Mass varying dark matter in effective GCG scenarios
aa r X i v : . [ g r- q c ] J a n Mass varying dark matter in effective GCG scenarios
A. E. Bernardini
Departamento de F´ısica, Universidade Federal de S˜ao Carlos,PO Box 676, 13565-905, S˜ao Carlos, SP, Brasil ∗ (Dated: December 6, 2018) Abstract
A unified treatment of mass varying dark matter coupled to cosmon- like dark energy is shownto result in effective generalized Chaplygin gas (GCG) scenarios. The mass varying mechanismis treated as a cosmon field inherent effect. Coupling dark matter with dark energy allows forreproducing the conditions for the present cosmic acceleration and for recovering the stabilityresulted from a positive squared speed of sound c s , as in the GCG scenario. The scalar fieldmediates the nontrivial coupling between the dark matter sector and the sector responsible forthe accelerated expansion of the universe. The equation of state of perturbations is the same asthat of the background cosmology so that all the effective results from the GCG paradigm aremaintained. Our results suggest the mass varying mechanism, when obtained from an exactlysoluble field theory, as the right responsible for the stability issue and for the cosmic accelerationof the universe. PACS numbers: 95.35.+d, 95.36.+x, 98.80.Cq ∗ Electronic address: [email protected], alexeb@ifi.unicamp.br . INTRODUCTION The ultimate nature of the dark sector of the universe is the most relevant issue relatedwith the negative pressure component required to understand why and how the universeis undergoing a period of accelerated expansion [1–5]. A natural and simplistic explana-tion for this is obtained in terms of a tiny positive cosmological constant introduced in theEinstein’s equation for the universe. Since the cosmological constant has a magnitude com-pletely different from that predicted by theoretical arguments, and it is often confrontedwith conceptual problems, physicists have been compelled to consider other explanationsfor that [6–12].Motivated by the high energy physics, an alternative for obtaining a negative pressureequation of state considers that the dark energy can be attributed to the dynamics of a scalarfield φ which realizes the present cosmic acceleration by evolving slowly down its potential V ( φ ) [13, 14]. These models assume that the vacuum energy can vary [15]. Followingtheoretical as well as phenomenological arguments, several possibilities have been proposed,such as k -essence [16, 17], phantom energy [18, 19], cosmon fields [20], and also several typesof modifications of gravity [21–23].One of the most challenging proposals concerns mass varying particles [24–27] coupled tothe dark energy through a dynamical mass dependence on a light scalar field which drivesthe dark energy evolution in a kind of unified cosmological fluid. The idea in the well-knownmass varying mechanism [12, 26–28] is to introduce a coupling between a relic particle andthe scalar field whose effective potential changes as a function of the relic particle density.This coupled fluid is either interpreted as dark energy plus neutrinos, or as dark energyplus dark matter [29, 30]. Such theories can possess an adiabatic regime in which the scalarfield always sits at the minimum of its effective potential, which is set by the local massvarying particle density. The relic particle mass is consequently generated from the vacuumexpectation value of the scalar field and becomes linked to its dynamics by m ( φ ) .A scenario which congregate dark energy and some kind of mass varying dark matter in aunified negative pressure fluid can explain the origin of the cosmic acceleration. In particu-lar, any background cosmological fluid with an effective behaviour as that of the generalizedChaplygin gas (GCG) [7–9] naturally offers this possibility. The GCG is particularly rel-evant in respect with other cosmological models as it is shown to be consistent with the2bservational constraints from CMB [31], supernova [32–34], gravitational lensing surveys[35], and gamma ray bursts [36]. Moreover, it has been shown that the GCG model can beaccommodated within the standard structure formation mechanism [7, 9, 33].In the scope of finding a natural explanation for the cosmic acceleration and the corre-sponding adequation to stability conditions for a background cosmological fluid, our purposeis to demonstrate that the GCG just corresponds to an effective description of a coupledfluid composed by dark energy with equation of state given by p ( φ ) = − ρ ( φ ) and a cold darkmatter (CDM) with a dynamical mass driven by the scalar field φ . Once one has consistentlyobtained the mass dependence on φ , which is model dependent, it can be noticed that thecosmological evolution of the composed fluid is governed by the same dynamics prescribedby cosmon field equations. It suggests that the mass varying mechanism embedded into thecosmon- like dynamics reproduces the effective behaviour of the GCG. In addition, couplingdark matter with dark energy by means of a dynamical mass driven by such a scalar fieldallows for reproducing the conditions for the present cosmic acceleration and for recoveringthe stability prescribed by a positive squared speed of sound c s . At least implicitly, it leadsto the conclusion that the dynamical mass behaviour is the main agent of the stability issueand of the cosmic acceleration of the universe.At our approach, the dark matter is approximated by a degenerate fermion gas (DFG).In order to introduce the mass varying behaviour, we analyze the consequences of couplingit with and underlying dark energy scalar field driven by a cosmon- type equation of motion.We discuss all the relevant constraints on this in section II. In section III, we obtain theenergy density and the equation of state for the unified fluid and compare them with thecorresponding quantities for the GCG. In section IV, we discuss the stability issue andthe accelerated expansion of the universe in the framework here proposed. The pertinentcomparisons with a GCG scenario are evaluated. We draw our conclusions in section V bysummarizing our findings and discussing their implications. II. MASS VARYING MECHANISM FOR A DFG COUPLED TO COSMON-
LIKE
SCALAR FIELDS
To understand how the mass varying mechanism takes place for different particle species,it is convenient to describe the corresponding particle density, energy density and pressure3s functionals of a statistical distribution. This counts the number of particles in a givenregion around a point of the phase space defined by the conjugate coordinates: momentum, β , and position, x . The statistical distribution can be defined by a function f ( q ) in terms ofa comoving variable, q = a | β | , where a is the scale factor (cosmological radius) for the flatFRW universe, for which the metrics is given by ds = dt − a ( t ) δ ij dx i dx j . In the flat FRWscenario, the corresponding particle density, energy density and pressure are thus given by n ( a ) = 1 π a Z ∞ dq q f ( q ) ,ρ m ( a, φ ) = 1 π a Z ∞ dq q ( q + m ( φ ) a ) / f ( q ) , (1) p m ( a, φ ) = 13 π a Z ∞ dq q ( q + m ( φ ) a ) − / f ( q ) . where the last two can be depicted from the Einstein’s energy-momentum tensor [37]. Forthe case where f ( q ) is a Fermi-Dirac distribution function, it can be written as f ( q ) = { exp [( q − q F ) /T ] + 1 } − , where T is the relic particle background temperature at present. In the limit where T tendsto 0, it becomes a step function that yields an elementary integral for the above equations,with the upper limit equal to the Fermi momentum here written as q F = a β ( a ) . It results inthe equations for a DFG [38]. The equation of state can be expressed in terms of elementaryfunctions of β ≡ β ( a ) and m ≡ m ( φ ( a )) , n ( a ) = 13 π β ,ρ m ( a ) = 18 π h β (2 β + m ) p β + m − arc sinh ( β/m ) i , (2) p m ( a ) = 18 π (cid:20) β ( 23 β − m ) p β + m + arc sinh ( β/m ) (cid:21) . One can notice that the DFG approach is useful for parameterizing the transition betweenultra-relativistic (UR) and non-relativistic (NR) thermodynamic regimes. It is not manda-tory for connecting the mass varying scenario with the GCG scenario.Simple mathematical manipulations allow one to easily demonstrate that n ( a ) ∂ρ m ( a ) ∂n ( a ) = ( ρ m ( a ) + p m ( a ) ) , (3)and m ( a ) ∂ρ m ( a ) ∂m ( a ) = ( ρ m ( a ) − p m ( a ) ) , (4)4oticing that the explicit dependence of ρ m on a is intermediated by β ( a ) and m ( a ) ≡ m ( φ ( a )) , one can take the derivative of the energy density with respect to time in order toobtain ˙ ρ m = ˙ β ( a ) ∂ρ m ( a ) ∂β ( a ) + ˙ m ( a ) ∂ρ m ( a ) ∂m ( a ) = ˙ n ( a ) ∂ρ m ( a ) ∂n ( a ) + ˙ m ( a ) ∂ρ m ( a ) ∂m ( a ) = − aa n ( a ) ∂ρ m ( a ) ∂n ( a ) + ˙ φ d m d φ ∂ρ m ( a ) ∂m ( a ) , (5)where the overdot denotes differentiation with respect to time ( · ≡ d/dt ). The substitutionof Eqs. (3-4) into the above equation results in the energy conservation equation given by˙ ρ m + 3 H ( ρ m + p m ) − ˙ φ d m d φ ( ρ m − p m ) = 0 , (6)where H = ˙ a/a is the expansion rate of the universe. If one performs the derivative withrespect to a directly from ρ m in the form given in Eq. (2), the same result can be obtained.The coupling between relic particles and the scalar field as described by Eq. (6) are effectivejust for NR fluids. Since the strength of the coupling is suppressed by the relativistic increaseof pressure ( ρ ∼ p ), as long as particles become relativistic ( T ( a ) = T /a >> m ( φ ( a )) )the matter fluid and the scalar field fluid tend to decouple and evolve adiabatically. Themass varying mechanism expressed by Eq. (6) translates the dependence of m on φ into adynamical behaviour. In particular, for a DFG, the consistent analytical transition betweenUR and NR regimes and their effects on coupling dark matter ( m ) and dark energy ( φ ) areevident from Eq. (6). The mass thus depends on the value of a slowly varying classical scalarfield [39, 40] which evolves like a cosmon field. The cosmon- type equation of motion for thescalar field φ is given by ¨ φ + 3 H ˙ φ + d V ( φ ) d φ = Q ( φ ) . (7)where, in the mass varying scenario, one identifies Q ( φ ) as − (d m/ d φ ) / ( ∂ρ m /∂m ). Thecorresponding equation for energy conservation can be written as˙ ρ φ + 3 H ( ρ φ + p φ ) + ˙ φ d m d φ ∂ρ m ∂m = 0 . (8)which, when added to Eq. (6), results in the equation for a unified fluid ( ρ, p ) with a darkenergy component and a mass varying dark matter component,˙ ρ + 3 H ( ρ + p ) = 0 , (9)5here ρ = ρ φ + ρ m and p = p φ + p m .As we shall notice in the following, this unified fluid corresponds to an effective descriptionof the universe parameterized by a GCG equation of state. III. DECOUPLING MASS VARYING DARK MATTER FROM THE EFFECTIVEGCG
Irrespective of its origin, several studies yield convincing evidences that the GCG sce-nario is phenomenologically consistent with the accelerated expansion of the universe. Thisscenario is introduced by means of an exotic equation of state [7, 9, 31] given by p = − A s (cid:18) ρ ρ (cid:19) α , (10)which can be obtained from a generalized Born-Infeld action [9]. The constants A s and α are positive and 0 < α ≤
1. Of course, α = 0 corresponds to the ΛCDM model and weare assuming that the GCG model has an underlying scalar field, actually real [7, 33] orcomplex [8, 9]. The case α = 1 corresponds to the equation of state of the Chaplygin gasscenario [7] and is already ruled out by data [31]. Notice that for A s = 0, GCG behavesalways as matter whereas for A s = 1, it behaves always as a cosmological constant. Henceto use it as a unified candidate for dark matter and dark energy one has to exclude thesetwo possibilities so that A s must lie in the range 0 < A s < ρ = ρ (cid:20) A s + (1 − A s ) a ( + α ) (cid:21) / ( α ) , (11)and p = − A s ρ (cid:20) A s + (1 − A s ) a ( + α ) (cid:21) − α / ( α ) . (12)One of the most striking features of the GCG fluid is that its energy density interpolatesbetween a dust dominated phase, ρ ∝ a − , in the past, and a de-Sitter phase, ρ = − p , atlate times. This property makes the GCG model an interesting candidate for the unificationof dark matter and dark energy. Indeed, it can be shown that the GCG model admitsinhomogeneities and that, in particular, in the context of the Zeldovich approximation,6hese evolve in a qualitatively similar way as in the ΛCDM model [9]. Furthermore, thisevolution is controlled by the model parameters, α and A s .Assuming the canonical parametrization of ρ and p in terms of a scalar field φ , ρ = 12 ˙ φ + V,p = 12 ˙ φ − V, (13)allows for obtaining the effective dependence of the scalar field φ on the scale factor, a , andexplicit expressions for ρ , p and V in terms of φ .Following Ref. [33], one can obtain through Eq. (8) the field dependence on a ,˙ φ ( a ) = ρ (1 − A s ) a ( α +1 ) (cid:20) A s + (1 − A s ) a ( α +1 ) (cid:21) − α / ( α +1 ) , (14)and assuming a flat evolving universe described by the Friedmann equation H = ρ (with H in units of H and ρ in units of ρ Crit = 3 H / πG ), one obtains φ ( a ) = − α + 1) ln " p − A s (1 − a ( α +1 ) ) − √ − A s p − A s (1 − a ( α +1 ) ) + √ − A s , (15)where it is assumed that φ = φ ( a = 1) = − α + 1) ln (cid:20) − √ − A s √ − A s (cid:21) . (16)One then readily finds the scalar field potential, V ( φ ) = 12 A αs ρ n [cosh (3 ( α + 1) φ/ α +1 + [cosh (3 ( α + 1) φ/ − αα +1 o . (17)If one supposes that energy density, ρ , may be decomposed into a mass varying CDM compo-nent, ρ m , and a dark energy component, ρ φ , connected by the scalar field equations (7)-(8),the equation of state (10) is just assumed as an effective description of the cosmologicalbackground fluid of the universe. Since the CDM pressure, p m , is null, the dark energycomponent of pressure, p φ , results in the GCG pressure, p = p φ . Assuming that dark energyobeys a de-Sitter phase equation of state, that is, ρ φ ( φ ) = − p φ ( φ ) , the dark energy densitycan be parameterized by a generic quintessence potential, ρ φ ( φ ) = U ( φ ) , since its kineticcomponent has to be null for a canonical formulation. It results in U ( φ ) = − p φ ( φ ) = p ,where p is the GCG pressure given by Eq. (12). By substituting the result of Eq.(15) into7he Eq.(12), and observing that H = ρ , with ρ given by Eq.(11), it is possible to rewritethe GCG pressure, p , in terms of φ . It results in the following analytical expression for U ( φ ) , U ( φ ) = ρ φ ( φ ) = − p φ ( φ ) = (cid:20) A s cosh (cid:18) ( α + 1) φ (cid:19)(cid:21) α α , (18)which is consistent with the result for V ( φ ) = (1 / ρ ( φ ) − p ( φ ) ) from Eq. (17). Since ρ φ ( φ ) + p φ ( φ ) = 0, the Eq. (8) is thus reduced tod U ( φ ) d φ + ∂ρ m ∂m d m ( φ ) d φ = 0 , (19)and the problem is then reduced to finding a relation between the scalar potential U ( φ ) and the variable mass m ( φ ) . From the above equation, the effective potential governing theevolution of the scalar field is naturally decomposed into a sum of two terms, one arisingfrom the original quintessence potential U ( φ ) , and other from the dynamical mass m ( φ ) . Forappropriate choices of potentials and coupling functions satisfying Eq. (19), the competitionbetween these terms leads to a minimum of the effective potential. For quasi -static regimes,it is possible to adiabatically track the position of this minimum, in a kind of stationarycondition. The timescale for φ to adjust itself to the dynamically modified minimum ofthe effective potential may be short compared to the timescale over which the backgrounddensity is changing. In the adiabatic regime, the matter and scalar field are tightly coupledtogether and evolve as one effective fluid. At our approach, once we have assumed thedark energy equation of state as p φ = − ρ φ , the stationary condition is a natural issuethat emerges without any additional constraint on cosmon- type equations. In the GCGcosmological scenario, the effective fluid description is valid for the background cosmologyand for linear perturbations. The equation of state of perturbations is the same as that of thebackground cosmology where all the effective results of the GCG paradigm are maintained.The Eq. (18) leads to ρ + p = ρ m + p m which, in the CDM limit, gives ρ ( a ) + p ( a ) = m ( a ) n ( a ) + p m ( ≡
0) = 13 π m ( a ) β ( a ) . (20)Since the dependence of m on a is exclusively intermediated by φ ( a ) , i. e. m ( a ) ≡ m ( φ ( a )) ,from Eqs. (11), (12) and (15), after some mathematical manipulations, one obtains m ( φ ) = m " tanh (cid:0) ( α + 1) φ (cid:1) tanh (cid:0) ( α + 1) φ (cid:1) α α (21)8hich is consistent with Eq. (19) once n ∝ a − . One can thus infer that the adequacy tothe adiabatic regime is left to the mass varying mechanism which drives the cosmologicalevolution of the dark matter component.To give the correct impression of the time evolution of the abovementioned dynamicalquantities driven by φ , in the Fig. 1 we observe the behaviour of m ( a ) and U ( a ) in confrontwith φ ( a ) and V ( a ) of the GCG. In the Fig. 2 we verify how the energy density ρ and thecorresponding equations of state ω for the unified fluid, ρ m + ρ φ , which imitates the GCG,deviates from the right GCG scenario. We assume that the mass varying dark matter behaveslike a DFG in a relativistic regime (hot dark matter (HDM)) and in a non-relativistic regime(CDM). For mass varying CDM coupled with dark energy with p φ = − ρ φ , the effective GCCleads to similar predictions for ω , independently of the scale parameter a . The same is nottrue for HDM which, in the DFG approach, when weakly coupled with dark energy, leadsto the same behavior of the GCG just for late time values of a ( a ∼ α = 1,the GCG model admits a d-brane connection as its Lagrangian density corresponds to theBorn-Infeld action plus some soft logarithmic corrections. Space-time is shown to evolvefrom a phase that is initially dominated, in the absence of other degrees of freedom on thebrane, by non-relativistic matter to a phase that is asymptotically De Sitter. The Chaplygingas reproduces such a behaviour. In this context, the explicit dependence of the mass on ascalar field is relevant in suggesting that the mass varying mechanism, when obtained froman exactly soluble field theory, can be the right responsible for the cosmological dynamics. IV. STABILITY AND ACCELERATED EXPANSION
Adiabatic instabilities in cosmological scenarios was predicted [41] in a context of a massvarying neutrino (MaVaN) model of dark energy. The dynamical dark energy, in this ap-proach, is obtained by coupling a light scalar field to neutrinos but not to dark matter. Theirconsequent effects have been extensively discussed in the context of mass varying neutrinos,9n which the light mass of the neutrino and the recent accelerative era are twinned togetherthrough a scalar field coupling. In the adiabatic regime, these models faces catastrophicinstabilities on small scales characterized by a negative squared speed of sound for the ef-fectively coupled fluid. Starting with a uniform fluid, such instabilities would give rise toexponential growth of small perturbations. The natural interpretation of this is that the Uni-verse becomes inhomogeneous with neutrino overdensities subject to nonlinear fluctuations[42] which eventually collapses into compact localized regions.In opposition, in the usual treatment where dark matter are just coupled to dark energy,cosmic expansion together with the gravitational drag due to CDM have a major impact onthe stability of the cosmological background fluid. Usually, for a general fluid for which weknow the equation of state, the dominant effect on the sound speed squared c s arises fromthe dark sector component and not by the neutrino component.For the models where the stationary condition (cf. Eq. (19)) implies a cosmologicalconstant type equation of state, p φ = − ρ φ , one obtains c s = − c s ≃ dp φ dρ φ >
0. The exact behavior for dark energy plus mass varying dark matter fluidin correspondence with the GCG is exhibited in the Fig. 3 for different GCG α parameters.A previous analysis of the stability conditions for the GCG in terms of the squared speedof sound was introduced in Ref. [33], from which positive c s implies that 0 ≤ α ≤
1. Theseresults are consistent with the accelerated expansion of the universe ruled by the dynamicalmass of Eq. (21) which sets positive values for (1 + 3( p φ + p m ) / ( ρ φ + ρ m )), as we can noticein the Fig. 3.For CDM ( p << m ) the unified fluid reproduces the GCG scenario. For HDM ( p >> m ),in spite of not reproducing the GCG, the conditions for stability and cosmic accelerationare maintained. Fig. 3 shows that the GCG can indeed be interpreted as the effective resultfor the coupling between mass varying dark matter and a kind of scalar field dark energywhich is cosmologically driven by a Λ-type equation of state, p φ = − ρ φ .10 . CONCLUSIONS The dynamics of the cosmology of mass varying dark matter coupled with dark en-ergy dynamically driven by cosmon- type equations was studied without introducing spe-cific quintessence potentials, but assuming that the cosmological background unified fluidpresents an effective behaviour similar to that of the GCG.We have comprehensively analyzed the stability characterized by a positive squared speedof sound and the cosmic acceleration conditions for such a dark matter coupled to dark energyfluid, that exists whenever such theories enter an adiabatic regime in which the scalar fieldfaithfully tracks the minimum of the effective potential, and the coupling strength is strongcompared to gravitational strength. The matter and scalar field are tightly coupled togetherand evolve as one effective fluid. The effective potential governing the evolution of thescalar field is decomposed into a sum of two terms, one arising from the original scalar fieldpotential U ( φ ) , and the other from the dynamical mass of the dark matter.The mass varying behaviour of the dark matter component was determined from theassumption of a kind of Λ-type dark energy dynamics embedded in an effective cosmologicalscenario which reproduces the cosmological effects of the GCG. It is equivalent to decouplingmass varying dark matter from the effective GCG concomitantly with assuming the darkenergy equation of state as p φ = − ρ φ . The adiabatic regime naturally occurs without anyadditional constraint on scalar field equations. The unified fluid description is valid for thebackground cosmology and for linear perturbations. The equation of state of perturbationsis the same as that of the background cosmology where all the effective results of the GCGparadigm are maintained.Unfortunately, we cannot provide a sharp criterion on the potential and on the massvarying dependence on the scalar field to discriminate between these two possibilities: theGCG scenario or an effective unified fluid imitating the GCG via scalar fields driven bycosmon- type equations. Many results for specific quintessence potentials that are found inthe literature, when they reproduce stability and cosmic acceleration, are recovered, andspeculative predictions for new scenarios featuring other mass dependencies on scalar fieldscan be made. It remains open if the present approach can lead to a natural solution ofthe cosmological constant problem. In the meanwhile we take the cosmon model analogyas an interesting phenomenological approach, through which we can reproduce the main11haracteristics of the GCG.Given the fundamental nature of the underlying physics behind the Chaplygin gas andits generalizations, it appears that it contains some of the key ingredients in the descriptionof the Universe dynamics at early as well as late times. Our results suggest that the massvarying mechanism, when eventually derived from an exactly soluble field theory, which isnoway trivial, can be the effective agent for the stability issue and for the cosmic accelerationof the universe, once it can effectively reproduce the main characteristics of a GCG scenario.It also stimulates our subsequent investigation of the evolution of density perturbations,instabilities and the structure formation in such scenarios. To summarize, we expect thatthe future precise data can provide more strong evidence to judge whether the dark energyis the cosmological constant and whether dark energy and dark matter can be unified intoone cosmological background effective component. Acknowledgments
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B663 , 160 (2008). .2 0.4 0.6 0.8 1a0.511.522.533.54 m H a L (cid:144) m , Φ H a L (cid:144) Φ , V H a L , U H a L (cid:144) (cid:144) (cid:144) Α m H a L (cid:144) m , Φ H a L (cid:144) Φ , V H a L , U H a L (cid:144) (cid:144) (cid:144) Α m H a L (cid:144) m , Φ H a L (cid:144) Φ , V H a L , U H a L (cid:144) (cid:144) (cid:144) Α m H a L (cid:144) m , Φ H a L (cid:144) Φ , V H a L , U H a L (cid:144) (cid:144) (cid:144) Α U H a L V H a L Φ H a L(cid:144) Φ m H a L(cid:144) m Φ (cid:144) Φ m H Φ L (cid:144) m , V H Φ L , U H Φ L U H Φ LL V H Φ L m H Φ L(cid:144) m FIG. 1: The cosmological evolution of m ( a ) and U ( a ) correlated with that of φ ( a ) and V ( a ) of theGCG as function of the scale factor (first plot) and the corresponding dependence on φ (secondplot). We are considering the GCG scenarios with A s = 3 / α = 1 , / , / , / .2 0.4 0.6 0.8 1a0.250.50.7511.251.51.752 E n e r g y D e n s it y p >> m 1 (cid:144) (cid:144) (cid:144) Α E n e r g y D e n s it y p >> m 1 (cid:144) (cid:144) (cid:144) Α E n e r g y D e n s it y p >> m 1 (cid:144) (cid:144) (cid:144) Α E n e r g y D e n s it y p >> m 1 (cid:144) (cid:144) (cid:144) Α DM + DEDM ® Ρ m DE ® Ρ Φ GCG ® Ρ << m0.2 0.4 0.6 0.8 1a - - Ω p >> m 1 (cid:144) (cid:144) (cid:144) Α - - Ω p >> m 1 (cid:144) (cid:144) (cid:144) Α - - Ω p >> m 1 (cid:144) (cid:144) (cid:144) Α - - Ω p >> m 1 (cid:144) (cid:144) (cid:144) Α H p Φ + p m L(cid:144)H Ρ Φ + Ρ m L DM - p m (cid:144) Ρ m DE - p Φ (cid:144) Ρ Φ GCG - p (cid:144) Ρ - - << m FIG. 2: Energy densities, ρ , and equations of state, ω = p/ρ , as function of the scale factorfor the effective GCG fluid and its components: mass varying dark matter ρ m and cosmon- like dark energy ρ φ . The results are compared with those of real GCG scenarios with A s = 3 / α = 1 , / , / , /
8. Dark matter is assumed to be described by a DFG in relativistic ( p >> m )and non-relativistic regimes ( p << m ) at present.) at present.
8. Dark matter is assumed to be described by a DFG in relativistic ( p >> m )and non-relativistic regimes ( p << m ) at present.) at present. .2 0.4 0.6 0.8 1a - - c s = dp (cid:144) d Ρ p >> m 1 (cid:144) (cid:144) (cid:144) Α - - c s = dp (cid:144) d Ρ p >> m 1 (cid:144) (cid:144) (cid:144) Α - - c s = dp (cid:144) d Ρ p >> m 1 (cid:144) (cid:144) (cid:144) Α - - c s = dp (cid:144) d Ρ p >> m 1 (cid:144) (cid:144) (cid:144) Α DM + DEDMDEGCG 0.2 0.4 0.6 0.8 1a - - << m0.2 0.4 0.6 0.8 1a - H a Ρ L - H d a (cid:144) d t L p >> m1 (cid:144) (cid:144) (cid:144) Α - H a Ρ L - H d a (cid:144) d t L p >> m1 (cid:144) (cid:144) (cid:144) Α - H a Ρ L - H d a (cid:144) d t L p >> m1 (cid:144) (cid:144) (cid:144) Α - H a Ρ L - H d a (cid:144) d t L p >> m1 (cid:144) (cid:144) (cid:144) Α DM + DEDMDEGCG 0.2 0.4 0.6 0.8 1a - << m FIG. 3: Squared speed of sound c s = d p/ d ρ and cosmic acceleration ¨ a/ ( a ρ ) as function of the scalefactor for the unified fluid decomposed into mass varying dark matter and cosmon- like dark energycomponents. Again the results are compared with those of real GCG scenarios for A s = 3 / α = 1 , / , / , /
8, for mass varying dark matter described by a DFG in relativistic ( p >> m )and non-relativistic regimes ( p << m ) at present.) at present.