Modified Chaplygin Gas Cosmology

Abstract

Modified Chaplygin gas as an exotic fluid has been introduced by H. B. Benaoum (2002). Essential features of the modified Chaplygin gas as a cosmological model are discussed. Observational constraints on the parameters of the model have been included. The relationship between the modified Chaplygin gas and a homogeneous minimally coupled scalar field is reevaluated by constructing its self-interacting potential. In addition, we study the role of the tachyonic field in the modified Chaplygin gas cosmological model and the mapping between scalar field and tachyonic field is also considered.

1. Introduction

Astronomical and cosmological observations, such as type Ia supernovae (SNe Ia) [1–4], large-scale redshift surveys structure (LSS) [5, 6], the cosmic microwave background (CMB) [7, 8], and Wilkinson Microwave Anisotropy Probe (WMAP) [9, 10], indicate that the observable universe experiences an accelerated expansion. These observations suggest also that the universe is nearly flat and dominated by a nonbaryonic substratum. The source of this acceleration is usually attributed to an exotic type of fluid with negative pressure called commonly dark energy.

Various kinds of dark energy models have been proposed such as cosmological constant [11], quintessence [12–14], 𝑘-essence [15–17], tachyon [18–20], phantom [21–23], Chaplygin gas [24], quintom [25], holographic dark energy [26], and extra dimensions [27]. The nature of the dark sector of the universe (i.e., dark energy and dark matter) remains a mystery. An economical and attractive idea to unify the dark sector of the universe is to consider it as a single component that acts as both dark energy and dark matter. One way to achieve the unification of dark energy and dark matter is by using the so-called Chaplygin gas. The pure Chaplygin gas or generalized Chaplygin gas is a perfect fluid which behaves like a pressureless fluid at an early stage and a cosmological constant at a later stage.

Pure Chaplygin gas with an exotic equation of state is characterized by a negative pressure [24], 𝐵𝑝=−𝜌, (1.1) where 𝑝 is the pressure, 𝜌 is the energy density, and 𝐵 is a positive parameter.

The pure Chaplygin gas has been extended to the so-called generalized Chaplygin gas with the following equation of state [28]: 𝐵𝑝=−𝜌^𝛼,where0≤𝛼≤1. (1.2) It is clear that the pure Chaplygin gas is recovered for the case 𝛼=1.

The interesting feature of the Chaplygin gas is its connection to the string theory. It can be obtained from the Nambu-Goto action for a 𝐷-brane moving in a (𝐷+2)-dimensional spacetime in the light cone parametrization [29–33].

The outline of this paper is as follows. In the next section, we study the cosmological model of the modified Chaplygin gas which was introduced by the author in [34]. We show that the modified Chaplygin gas model interpolates between an epoch with a soft equation of state and a de Sitter phase. In Section 3, the relationships between the modified Chaplygin gas and a cosmological scalar field are reevaluated. In Section 4, a tachyonic field is considered as a candidate for the modified Chaplygin gas model. The correspondence between minimally coupled scalar field and the tachyon field is also investigated.

2. FRW Cosmology from the Modified Chaplygin Gas Model

Within the framework of Friedmann-Robertson-Walker (FRW) cosmology, a model called modified Chaplygin gas has been proposed by the author [34]. This model includes an initial phase of radiation and it is based on the following equation of state: 𝐵𝑝=𝐴𝜌−𝜌^𝛼, (2.1) where 𝐴,𝐵, and 𝛼 are constant parameters.

When 𝐵=0, we recover the equation of state of perfect fluid, that is, 𝑝=𝐴𝜌. For 𝐴=0, it reduces to the generalized Chaplygin gas.

In (2.1), the two terms start to be of the same order when the pressure vanishes (i.e., 𝑝=0). In this case, the fluid has pressureless density 𝜌₀, corresponding to some scale factor 𝑎₀, 𝜌₀=𝜌^𝛼+1𝑎₀=𝐵𝐴. (2.2) The metric of 𝐷-dimensional FRW spacetime is 𝑑𝑠²=−𝑑𝑡²+𝑎²(𝑡)𝑑Ω²_𝑘, (2.3) where 𝑎(𝑡) is the scale factor and 𝑑Ω²_𝑘 is the metric of the maximally symmetric (𝐷−1)-space with curvature 𝑘=0,±1.

The Friedmann equation which governs the evolution of the scale factor is given by 𝐻²=𝑎̇𝑎²=2𝜌()−𝑘𝐷−1)(𝐷−2𝑎², (2.4) where 𝐻 is the Hubble parameter.

In the FRW framework, a fluid with an energy density 𝜌 and a pressure 𝑝 must satisfy the conservation law: 𝜌̇+(𝐷−1)𝐻𝜌+𝑝=0. (2.5)

The latter two equations imply that 𝑎̈𝑎=𝐻̇+𝐻²()()𝜌=−𝐷−1𝑝+𝐷−3().𝐷−1)(𝐷−2 (2.6)

By defining 𝑊=𝑎^{(𝐷−1)(𝐴+1)} and a rescaled density 𝜌=𝜌𝑊, (2.5) becomes 𝜌̇‌−𝐵𝑊𝐴+1^𝛼𝑊̇𝜌^𝛼=0. (2.7) This equation can be easily integrated leading to 𝜌^𝛼+1=𝐵𝛼+1𝑊𝐴+1^𝛼+1+𝐶𝛼+1,𝛼+1 (2.8) where 𝐶 is an integration constant.

The density will be 𝐵𝜌=+𝐶𝐴+1𝑊^{𝛼+11/(𝛼+1)}. (2.9) The constant of integration 𝐶 can be expressed in terms of the cosmological scale 𝑎₀, (i.e.,𝑊₀=𝑎₀^{(𝐷−1)(𝐴+1)}) where the fluid has a vanishing pressure: 𝐵𝐶=𝑊𝐴+1𝐴^0𝛼+1. (2.10) The energy density 𝜌 will be 𝐵𝜌=𝐴+1^1/(𝛼+1)11+𝐴𝑊_𝑟^{𝛼+11/(𝛼+1)}, (2.11) where 𝑊_𝑟=𝑊/𝑊₀.

For large-scale factor 𝑎, that is, 𝑊_𝑟≫1, we have 𝐵𝜌≃𝐴+1^1/(𝛼+1),𝐵𝑝≃−𝐴+1^1/(𝛼+1)=−𝜌, (2.12) which correspond to an empty universe with a cosmological constant (𝐵/(𝐴+1))^1/(𝛼+1) (i.e., a de Sitter space).

Also for small-scale factor 𝑎, that is, 𝑊_𝑟≪1, we have 𝐵𝜌≃𝐴(𝐴+1)^1/(𝛼+1)1𝑊_𝑟,𝐴𝑝≃𝐴𝐵(𝐵+1)^1/(𝛼+1)1𝑊_𝑟=𝐴𝜌, (2.13) which correspond to universe dominated by an equation of state 𝑝=𝐴𝜌. This shows that this model interpolates between a universe dominated by matter phase with equation of state 𝑝=𝐴𝜌 and a de Sitter phase 𝑝≃−𝜌.

Moreover, expanding (2.11) and (2.1) to the subleading terms at large cosmological constant, we obtain the following expressions for the energy and the pressure: 𝐵𝜌≃𝐴+1^1/(𝛼+1)11+(𝛼+1)𝐴𝑊_𝑟^𝛼+1,𝐵𝑝≃𝐴+1^1/(𝛼+1)()−1+𝐴+𝛼𝐴+1(𝛼+1)𝐴𝑊_𝑟^𝛼+1. (2.14) These correspond to the mixture of a cosmological constant (𝐵/(𝐴+1))^1/(𝛼+1) and a type of matter described by an equation of state: 𝑝=(𝛼+𝐴(𝛼+1))𝜌. (2.15) The equation of state parameter takes the form: 𝑝𝜔=𝜌=−1−1/𝑊_𝑟^𝛼+11+1/𝐴𝑊_𝑟^𝛼+1, (2.16) which ranges over −1<𝜔<𝐴, depending on the cosmological scale 𝑎, 𝜔=−1for𝑊_𝑟≫1,0for𝑊_𝑟=1,𝐴for𝑊_𝑟≪1. (2.17) The speed of sound 𝑐_𝑠 is defined as 𝑐²_𝑠=𝜕𝑝=𝑝̇𝜕𝜌𝜌̇. (2.18) Now by computing 𝜔̇, we obtain 𝜔̇=𝑐²_𝑠𝜌̇−𝜔𝜌, (2.19) which for the modified Chaplygin gas gives the following expression for the speed of sound: 𝑐²_𝑠=𝜔+𝜌𝑑𝜔=()𝑑𝜌𝛼+𝛼+1𝐴+1/𝑊_𝑟^𝛼+11+1/𝐴𝑊_𝑟^𝛼+1, (2.20) implying that 𝑐²_𝑠 is always positive and hence there is no concern about imaginary speed of sound.

Moreover, it has the following asymptotic limit, 𝑐²_𝑠=𝛼+(𝛼+1)𝐴for𝑊_𝑟()≫1,𝛼+1𝐴for𝑊_𝑟=1,𝐴for𝑊_𝑟≪1. (2.21) The speed of sound never exceeds that of light for smaller-scale 𝑎 or scale of the order of 𝑎₀ where the pressure vanishes, provided that 𝐴<1/(𝛼+1) and will exceed it for large scale compared to 𝑎₀.

The constraints from the astrophysical and cosmological observables on the modified Chaplygin gas have been studied by many authors [35–40]. The permissible values of the parameters 𝐴,𝐵, and 𝛼 have been explored from the observed data.

Using data from different observations, namely, observational Hubble Data (OHD), baryon acoustic oscillation (BAO), and CMB shift parameters data, the allowed values for some of the parameters of the modified Chaplygin gas have been extracted. In the light of the 182 Gold SNe Ia, the 3-year WMAP and the SDSS data, the best fits correspond to 𝐵=−0.085 and 𝛼=1.724 [35]. This result was obtained by decomposing the modified Chaplygin gas into two components, that is, dark matter and dark energy component.

However, by using Markov chain Monte Carlo method with the observational data from the SN Ia Union 2, OHD, cluster X-ray gas mass fraction (CBF), BAO, and CMB data, the best fits of the modified Chaplygin gas parameters give 𝐵=0.00189 and 𝛼=0.1079 [36].

Moreover, a perturbative analysis of the modified Chaplygin gas shows that the power spectrum observational data restricts the value of |𝐵|<10⁻⁶ [40] such that the modified Chaplygin gas is disfavored.

3. Modified Chaplygin Gas as a Scalar Field

Following [41–45], we describe the modified Chaplygin gas cosmological model by introducing a scalar fields having a self-interacting potential 𝑈(𝜑) with the Lagrangian: ℒ_𝜑=𝜑̇²2−𝑈𝜑. (3.1) Both the energy density and the pressure of the modified Chaplygin gas can be related to the scalar 𝜑 through the following transformation equations: 𝜌_𝜑=𝜑̇²2𝜑𝑝+𝑈=𝜌,_𝜑=𝜑̇²2𝜑𝐵−𝑈=𝐴𝜌−𝜌^𝛼. (3.2) The kinetic energy of the scalar field 𝜑 and its corresponding potential 𝑈(𝜑) are 𝜑̇²=1+𝜔_𝜑𝜌_𝜑,𝑈𝜑=121−𝜔_𝜑𝜌_𝜑, (3.3) where 𝜔_𝜑=𝑝_𝜑/𝜌_𝜑.

Now since 𝜑̇=𝜑^′𝑊̇_𝑟 where the prime denotes derivation with respect to 𝑊_𝑟 and 𝑊̇_𝑟=(𝐷−1)(𝐴+1)𝐻𝑊_𝑟, we get 𝜑^′2=𝐷−22(𝐷−1)(𝐴+1)²1+𝜔_𝜑𝑊²_𝑟. (3.4) Here, we have used (2.4) for the Hubble constant and guided by the cosmic microwave background CMB data which is strongly consistent with a flat universe and restricted ourselves to the flat case 𝑘=0.

By using (2.16), we get 𝜑^′=𝐷−22()𝐴()1𝐷−1𝐴+1𝑊_𝑟^(𝛼+3)/21+1/𝐴𝑊_𝑟^𝛼+1,𝑈𝜑=12𝐵𝐴+1^1/22+(1−𝐴)/𝐴𝑊_𝑟^𝛼+11+1/𝐴𝑊_𝑟^𝛼+1. (3.5) The first equation can be integrated easily which gives 𝐴𝑊_𝑟^𝛼+1=1sinh²𝜛(),𝛼+1Δ𝜑 (3.6) where 𝜛=(𝐷−1)(𝐴+1)/2(𝐷−2) and Δ𝜑=𝜑−𝜑₀.

We note that for larger scales, the scalar field asymptotically approaches to the constant field 𝜑₀ and becomes infinite (i.e., 𝜙→+∞) for small scales.

Next, by substituting the latter expression in (3.2), we can write all our physical quantities 𝜌_𝜑,𝑝_𝜑, and 𝜔_𝜑 in terms of the scalar field 𝜑 as 𝜌_𝜑=𝐵𝐴+1^1/(𝛼+1)cosh^2/(𝛼+1)𝜛(),𝑝𝛼+1Δ𝜑_𝜑=𝐵𝐴+1^1/(𝛼+1)𝐴cosh^2/(𝛼+1)𝜛(𝛼+1)Δ𝜑−𝐴+1cosh^{2𝛼/(𝛼+1)}𝜛(),𝜔𝛼+1Δ𝜑_𝜑=−1−𝐴sinh²𝜛(𝛼+1)Δ𝜑cosh²𝜛().𝛼+1Δ𝜑 (3.7) Notice that these physical quantities do not depend on the intermediate constant 𝑊₀ (i.e., constant of integration 𝐶).

Finally, we get the following potential which has a simple form: 𝑈𝜑=12𝐵𝐴+1^1/(𝛼+1)1+𝐴cosh^{2𝛼/(𝛼+1)}𝜛()+()𝛼+1Δ𝜑1−𝐴cosh^2/(𝛼+1)𝜛(𝛼+1)Δ𝜑. (3.8)

4. Modified Chaplygin Gas as Tachyonic Field

The importance of tachyon in cosmology is inspired by string theory [18, 19]. The action of the homogeneous tachyon condensate of string theory in a gravitational background is given by 𝑆=𝑑^𝐷𝑥√−𝑔𝑅+ℒ_tachyon, (4.1) where 𝑅 is the scalar curvature. For a tachyonic field 𝑇 with tachyonic potential 𝑉(𝑇), the relativistic Lagrangian can be expressed as ℒ_tachyon=−𝑉(𝑇)1+𝑔^𝜇𝜈⋅∇_𝜇𝑇∇_𝜈𝑇. (4.2) The corresponding energy-momentum tensor for the tachyonic field is 𝑇_𝜇𝜈=−2𝛿𝑆√−𝑔𝛿𝑔^𝜇𝜈=𝑝_𝑇𝑔_𝜇𝜈+𝑝_𝑇+𝜌_𝑇𝑢_𝜇𝑢_𝜈, (4.3) where the velocity 𝑢_𝜇 is 𝑢_𝜇=−∇_𝜇𝑇−𝑔^𝜇𝜈∇_𝜇𝑇∇_𝜈𝑇 (4.4) with 𝑢^𝜇𝑢_𝜇=−1.

It follows that the energy density 𝜌_𝑇 and the pressure 𝑝_𝑇 of the tachyonic field are given by 𝜌_𝑇=𝑉(𝑇)1−𝑇̇²,𝑝_𝑇=−𝑉(𝑇)1−𝑇̇². (4.5) The equation of state parameter is 𝜔_𝑇=𝑝_𝑇𝜌_𝑇=−1−𝑇̇². (4.6) The condition of accelerating universe (i.e.,𝑎/̈𝑎>0) requires that 𝑇̇²<2,𝐷−1 (4.7) and the rolling tachyon has an interesting equation of state whose parameter 𝜔_𝑇 interpolates between −1 and (−𝐷+3)/(𝐷−1).

The evolution of the tachyonic field is driven by 𝑇̈+1−𝑇̇²(𝐷−1)𝐻𝑇̇+1𝑉(𝑇)𝑑𝑉(𝑇)𝑑𝑇=0 (4.8) with the constraint equation for the Hubble parameter given by 𝐻²=2()𝑉(𝑇)𝐷−1)(𝐷−21−𝑇̇²,𝐻̇=−1𝑇̇𝐷−1²𝑉(𝑇)1−𝑇̇². (4.9) By combining the last two equations, the tachyonic field 𝑇 and the potential 𝑉(𝑇) can be expressed as 𝑇(𝑡)=−2𝑑𝑡𝐻̇𝐷−1𝐻^21/2,𝑉(𝑇)=(𝐷−1)(𝐷−2)2𝐻²21+𝐻̇𝐷−1𝐻^21/2. (4.10) Note that the knowledge of 𝐻 and 𝐻̇ (i.e., cosmological scale factor 𝑎(𝑡)) completely determines the tachyonic field 𝑇 and its corresponding potential 𝑉(𝑇).

By mapping the pressure 𝑝_𝜑 and the energy density 𝜌_𝜑 for the scalar field with the corresponding tachyon field 𝑝_𝑇 and 𝜌_𝑇, we obtain that 𝑑𝑇=𝑑𝜑𝜌_𝜑,𝑉(𝑇)=−𝜔_𝜑𝜌_𝜑. (4.11) Cosmological correspondence between the tachyonic field and a minimally coupled scalar field has been noted by [34, 46]. Such correspondence was also investigated by [47] where it was demonstrated explicitly that distinct scalar field and tachyonic models may give rise to the same cosmological evolution for a particular choice of initial conditions.

Now by using (3.7), an exact integration to the first equation in (4.11) can be performed without any approximation as follows: 𝑇−𝑇₀=cosh^{𝛼/(𝛼+1)}𝜛(𝛼+1)Δ𝜑𝛼𝜛(𝐵/(𝐴+1))^21/2(𝛼+1)𝐹₁12,12−12()3𝛼+12−12(𝛼+1);−cosh²𝜛(𝛼+1)Δ𝜑, (4.12) where₂𝐹₁ is the hypergeometric function given by ₂𝐹₁𝑐=𝑎,𝑏;𝑥^[∞_𝑘=0(𝑎)_𝑘(𝑏)_𝑘(𝑐)_𝑘𝑥^𝑘𝑘! (4.13) with (𝑎)_𝑘=𝑎(𝑎+𝑠)…(𝑎+(𝑛−1)𝑠) as the Pochhammer symbol.

Furthermore, by replacing the first equation in (3.7), (4.12) for tachyonic field 𝑇 can be written in terms of the energy density 𝜌_𝜑 as 𝑇−𝑇₀=𝜌_𝜑^𝛼/2𝛼𝜛(𝐵/(𝐴+1))^21/2𝐹₁12,12−12()3𝛼+12−12(𝛼+1);−𝐴+1𝐵𝜌_𝜑^𝛼+1. (4.14)

To find the scalar field 𝜑 and its potential 𝑈(𝜑) in terms of the tachyon field and its potential 𝑉(𝑇), one has to perform the inverse transformation: 𝑑𝜑=√𝜌_𝑇𝑈𝜑=1𝑑𝑇,21−𝜔_𝑇𝜌_𝑇. (4.15)

In the slow-rolling approximation, the tachyon field potential 𝑉(𝑇) and the scalar field potential 𝑈(𝜑) are approximately the same. To see this, we expand (4.10) up to the first order in 𝐻̇/𝐻² and use (2.4) and (2.6) to get: 𝑉(𝑇)≃(𝐷−1)(𝐷−2)2𝐻²+1̇𝐻=1𝐷−121−𝜔_𝜑𝜌_𝜑=𝑈𝜑. (4.16) Now to rewrite the potential 𝑉(𝑇) in terms of 𝑇, we first expand (4.14) to the first order in 𝜌_𝜑: 𝑇−𝑇₀≃𝜌_𝜑^𝛼/2𝛼𝜛(𝐵/(𝐴+1))^1/2 (4.17) and then substitute it in (4.16) to finally get: 𝑉(𝑇)≃𝑐₁𝑇−𝑇₀²+𝑐₂𝑇−𝑇₀², (4.18) where 𝑐₁ and 𝑐₂ are constant parameters depending on 𝐷,𝛼,𝐴, and 𝐵.

5. Conclusions

One approach in modern cosmology consists in supposing that dark energy and dark matter are different manifestations of a single entity. Following such an idea, this work (see also [34]) presents a cosmological model based on the modified Chaplygin gas that acts as a single component. It is shown that the equation of state of the modified Chaplygin gas interpolates from matter-dominated era to a cosmological constant-dominated era. Astronomical and cosmological constraints on the parameters 𝐴,𝐵, and 𝛼 of the modified Chaplygin have been investigated and still no basic agreement is reached on their values.

In addition, from the theoretical point of view, the modified Chaplygin gas model is equivalent to that of a scalar field 𝜑 having self-interacting potential 𝑈(𝜑). Such a description has been explored and its self-interacting potential has been determined.

Furthermore, it has been shown that the modified Chaplygin gas can also be described by a tachyonic field 𝑇 having a potential 𝑉(𝑇). Correspondence between scalar field and tachyonic field descriptions has been investigated. The transformations between the scalar field and tachyonic field and between their corresponding potentials have been determined. Such a correspondence has been applied to derive the exact expression of the tachyonic field 𝑇 in terms of the scalar field 𝜑 for the modified Chaplygin gas. Finally, in the slow-rolling approximation, we have expressed the the potential 𝑉(𝑇) in terms of the tachyonic field 𝑇.

Acknowledgment

The author would like to express his gratitude to the anonymous referees for very helpful and detailed comments.