A closer look at the Barboza-Alcaniz equation of state parametrization

A closer look at the Barboza-Alcaniz equation of state parametrization

Anindita Mondal111Electronic Address: anindita12@bose.res.in Department of Astrophysics & Cosmology,
S. N. Bose National Centre for Basic Sciences, Kolkata 700106, West Bengal, India.
   Subhajit Saha222Electronic Address: subhajit1729@gmail.com Department of Mathematics,
Panihati Mahavidyalaya,
Sodepur 700110, West Bengal, India.

The Barboza-Alcaniz EoS parametrization has been considered and its -parametrization has been investigated in search for a thermodynamic motivation. For this, we have studied the validity of the generalized second law of thermodynamics as well as the thermodynamic equilibrium considering the cosmological apparent horizon as the boundary. Also, an expression for the particle creation rate has been obtained in terms of assuming an adiabatic particle creation scenario and its behavior has been studied for consistency during various phases of evolution of the Universe as suggested by various thermodynamic arguments found in the literature.

Keywords: FLRW universe; Barboza-Alcaniz EoS; Generalized second law; Thermodynamic equilibrium; Adiabatic particle creation

PACS Numbers: 98.80.-k

I Introduction

Parametrization of the deceleration parameter has been found to be a useful tool towards a more complete characterization of the evolutionary history of the Universe. In the literature, several well-known -parametrizations have been proposed q1 (); q2 (); q3 (); q0 (); q4 (); q5 (); q6 (); q7 (); q8 (); q9 (); q10 (); q11 (); q12 (); q13 (). In this short paper, we shall undertake a study of the Barboza-Alcaniz EoS parametrization Barboza1 (). With the corresponding -parametrization, we shall study the validity of the generalized second law (GSL) of thermodynamics and thermodynamic equilibrium (TE). Moreover, we shall obtain the expression for the particle creation rate in the gravitationally induced isentropic333The words ”isentropic” and ”adiabatic” can be used interchangeably. particle creation scenario and investigate the consistency in its behavior based on thermodynamic arguments found in the literature.

We shall consider a universe governed by the flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in comoving coordinates (, , , ) as


where is the scale factor of the Universe. Assuming the energy-momentum tensor of the form


with as the 4-velocity of the fluid. The Friedmann and the acceleration equations can be derived from the Einstein’s field equations as


respectively, where is the Hubble parameter, whereas and denote, respectively, the energy density and the pressure of the cosmic fluid. The energy-momentum conservation equation can be obtained using the expressions in Eq. (3) and it has the form


The deceleration parameter is defined as


is the equation of state (EoS) of the cosmic fluid chosen. It may be either a constant or a variable function of the scale factor , or equivalently, the redshift parameter (). A positive indicates deceleration while implies acceleration.

As stated earlier, in the present work, we shall focus our investigation on a particular form of the EoS parameter , popularly known as the Barboza-Alcaniz parametrization. The explicit form is given by Barboza1 ()


Plugging into the last equality in Eq. (5), the parametrization for the deceleration parameter becomes


with and . They performed statistical analysis using observational data from Supernova Type Ia (SNLS), Baryon Acoustic Oscillations (SDSS), Cosmic Microwave Background shift parameter (WMAP), and estimates of (from the ages of high- galaxies), and obtained the best fit values of the parameters to be and at confidence level.

Ii Thermodynamic Consequences

Our first task is to study the thermodynamics, particularly the GSL and the TE for the above system considering cosmological (or dynamical) apparent horizon as the boundary, since we are interested in the thermodynamic behavior of the local Universe. This type of thermodynamic study was first introduced in their pioneering work by Wang, Gong, and Abdalla Wang1 (). It is worthwhile to mention that while GSL444The idea of incorporating the GSL in cosmology was first developed by Ram Brustein Brustein1 (). This second law is based on the conjecture that causal boundaries and not only event horizons have geometric entropies proportional to their area. should hold throughout the evolution of the Universe, TE is required to be satisfied at least during the final phases of its evolution. Now, if be the entropy of our system (horizoncosmic fluid inside), then GSL and TE, respectively, refer to the inequalities and . The total entropy will be , where and are the entropies of the apparent horizon and the fluid bounded by it respectively. The above two inequalities then get modified as and . As the horizon entropy scales with its surface area, we have


where is the location of the cosmological apparent horizon. The temperature of the horizon is found to be proportional to its surface gravity and goes by the expression


The entropy of the fluid can be obtained from the Gibbs equation


where is the total entropy of the fluid inside the apparent horizon, is the volume of the fluid, and and are, respectively, the energy density and the pressure of the cosmic fluid respectively. The fluid temperature is assumed to be equal to the horizon temperature in these kinds of thermodynamic studies, i.e., .

Let us set and the fundamental constants to unity, without any loss of generality. The the first order time-derivative of the total entropy can be evaluated as Wang1 (); Saha0 (); Saha00 ()


which shows that the GSL always holds irrespective of the nature of cosmic fluid chosen. Thus, GSL is satisfied throughout the evolution of the Universe.

Figure 1: The variation of against . We have used the best fit values of and as obtained in Ref. Barboza1 ().

The second derivative of was obtained in the following form Saha0 (); Saha00 (); Saha000 ()


which is not useful for our present investigation. Noting that

it is remarkable to see that Eq. (12) can be entirely expressed in terms of and as


using the Einstein field equations and the definition of the deceleration parameter, i.e., the first equality of Eq. (5). In order to know how behaves during the last stages of the evolution, we have plotted it against with the help of Maple plotting software (see Figure 1). It is clearly visible from the figure that our physical system remains in TE during the final phases of evolution of the Universe. Thus, the Barboza-Alcaniz EoS paramatrization and the resulting -paramterization is well consistent with GSL and TE and therefore can be considered to be thermodynamically motivated. Moreover, our deductions are perfectly in line with previous studies undertaken in this direction Radicella1 (); Mimoso1 (); Pavon1 ().

Iii Implications of isentropic particle creation mechanism

We now introduce a dissipative effect (in the FLRW Universe) in the form of a bulk viscous pressure which occurs in the expression for the energy-momentum tensor in the following manner —


In Cosmology, it is customary to assume that the dissipation arises solely due to the nonconservation of the (quantum) particle number Zeldovich1 (); Murphy1 (); Hu1 (), in other words, due to isentropic (or adiabatic) production of perfect fluid particles Prigogine1 (); Calvao1 (). The associated Friedmann and acceleration equations are


The conservation laws, namely,


have the explicit expressions


Here, is the fluid expansion, is the particle flow vector, is the particle number density, and .

The phenomenon of nonconservation of the total number of particles in an open thermodynamic system modifies the second expression in Eq. (17) as


where denotes the rate of change of the number of particles () in a comoving volume . Hence, refers to creation of particles while indicates particle annihilation. In this scenario, the energy density and the thermostatic pressure of the cosmic fluid is assumed to be related by the EoS with . The bounds on ensures that the fluid EoS does not become exotic.

Now, using the first expression in Eq. (17), Eq. (18), and the Gibbs relation ( is the temperature and is the entropy per particle)


we obtain


under the assumption of isentropicity555Note that although the entropy per particle is constant, there is entropy production due to the enlargement of the phase space of the system., i.e., . Plugging the above expression for in the Einstein field equation (15), we get


It is evident from the first equality in Eq. (5) that the particle creation rate can be written in terms of the deceleration parameter as Saha1 ()


where the Hubble parameter is given by


where is the value of the Hubble parameter at the present epoch. Note that in a FLRW universe.

Now, taking the form of obtained with the Barboza-Alcaniz EoS parametrization, we have obtained the expressions for and in terms of the redshift as

Figure 2: Left Panel: Variation of against and Right Panel: Variation of against for (red solid curve), (blue dashed curve), and (green dashdot curve). We have used the best fit values of and as obtained in Ref. Barboza1 (). has been set to .

The behavior of the particle creation rate is similar for all the three choices of the perfect fluid EoS parameter . Now, thermodynamic arguments tell us that Gunzig1 () in the very early Universe, starting from a regular vacuum, most of the particle creation effectively took place and the creation rate was greater than the expansion rate, while particle creation was strongly suppressed during the radiation era Parker1 (). Also, the creation rate again dominates during the late time acceleration phase. The right panel of Figure 2 exactly demonstrates the above situations during the various phases of evolution of the Universe. For the creation rate is greater and it boils down to a very small value during the deceleration phase. The recent acceleration phase again sees a dominance of the creation rate. Further, it predicts a significant increase in the creation of particles during future phases of evolution. Thus, the Barboza-Alcaniz EoS parametrization is consistent with the adiabatic particle creation mechanism.

Iv Short discussion

The paper dealt with an investigation of the Barboza-Alcaniz EoS parametrization in terms of thermodynamic motivation as well as correspondence with the gravitationally induced adiabatic particle creation scenario. It has been found that the associated -parametrization is well consistent with the GSL (throughout the evolution) as well as TE (during the final phases of evolution). Also, our study is perfectly in line with previous investigations in this direction. From the expression connecing the deceleration parameter and the particle creation rate in an adiabatic particle creation mechanism, we have expressed in terms of the deceleration parameter . We have plotted the variation of for three values of the perfect fluid EoS parameter using Maple and found that it’s behavior is consistent during each phase of evolution of the Universe as speculated from thermodynamic arguments. Our analysis has also predicted a significant increase in the creation of particles during the final phases of evolution of the Universe.

AM wishes to thank DST, Govt. of India for providing a research fellowship. SS was partially supported by SERB, Govt. of India under National Post-doctoral Fellowship Scheme [File No. PDF/2015/000906].


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