can be tested [18]The slow-roll parameters and the inflationary parameters likescalar spectral index 119899119904 and tensor-to-scalar ratio 119903 of severalforms of the inflationary potentials were investigated in [19]Further investigations of the parameters 119899119904 and 119903 can be foundin [20ndash22]

The prediction of the statistical isotropy claiming thatinflation removes the classical anisotropy of the Universehas been a central prediction of inflationary scenarios alsostrongly supported by the cosmic no-hair conjecture How-ever recently several observations of the large scale structureof the Universe have raised the possibility that the principlesof homogeneity and isotropy may not be valid at all scalesand that the presence of an intrinsic large scale anisotropyin the Universe cannot be ruled out a priori [23] Althoughgenerally the recent Planck observations tend to confirmthe fundamental principles of the inflationary paradigmthe CMB data seem to show the existence of some tensionbetween the models and the observations For example thecombined Planck andWMAPpolarization data show that theindex of the power spectrum is 119899119904 = 09603 plusmn 00073 [14 15]at the pivot scale 1198960 = 005Mpcminus1 a result which definitelyrules out the exact scale-invariance (119899119904 = 1) as predicted bysome inflationarymodels [24 25] atmore than 5120590Moreoverthis case can also be ruled out based on some fundamentaltheoretical considerations since it is connected with a deSitter phase and more importantly it does not lead to aconsistent inflationary model mainly due to the gracefulexit problem On the other hand the joint constraints on 119903(tensor-to-scalar ratio) and 119899119904 can put strong restrictions onthe inflationary scenarios For example inflationary modelswith power law potentials of the form 1206014 cannot provide anacceptable number of 119890-folds (of the order of 50ndash60) in therestricted space 119903 minus 119899119904 at a 2120590 level [15] In actuality thetheories of inflation have not yet reached a general consensusbecause not only the overall inflationary picture but also thedetails of the theoretical models are based on physics beyondthe Standard Model of particle physics

During inflation the exponential growth of the Universeled to a homogeneous isotropic and empty Universe Ele-mentary particles are believed to be created in the period ofreheating at the end of inflation defrosting the Universe andtransferring energy from the inflationary scalar field to mat-ter Reheating was suggested along with the new inflationaryscenario [6] and further developed in [26ndash28] When the

inflationary scalar field reaches its minimum after the expan-sion of the Universe it oscillates around the minimum ofthe potential and decays into Standard Model particles Thenewly created particles interact with each other and reach athermal equilibrium state of temperature 119879 However quan-tum field theoretical investigations indicate that the reheat-ing period is characterised by complicated nonequilibriumprocesses the main characteristic of which are initial violentparticle production via parametric resonance and inflatondecay (preheating) with a highly nonequilibrium distribu-tion of the produced particles subsequently relaxing to anequilibrium state Such explosive particle production is dueto parametric amplification of quantum fluctuations for thenonbroken symmetry case (appropriate for chaotic inflation)or spinodal instabilities in the broken symmetry phase [29ndash32]

The reheating temperature 119879reh is thought to be the max-imum temperature of the radiation dominated Universe afterreheating However it was argued that the reheating tem-perature is not necessarily the maximum temperature of theUniverse after reheating [33] In other words reheatingmightbe just one of the stages of the matter creation and othersubstance with larger freeze-out temperature may be createdafter inflation through other process [34]

In the study of reheating the phenomenological approachintroduced in [26] has been widely investigated Thisapproach is based on the introduction of a suitable loss termin the scalar field equation which also appears as a sourceterm for the energy density of the newly created matterfluid It is this source term that if chosen appropriately isresponsible for the reheating process that follows adiabaticsupercooling during the de Sitter phase In this simple modelthe corresponding dynamics is considered within a twocomponent model The first component is a scalar field withenergy density and pressure 120588120601 and 119901120601 respectively and withenergy-momentum tensor (120601)119879]120583 = (120588120601 + 119901120601)119906120583119906] minus 119901120601120575]120583The second component is represented by normal matterwith energy density and pressure 120588119898 and 119901119898 and withenergy-momentum tensor (119898)119879]120583 = (120588119898 + 119901119898)119906120583119906] minus 119901119898120575]120583In the above equations for simplicity we assume that bothcosmological fluid components share the same four velocities119906120583 normalized as 119906120583119906120583 = 1 Einsteinrsquos field equations implythe relation nabla120583((120601)119879]120583+(119898)119879]120583) = 0 which in the case of a flatgeometry reduces to

120601 120601 + 3119867 1206012 + 1198811015840 (120601) 120601 + 120588119898 + 3119867 (120588119898 + 119901119898) = 0 (1)

where 119867 = 119886119886 is the Hubble function and 119886 is the scalefactor of the Universe In order to describe the transitionprocess between the scalar field and the matter componentit is convenient to introduce phenomenologically a frictionterm Γ describing the decay of the scalar field componentand acting as a source term for the matter fluid Hence (1)is assumed to decompose into two separate equations

120601 + 3119867 120601 + 1198811015840 (120601) + Γ 120601 = 0 (2)

120588119898 + 3119867 (120588119898 + 119901119898) minus Γ 1206012 = 0 (3)


















d(120588119899) = d119902 minus 119901d(1119899) (5)


d (120588V) + 119901dV = 0 (6)


d (120588V) = dQ minus 119901dV + ℎ119899d (119899V) (7)


d (120588V) + 119901dV minus ℎ119899d (119899V) = 0 (8)


120588 = (120588 + 119901) 119899119899 (9)





d (120588V) = minus (119901 + 119901119888) dV (10)


119901119888 = minus120588 + 119901119899 d (119899V)dV

(11)


dS = d119890S + d119894S ge 0 (12)




(13)



(14)




T120583] = (120588 + 119901 + 119901119888) 119906120583119906] minus (119901 + 119901119888) 119892120583] (15)

119904120583 = 119899120590119906120583N120583 = 119899119906120583 (16)



nabla]T120583] = 0 (17)


nabla120583119904120583 ge 0 (18)


nabla120583N120583 = Ψ (19)


119899119879d120590 = d120588 minus 120588 + 119901119899 d119899 (20)



= 120588 + (120588 + 119901 + 119901119888) nabla]119906] = 0(21)


120588 + (120588 + 119901 + 119901119888) nabla120583119906120583 = 0 (22)










minus119901119888Θ119879 minus 120583Ψ119879 = 120590Ψ (26)


119901119888 = minus120588 + 119901119899 ΨΘ (27)


d1199042 = d1199052 minus 1198862 (119905) (d1199092 + d1199102 + d1199112) (28)


120583)120597119909120583 = 11198863 120597 (1198863119906120583)120597119909120583 = V

V= 3119867 (29)


T00 = 120588

T11 = T

22 = T

33 = minus (119901 + 119901119888) (30)


Ψ = 119906120583nabla120583119899 + 119899nabla120583119906120583 = 119899 + 119899VV (31)


S = S0 exp(int1199051199050

Ψ (119905)119899 d119905) (32)






120588 = 120588120601 + 120588DM + 120588m (33)


119901 = 119901120601 + 119901DM + 119901m119899 = 119899120601 + 119899DM + 119899m (34)






Ψ119886 = 119899119886 + 119899119886nabla120583119906120583 = 119899119886 + 3119867119899119886 119886 = 120601mDM (36)



120588120601119899m + 3119867119899m = Γm119898m

120588120601(37)


119899 + 3119867119899 = (Γm + ΓDM minus Γ120601) 120588120601 (38)


120588120601 + 3119867(120588120601 + 119901120601) = minusΓ120601 (120588120601 + 119901120601) 120588120601119898120601119899120601 (39)


120588m + 3119867 (120588m + 119901m) = Γm (120588m + 119901m) 120588120601119898m119899m (41)


(42)

respectively


119901(119886)119888 = minus120588(119886) + 119901(119886)119899(119886) Ψ(119886)Θ(119886) = minus(120588(119886) + 119901(119886))Ψ(119886)3119867119899(119886) (43)



(44)



(45)





Sm (119905)Sm (1199050) = exp(int119905

1199050

Γm 120588120601119898m119899m d119905) (47)





120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)


119899 120588 = 120588 119899 (49)



31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)



119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)


form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)


d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)



119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




















d(120588119899) = d119902 minus 119901d(1119899) (5)


d (120588V) + 119901dV = 0 (6)


d (120588V) = dQ minus 119901dV + ℎ119899d (119899V) (7)


d (120588V) + 119901dV minus ℎ119899d (119899V) = 0 (8)


120588 = (120588 + 119901) 119899119899 (9)





d (120588V) = minus (119901 + 119901119888) dV (10)


119901119888 = minus120588 + 119901119899 d (119899V)dV

(11)


dS = d119890S + d119894S ge 0 (12)




(13)



(14)




T120583] = (120588 + 119901 + 119901119888) 119906120583119906] minus (119901 + 119901119888) 119892120583] (15)

119904120583 = 119899120590119906120583N120583 = 119899119906120583 (16)



nabla]T120583] = 0 (17)


nabla120583119904120583 ge 0 (18)


nabla120583N120583 = Ψ (19)


119899119879d120590 = d120588 minus 120588 + 119901119899 d119899 (20)



= 120588 + (120588 + 119901 + 119901119888) nabla]119906] = 0(21)


120588 + (120588 + 119901 + 119901119888) nabla120583119906120583 = 0 (22)










minus119901119888Θ119879 minus 120583Ψ119879 = 120590Ψ (26)


119901119888 = minus120588 + 119901119899 ΨΘ (27)


d1199042 = d1199052 minus 1198862 (119905) (d1199092 + d1199102 + d1199112) (28)


120583)120597119909120583 = 11198863 120597 (1198863119906120583)120597119909120583 = V

V= 3119867 (29)


T00 = 120588

T11 = T

22 = T

33 = minus (119901 + 119901119888) (30)


Ψ = 119906120583nabla120583119899 + 119899nabla120583119906120583 = 119899 + 119899VV (31)


S = S0 exp(int1199051199050

Ψ (119905)119899 d119905) (32)






120588 = 120588120601 + 120588DM + 120588m (33)


119901 = 119901120601 + 119901DM + 119901m119899 = 119899120601 + 119899DM + 119899m (34)






Ψ119886 = 119899119886 + 119899119886nabla120583119906120583 = 119899119886 + 3119867119899119886 119886 = 120601mDM (36)



120588120601119899m + 3119867119899m = Γm119898m

120588120601(37)


119899 + 3119867119899 = (Γm + ΓDM minus Γ120601) 120588120601 (38)


120588120601 + 3119867(120588120601 + 119901120601) = minusΓ120601 (120588120601 + 119901120601) 120588120601119898120601119899120601 (39)


120588m + 3119867 (120588m + 119901m) = Γm (120588m + 119901m) 120588120601119898m119899m (41)


(42)

respectively


119901(119886)119888 = minus120588(119886) + 119901(119886)119899(119886) Ψ(119886)Θ(119886) = minus(120588(119886) + 119901(119886))Ψ(119886)3119867119899(119886) (43)



(44)



(45)





Sm (119905)Sm (1199050) = exp(int119905

1199050

Γm 120588120601119898m119899m d119905) (47)





120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)


119899 120588 = 120588 119899 (49)



31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)



119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)


form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)


d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)



119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






d (120588V) = minus (119901 + 119901119888) dV (10)


119901119888 = minus120588 + 119901119899 d (119899V)dV

(11)


dS = d119890S + d119894S ge 0 (12)




(13)



(14)




T120583] = (120588 + 119901 + 119901119888) 119906120583119906] minus (119901 + 119901119888) 119892120583] (15)

119904120583 = 119899120590119906120583N120583 = 119899119906120583 (16)



nabla]T120583] = 0 (17)


nabla120583119904120583 ge 0 (18)


nabla120583N120583 = Ψ (19)


119899119879d120590 = d120588 minus 120588 + 119901119899 d119899 (20)



= 120588 + (120588 + 119901 + 119901119888) nabla]119906] = 0(21)


120588 + (120588 + 119901 + 119901119888) nabla120583119906120583 = 0 (22)










minus119901119888Θ119879 minus 120583Ψ119879 = 120590Ψ (26)


119901119888 = minus120588 + 119901119899 ΨΘ (27)


d1199042 = d1199052 minus 1198862 (119905) (d1199092 + d1199102 + d1199112) (28)


120583)120597119909120583 = 11198863 120597 (1198863119906120583)120597119909120583 = V

V= 3119867 (29)


T00 = 120588

T11 = T

22 = T

33 = minus (119901 + 119901119888) (30)


Ψ = 119906120583nabla120583119899 + 119899nabla120583119906120583 = 119899 + 119899VV (31)


S = S0 exp(int1199051199050

Ψ (119905)119899 d119905) (32)






120588 = 120588120601 + 120588DM + 120588m (33)


119901 = 119901120601 + 119901DM + 119901m119899 = 119899120601 + 119899DM + 119899m (34)






Ψ119886 = 119899119886 + 119899119886nabla120583119906120583 = 119899119886 + 3119867119899119886 119886 = 120601mDM (36)



120588120601119899m + 3119867119899m = Γm119898m

120588120601(37)


119899 + 3119867119899 = (Γm + ΓDM minus Γ120601) 120588120601 (38)


120588120601 + 3119867(120588120601 + 119901120601) = minusΓ120601 (120588120601 + 119901120601) 120588120601119898120601119899120601 (39)


120588m + 3119867 (120588m + 119901m) = Γm (120588m + 119901m) 120588120601119898m119899m (41)


(42)

respectively


119901(119886)119888 = minus120588(119886) + 119901(119886)119899(119886) Ψ(119886)Θ(119886) = minus(120588(119886) + 119901(119886))Ψ(119886)3119867119899(119886) (43)



(44)



(45)





Sm (119905)Sm (1199050) = exp(int119905

1199050

Γm 120588120601119898m119899m d119905) (47)





120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)


119899 120588 = 120588 119899 (49)



31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)



119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)


form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)


d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)



119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












minus119901119888Θ119879 minus 120583Ψ119879 = 120590Ψ (26)


119901119888 = minus120588 + 119901119899 ΨΘ (27)


d1199042 = d1199052 minus 1198862 (119905) (d1199092 + d1199102 + d1199112) (28)


120583)120597119909120583 = 11198863 120597 (1198863119906120583)120597119909120583 = V

V= 3119867 (29)


T00 = 120588

T11 = T

22 = T

33 = minus (119901 + 119901119888) (30)


Ψ = 119906120583nabla120583119899 + 119899nabla120583119906120583 = 119899 + 119899VV (31)


S = S0 exp(int1199051199050

Ψ (119905)119899 d119905) (32)






120588 = 120588120601 + 120588DM + 120588m (33)


119901 = 119901120601 + 119901DM + 119901m119899 = 119899120601 + 119899DM + 119899m (34)






Ψ119886 = 119899119886 + 119899119886nabla120583119906120583 = 119899119886 + 3119867119899119886 119886 = 120601mDM (36)



120588120601119899m + 3119867119899m = Γm119898m

120588120601(37)


119899 + 3119867119899 = (Γm + ΓDM minus Γ120601) 120588120601 (38)


120588120601 + 3119867(120588120601 + 119901120601) = minusΓ120601 (120588120601 + 119901120601) 120588120601119898120601119899120601 (39)


120588m + 3119867 (120588m + 119901m) = Γm (120588m + 119901m) 120588120601119898m119899m (41)


(42)

respectively


119901(119886)119888 = minus120588(119886) + 119901(119886)119899(119886) Ψ(119886)Θ(119886) = minus(120588(119886) + 119901(119886))Ψ(119886)3119867119899(119886) (43)



(44)



(45)





Sm (119905)Sm (1199050) = exp(int119905

1199050

Γm 120588120601119898m119899m d119905) (47)





120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)


119899 120588 = 120588 119899 (49)



31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)



119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)


form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)


d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)



119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Ψ119886 = 119899119886 + 119899119886nabla120583119906120583 = 119899119886 + 3119867119899119886 119886 = 120601mDM (36)



120588120601119899m + 3119867119899m = Γm119898m

120588120601(37)


119899 + 3119867119899 = (Γm + ΓDM minus Γ120601) 120588120601 (38)


120588120601 + 3119867(120588120601 + 119901120601) = minusΓ120601 (120588120601 + 119901120601) 120588120601119898120601119899120601 (39)


120588m + 3119867 (120588m + 119901m) = Γm (120588m + 119901m) 120588120601119898m119899m (41)


(42)

respectively


119901(119886)119888 = minus120588(119886) + 119901(119886)119899(119886) Ψ(119886)Θ(119886) = minus(120588(119886) + 119901(119886))Ψ(119886)3119867119899(119886) (43)



(44)



(45)





Sm (119905)Sm (1199050) = exp(int119905

1199050

Γm 120588120601119898m119899m d119905) (47)





120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)


119899 120588 = 120588 119899 (49)



31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)



119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)


form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)


d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)



119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120588120601 = 12 1206012 + 119881 (120601) 119901120601 = 12 1206012 minus 119881 (120601)

(48)


119899 120588 = 120588 119899 (49)



31198672 = 11198982Pl (120588120601 + 120588DM + 120588m) (50)

2 + 31198672 = minus 11198982Pl (119901120601 + 119901m + 119901(total)119888 ) (51)

120601 + 1198811015840 (120601) + 3119867 120601 + Γ120601120588120601119898120601119899120601 120601 = 0 (52)

120588DM + 3119867120588DM = ΓDM120588120601120588m + 4119867120588m = Γm120588120601 (53)

119899120601 + 3119867119899120601 + Γ120601119898120601 120588120601 = 0 (54)



119905 = 1119872120601 120591119867 = 119872120601ℎ120588119886 = 11987221206011198982Pl119903119886119901119886 = 11987221206011198982Pl119875119886119886 = 120601DMm 119888

V (120601) = 11198722120601119881 (120601) 119899119886 = 1198722120601119873119886119886 = 120601DMmΓ119887119898119887 = 119872120601120574119887119887 = 120601DMm

(55)


form

3ℎ2 = 119903120601 + 119903DM + 119903m (56)


d2120601d1205912 + V1015840 (120601) + 3ℎd120601d120591 + 120574120601119903120601119873120601 d120601

d120591 = 0 (58)

d119903DMd120591 + 3ℎ119903DM = 120574DM119903120601

d119903md120591 + 4ℎ119903m = 120574m119903120601

(59)

d119873120601d120591 + 3ℎ119873120601 + 120574120601119903120601 = 0 (60)



119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119873120601 (120591) = 119873120601 (0) 119886 (0)1198863 (120591) minus int 1205741206011205881206011198863 (120591) d1205911198863 (120591)

119903119894 = int 1205741198861205881206011198863 (120591) d120591119886119899 (120591) 119894 = DM 119899 = 3 119894 = m 119899 = 4

(61)


119903DM = 119903DM01198863 119903m = 119903m01198864

(62)



giving



119902 (119886) = 3119903DM0119886 + 4119903m02 (119886119903DM0119886 + 119903m0) minus 1 (65)





31198672 = 11198982Pl 120588120601 = 11198982Pl (12 1206012 + 119881 (120601)) 120601 + 1198811015840 (120601) + 3119867 120601 = 0 (66)


31198672 ≃ 11198982Pl119881 (120601) 3119867 120601 ≃ minus 11198982Pl1198811015840 (120601)

(67)


119873 = int119905end119905119867 d119905 ≃ int120601CMB

120601end

1198811198811015840 d120601 = int120601CMB

120601end

1radic120598 (1206011015840)d1206011015840 (68)




119902 = dd119905 ( 1119867) minus 1 = minus119886 1198862 (69)























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics























119879 sim 0511989812Pl 11987212120601 (73)


119879reh sim 329 times 107 GeV (74)






120588120601 = 119898120601119899120601 (76)


120588120601 = 1206012119901120601 = 0 (77)

giving

119881 (120601) = 12 1206012 (78)


120588DM 120588m ≪ 120588120601119899DM 119899m ≪ 119899120601119901DM 119901m ≪ 119901120601

(79)


2 + 31198672 = Γ120601119898120601119867119899DM + 3119867119899DM = 3ΓDM1198672119899m + 3119867119899m = 3Γm1198672

(80)


119905 = 2Γ120601119898120601 120591119867 = Γ1206011198981206013 ℎ

119899DM = 2Γ120601119898120601ΓDM3 119873DM119899m = 2Γ120601119898120601Γm3 119873m

(81)


dℎd120591 + ℎ2 + ℎ = 0

d119873DMd120591 + 2ℎ119873DM = ℎ2d119873md120591 + 2ℎ119873m = ℎ2

(82)



ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ℎ (120591) = 1119890120591 minus 1 (83)



119886 = DMm(84)


119886 (120591) = 1198860 (119890120591 minus 1119890120591 )23 (85)


119902 = 32119890120591 minus 1 (86)



ℎ ≃ 1120591 119886 ≃ 119886012059123120588120601 ≃ 11205912 119873119886 ≃ 120591 minus 1205910 + 1198731198860120591201205912 119886 = DMm

(87)


119899(max)119886 = 4Γ211988612Γ1198861199050 minus 9119899119886011990520 119886 = DMm (88)



Sm (119905)119878m (1199050) = 120591 minus 1205910119873012059120 + 1 (89)




119881 (120601) = 1198722120601 (120601120583)119901 (90)



119903120601 = 12 (d120601d120591)2 + (120601120583)

119901 119875120601 = 12 (d120601d120591)

2 minus (120601120583)119901

(91)


d120601d120591 = 119906d119906d120591 = minus119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + (120601120583)119901] 119906


2

2 + (120601120583)119901 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + (120601120583)119901]


2

2 + (120601120583)119901 + 119903DM + 119903m119903DM

+ 120574DM [11990622 + (120601120583)119901]


0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0000

0002

0004

0006

0008

0010

r ＄－()

05 10 15 2000

(a)

00000

00005

00010

00015

00020

00025

00030

r m()

05 10 15 2000

(b)


01

02

03

04

05

06

r ()

05 10 15 2000

(a)

minus025

minus020

minus015

minus010

minus005

q()

05 10 15 2000

(b)



2

2 + (120601120583)119901 + 119903DM + 119903m119903m

+ 120574m [11990622 + (120601120583)119901]

(92)












or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









or equivalently


14

times11987212120601 GeV(94)


120572120601119903120601 (120591max)ℎ (120591max) ]minus14

GeV (95)


[ 241205872119892reh120572120601119903120601 (120591max)ℎ (120591max) ]




119881 (120601) = 1198722120601 [1 minus (120601120583)119901] (97)


119903120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

119901] 119875120601 = 12 (d120601d120591)

2 minus [1 minus (120601120583)119901]

(98)


d120601d120591 = 119906d119906d120591 = 119901120583 (120601120583)

119901minus1


minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)119901] 119906


000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




000

005

010

015

020

025

030

r ＄－()

02 04 06 08 1000

(a)

000

002

004

006

008

r m()

02 04 06 08 1000

(b)



2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)119901]


119901] + 119903DM + 119903m119903M+ 120572DM [11990622 + 1 minus (120601120583)


2

2 + [1 minus (120601120583)119901] + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)119901]






119881 (120601) = 1198722120601 exp(plusmnradic 2119901120601) (100)



0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

2

4

6

8r

()

02 04 06 08 1000

(a)

minus10

minus08

minus06

q() minus04

minus02

00

02 04 06 08 1000

(b)




119903120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

119875120601 = 12 (d120601d120591)2 + 119890plusmnradic2119901120601

(101)




2


2

2 + 119890plusmnradic2119901120601 + 119903DM + 119903m119873120601



2


(102)




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

02

04

06

08

r ＄－()

05 10 15 2000

(a)

000

005

010

015

020

025

r m()

05 10 15 2000

(b)


05 10 15 2000

0

5

10

15

r ()

(a)

05 10 15 2000

minus04

minus03

q()

minus02

minus01

00

(b)





119881 (120601) = 1198722120601 [1 minus (120601120583)2 + (120601

])4] (103)





120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120588120601 = 12 (d120601d120591)2 + [1 minus (120601120583)

2 + (120601])4]

119875120601 = 12 (d120601d120591)2 minus [1 minus (120601120583)

2 + (120601])4]

(104)


d120601d120591 = 119906d119906d120591= 120601( 21205832 minus 4120601

2

]4)


])4 + 119903DM + 119903m119906

minus 120572120601119873120601 [1199062

2 + 1 minus (120601120583)2 + (120601

])4] 119906


2 + (120601])4 + 119903DM + 119903m119873120601

minus 120572120601 [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903M

+ 120572DM [11990622 + 1 minus (120601120583)2 + (120601

])4]


2 + (120601])4 + 119903DM + 119903m119903m

+ 120572m [11990622 + 1 minus (120601120583)2 + (120601

])4]

(105)







0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

5

10

15

20

r ＄－()

1004 06 080200

(a)

0

2

4

6

r m()

02 04 06 08 1000

(b)


0

100

200

300

400

500

600

700

r ()

02 04 06 08 1000

(a)

minus100

minus095

minus090

minus085

q()

minus080

minus075

minus070

02 04 06 08 1000

(b)











31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









31198672 = 11198982Pl (sum119894 120588119894 12 1206012 + 119881 (120601)) 120588119894 + 3119867 (120588119894 + 119901119894) = 0 119894 = 1 2 120601 + 3119867 120601 + 1198811015840 (120601) = 0

(106)










Acknowledgments


References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











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



irreversible thermodynamic description of dark matter and

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