{'fraction_non_alphanumeric': 0.03718155554721795, 'fraction_numerical': 0.019259870683725408, 'mean_word_length': 4.37763131347098, 'pattern_counts': {'":': 0, '<': 34, '<?xml version=': 0, '>': 18, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Accelerating Expansion of the Universe

This thesis concentrates on the accelerated expansion of the Universe recently explored by measurements of redshift and luminosity-distance relations of type Ia Supernovae. We have considered a model of the universe filled with modified Chaplygin gas and barotropic fluid. The role of dynamical cosmological constant has been explored with Modified Chaplygin Gas as the background fluid. Various phenomenological models for \Lambda have been studied in presence of the gravitational constant G to be constant or time dependent. A new form of the well known Chaplygin gas model has been presented by introducing inhomogeneity in the EOS. This model explains w=-1 crossing. An interaction of this model with the scalar field has also been investigated through a phenomenological coupling function. Tachyonic field has been depicted as dark energy model to represent the present acceleration of the Universe. A mixture of the tachyonic fluid has been considered with Generalized Chaplygin Gas to show the role of the later as a dark energy candidate in presence of tachyonic matter. A model of interaction has been studied with scalar field and the inhomogeneous ideal fluid. Two forms of the ideal fluid have been analysed. A power law expansion for the scale factor has been assumed to solve the equations for the energy densities. Brans-Dicke theory has been used to investigate the possibility of obtaining cosmic acceleration. For this purpose a constant and a variable \omega (Brans-Dicke parameter) have been considered. A self-interacting potential has been introduced to show its role in the evolution of the Universe. This model has been studied in presence of barotropic fluid and Generalized Chaplygin Gas.


Standard Cosmology and FRW Model
The standard Cosmology assumes that at the beginning (approximately 13 billion years ago) there must have been an initial singularity from which the space time suddenly started evolving. Since then the Universe has more or less gone through a process of expansion and cooling from an extremely hot and dense state to the present day cool state. In the first few seconds or so there was a very fast expansion, known as Inflation [http://cosmology.berkeley.edu], which is responsible for the present homogeneous and isotropic Universe. Following this inflationary phase, further expansion cooled down the Universe and matter was produced in the process called baryogenesis. Various light elements like deuterium, helium, lithium-7 were created in a process called Big Bang Nucleosynthesis. The universe was still very hot for the nuclei to bind electrons and form atoms, therefore causing the Universe to be opaque to photons and other electromagnetic radiation. Eventually the temperature drops enough for free nuclei and electrons to combine into atoms in a process called recombination. After the formation of atoms photons could travel freely without being scattered. This caused the emission of Cosmic Microwave Background Radiation (CMBR) which gives us the information about the Universe at that time. Galaxies and stars began to form after a few hundred million of years as a result of gravitational collapse.
Modern Cosmology with the help of observational evidences has within its reach billions of galaxies and all the heavenly bodies spread all across the vast distances.
Advanced observational techniques have strengthened the particular branch of science, sometimes supporting the conventional theories and sometimes producing reverse results. As a consequence of these observational advances cosmology has become more or less data driven, so that all the theories need to be fitted with array of observations, although there are still doubts and debates about the reliability and interpretation of such data [Frieman, 1994]. The standard cosmological model which is very successful in describing the evolution of the Universe, is based on homogeneous and isotropic Friedmann-Robertson-Walker (FRW) spacetime. Homogeneity and isotropy that we assume for this model is supported by strong observational data [Smoot etal, 1992;Maddox etal, 1990;Collins etal, 1992], CMBR measurements and galaxy redshift surveys [Fisher etal, 1993;Geller etal, 1989].
This global isotropy and homogeneity which is the foundation of the standard cosmology is called Cosmological Principle. Cosmological Principle leads to Hubble's Law, which says that the recession velocity of galaxy is proportional to the distance from us, i.e., V = HD. The proportionality constant H =˙a a is known as Hubble constant and a(t) is the scale factor.
We now look at Einstein's model of the Universe. In 1932, Einstein and de Sitter presented the Standard Cosmological Model of the Universe, which has been the most favourite among the cosmologists till 1980.
Initially Einstein assumed homogeneity and isotropy in his cosmological problem. He chose a time coordinate t such that the line element of static space-time could be described by [Narlikar, An Introduction to Cosmology], where g µν are functions of space coordinates x µ (µ, ν = 1, 2, 3) only.
Also the equation of motion describing the Universe, known as Einstein field equations in general relativity are where, G ik = Einstein Tensor, R ik = Ricci Tensor, R = Ricci Scalar.
Thus for a static (ȧ = 0), dust filled (p = 0) and closed (k = +1) model of the universe, the field equations yield, (choosing 8πG = c = 1) 3 a 2 = ρ, 1 a 2 = 0 (1. 10) Clearly no feasible solution is possible from these equations, thus suggesting that no static homogeneous isotropic and dense model of the Universe is possible under the regime of Einstein equations stated above.
For this reason Einstein later modified his field equations as Thus introducing the famous Λ-term, known as Cosmological Constant. With this, the picture changes to, 3 a 2 = ρ + Λ, 1 a 2 = Λ, so that a = 1 This Λ is universal constant like G, c, etc. To estimate the value of Λ the value of ρ was used in 1917, which are given as follows, a ∼ 10 29 cm, ρ ∼ 10 −31 gm cm −3 , Λ ∼ 10 −58 cm −2 .
The Λ -term introduces a force of repulsion between two bodies that increases in proportion to the distances between them.
Einstein first proposed the cosmological constant as a mathematical fix to the theorẏ a 2 a 2 + k a 2 = 1 3 ρ (1.12) a a = − 1 6 (ρ + 3p) (1.13) in general relativity. In its simplest form, general relativity predicted that the universe must either expand or contract. Einstein thought the universe was static, so he added this new term to stop the expansion. Friedmann, a Russian mathematician, realized that this was an unstable fix and proposed an expanding universe model called Friedmann model of the Universe.
For expanding Universe,ȧ > 0. But since for normal matter ρ > 0, p ≥ 0, hence the second equation givesä < 0. So that,ȧ is decreasing, i.e., expansion of Universe is decelerated. This model is known as Standard Cosmological Model (SCM).
Teams of prominent American and European Scientists using both Hubble Space Telescope (HST) and Earth based Telescopes had announced in 1998 the results of their many years observation and measurement of the expansion of the Universe. Their collective announcement was that the Universe is not just expanding, it is in fact, expanding with ever increasing speed. This combined discovery has been a total surprise for Cosmology! The SCM states that Universe is decelerating but recent high redshift type Ia Supernovae (explosion) observation suggests the Universe is accelerating [Perlmutter et al, 1998Riess et al, 1998;Garnavich et al, 1998]. So there must be some matter field, either neglected or unknown, which is responsible for accelerating Universe.
For accelerating Universe,ä > 0, i.e., ρ + 3p < 0, i.e., p < − ρ 3 . Hence, the matter has the property −ve pressure. This type of matter is called Quintessence matter (Q-matter) and the problem is called Quintessence problem. The missing energy in Quintessence problem can be associated to a dynamical time dependent and spatially homogeneous / inhomogeneous scalar field evolving slowly down its potential V (φ). These types of cosmological models are known as quintessence models. In this models the scalar field can be seen as a perfect fluid with a negative pressure given by p = γρ, (−1 ≤ γ ≤ 1).
For normal fluid, relation between p and ρ is given by, p = γρ, (0 ≤ γ ≤ 1) which is called equation of state.
For dust, γ = 0; for radiation, γ = 1 3 . Here Λ satisfies an equation of state p = −ρ, so pressure is negative. Therefore for accelerating Universe we need such type of fluid which generates negative pressure. The most puzzling questions that remain to be explained in cosmology are the questions about the nature of these types of matter or the mystery of "missing mass", that is, the "dark energy" and "dark matter" problem.

Dark Energy and Dark Matter
In 1998, two teams studying distant Type Ia supernovae presented independent results of observation that the expansion of the Universe is accelerating [Perlmutter et al, 1998Riess et al, 1998;Garnavich et al, 1998]. For the last few decades cosmologists had been trying to measure the deceleration of the Universe caused by the gravitational attraction of matter, characterized by the deceleration parameter q = − aä a 2 , as suggested by SCM. Therefore, discovery of cosmic acceleration has been proved to be one of the most challenging and important development in Cosmology, which evidently indicates existence of some matter field, either unknown or neglected so far, responsible for this accelerated Universe [Bachall et al, 1999].
Observations suggest that Ω baryon ≃ 0.04 and Ω total = 1.02 ± 0.02. That is ordinary matter or baryons (atoms) described by the standard model of particle Physics is only approximately 4% of the total energy of the Universe, another 23% is some dark matter and 73% is dark energy, yet to be discovered. Thus dark matter and dark energy are considered to be the missing pieces in the cosmic jigsaw puzzle [Sahni, 2004] Ω total − Ω baryons =?
Although the nature of neither dark matter (DM) nor dark energy (DE) is currently known, it is felt that both DM and DE are non-baryonic in nature. Non-baryonic dark matter does not emit, absorb or scatter light and also it has negligible random motion [Springel et al, Bennett, 2006]. Thereafter it is called cold dark matter (CDM). Although, depending on the mass of the particle it is usually differentiated between hot dark matter and cold dark matter. Non-baryonic Hot Dark Matter (HDM) particles are assumed to be relativistic while decoupling from the masses around and so have a very large velocity dispersion (hence called hot), whereas, CDM particles have a very small velocity dispersion and decouple from the masses around when they are non-relativistic. Non-baryonic dark matters do not seem to interact with light or any known baryonic matter, therefore they are called weakly interacting massive particles (WIMP). The standard cold dark matter (SCDM) paradigm assumes Ω m = 1, whereas, LCDM, a modification of SCDM, which consists of CDM and a Lambda-term or cosmological constant Ω m = 0.3 and the hot dark matter scenario is constrained due to structure formation calculations. A variety of dark matter candidates are present in literature, among which a WIMP called nutralino is one of the most favoured one. This is a 100-1000 GeV particle and fermionic partner to the gauge and Higgs bosons [Sahni, 2004]. Another candidate for CDM is an ultra-light pseudo-Goldstone boson called an axion, which arises in the solution of the strong CP problem of particle physics [Masso, 2003]. Other candidates for CDM are string theory motivated modulii fields [Brustein, 1998]; non-thermally produced superheavy particles having mass ∼ 10 14 GeV and dubbed Wimpzillas [Maartens, 2003 ]; axino (with a mass m ∼ 100keV and a reheating temperature of 106K) and gravitino, superpartners of the axion and graviton respectively [Roszkowski, 1999]. Although dark matter is assumed to be comprised of particles which are pressureless cold dark matter with an equation of state ω CDM = 0, nature of dark matter is yet to be discovered.
Existence of dark energy has been driven by the recently discovered accelerated expansion of the Universe. Equation (1.13) shows that acceleration of the Universe is to be expected when pressure is sufficiently negative. Also expansion of the Universe should decelerate if it is dominated by baryonic matter and CDM. Two independent groups observed SNe Ia to measure the expected deceleration of the Universe in 1998. Instead, both the groups surprisingly discovered that the expansion of the Universe is accelerating. This discovery hinted at a new negative pressure contributing to the mass-energy of the Universe in equation (1.13). Now, equation (1.13) that a universe consisting of only a single component will accelerate if ω < −1/3. Fluids satisfying ρ + 3p ≥ 0 or ω ≥ −1/3 are said to satisfy the strong energy condition (SEC). We therefore find that, in order to accelerate, dark energy must violate the SEC. Another condition is the weak energy condition (WEC) ρ + p ≥ 0 or ω ≥ −1. If WEC is not satisfied, Universe can expand faster than the exponential rate causing cosmic Big Rip. Now the greatest challenge for cosmologists was to find an explanation for this accelerated expansion of the Universe. One modification was made by Einstein himself in1917 by introducing the cosmological constant term, to act as a gravitational repulsive term, to achieve a static Universe, as seen in the previous section. Later it was dropped after Hubble's discovery of the expansion of the Universe in 1929. In some models of the dark energy, it is identified with this cosmological constant Λ. However, particle physics looks at cosmological constant as an energy density of the vacuum. Moreover, the energy scale of Λ should be much larger than that of the present Hubble constant H 0 , if it originates from the vacuum energy density. This is the cosmological constant problem [Weinberg, 1989].
Thus cosmological constant with equation of state ω = −1 may play the role to drive the recent cosmic acceleration. Another possibility is that there exists a universal evolving scalar field with equation of state ω < −1/3, called quintessence field. Also a few more models have been proposed recently in support of dark energy analysis. We discuss a few of these models below.

Cosmological Constant
Cosmological constant is the simplest form of dark energy (ω = −1). Also several cosmological observations suggest that cosmological constant should be considered as the most natural candidate for dark energy [Padmanabhan, 2006]. As discussed above Einstein [1917] introduced cosmological constant Λ in order to make the field equations of GR compatible with Mach's Principle and thus producing a static Einstein Universe.
Later in 1922, Friedmann constructed a matter dominated expanding universe without a cosmological constant [Sahni etal, 2000]. Friedmann cosmological model was accepted as a standard cosmological model after Hubble's discovery of expansion of Universe [Weinberg, 1989] and thus Einstein dropped the idea of a static Universe and henceforth positive Λ-term. Later Bondi [1960] and McCrea [1971] realized that if the energy density of the cosmological constant is comparable to the energy density of the present epoch, the cosmological model takes a very reliable form [Padmanabhan, 2003]. But the importance of cosmological constant was first noticed when Zeldovich [1968] showed that zero-point vacuum fluctuations have Lorentz invariant of the form p vac = −ρ vac , which is the equation of state for Λ, therefore T vac µν = Λg µν , which shows that the stress-energy of vacuum is mathematically equivalent to Λ. Now we define [Carroll etal, 1992], and thus the relation between Λ and vacuum energy density is, Thus the deceleration parameter (q = − aä a 2 ) becomes, so that, for the present universe consisting of pressureless dust (dark matter) and Λ, the deceleration parameter takes the form,  [Sahni etal, 2000]. This is evidence enough for a non-zero Λ-term. Now, the vacuum energy density ρ vac , proportional to Λ, obeys [Cohn, 1998], This disparity between the expected and the observed value of Λ poses the famous cosmological constant problem.
One solution to this problem of very large Λ-term (predicted by field theory) and a small one (suggested by observations) is to make Λ time-dependent. An initial large value of Λ would explain inflation and galaxy formation, while subsequent slow decay of Λ(t) would produce a small present value Λ(t 0 ) to be reconciled with observations suggesting Ω Λ ∼ 0.7 [Sahni etal, 2000]. For this purpose a few phenomenological models have been introduced, which Sahni has classified into three categories , viz, This implies that the effective energy-momentum tensor is described by effective pressurep = p − Λ 8πG and effective energy-densityρ = ρ + Λ 8πG . Thus the energy conservation law (1.19) becomes,   Waga (1994, 1997); Torres and Waga (1996) Λ ∝ f (H) Lima and Maia (1994); Lima and Trodden (1996) The Λ-decay laws that have been analyzed in the theory are mostly based on scalar fields or derived from the modified version of the Einstein action principle and they show the decay of the cosmological term is consistent with the cosmological observations. Many of these works do not exhibit the energy transfer between the decaying Λ-term and other forms of matter . Some models overlook this energy-exchange process assuming that equal amounts of matter and antimatter are being produced (if the decay process does not violate the baryon number) [Freese etal, 1987]. These models are constrained by diffuse gamma-ray background observations [Matyjasek, 1995]. Some models refer to this energy exchange process as production of baryons or radiation. These models are constrained by CMB anisotropies Waga, 1994, 1997] and nucleosynthesis arguments [Freese etal, 1987]. Irrespective of the sources, these models (many of which are independently motivated), in general, solve the cosmological problems in a simpler way and satisfy the cosmological observational constraints [Overduin and Cooperstock, 1998].
Some authors have incorporated a variable gravitational constant also while retaining the usual energy conservation law [Arbab, 1997[Arbab, , 2002. A possible time-variable G was suggested by Dirac [1988] on the basis of his large number hypothesis and since then many workers have extended GR with G = G(t) to obtain a satisfactory explanation for the the present day acceleration [Abdel-Rahman, 1990; Abbussattar and Vishwakarma, 1997; Kalligas etal, 1992]. Since G couples geometry to matter, a time-dependent G as well as a time-dependent Λ can explain the evolution of the Universe, as the variation of G cancels the variation of Λ, thus retaining the energy conservation law. There also have been works considering G and Λ to be coupled scalar fields. This approach is similar to that of Brans-Dicke theory [Brans and Dicke, 1961]. This keeps the Einstein field equations formally unchanged [Vishwakarma, 2008].
In FRW model the Einstein field equations with variable G and Λ take the forms [Abdel- Rahman, 1990]: Elimination ofä and k gives, Assuming that the usual conservation law holds we can split this equation as, and equation (1.20).
This also shows that gravitation interaction increases as Λ decreases and thus causing the accelerated expansion of the Universe, i.e., while Λ decreases with time, the gravitational constant is found to increase with time, causing the Universe to have accelerated expansion in order to overcome the increasing gravity [Arbab, 1999] [Zee, 1979;Smolin, 1979;Adler, 1980;Vishwakarma, 2008]. Another theory uses a renormalization group flow near an infrared attractive fixed point [Bonanno and Reuter, 2002;Vishwakarma, 2008]. Mostly the varying G models are consistent with redshift SNe Ia observations. Some of these models solve the horizon and flatness problem also without any unnatural fine tuning of the parameters [Bonanno and Reuter, 2002].

Quintessence Scalar Field
The fine tuning problem facing dark energy models with a constant equation of state can be avoided if the equation of state is assumed to be time dependent. An important class of models having this property is scalar quintessence field proposed by Wetterich [1988] and Ratra and Peebles [1988] which slowly rolls down its potential such that the potential term dominates over the kinetic term and thus generates sufficient negative pressure to drive the acceleration. This Q-field couple minimally to gravity so that the action for this field is given by where, V (φ) is the potential energy and the field φ is assumed to be spatially homogeneous. Thus the energy-momentum tensor is given by, [Copeland etal,2006], For a scalar field φ, with Lagrangian density L = 1 2 ∂ µ φ∂ µ φ − V (φ) in the background of flat FRW Universe, we have the pressure and energy density are respectively, Hence the equation of state (EOS) parameter is, where an overdot means derivative with respect to cosmic time and prime denotes differentiation w.r.t. φ. Thus ifφ 2 << V (φ), that is, Q-field varies very slowly with time, it behaves as a cosmological constant, as Now the equation of motion for the quintessence field is given by the Klein-Gordon From equation (1.40) we see that ω can take any value between −1 (rolling very slowly) and +1 (evolving very rapidly) and varies with time [Frieman etal, 2008]. Also, Various quintessence potentials have been introduced in order to explain recent cosmic acceleration. Inverse Power Law Potential, given by V (φ) = V 0 φ −α is one of the simplest of the lot [Ratra etal, 1988] . Exponential Potential [Wetterich, 1995] given M P ), where M P is the reduced Planck mass, is one of the most favoured one among cosmologists. But in this case ρ φ ρ total < 0.02, suggesting that exponential potential cannot make the transition from subdominant to dominant energy density component of the universe in late times. Sahni and Wang proposed a model in 2000 as V (φ) = V 0 (cosh αφ − 1) p . This model interpolates from V ∝ exp(pαφ) to V ∝ (αφ) 2p , thereby preserving some of the properties of the simpler exponential potential but allowing a different late time behavior. This potential describes quintessence for p ≤ 1/2 and pressureless CDM for p = 1. Thus the cosine hyperbolic potential is able to describe both dark matter and dark energy within a tracker framework [Sahni etal, 2000]. Sahni [2004] has presented a few quintessence potentials proposed in literature in a tabular form as below:  V 0 (expM P /φ − 1) Zlatev, Wang and Steinhardt (1999) Albrecht and Skordis (2000) In order to obtain a feasible dark energy model, the energy density of the scalar field should be subdominant during the radiation and matter dominating eras, emerging only at late times to give rise to the current observed acceleration of the universe [Copeland etal, 2006]. Therefore, we introduce a barotropic fluid in the background, with EOS given by, ω m = pm ρm , where p m and ρ m are the pressure and energy density of the barotropic fluid respectively. A homogeneous and isotropic universe is characterized by the Friedmann-Robertson-Walker (FRW) line element is given by equation (1.7) with c = 1. Thus, the over all stress-energy tensor of the scalar field φ in presence of barotropic fluid the has the form, where ρ = ρ m + ρ φ and p = p m + p φ . Here ρ φ and p φ are the energy density and pressure of the Q-field given by equations (1.39) and (1.38) respectively. If ω m is assumed to be a constant, the fluid energy will be given by ρ m = ρ 0 a −3(1+ωm) and ω φ dynamically changes in general.
The Friedmann equations together with the energy conservation of the normal matter fluid and quintessence (Klein-Gordon equation) are, where H ≡˙a a denotes the Hubble factor. Introducing Ω m ≡ ρm ρc , and Ω = Ω m + Ω ψ with ρ c ≡ 3H 2 as the critical density, Ellis et al [1997] showed that the matter violates the strong energy condition ρ+3p ≤ 0 and so the deceleration parameter q = − aä a 2 < 0. Hence as a consequence the universe accelerates its expansion.
In the investigation of cosmological scenarios we are also interested about those solutions in which the energy density of the scalar field mimics the background fluid energy density, called scaling solutions [Copeland etal, 1998]. Exponential potentials have been proved to give rise to scaling solutions and so can play an important role in quintessence scenarios, allowing the field energy density to mimic the background being sub-dominant during radiation and matter dominating eras. In this case, as long as the scaling solution is the attractor, then for any generic initial conditions, the field would sooner or later enter the scaling regime, thereby opening up a new line of attack on the fine tuning problem of dark energy [Copeland etal, 2006].
Going on with the analysis, very interesting question is whether it is possible to construct a successful common scheme for the two cosmological mechanisms involving rolling scalar fields i.e., quintessence and inflation. This perspective has the appealing feature of providing a unified view of the past and recent history of the universe, but can also remove some weak points of the two mechanisms when considered separately. In general, scalar fields tend to be of heavy (high energy). When renormalized, scalar fields tend to become even heavier. This is acceptable for inflation, because Universe was in a very high energy state at that epoch, but it seems highly unnatural for the recent Universe [Bennet, 2006]. Inflation could provide the initial conditions for quintessence without any need to fix them by hand and quintessence could hope to give some more hints in constraining the inflation potential on observational grounds.

Chaplygin Gas
In recent years a lot of other models, other than cosmological constant and quintessence scalar fields, have also been proved to provide plausible explanation for dark energy.
One of the most popular among these models is Chaplygin gas. Chaplygin Gas unifies dark matter and dark energy under same EOS given by, where A is a positive constant.
Chaplygin [1904] introduced this EOS to calculate the lifting force on a wing of an airplane in aerodynamics. Later Tsien [1939] and Karman [1941] proposed the same model.
Also Stanyukovich [1960] showed that the same EOS can describe certain deformable solids. Jackiw [2000] showed that this is the only gas to admit a supersymmetric generalization. This invokes interest in string theory as well since in a D-brane configuration in the D+2 Nambu-Goto action, the employment of the light-cone parameterization gives rise to the EOS (1.45).
As seen before, the metric of a homogeneous and isotropic universe in FRW model  Now to find a homogeneous scalar field φ(t) and a self-interacting potential V (φ) corresponding to the Chaplygin gas cosmology, we consider the Lagrangian of the scalar field as, The analogous energy density ρ φ and pressure p φ for the scalar field then read, and (1.50) Hence for flat universe (i.e., k = 0) we havė This is obtained from the generalized Born-Infeld Lagrangian density given by, Now we look at the field theoretic approach of this model. Considering the Lagrangian of the scalar field φ with potential V (φ), the corresponding energy density and pressure will be given by, Thus for flat Universe (k = 0), the scalar field and the potential will be given by, and which reduces to that corresponding to pure Chaplygin gas model if α = 1.
Introducing inhomogeneities, Bento etal [2002] have shown that the model evolves consistently with the observations (specially, CMBR peaks, such as Archeops for the location of the first peak and BOOMERANG for the location of the third peak, supernova and high-redshift observations [Bento etal, 2003]) and that the density contrast in this model is less than the CDM model and even gives a better approximation of the ΛCDM With this EOS the energy density takes a much generalized form, where C is an arbitrary integration constant.
Substituting A = 0 and B = A we get back the results of GCG. Debnath etal [2004] have shown that in this model for small value of scale factor Universe will decelerate and for large values of scale factor Universe will accelerate and the transition occurs when a = C(1+A)(1+3A) 2B 1 3(1+α)(1+A) . They have also examined that this model can describe the evolution of the Universe from radiation era (A = 1 3 and ρ is very large) to ΛCDM (ρ is very small constant) model and thus can explain the evolution of the Universe to a larger extent than the pure Chaplygin Gas or GCG models.
Considering a scalar field φ with self-interacting potential V (φ), the corresponding energy density and pressure will be, Hence for flat Universe, the scalar field and the potential will be given by, For small values of the scale factor a(t) and large values of ρ, Debnath etal [2004] obtained two different qualitative nature of the potential for A = 1 and A = 1. For

Tachyonic Field
Recently rolling tachyon condensate has been studied as a source of dark energy. This is a scalar field of non-standard form motivated by string theory as the negative-  . The unstable vacuum corresponds to rolling tachyon. Sen showed that this tachyonic state is associated with the condensation of electric flux tubes of closed strings described by Dirac-Born-Infeld action. For strings attached to Dirichlet D-branes such tachyonic modes reflect D-brane instability [Das etal, 2005]. Tachyonic field has a potential which rolls down from an unstable maximum to a stable minimum with a stable vacuum expectation value. This is known as tachyon condensate [Das etal, 2005]. Sen [2002] has shown that the energy momentum tensor for rolling tachyon solution in D-branes in bosonic string theory is described by a pressureless gas with non-zero energy density, which is stored in open string field, although there are no open string degrees of freedom around the minimum of tachyonic potential.
Thus it represents dust, which can be considered as a candidate for CDM. Also the energy-momentum tensor of tachyon condensate can be split into two parts, one with ω = 0 and the other with ω = −1. This has led a lot of authors to construct cosmological models with tachyonic field as a candidate for dark energy, as dark matter and dark energy thus can be described by a single scalar field. Hence in cosmology rolling of tachyon is analogous to the expansion of the Universe . Also  showed that the supersymmetry breaking by tachyon matter can be adjusted since the total energy of the tachyon matter is adjustable and is determined by the initial position and velocity of tachyon. Now let us move to the dynamics of the tachyon condensate. The Lagrangian density of tachyon condensate is given by the Born-Infeld action where T is the tachyonic field, V (T ) is the tachyon potential having a local maximum at the origin and a global minimum at T = ∞ [Gibbons, 2003] where the potential vanishes.
The tachyon metric is thus given by, which has been proved to be more effective as to explore more physical situations than quintessence [Chimento, 2003;Srivastava, 2005].
Thus the stress tensor of the tachyonic field is given in the form of a perfect fluid by, which for a homogeneous and time dependent tachyonic field reduce to and the EOS parameter reads, As for the strong energy condition ρ As seen from the above equations |Ṫ |< 1.

The equation of motion is
Considering the gravitational field generated by tachyon condensate and assuming that the cosmological constant term vanishes in the tachyon ground state the action becomes [Gibbons, 2003], Using this the Raychaudhuri and Friedmann equations give, The equation of motion reads, Also the conservation equation becomes, These equations show that tachyon field rolls down hill with an accelerated motion and the universe expands . Raychaudhuri equation shows that initially for small T , i. e., when | T |< 2 3 ,ä > 0, i.e., the Universe accelerates and starts decelerating once | T |> 2 3 . For flat space-time, i. e., for k = 0, a(t) approaches a constant value, for k = −1, a → t and for k = 1 re-collapse will take place .
Tachyonic field can be treated as dark energy or dark matter depending on the form of the potential associated with it. Tachyon potential is usually assumed to be exponentially decaying or inversely quadratic. However, Copeland etal [2005] and Calacagni etal [2006] have carried out the analysis for a wide range of potentials, as given below, although more or less all these models face the problem of fine tuning or are constrained by observational data. 1) V = V 0 T −n : For n < 0, the model shows instability; for 0 < n < 2, there is s stable late time attractor solution and V 0 does not need to be fine tuned as super-Planckian problem does not affect this model; for n = 2, one gets a power law solution of the form a = t m and V 0 needs to be fine tuned in order to get present day acceleration, and hence is not a good candidate for dark energy; for n > 2, the model has a dust attractor.
2) V = V 0 e 1/αT , α > 0: This model gives an asymptotic de-Sitter solution with V 0 representing the effective cosmological constant. of non-relativistic matter and radiation, choosing the inverse square potential and the exponential potential for the tachyonic field and showed that for both these models the density parameter for matter and the tachyons are comparable even in the matter dominated phase. For the exponential potential, we get a phase where a ∝ t 2 3 as t → ∞ preceded by an accelerated expansion and thus eliminating the event horizon present in ΛCDM model. They also carried a supernova Ia data analysis and showed that both the potentials present models where the Universe accelerates at low redshifts and are also consistent with requirements of structure formation.
To study the tachyon driven cosmology Sen [2003] has considered the effective action, taking into account the cosmological constant term, to be, Implementing the FRW line element the Einstein's field equations and the equations of motion of T are,ä Sen has used time reversal symmetry and concluded that if Λ is comparatively small rather negligible the universe begins with a big bang and ends in a big crunch. Whereas, in presence of a bulk cosmological constant Universe expands without any singularity for some special range of initial conditions on the tachyon.
where, A, B, α, β are constant and f is a function of its arguments.
They present the likelihood analysis to show that both models fit the data given by

Brans-Dicke Cosmology
Brans-Dicke (BD) theory has been proved to be very effective regarding the recent study of cosmic acceleration. This theory has very effectively solved the problems of inflation and the early and the late time behaviour of the Universe. The starting point of Brans-Dicke Theory [Brans et al, 1961] is Mach's Principle, that the phenomenon of inertia ought to arise from acceleration w.r.t. the general mass distribution of the Universe.
Therefore gravitational acceleration should be used to measure the absolute scale of the elementary particle masses, as they are not constants, rather, represents the particles' interaction with some cosmic field [Weinberg, Gravitation and Cosmology]. Thus the Gravitational Constant G is related to the mass distribution in an expanding Universe by the relation GM Rc 2 ∼ 1, where R is the radius of the Universe and M is the mass of the Universe, or rather, where the sum is over all the matter that contribute to the inertial reaction, since both nearby and distant matter should contribute to the inertial reaction. Now if G is to vary it should be a function of some scalar field variable. Thus if φ represents the scalar field coupled to the mass density of the Universe, G should be related to φ in some manner.
Since a wave equation for φ with a scalar matter density as source gives an equation same as equation (1.86), a suitable relation between G and φ could be given by φ ≃ 1 G . Brans and Dicke proposed a theory in which the correct field equations for gravitation are obtained by replacing G with 1 φ . Thus Brans-Dicke Theory is a generalization of the theory of general relativity. Here gravitation effects are described by a scalar field in Riemannian manifold, thus expressing the gravitational effects as both geometrical and due to scalar interactions. They generalize the usual variational principle of general relativity to obtain equations of motion of matter and non-gravitational fields using Einstein field equations, by where R is the scalar curvature and L is the Lagrangian density of matter including all non-gravitational fields (and not of φ) and ω is a dimensionless constant.
Now the conservation laws give, where T ij is the energy momentum tensor of matter (excluding φ).
The wave equation for φ is given by (varying φ and φ, i in equation (1.87)) where the generally covariant D'Alembertian is defined to be the covariant diver- Varying the components of the metric tensor and the first derivatives in equation (1.87) the field equations for the metric field are obtained as, Combining this equation with equation (1.89), the wave equation for φ is given by, Now for a perfect fluid the energy-momentum tensor given by equation (1.42). Thus we have, where, ρ and p are respectively energy density and pressure of the fluid.
Now to apply the Brans-Dicke theory to cosmology, we consider the Universe to be homogeneous and isotropic.
The Robertson-Walker form of the metric is given by equation (1.7).
We write the gravitational field equations (1.91) as, The time-time component of equation (1.95) gives, Equations ( shown that this can be solved by a varying ω theory where ω is considered to be a function of the scalar field φ [Nordtvedt, 1970;Bergmann, 1968;Wagoner, 1970]. Therefore, the action integral for this general class of scalar-tensor gravitational theory is, Thus the field equations for the tensor and scalar fields become, Thus these equations can be combined to get, This modifies equation (1.100) by, Now the effect of ω ′ = 0 will be in g µν only in the non-linear order in the mass This self-interacting Brans-Dicke Theory is described by the action, where, V (φ) is the self-interacting potential for the Brans-Dicke scalar field φ. Thus the field equations are obtained as, Bertolami and Martins have obtained the solution for accelerated expansion with a quadratic self-coupling potential (V (φ) ∼ φ 2 ) and a negative coupling constant ω, although they have not considered the positive energy conditions for the matter and scalar field. Amendola [1999] has shown that coupled quintessence models are conformally equivalent to Brans-Dicke Lagrangians with power-law potential given by, [2003] extended the gravity theories and obtained a BD theory with potential for BD scalar field, but found that this is not compatible with solar system experiments if the field is very light.

Statefinder Diagnostics
Over the years a lot of models have proved to be viable candidates of Dark Energy, thus leading to the problem of discriminating between these models. For this purpose Sahni etal [2003] proposed a new geometric diagnosis (dimensionless) to characterize the properties of dark energy in a model independent manner. They introduced a pair of parameters called statefinder parameters depending on the scale factor and its derivatives, where q = − aä a 2 is the deceleration parameter. The parameter r forms the next step in the hierarchy of geometrical cosmological parameters after H and q. In fact trajectories in the {s, r} plane corresponding to different cosmological models demonstrate qualitatively different behaviour, for example ΛCDM model correspond to the fixed point For spatially flat space-time (k = 0), considering the Universe to be consisted of non-relativistic matter Ω m , i.e., CDM and baryons, and dark energy Ω x = 1 − Ω m , the statefinder pair {r, s} takes the form, [Sahni etal, 2003] Thus for ΛCDM model with a non-zero Λ (ω = −1), r = 1 and s = 0.
If ω is constant these parameters reduce to Degeneracy occurs when ω = −1/3 or ω = −2/3, as r → 1 at earlier stage and r → 0 at later age, with r ≃ 0.3 for present age (Ω x ≃ 0.7) at the present time.
For a quintessence scalar field these parameters take the forms Gas, we get respectively, [Gorini etal, 2002] For modified Chaplygin gas (p = Aρ − B ρ α with 0 ≤ α ≤ 1), these parameters have rather an implicit form given by, [Debnath etal, 2004] In general, for one fluid model, these {r, s} can be written as  In this chapter, we have generalized the model proposed by Gorini etal [2003]. We have considered a two fluid model consisting of modified Chaplygin gas and barotropic fluid.
We have analysed this model to study the cosmological evolution of the Universe. Also we have carried out the statefinder analysis to describe the different phases of the evolution and the significance of this model in comparison with the one fluid model of MCG.

Field Equations and Solutions
The metric of a homogeneous and isotropic universe in FRW model is given by equation where d is an integration constant.

Field Theoretical Approach
We can describe this two fluid cosmological model from the field theoretical point of view by introducing a scalar field φ and a self-interacting potential V (φ) with the effective Lagrangian is given by (1.48). The analogous energy density ρ φ and pressure p φ corresponding scalar field φ having a self-interacting potential V (φ) are the following: and For flat Universe (k = 0) and by the choice γ = A, we have the expression for φ and and The graphical representation of φ against a and V (φ) against a and φ respectively have been shown in figures 2.1 -2.3 for A = 1/3 and α = 1. From figure 2.1 we have seen that scalar field φ decreases when scale factor a(t) increases for A = 1/3. In figure   2.2, we see that potential function V (φ) sharply decreases from extremely large value

Statefinder Diagnosis
We have already studied the significance of statefinder diagnosis of the models. Let us now analyse our model using statefinder parameters. The statefinder diagnostic pair has the form given by equation (1.113). For one fluid model, these {r, s} can be given by equations (1.122) and (1.123).
For the two component fluids, these equations take the following form: The deceleration parameter q has the form: For modified gas and barotropic equation states, we can set: Thus equations (2.6) and (2.7) can be written as      x + γy 1 + y (2.11) and

Discussion
In this chapter, we have analysed a model consisting of modified Chaplygin gas and barotropic fluid. As We have shown that the mixture of these two fluid models is valid from (i) the radiation era to ΛCDM for −1 ≤ γ ≤ 1 and (ii) the radiation era to   [Wesson, 1978[Wesson, , 1980. Dirac [1979] proposed a theory with variable G motivated by the occurrence of large numbers discovered by Weyl, Eddington and Dirac himself.
It is widely believed that the value of Λ was large during the early stages of evolution and strongly influenced its expansion, whereas its present value is incredibly small [Weinberg, 1989;Carroll etal, 1992]. We have already discussed in the introduction that several authors [Freese etal, 1987;Ozer and Taha, 1987;Gasperini, 1987Gasperini, , 1998Chen and Wu, 1990] have advocated a variable Λ in the framework of Einstein's theory to account for this fact. Λ as a function of time has also been considered in various variable G theories in different contexts [Banerjee etal, 1985;; Abdussattar and Vishwakarma, 1997; Kalligas etal, 1992]. For these variations, the energy-momentum tensor of matter leaves the form of the Einstein's field equations unchanged.
In attempt to modify the General Theory of Relativity, Al-Rawaf and Taha [1996] related the cosmological constant to the Ricci Scalar R. This is written as a built-incosmological constant, i.e., Λ ∝ R. Since the Ricci Scalar contains a term of the form a a , one adopts this variation for Λ. We parameterized this as Λ ∝ä a [Arbab, 2003[Arbab, , 2004. Similarly, we have chosen another two forms for Λ : Λ ∝ ρ and Λ ∝˙a 2 a 2 [Carvalho, 1992]; where ρ is the energy density.
In this chapter we have considered the Universe to be filled with Modified Gas and the Cosmological Constant Λ to be time-dependent with or without the Gravitational Constant G to be time-dependent. We have considered various phenomenological models for Λ , viz., Λ ∝ ρ, Λ ∝˙a 2 a 2 and Λ ∝ä a . Also we have shown the natures of G and Λ over the total age of the Universe and analysed our models in the viewpoint of satefinder diagnostics.

Einstein Field Equations with Dynamic Cosmological Constant
We consider the spherically symmetric FRW metric (1. where ρ and p are the energy density and isotropic pressure respectively. Let us choose MCG with EOS given by equation (1.60). Here, we consider the phenomenological models for Λ(t) of the forms Λ ∝ ρ, Λ ∝˙a 2 a 2 and Λ ∝ä a .
First we will consider G to be constant and try to find out the solutions for density ρ and the scale factor a(t) and hence study the cosmological models in terms of the statefinder parameters r, s. Secondly we will consider G to be variable as well and study the various phases of the Universe represented by the models.

Models keeping G constant and Λ variable
Taking G to be constant and Λ to be time dependent, the energy conservation equation
Again for this model

Model with
where β 2 is a constant and proceeding as above, we obtain the solutions for ρ, a(t), Λ as, Here for β 2 < 3 we can check the consistency of the result by showing p ≃ Aρ at small values of a(t) and p = −ρ for large values of a(t). But if we take β 2 > 3 we get opposite results which contradict our previous notions of the nature of the EOS (1.61). Again for β 2 = 3, we get only ΛCDM point ,i.e., we get a discontinuity. Therefore, we restrict our choice for β 2 in this case to be β 2 < 3.
Computing the state-finder parameters given by equation (1.113), we get the equations for r and s to be,  (where y = p ρ ), which can still be resolved into a single relation and can be plotted in the r, s plane. Here q = 1 2 [(3 − β 2 )y − (β 2 − 1)]. Hence the Universe will accelerate if q < 0 ⇒ y < β 2 −1 3−β 2 . Again for the present Universe q = − 1 2 ⇒ y = β 2 −2 3−β 2 . Assuming the present Universe to be dust dominated, i.e., y = 0 we get the best fit value for β 2 to be 2. Taking A = 1 3 , α = 1, 8πG = 1, β 2 = 2 and y = 0 (i.e., dust dominated present Universe) we get the present value to be r = 1/3, also the same values with β 2 = 1 gives the present values to be y = − 1 2 , r = 5 3 .This is shown in figure 3.3 (A = 1 3 , α = 1, β 2 = 1 and 2, 8πG = 1), which explains the evolution of the Universe from radiation era to ΛCDM model. Again variation of Λ against time is shown in figure 3.4, where we can see that Λ decreases with time for whatever the value of β 2 be.

Model with Λ ∝ä a
Taking Λ = β 3ä a (3.16) (where β 3 is a constant), and proceeding as above we get a relation for ρ as, Unlike the previous two cases here we get a far more restricted solution. Here the only choice of β 3 for which we get the feasible solution satisfying p ≃ Aρ for small values of a(t) and p ≃ −ρ for large values of a(t) is  Using equations (1.113) we get the statefinder parameters as, Here we have taken A = 1 3 , α = 1, 8πG = 1, β 3 = 1 2 , 4 and 1 3 , combining two cases. Taking β 3 = 1 2 , 1 3 we can explain the evolution of the Universe starting from p ρ = − 1 3 to ΛCDM model and β 3 = 4 explains the evolution of the Universe starting from radiation era to y = − 1 3 , as seen from the expression for q. Considering the present epoch to be dust-dominated, the present value of r is given for β 3 = 4 to be − 1 7 . As follows, the former two cases cannot give the present value of r, as y = 0 > − 1 3 for the present epoch. Here we have an infinite discontinuity at p ρ = − 1 3 , i.e., when ρ+3p = 0. Also since we do not get a closed from of ρ here, it is difficult to plot Λ against the scale factor a(t) .

Models with G and Λ both variable
Now we consider G as well as Λ to be variable. With this the conservation law reads, where C is an arbitrary constant. This result is consistent with the results already obtained [Gorini etal, 2003].

Model with Λ ∝ ρ
Here we consider where C 1 is a constant and ρ is given by equation ( where y = p ρ . Equation (3.27) cannot be resolved to get a single relation between r and s, rather we obtain a parametric relation between the same with y = p ρ as the parameter. This can be represented diagrammatically in the r, s plane, which is shown in figure 3.6 taking γ 1 = 1, 3 and 3.5 and A = 1 3 , α = 1, B = 1, C 1 = 1. Now q = 4πG(1+3y)−γ 1 8πG+γ 1 . Taking into account that q = − 1 2 for the present epoch, we get y = γ 1 8πG − 2 3 . Therefore, for the present dust-dominated era y = 0 and γ 1 = 16πG 3 . hence for the This models represents the Universe starting from the radiation era to ΛCDM model. Again figure 3.7 represents the variation of Λ against the scale factor a(t) with γ 1 = 1, 3, 3.5 and figure 3.8 represents the variation G against the scale factor a(t). These figures show that for this particular phenomenological model of Λ, G starting from very low initial value increases largely and becomes constant after a certain period of time, whereas Λ starting from a very large decreases largely to reach a very low value and becomes constant.

Model with Λ ∝ H 2
We consider Λ = γ 2 H 2 (3.28) Proceeding as above we get, where γ 2 is a constant.   ___ ___ _ γ 2 =3.5 Using equations (1.113), (3.1) and (3.23), we find the state-finder parameters as, These equations can further be resolved into a single relation of r and s, which can be plotted diagrammatically in the r, s plane. Here we get a discontinuity at γ 2 = 3. We have plotted these values in the r, s plane taking γ 2 = 1 and 3.5 in figure 3.9 (A = 1 3 , α = 1). This case explains the present acceleration of the Universe, starting from radiation era to ΛCDM model. Also figures 3.10 and 3.11 show respectively the variation of G and Λ against the scale factor for the same values of the constants. Here also like the previous case G starting from a very low initial value increases largely and then continues to be constant near ___ ___ _ γ 2 =3.5  ___ ___ _ γ 2 =3.5 unity. On the other hand Λ starting from a large value decreases largely and continues to be constant after a certain period of time.

Model with Λ ∝ä a
Here we consider where γ 3 is a constant.
Using equation (3.33) in equations (3.1) and (3.2), we get, Also, G can be solved to be, where y = p ρ and C 3 is a constant. Equations (3.36) and (3.37) can further be resolved to get one single relation between r and s and plotted diagrammatically taking γ 3 = 2 and 3.5 (figure 3.12). Since deceleration parameter q = −ä aH 2 is negative in the present epoch, we get 3 − γ 3 < 0, i.e., γ 3 > 3. Also for γ 3 = 3 we  get discontinuity. Both the models represent the phases of the Universe starting from radiation era to ΛCDM model. Again G and Λ can be plotted against a (figures 3.13 and 3.14 respectively) . Unlike the previous cases this model an opposite nature of G and Λ, as G decreases with time and Λ increases with time.
Here also we get cosmic acceleration and the nature of variation G and Λ as well. We The EOS of this model is where C = A 1+ω − 1 with A a constant which can take on both positive and negative values and −l < ω < 0, l being a positive definite constant which can take on values larger than unity.
The EOS reduces to that of current Chaplygin unified models for dark matter and dark energy in the limit ω → 0 and satisfies the conditions: (i) it becomes a de Sitter fluid at late time and when ω = −1, (ii) it reduces to p = wρ in the limit that the Chaplygin parameter A → 0, (iii) it also reduces to the EOS of current Chaplygin unified dark matter models at high energy density and (iv) the evolution of density perturbations derived from the chosen EOS becomes free from the pathological behaviour of the matter power spectrum for physically reasonable values of the involved parameters at late time. This EOS shows dust era in the past and ΛCDM in the future.
In this chapter, we consider the Universe is filled with the mixture of radiation and GCCG. We also perform a statefinder diagnostic to this model without and with interaction in different cases. The energy conservation equation (T ν µ;ν = 0) iṡ

GCCG in presence of radiation
where, ρ tot and p tot are the total energy density and the pressure of the Universe, given by, and p tot = p + p r (4.6) with ρ and p are respectively the energy density and pressure due to the GCCG satisfying the EOS (4.1) and ρ r and p r are the energy density and the pressure corresponding to the radiation fluid with EOS, where γ = 1 3 .
Since GCCG can explain the evolution of the Universe starting from dust era to ΛCDM, considering the mixture of GCCG with radiation would make it possible to explain the evolution of the Universe from radiation to ΛCDM.

Interacting Model
We consider the GCCG interacting with radiation fluid through an energy exchange between them. The equations of motion can be written as,  andρ r + 3ȧ a (ρ r + p r ) = 3Hδ (4.18) where δ is a coupling function.
Let us choose, Now equation (4.17) together with equation (4.1) gives,  Also equations (4.7), (4.18) and (4.20) give and and s = 2(r − 1)(ρ + ρ r ) 9(p + p r ) (4.25) Also the deceleration parameter q has the form: Now substituting u = ρ 1+α , y = ρr ρ , equation (4.12) and (4.13) can be written as, where, Now we find the exact solution for the r, s parameters for the following particular choices of ω: , equation (4.21) can be written as Normalizing the parameters, the corresponding statefinder parameters are given in figure , equation (4.21) can be written as Normalizing the parameters, the corresponding statefinder parameters are given in (iii) If ω = −2, equation (4.21) can be written as Normalizing the parameters, the corresponding statefinder parameters are given in

Discussion
Recently developed Generalized Cosmic Chaplygin gas (GCCG) is studied as an unified model of dark matter and dark energy. In this chapter, we have considered the matter in our Universe as a mixture of the GCCG and radiation as GCCG can explain the evolution  Universe. This kind of interaction term has been studied in ref. [Cai and Wang, 2005].

Field Equations and Solutions
The metric of a spatially flat homogeneous and isotropic universe in FRW model is con- As we can see this is an inhomogeneous EOS [Brevik etal, 2007] where the pressure is where Γ(a, x) is the upper incomplete gamma function and C 0 is an integration constant .

Now, considering the equation of state
for this fluid, we have, For small values of the scale factor a(t), ρ is very large and where A * is a constant,
We have plotted the {r, s} parameters in figure 5.2 normalizing the parameters and varying the scale factor a(t). We can see that the model starts from radiation era. Then

New modified Chaplygin gas and interacting scalar field
Where the total energy density and pressure of the universe are given by, and where, ρ and p are the energy density and pressure of the extended modified Chaplygin gas model given by equations (5.1), (5.2), (5.3), (5.4) and ρ φ and p φ are the energy density and pressure due to the scalar field given by, and where, V (φ) is the relevant potential for the scalar field φ.
Thus from the field equations (4.2) and (4.3) and the conservation equation (4.3), we get the solution for ρ as where C 0 is an integration constant.
Further substitution in the above equations give, To get an explicit form of the energy density and the potential corresponding to the scalar field we consider a power law expansion of the scale factor a(t) as, so that, for β > 1 we get accelerated expansion of the Universe thus satisfying the observational constrains. If β = 1 or β < 1 we get constant and decelerated expansion respectively.
Also the potential takes the form, The graphical representation of V against time is shown in figure 5.3 normalizing the parameters. We see that the potential decays with time.

Discussion
Here show the diagrammatical representation of {r, s} for our model (in Fig. 5.2), varying the scale factor. We see that starting from the radiation era it crosses ω = −1 and extends till phantom era. Also we can see that the deceleration parameter starting from a positive point becomes negative, indicating deceleration initially and acceleration at later times. Again we have considered an interaction of this fluid with that of scalar field by introducing a phenomenological coupling term, so that there is a flow of energy between the field and the fluid which decays with time, as in the initial stage the fluid behaves more like dark matter and the field that of dark energy, whereas in the later stage both explain the dark energy present in the Universe. In Fig. 5.3, we have shown the nature of the potential due to the scalar field considering a power law expansion of the Universe to keep the recent observational support of cosmic acceleration, and we see that the potential decays with time.

Chapter 6
Dynamics of Tachyonic Field in Presence of Perfect Fluid

Prelude
Recently tachyonic field with Lagrangian L = −V (T ) 1 + g µν ∂ µ T ∂ ν T  has gained a lot of importance as dark energy model. The energy-momentum tensor of the tachyonic field can be seen as a combination of two fluids, dust with pressure zero and a cosmological constant with p = −ρ, thus generating enough negative pressure such as to drive acceleration. Also the tachyonic field has a potential which has an unstable maximum at the origin and decays to almost zero as the field goes to infinity. Depending on various forms of this potential following this asymptotic behaviour a lot of works have been carried out on tachyonic dark energy [Bagla etal, 2003;Copeland etal, 2005Copeland etal, , 2006], tachyonic dark matter [Padmanabhan, 2002;Das etal, 2005] and inflationary models [Sami, 2003]. Recently, interacting tachyonic-dark matter model has also been studied [Herrera etal, 2004].
In this chapter, we consider a model which comprises of a two component mixture.
Firstly we consider a mixture of barotropic fluid with tachyonic field without any interaction between them, so that both of them retain their properties separately. Then we consider an energy flow between them by introducing an interaction term which is proportional to the product of the Hubble parameter and the density of the barotropic fluid. We show that the energy flow being considerably high at the beginning falls down noticeably with the evolution of the Universe indicating a more stable situation. Also in both the cases we find the exact solutions for the tachyonic field and the tachyonic potential and show that the tachyonic potential follows the asymptotic behaviour discussed above. Here the tachyonic field behaves as the dark energy component whereas the dust acts as the cold dark matter. Next we consider tachyonic dark matter, the Gen- such that the deceleration parameter reads q = − aä a 2 = −(1 − 1 n ), where a is the scale factor. Hence for n > 1 we always get an accelerated expansion and for n = 1 we get a constant expansion of the Universe. This kind of recipe has been studied in ref. [Padmanabhan, 2002].

Field Equations
The action for the homogeneous tachyon condensate of string theory in a gravitational background is given by, where L is the Lagrangian density given by equation (1.66), where T is the tachyonic field, V (T ) is the tachyonic potential and R is the Ricci Scalar. The energy-momentum tensor for the tachyonic field is, where the velocity u µ is : The energy density ρ T and the pressure p T of the tachyonic field therefore are given by

Tachyonic Dark Energy in presence of Barotropic Fluid
Now we consider a two fluid model consisting of tachyonic field and barotropic fluid.
The EOS of the barotropic fluid is given by, where p b and ρ b are the pressure and energy density of the barotropic fluid. Hence the total energy density and pressure are respectively given by, and p tot = p b + p T (6.7)

Without Interaction
First we consider that the two fluids do not interact with each other so that they are conserved separately. Therefore, the conservation equation (4.4) reduces to, Equation (6.9) together with equation (6.5) gives, Now, we consider a power law expansion of the scale factor a(t) given by equation (6.1).
Using (6.1), equation (6.10) reduces to, Also the energy density corresponding to the tachyonic field becomes, Solving the equations the tachyonic field is obtained as, where, Appell F 1[a, b 1 , b 2 , c, x, y] is the Appell Hypergeometric function of two variables x and y.
Also the potential will be of the form, We can show the graphical representation of the potential against time in figure 6.1.
We can see that V → 0 with time, thus retaining the original property of the tachyon potential.

With Interaction
Now we consider an interaction between the tachyonic field and the barotropic fluid where δ is a coupling constant.
Solving equation (6.16) with the help of equation (6.5), we get, Considering the power law expansion (6.1), we get  Solving the equations the tachyonic field is obtained as, Also the potential will be of the form, In this case also V → 0 with time as shown in the graphical representation of V in figure 6.2.

Tachyonic Dark Matter in presence of GCG
Now we consider a two fluid model consisting of tachyonic field and GCG. The EOS of GCG is given by, where p ch and ρ ch are the pressure and energy density of GCG. Hence the total energy density and pressure are respectively given by, and p tot = p ch + p T (6.24)

Without Interaction
First we consider that the two fluids do not interact with each other so that they are conserved separately. Therefore, the conservation equation (4.4) reduces to, andρ ch + 3ȧ a (ρ ch + p ch ) = 0 (6.26) Equation (6.26) together with equation (6.22) give, Using (6.1), equation (6.27) reduces to Hence the energy density of the tachyonic fluid is, Solving the equations the tachyonic field and the tachyonic potential are obtained as, Also the potential will be of the form, Like the mixture of tachyonic fluid with barotropic fluid in this case also the potential V starting from a low value increases largely and then decreases to 0 with time as shown in figure 6.3.

With Interaction
Now we consider an interaction between the tachyonic fluid and GCG by phenomenologically introducing an interaction term as a product of the Hubble parameter and the energy density of the Chaplygin gas. Thus there is an energy flow between the two fluids.
Solving equation (6.33) with the help of equation (6.22) and (6.1), we get, Also the energy density of the tachyonic field will be read as, Solving the equations the tachyonic field is obtained as, Also the potential will be of the form, In this case the potential starting from a large value tends to 0 (figure 6.4).

Discussion
We have considered the flat FRW Universe driven by a mixture of tachyonic field and a perfect fluid. We have considered barotropic fluid and Chaplygin gas for this purpose.
We have presented accelerating expansion of our Universe due to interaction/without interaction of the mixture of these fluids. We have found the exact solution of the density and potential by considering a power law expansion of the scale factor. We show that these potentials represent the same decaying nature regardless the interaction between the concerned fluids. Since we have considered a power law expansion of the scale factor of the form a = t n , we see that for the present acceleration of the Universe to support the observational data we need n > 1. Now we consider the interaction terms between these fluids. For the mixture of barotropic fluid with tachyonic fluid, we see that the interaction term reduces the potential. Also for the mixture of GCG with tachyonic fluid the interaction parameter ǫ satisfying 0 < ǫ < 1 so that equation (6.34) exists for smaller values of t. In this case also the interaction reduces the potential. Also if we consider only tachyonic fluid with the power law expansion, we see that the potential (which is obtained to be V = 3n 2 t 2 1 − 2 3n ) is greater than that we get in mixtures. Also the potentials differ in the two cases we have considered. For the mixture with GCG the potential decreases faster than that in case of mixture with barotropic fluid.

Chapter 7
Interacting Model of Inhomogeneous EOS and Scalar Field

Prelude
Presently we live in an epoch where the densities of the dark energy and the dark matter are comparable. It becomes difficult to solve this coincidence problem without a suitable interaction. Generally interacting dark energy models are studied to explain the cosmic coincidence problem [Cai and Wang, 2005]. Also the transition from matter domination to dark energy domination can be explained through an appropriate energy exchange rate. Therefore, to obtain a suitable evolution of the Universe an interaction is assumed and the decay rate should be proportional to the present value of the Hubble parameter for good fit to the expansion history of the Universe as determined by the Supernovae and CMB data [Berger etal, 2006]. A variety of interacting dark energy models have been proposed and studied for this purpose [Zimdahl, 2005;Hu and Ling, 2006].
Although a lot of models have been proposed to examine the nature of the dark energy, it is not known what is the fundamental nature of the dark energy. Usually models mentioned above are considered for producing the present day acceleration. Also there is modified gravity theories where the EOS depends on geometry, such as Hubble parameter. It is therefore interesting to investigate models that involve EOS different from the usual ones, and whether these EOS is able to give rise to cosmological models meeting the present day dark energy problem. In this chapter, we consider model of were studied in [Brevik etal, 2004[Brevik etal, , 2007Capozziello, 2006] with a linear inhomogeneous EOS. Here we take the inhomogeneous EOS to be in polynomial form to generalize the case. Also, the ideal fluid present here behaves more like dark matter dominated by the scalar field so that the total energy density and pressure of the Universe decreases with time. Also the potential corresponding to the scalar field shows a decaying nature. Here we have considered a power law expansion of the scale factor, so that we always get a non-decelerated expansion of the Universe for the power being greater than or equal to unity. We have solved the energy densities of both the scalar field and ideal fluid and the potential of the scalar field. Also a decaying nature of the interaction parameter is shown.

Field Equations
The metric of a spatially flat isotropic and homogeneous Universe in FRW model is given and with ρ φ and p φ are respectively the energy density and pressure due to the scalar field given by equations (5.14) and (5.15) respectively. Also, ρ d and p d are the energy density and the pressure corresponding to the ideal fluid with an inhomogeneous EOS, where, ω(t) is a function of t and f (H, t) is a function of H and t (H is the Hubble parameter =˙a a ).
Now we consider the scalar field interacting with the ideal fluid with inhomogeneous EOS through an energy exchange between them. The equations of motion of the scalar field and the ideal fluid can be written as, where δ is coupling constant.

Case I: Model with EOS in power law form
Taking into account the recent cosmological considerations of variations of fundamental constants, one may start from the case that the pressure depends on the time t [Brevik etal, 2004]. Unlike the EOS studied in [Brevik etal, 2007] where the parameters involved in EOS are linear in t, we consider rather a polynomial form. First, we choose the EOS of the ideal fluid to be, where, a 1 , c, α, β are constants.
Here, we see that initially the pressure is very large and as time increases pressure falls down, which is very much compatible with the recent observational data.
We consider a Universe with power law expansion given by equation (6.1), so as to get a non-decelerated expansion for n ≥ 1, as the deceleration parameter reduces to q = − aä a 2 = 1−n n < 0.
Now equation (7.4) together with (7.6) and (6.1) gives the solution for ρ d to be, where, Γ(a, x) is upper incomplete Gamma function.
Further substitution in the above equations give the solution for ρ φ ,φ 2 and V (φ) to be, so that,

Case II: Model with EOS depending on Hubble parameter
Inhomogeneous dark energy EOS coming from geometry, for example, H can yield cosmological models which can avoid shortcomings coming from coincidence problem and a fine-tuned sudden evolution of the Universe from the early phase of deceleration driven by dark matter to the present phase of acceleration driven by dark energy. Furthermore, such models allow to recover also early accelerated regimes with the meaning of inflationary behaviors [Capozziello etal, 2006]. The following model is often referred to as Increased Matter Model where the pressure depends on energy density and H. A detailed discussion of this kind of EOS can be found in ref. [Capozziello etal, 2006]. Now we choose the EOS of the ideal fluid to be, where, A and B are constants.
Considering the power law expansion (6.1) and using (7.4) and (7.12), we get the solution for ρ d to be, Further substitution in the related equations yields the solution for ρ φ , φ, V (φ) to be, where, Equation (7.15) shows that K 1 must be positive and hence K 2 also must be positive for a valid expression. Also equation (7.13) says that C 0 must be positive, otherwise ρ d becomes negative initially. Therefore expression of K 2 says that A must be nega-

Discussion
In this chapter, we study a cosmological model of the Universe in which the scalar field has an interaction with an ideal fluid with inhomogeneous EOS. The interaction is in- 7.4, we see that, the potential function V decreases for both decelerating (n < 1) and accelerating phase (n > 1). Also from the values of density and pressure terms, it can be shown that the individual fluids and their mixtures satisfy strong energy condition for n < 1 and violate for n > 1. A detailed discussion of the potential of a scalar field can be found in ref. [Cardenas etal, 2004]. We see that the coupling parameter shows a decaying nature in both the cases implying strong interaction at the early times and weak interaction later. Thus following the recipe provided in ref. [Padmanabhan, 2002] we can establish a model which can be a suitable alternative to dark energy explaining the decaying energy flow between the scalar field and the fluid and giving rise to a decaying potential. As a scalar field with potential to drive acceleration is a common practice in cosmology [Padmanabhan, 2002], the potential presented here can reproduce enough acceleration together with the ideal fluid, thus explaining the evolution of the Universe. Also we have considered inhomogeneous EOS interacting with the scalar field which can represent an alternative to the usual dark energy model. However, stability analysis and spatial inhomogeneity analysis [Peebles etal, 1988;Rara etal, 1988] are more complicated for our investigation, since we are considering the ideal fluid with two types of equation of states and are analysing whether they can be considered as an alternative to dark energy. Also we have seen that the conservation equation ( Though this is the backward approach, but otherwise if we start from V (φ) i.e., say In this chapter, we investigate the possibilities of obtaining accelerated expansion of the Universe in BD theory where we have considered a self-interacting potential V which is a function of the BD scalar field φ itself and a variable BD parameter which is also a function of φ. We show all the cases of ω = constant, ω = ω(φ), V = 0 and V = V (φ) to consider all the possible solutions. We examine these solutions for both barotropic fluid and the GCG, to get a generalized view of the results in the later case. We analyze the conditions under which we get a negative q (deceleration parameter, − aä a 2 ) in all the models of the Universe. For this purpose we have shown the graphical representations of these scenario for further discussion.

Field Equations
The self-interacting BD theory is described by the action (choosing 8πG 0 = c = 1), given by equation (1.107), where V (φ) is the self-interacting potential for the BD scalar field φ and ω(φ) is modified version of the BD parameter which is a function of φ [Sahoo and Singh, 2003]. The matter content of the Universe is composed of perfect fluid given by equation (1.42). From the action (1.107), we obtain the field equations (1.108), where, and T = T µν g µν .
The line element for Friedman-Robertson-Walker space-time is given by equation (1.7).
The Einstein field equations for the metric (1.7) and the wave equation for the BD scalar field φ are respectively given by equation (1.111), (1.112) and The energy conservation equation is (1.20).
Now we consider two types of fluids, first one being the barotropic perfect fluid and the second one is GCG.

Model with Barotropic Fluid in the Background
Here we consider the Universe to be filled with barotropic fluid with EOS The conservation equation (1.20) yields the solution for ρ as, where ρ 0 (> 0) is an integration constant.
For k = 0 we get from the field equations (1.111) and (1.112), the value of α = 1 and We have seen that γ = − 1 3 and we have Since ω must be negative for − 1 3 < γ < 1, we have seen that for this case the deceleration parameter q = 0, i.e., the universe is in a state of uniform expansion. For k = 0, the field equations yield From equation (8.9) we have two possible solutions for α: For these values of α, we have seen that ω < 0 and the deceleration parameter q < 0.
Thus for k = 0 with the power law form of the scale factor a = a 0 t α it is possible to get the accelerated expansion of the Universe.
Case II: Now we choose ω = ω(φ) to be variable. Here we consider the power law form of φ as  with the power law from of a(t) given by equation (8.5).
Case II: Now we choose ω(φ) to be dependent on φ. Again we consider the power law forms (8.5) and (8.10). Solving the equations in a similar manner, we get and Substituting these values in equation (8.2), we get Therefore for cosmic acceleration q < 0 ⇒ α > 1 and β ≤ −2.
Therefore for the present era, Also for vacuum dominated era,

Model with GCG in the Background
Here we consider the Universe to be filed with Generalized Chaplygin Gas with EOS Here the conservation equation (1.20) yields the solution for ρ as, where C is an integration constant.
We consider the power law form where P = ωα 2 +4ωα−6 ωα−3 and K 0 is an integration constant.
First we consider P > 0. Multiplying both sides of equation (8.24) by a P after squaring it, we get K 0 = 0, therefore giving, a = 6k (ωα−3)P t.
Hence for flat Universe, we get, a = constant.
Also the power law forms considered will be (8.5) and (8.10). Solving the equations we get,
We again consider the power law forms (8.5) and (8.10). We get the solution for V (φ) to be Substituting these values in the other equations we get that n = −1, i.e., the equation of state of Generalized Chaplygin Gas takes the form of that of barotropic fluid. Also we get, α = 1, which implies q = 0, i.e., uniform expansion of the Universe.
Case II: Now we choose ω(φ) to be dependent on φ.
Again we consider the power law forms, (8.5) and (8.10). Solving the equations we get the solutions for Brans-Dicke parameter and self-interacting potential as same as equations (8.25) and (8.26) respectively.
Therefore for the dust dominated era,

Discussion
We are considering Friedman-Robertson-Walker model in Brans-Dicke Theory with and without potential (V ). Also we have considered the Brans-Dicke parameter (ω) to be constant and variable. We take barotropic fluid and Generalized Chaplygin Gas as the concerned fluid.
Using barotropic equation of state, we get, (i) for V = 0 and ω =constant, ω < 0 and q < 0 for some values of α, giving rise to cosmic acceleration, (ii) for V = 0 and ω = ω(φ), we obtain cosmic acceleration depending on some values of α and β. In this case we get acceleration for closed model also at the radiation phase. For β = −2, the variation of ω against φ is same as figure 8.1 and that for closed and  Here we can see that only for the closed model the potential increases positively, in the other two cases the potential becomes negative after a certain point.

Short Discussions and Concluding Remarks
This thesis concentrates on the accelerated expansion of the Universe recently explored by measurements of redshift and luminosity-distance relations of type Ia Supernovae.
This work also deals with the dark energy problem, which is recently one of the most widely investigated problems in Cosmology. A few dark energy models have been considered for this purpose. These models have been discussed, the equations have been solved for exact solutions to show the significance of these models to solve the dark energy problem. Statefinder diagnostics play very important role here, as the statefinder parameters have been solved and plotted to show the evolution of the Universe.
In chapter 1, standard cosmology and the FRW model of the Universe have been described. It also discusses dark matter and dark energy and various candidates of dark energy. A brief introduction to Brans-Dicke cosmology has also been given. This chapter ends with a short note on the statefinder parameters. In chapter 6 tachyonic field has been depicted as dark energy model to represent the present acceleration of the Universe. For this purpose a mixture of tachyonic fluid with barotropic fluid has been assumed. Also a mixture of the tachyonic fluid has been considered with Generalized Chaplygin Gas to show the role of the later as a dark energy candidate in presence of tachyonic matter. A particular form of the scale factor has been assumed to solve the equations of motion and get the exact solutions of the density, tachyonic potential and the tachyonic field. A coupling term has also been introduced in both the models to represent the energy transfer between the two fluids. The interaction term and the tachyonic potential has been analysed and plotted to show their nature in the evolution of the Universe.
Chapter 7 deals with inhomogeneous EOS. A model of interaction has been studied with scalar field and the inhomogeneous ideal fluid. Two forms of the ideal fluid have been analysed. A power law expansion for the scale factor has been assumed to solve the equations for the energy densities. This model shows a decaying nature of the scalar field potential and the interaction parameter.
In chapter 8, Brans-Dicke theory has been used to investigate the possibility of obtaining cosmic acceleration. For this purpose a constant and a variable ω (Brans-Dicke parameter) have been considered. A self-interacting potential has been introduced to show its role in the evolution of the Universe. This model has been studied in presence of barotropic fluid and Generalized Chaplygin Gas. Power law forms of the scale factor and the scalar field have been assumed to solve the field equations. It has been shown that accelerated expansion can also be achieved for high values of ω for closed Universe.