A UNIFIED DARK ENERGY MODEL FROM A VANISHING SPEED OF SOUND WITH EMERGENT COSMOLOGICAL CONSTANT

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January 22, 2014 14:16 WSPC/S0218-2718 142-IJMPD 1450012

International Journal of Modern Physics DVol. 23, No. 2 (2014) 1450012 (15 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218271814500126

A UNIFIED DARK ENERGY MODEL FROM A VANISHINGSPEED OF SOUND WITH EMERGENT

COSMOLOGICAL CONSTANT

ORLANDO LUONGO∗,‡,¶,‖ and HERNANDO QUEVEDO†,‡,§,∗∗

∗Dipartimento di Scienze Fisiche,Universita di Napoli “Federico II”,Via Cinthia, I-80126, Napoli, Italy

†Dipartimento di Fisica and Icra,Universita di Roma “La Sapienza”,

Piazzale Aldo Moro 5, I-00185, Roma, Italy

‡Instituto de Ciencias Nucleares,Universidad Nacional Autonoma de Mexico,

AP 70543, Mexico DF 04510, Mexico

§Instituto de Cosmologia,Relatividade e Astrofisica Icra-CBPF Rua, Dr. Xavier Sigaud,

150, CEP 22290-180, Rio de Janeiro, Brazil

¶INFN Sezione di Napoli,Complesso Universitario di Monte S. Angelo Edificio,

N Via Cinthia, I-80126, Napoli, Italy‖[email protected]∗∗[email protected]

Received 20 March 2013Revised 5 October 2013Accepted 7 October 2013

Published 12 November 2013

The problem of the cosmic acceleration is here revisited by using the fact that theadiabatic speed of sound can be assumed to be negligible small. Within the context ofgeneral relativity, the total energy budget is recovered under the hypothesis of a vanishingspeed of sound by assuming the existence of one fluid only. We find a cosmologicalmodel which reproduces the main results of the ΛCDM paradigm at late-times, showingan emergent cosmological constant, which is not at all related with the vacuum energyterm. As a consequence, the model presented here behaves as a unified dark energy (DE)model.

Keywords: Dark energy; sound speed; emergent cosmological constant.

PACS Number(s): 98.80.−k, 98.80.Jk, 98.80.Es

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http://dx.doi.org/10.1142/S0218271814500126

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O. Luongo and H. Quevedo

1. Introduction

To explain the observed cosmological acceleration,1,2 it is usually assumed that,besides a dust-like component, the universe is filled by an additional exotic fluidthat accounts for about 75% of the total energy density.3 Many models are knownin the literature that intend to describe the nature of this additional component.4

The simplest explanation is obtained by assuming the existence of a cosmologicalconstant that is the basic ingredient of the well-known ΛCDM model.5 Despite itssimplicity, the ΛCDM model suffers from various theoretical shortcomings6,7 so thatit seems inadequate to consider it as the definitive paradigm.8 Thus, an additionalfluid, namely dark energy (DE), is thought to be responsible of the observed acceler-ation.9–13 In this paper, we propose a different approach in which we use Einstein’sgeneral relativity without cosmological constant and assume that the dynamics ofthe universe satisfies the laws of ordinary thermodynamics with a vanishing adia-batic sound speed.

A perfectly homogeneous and isotropic background is described by theFriedman–Lemaıtre–Robertson–Walker (FLRW) metric

ds2 = dt2 − a(t)2(dr2 + r2 sin2 θdφ2 + r2dθ2), (1)

where we limit ourselves to the case of zero spatial curvature to be in accordancewith observations.a

Moreover, the gravitational source is assumed to be described by the energy–momentum tensor of a perfect fluid, i.e. Tµν = diag(ρ(t),−P (t),−P (t),−P (t)),so that the Friedmann equations, i.e. H2 ≡ ( a

a )2 = 8πG3 ρ and H + H2 = a

a =− 4πG

3 (3P + ρ), determine the dynamics of the model and the conservation lawρ + 3H(P + ρ) = 0.

In a rest frame, the pressure P (t) can be related to the overall energy densityand entropy of the universe, becoming a function of them, i.e. P = P (ρ, S); thecorresponding perturbations are written as

δP = c2aδρ + σδS. (2)

However, it is a well-consolidate convention to consider adiabatic perturbationsonly, along the universe time evolution; then, for considering a nonvanishing pres-sure perturbation, the entropy perturbations can be neglected at a first approx-imation. This is the case of a perfect fluid in which the definition of the soundspeed, i.e. ca in Eq. (2), depends only on the adiabatic perturbations of pressureand energy density.14 In general, circumscribing our analysis in a rest frame, theentropy perturbations are suppressed at early-times,15 while at late-times, we canonly approximate it as δS ≈ 0; therefore the adiabatic case can be reviewed as alimiting case in which δS → 0.16,17 However, by assuming that inside the Hubblehorizon the entropy perturbations are negligible small,18,19 and the fluid behaves

aThe generalization to the case of nonzero spatial curvature is straightforward.

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A Unified DE Model from a Vanishing Speed of Sound

adiabatically, it is ilicit to assume δP ∝ δρ. Consequently, in this case only anadiabatic sound speed can be defined as20–22

c2a ≡

(∂P

∂ρ

)S

=P

ρ, (3)

which is a scale independent and gauge invariant quantity,23 evaluated at constantentropy S. Written in terms of the redshift z, defined by dz/dt = −(1 + z)H , wenotice that the condition of a vanishing adiabatic sound speed, c2

a = dPdz (dρ

dz )−1 = 0,implies that the pressure must be constant along the expansion history of the uni-verse. This condition guarantees that the structures in the universe can be formedat all scales. This is clear because the pressure perturbations and the Jeans lengthidentically vanish, i.e. δP ∝ c2

aδρ = 0 and λJ ≡ ca

√π

Gρ = 0.24–26 In other words, if

the adiabatic pressure perturbations are assumed to vanish along all the expansionhistory, it is possible to reproduce the observed structures at all scales. Thus, wewonder whether the cosmological model, which arises from the simple assumptionof ca = 0, is capable to reproduce the current observations of our universe.27,28 Itis the aim of this paper to answer this question in the affirmative.

The paper is then organized as follows: in Secs. 2 and 3, we describe the theoret-ical features of our approach by using the mentioned information following from theassumption of a vanishing speed of sound. In Sec. 4, we propose a thermodynami-cal interpretation of a constant (negative) pressure and in Sec. 5, we confront thepredictions of the model with the supernovae Ia (SNeIa), baryonic acoustic oscilla-tion (BAO) and cosmic microwave background (CMB) data; moreover we compareour model with ΛCDM and the Chevallier–Polarski–Linder (CPL) parametrization.Finally, in Sec. 6, we present the conclusions and discuss future perspectives of ourpaper.

2. The Equation of State from a Vanishing Sound Speed

In this section, we assume a general equation of state (EoS) of the form ω ≡ P (z)ρ(z) .

By combining the continuity equation with the vanishing of the sound speed, weobtain a differential equation for ω

dω

da= 3ω

(1 + ω)a

. (4)

Equation (4) is a nonlinear differential equation, whose general solution is

ω = − 11 − ξa−3

, (5)

where ξ is an integration constant to be fixed in the next sections. We see thatthe assumption of a vanishing adiabatic sound speed fixes almost completely theEoS and, consequently, the energy density. Notice that Eq. (4) is also satisfied bythe constant solutions ω = 0 (dust-like case) and ω = −1 (cosmological constantcase); however, we do not consider these cases here because a universe dominated

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only by dust is ruled out by observations29,30 and the cosmological constant is nottreated in this work. Notice, however, that Eq. (5) is degenerate in the sense thatit leads to a ΛCDM-like model when the constant ξ = −Ωm

ΩΛ, i.e. the ratio between

the matter (Ωm) and the cosmological constant (ΩΛ) densities. Since we startedfrom the assumption that no cosmological constant is taken into account a prioriin Einstein’s equations, it is important to determine the cosmological model whichfollows from Eq. (4).

First, as noticed above, the case of a dust-like fluid is also allowed as a particularsolution of Eq. (4), i.e. ρ1 ≡ ρb = ρb0a

−3, (ρb0 = const.). According to Eq. (5),the fluid thought to drive the cosmic speed up corresponds to the energy densityρ2 = ρ

X−ρ

Xξa−3, (ρ

X= const.). By using the property that the continuity equation

is linear in terms of total density, to study both solutions simultaneously, it isconvenient to write the general solution as

ρ(z) = (ρb0 − ρX

ξ)a−3 + ρX

. (6)

Equation (6) shows that at small z (large a), ρ ρX , while for large z (small a),ρ ∝ (1+ z)3 ≡ a−3. This suggests that the model can be viewed as a unified model,since it predicts the existence of either dark matter or cosmological acceleration,when ρb0 is assumed to be the total baryon density.

Equation (6) corresponds to a solution of the conservation law equation with atotal EoS of the form

ω(z) = − 1

1 −(

ξ − ρb0

ρX

)a−3

, (7)

which is functionally equivalent to the one obtained in Eq. (5) from the conditionof vanishing sound speed. This general solution involves three constant parameters,namely, ξ, ρb0 and ρ

X. Furthermore, by using Eqs. (6) and (7), we get

P = ω(z)ρ(z) = −ρX

. (8)

We see that the pressure is constant, as expected, since ca = 0. This resemblesthe behavior of the pressure in the ΛCDM model. However, in our picture, thedifference lies on the fact that a constant pressure does not imply necessarily aconstant density, but it is also possible to have an evolving barotropic factor andan evolving density whose product remains constant. Our model does not need theintroduction of a cosmological constant and has the advantage that it contains onefluid only, defined by Eq. (4).

Summing up, a constant pressure leads either to a constant density or to thecase presented here. Moreover, if we choose the value of ρ

Xto be positive, the

resulting pressure guarantees an accelerating universe, as in the ΛCDM case, butwithout invoking a cosmological constant a priori.

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3. The Emergent Cosmological Constant

The key feature of our approach is that from the condition of a vanishing speed ofsound, it follows naturally a constant pressure. We now wonder whether, withoutputting any further information into Einstein’s equations, we may expect as a resultthat a cosmological model is able to describe the cosmological acceleration.

Introducing ρ(z) into the first Friedman equation H2 = (8πG/3)ρ(z), we obtainthe generic normalized Hubble rate in terms of z, i.e.

E ≡ H

H0=

√(1 − ΩX)(1 + z)3 + ΩX , (9)

where we adopt the convention ΩX ≡ ρX

ρc, with the critical density defined as

ρc ≡ 3H20

8πG . In Eq. (9), we used the condition E(z = 0) = 1, which gives

ξ =Ωb + ΩX − 1

ΩX, (10)

where Ωb ≡ ρb0ρc

. Therefore, to determine the Hubble rate E(z) we need the con-stant ΩX , whereas to determine the barotropic factor ω(z) in Eq. (7) it is necessaryto know the two constants Ωb and ΩX . Since the amount of baryons can be con-strained independently of the measurements of the visible matter in the universe,31

we conclude that the model depends on one parameter only, i.e. ΩX .In addition, for comparison with the observational data, it is convenient to use

the relationship ΩX = Ωb−1ξ−1 , which follows from Eq. (10), to rewrite Eq. (9) as

E =√

Ωm(1 + z)3 + ΩΛ, (11)

where Ωm ≡ ξ−Ωb

ξ−1 ; moreover, we emphasize here the appearance of an emergentcosmological constant, i.e. ΩΛ ≡ 1 − Ωm. In other words, by recasting Eq. (10) asin Eq. (11) one infers that our model predicts the existence of a constant termthat drives the acceleration, is emergent, and is not related to the vacuum energy.As pointed out in Sec. 2, this is a consequence of the presence of a dynamic EoSfollowing from the physical assumption that the universe is made of only one matterfluid with vanishing adiabatic speed of sound.

From the explicit expression of ω(z) and Eq. (10), it is possible to infer thelimits of the evolution of w(z) as

ω0 = −ΩX , for z = 0;

ω∞ = 0, for z → ∞.(12)

Equation (12) show that at the redshift z = 0, ω0 is a constant which predictsΩX ≤ 1 in order to get −1 ≤ ω0 < 0. Thus, the parameter ΩX can be chosen suchthat the present acceleration of the universe can be described. At higher redshift,the usual dust-like component dominates, because ω∞ = 0.

By considering Eqs. (7) and (12), it is clear that our model reproduces the effectsof dark matter and DE, under a unified scheme; it is therefore easy to show that

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our approach represents a unified DE model, which naturally predicts the amountof both dark matter and DE densities in the universe, without postulating theseabundances a priori.

Our model mimics the EoS of dark matter as well as the Chaplygin gasparadigm. An accurate study of early perturbations is therefore necessary in orderto establish if our model is well-behaved at higher redshifts. However, while theΛCDM model predicts a vanishing speed of sound at early-times, the Chaplygingas seems to be compatible with ca > 0 only.32,33 Hence, we expect that as the red-shift increases our approach will behave better than the Chaplygin gas, because ourDE term is an emergent constant, in analogy with the ΛCDM model. In any case,a direct analysis of cosmological perturbations is needed to compare our proposalwith the Chaplygin and concordance models.

3.1. The kinematics of the acceleration

If the universe accelerates, the so-called acceleration parameter, defined as

q = −1 − H

H2, (13)

must be negative. In particular, at our time q should be confined in the range−1 < q0 < 0. The case q → −1 corresponds to a cosmological inflation and so it isruled out by current observations.34

From Eq. (13), for our model we obtain

q = −1 +3(1 − ΩX)(1 + z)3

2 + 2(1 − ΩX)z[3 + z(3 + z)], (14)

so that at z = 0 it reduces to

q0 =12(1 − 3ΩX). (15)

For the particular value ΩX = 0.712, we get q0 ≈ −0.57 which is in agreement withobservations.35,36 Moreover, since at z = 0 the EoS of the universe can be recast as

ω0 = −13(1 − 2q0), (16)

by using Eqs. (16) and (15), it is convenient to write

ΩX = −ω0, (17)

which represents the relation between ΩX and the total EoS of the universe ω0.Moreover, at the moment in which the acceleration starts (q = 0), the corre-

sponding redshift reads

zacc = −1 +[2(ΩX − 2Ω2

X + Ω3X)]

13

1 − ΩX, (18)

so that for ΩX = 0.712 we have zacc ≈ 0.7. In Fig. 1, we compare the evolution of theacceleration parameter for our model, with that of the ΛCDM model. The qualita-tive agreement of the predictions of both models is obvious. An interesting property

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Fig. 1. (Color online) This graph is plotted q(z) for our model (dashed line) and ΛCDM (redline) versus z. The indicative values for the constants are Ωm = 0.274 for ΛCDM and ΩX = 0.690for our model.

of ω is that for future redsfhits (z → −1) the form of ω reduces to ω−1 = −1, indi-cating the presence of a de Sitter behavior; in other words, the universe is thoughtto evolve with an endless constant acceleration parameter q−1 = −1. This pecu-liarity overcomes the recent problem of finding phenomenological parametrizationsof ω, able to reproduce both the future and the past evolutions.37–43

In Sec. 5, we will derive more accurate bounds on the free parameters ΩX and ξ

by performing cosmological tests based on SNeIa, BAO and CMB data.

4. A Thermodynamic Interpretation

The results presented in Secs. 2 and 3 show that the condition of a vanishingspeed of sound leads to a model in which the total density is proportional to a−3.Section 3 underlines the key features of the cosmological model, emphasizing thefact that a constant (negative) EoS follows immediately from ca = 0. In our model,an important consequence of the assumption ca = 0 is that it implies ω(z)×ρ(z) =const., extending the ΛCDM paradigm in which ωρ = const. with ω and ρ beingconstants as well.44 Whereas in the ΛCDM model it is clear that a constant densitynaturally implies a negative pressure, in our model, however, the sign of Eq. (8) isundetermined.

Hence, in order to understand the physical mechanism for which the pressureshould be negative, let us consider a reversible and frictionless universe, satisfyingthe first law of thermodynamics, dQ = dU + PdV , and the EoS of an ideal gasP = ρ(Cp − CV )T , where the heat capacities are defined as CV = (∂U

∂T )V andCp = ( ∂h

∂T )p, with h = U + PV being the enthalpy of the system. In the specialcase of a homogeneous and isotropic cosmological model, it is possible to apply in aconsistent manner the laws of ordinary thermodynamics45 and then we can assumethat the evolution of the universe is adiabatic and reversible. For any adiabatic andreversible system, the pressure can be conventionally written as in the polytropic

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law, i.e. P = pργ ,46 where γ = Cp/CV and p a constant. By comparing this lawwith Eq. (8), we can get information about the sign of P . Indeed, from the abovethermodynamic assumptions we obtain that

ca =

√γP

ρ=

√γ(Cp − CV )T . (19)

Then, for a vanishing sound speed, the allowed solutions are formally Cp −CV = 0or γ = 0. The first solution implies a pressureless universe whereas the secondsolution leads to a vanishing heat capacity Cp = 0, equivalent to h = const., andpressure P = −ρCV T which is negative for realistic positive values of the heatcapacity CV .

We conclude that the universe is dominated by an ideal-gas component withvanishing speed of sound leads to a pressure which can be zero or negative.b More-over, this can be also viewed as a consequence of the fact that each particle ofa given fluid undergoes an early isentropic process, when the wave amplitude isinfinitesimal. In particular, this condition follows from the fact that, in general, theentropy is proportional to the square of the velocity and temperature gradients.46,49

5. Experimental Constraints

The aim of this paragraph is to perform experimental tests to constrain the valuesof the constants ξ and ΩX by using Eqs. (9) and (11). In particular, we employ thethree most common fitting procedures: SNeIa, BAO and CMB. We use the mostrecent dataset, namely Union 2.1 compilation,50 which updates the old one, i.e.Union 251; the use of the Union 2.1 compilation alleviates the problem of systematicerrors, afflicting older datasets (see Refs. 52 and 53).

Thus, associating to each Supernova modulus µi the corresponding 1σ error,denoted by σi, we define the distance modulus µ = 25 + 5 log10

dL

Mpc , where dL(z)is the luminosity distance

dL(z) = (1 + z)∫ z

0

dz′

H(z′)(20)

and we minimize the chi square, defined as follows

χ2SN =

1ν − 1

∑i

(µtheori − µobs

i )2

σ2i

, (21)

where ν is the overall number of data; for Union 2.1 ν = 580.The second test that we perform is related to the observations of large scale

galaxy clusterings, which provide the signatures of the BAO.54 We use the mea-surement of the peak of luminous red galaxies observed in Sloan Digital Sky Survey

bSimilar results were obtained by analyzing the ideal gas in the context of geometrothermody-namics.47,48

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(SDSS), denoted by A

A =√

Ωm,t

[H0

H(zBAO)

] 13

[1

zBAO

∫ zBAO

0

H0

H(z)dz

] 23

, (22)

with zBAO = 0.35. Let us notice that Ωm,t = 1 − ΩX and Ωm,t = ξ−Ωb

ξ−1 for themodel of Eqs. (9) and (11), respectively. In addition, the observed A is estimatedto be

Aobs = 0.469(

0.950.98

)−0.35

, (23)

with an error σA = 0.017. In the case of the BAO measurement, we minimize thechi square

χ2BAO =

(A − Aobs

σA

)2

. (24)

Finally, for the CMB test we define the so-called CMB shift parameter. Insteadof the standard definition of the CMB shift

R = H0

√Ωm,t

∫ zrec

0

dz

H(z), (25)

which presents some difficulties, we use the alternative definition55

R ≡ 2l1l′1

, (26)

where l1 is the position of the first peak on the CMB TT power spectrum of themodel under consideration and l′1 is the first peak in a flat FRW universe withΩm,t = 1.

In particular, for approaches providing a unified description of both DE anddark matter, which is our case, the latter expression is strongly requested. Thus, l1is defined as

l1 =DA(zrec)s(zrec)

, (27)

where DA(zrec) is the comoving angular distance at recombination, i.e.

DA(zrec) =∫ zrec

0

(1 + z)dz (28)

and s(zrec) represents the sound horizon at recombination

s(zrec) =∫ ∞

zrec

cs(z)H(z)

dz, (29)

where, conventionally, cs(z) is the sound speed of the photon-to-baryon fluid that isusually given as cs(z) = 3−1/2(1 + 4ρb0/3ργ)−1/2, with ργ the photon density. Thisprocedure gives a complementary bound to the SNeIa data and BAO, because the

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SNeIa redshift is z < 2, zBAO = 0.35, while here zCMB = 1091.36.56 Accordingly,we minimize the chi square

χ2CMB =

(R−Robs

σR

)2

. (30)

It is important to note that the data of BAO and CMB do not depend on the valuesof H0. We summarize the numerical results in Table 1 and show the correspondingplots in Figs. 2–4.

The values of Ωm, ΩX and ξ of Table 1 are in agreement with the theoreticalresults previously shown. The results of Table 1 confirm that our model behaves as

ξ

Fig. 2. Two-dimensional marginalized contour plots for the parameters explored with theBayesian analysis. Solid lines represent the 1-σ confidential levels. Dashed lines represent respec-tively the 2 and 3 sigma confidential levels.

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Ω

Fig. 3. (Color online) This graph is plotted ω(z) (Y-axis) for our model (dashed line) and CPL(red line) versus the redshift z. The indicative values for the constants are Ωm = 0.274, ξ = −0.35,w0 = −0.93 and wa = 0.58. The redshift range spans from future time (z = −1) to past timez ≥ 0.

Fig. 4. (Color online) The graph is plotted for the expansion history of t(a) versus H(a) for ourmodel (dashed line), ΛCDM (red line) and CPL (black line). The indicative values are ΩX = 0.600,w0 = −0.93, wa = 0.58 and Ωm = 0.274. Significative deviations from CPL have been obtainedfor high redshift.

a ΛCDM-like model, in which the cold dark matter term naturally arises, while thecosmological constant emerges as a result of ca = 0.

5.1. Comparison with other models

During the last few decades, different parametrizations of ω(z) were proposed.3

For instance, ω = w1 + w2z, ω = wα + wβ log(1 + z) or ω = w0 + wa(1 − a).In particular, the third case was introduced by Chevallier, Polarski and Linderand it is referred to as the CPL parametrization.57,58 The CPL parametrizationhas the advantages that at low and very high redshifts it reduces to constant

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Table 1. Table of best fits given by the maximum of the likelihood function. The

panel contains the parameter spaces analyzed by the use of a Bayesian analysis,for each one of the three datasets. The data used are the union 2.1 supernovacompilation, observations of the BAO and the CMB measurements. The parametersΩb and ξ have been fitted by using Eq. (11) and the parameter ΩX by using Eq. (9);the error propagations on Ωm have been obtained by using the standard logarithmicpropagation. The mean values for the three parameters are Ωm (mean) = 0.280,ΩX (mean) = 0.727 and ξmean = −0.313 in excellent agreement with theoretical

predictions. The H0 value for the SNeIa tests is H0 = 69.966+1.285−1.262 Kms−1 Mpc−1.

The reported error corresponds to the 68% confidence level.

Parameters SN SN+BAO SN+BAO+CMBχ2 = 0.9727 χ2 = 0.9726 χ2 = 0.9727

ξ −0.318+0.139−0.102 −0.291+0.135

−0.101 −0.329+0.127−0.092

Ωm 0.279+0.083−0.062 0.280+0.084

−0.063 0.281+0.066−0.084

ΩX 0.721+0.068−0.068 0.732+0.065

−0.064 0.729+0.063−0.061

Note: χ2 ≡ −2 logL is given by the pseudo-chisquared analysis.

values, respectively ω(z → 0) = w0, and ω(z → ∞) = w0 + wa. However, all theparametrizations suggested so far for ω(z) are either ad hoc proposals or the resultof phenomenological assumptions only. Our model predicts a theoretical EoS ω(z),and it is also able to reproduce previous results (see, for instance, Ref. 44).

Moreover, the barotropic factor is also connected to an interesting quantity, thatone might consider as a natural measure of time variation, namely, dω

d ln(1+z) | z=1. Inmodels involving a scalar field ϕ with potential V (ϕ), this quantity is related to theslow-roll potential ∝ V ′

V , in the region z = 1, where the scalar field is most likelyto be evolving as the epoch of matter domination changes over to DE domination.For the CPL parametrization for example dω

d ln(1+z) | z=1 = wa

2 , while for our modelit is

dω

d ln(1 + z)

∣∣∣∣z=1

= −24ΩX1 + ΩX − Ωb

(8 + 7ΩX − 8Ωb)2. (31)

By comparing our result with CPL, we infer two solutions. One of these solu-tions is physically compatible with the accelerating scenario; in fact, by consideringΩb(mean) and the indicative value ΩX ≈ 0.716, we find wa in agreement with theobservational results.57,58

We present below the graphics of the evolution of ω(z), Fig. 2, and the expansionhistory for a(t), Fig. 3, given by57–59

H0t(a) =∫ 1

a

da′

a′E(a′). (32)

6. Conclusions

The presence of baryonic and dark matter is generally thought to be intertwinedwith the addition of an exotic fluid which drives the acceleration. Unfortunately,all the attempts made to describe this unexpected scenario suffer from various

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shortcomings. Moreover, ΛCDM remains the favorite fitting model to describe theuniverse dynamics by including within the Einstein equations a second fluid, i.e.the cosmological constant. We showed that it is possible to discard the existenceof a second fluid and, instead, to use ordinary thermodynamics to obtain a unifiedcosmological model. This model naturally predicts the cosmic speed up and thepresence of dark matter in the universe. In fact, by assuming a vanishing speedof sound, which guarantees that small pressure perturbations do not propagate,we obtain a theoretical parametrization for ω and a corresponding varying density,generalizing the ΛCDM model, because the inferred EoS is a negative function.Our approach reproduces the behavior of the ΛCDM model and is able to avoidthe problems derived from assuming the existence of a cosmological constant. Anadditional result from our thermodynamic analysis is that the pressure must benegative and so one does need to fix its sign a priori. In other words, a constantpressure is found to be negative by simply assuming a frictionless and isentropicuniverse with a varying EoS and no cosmological constant.

Our model unifies DE and dark matter under one fluid only, i.e. the one derivedfrom the condition ca = 0. Moreover, the corresponding EoS of such a fluid behavesas a de Sitter phase at future evolution, while at higher redshift it behaves asa pressureless term. The fluid is characterized by the existence of an emergentcosmological constant that at our time dominates over the term ∝ a−3.

This mimes the ΛCDM behavior, reproducing the main results of the standardmodel, without assuming a priori the existence of a cosmological constant. Thedifficulties related to the well-known problems of coincidence and of fine tuning3 arehowever not completely solved in the context of the present model. The presence of avariable barotropic factor ω(z) cannot guarantee that a vacuum energy cosmologicalconstant does not exist, since any field placed in vacuum will contribute to theenergy which enters the Einstein equations. Our model can describe the universedynamics, although the fine tuning problem due to quantum vacuum fluctuationsstill remains.

Using the cosmological tests of SNeIa, BAO and CMB, it has been shown thatthe present model is able to reproduce the observable universe and remains inagreement with the theoretical limits as well. From these conclusions, it is clearthat our approach can be considered as a serious alternative to ΛCDM. In futureworks, we will investigate how perturbations behave in the framework of our model.Only then, we will understand if our approach is also compatible with early phasesof the universe evolution.

Acknowledgments

This work was supported in part by DGAPA-UNAM, Grant No. IN106110, andConacyt-Mexico, Grant No. 166391. One of us (H.Q.) would like to thank theICRANet-Rome for hospitality and support. One of the authors (O.L.) is gratefulto B. Luongo and G. Capasso.

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