A revised model of the inflationary universe under the framework of Brans-Dicke theory is proposed. It is shown that the scale factor of the universe during the inflation varies as R approximately tw+1/2 and not exponentially, but still it gives rise to enough inflation to solve the cosmological puzzles. It is found that the quantum gravity effects come into play in the same scale as that of grand unified theories (i.e. at T approximately 1014 GeV) rather than at T approximately 1019 GeV, hence providing a natural solution to the 'milder version' of the flatness problem inherent in the conventional inflationary scenarios. This scenario also provides a simple way to estimate theoretically the present value of the gravitational constant. The estimated value agrees well with the actual value despite many approximations and the ignorance of quantum gravity; the difference is only by a factor of the order of 10-2.

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The dynamics of a minimally coupled scalar field in the expanding universe is discussed with special reference to phantom cosmology. The evolution of the universe with a phantom field vis-a-vis a quintessence field is compared. Phantom cosmologies are found to have two special features i) occurrence of a singularity where the scale factor, the energy density and Ricci curvature scalar diverge to infinity. This singularity occurs at a finite timexs, depending on the value of w during cosmic evolution, ii) degeneracy in the determination of w(z m ) for a given transition redshift z m which seems to impart similar observational properties to corresponding phantom and quintessence models and makes both of them compatible with the cosmological observations. Although due to the uncertainties in the measurement of the Hubble constant H 0 , the Hubble dependent observational parameters yield only loose constraints over the range of w, the duality in the determination of w with respect to transition redshift may be used to constrain w. An observational test, based upon the observations of low redshift galactic clusters, is suggested to discriminate between the quintessence and phantom dark energy. PACS numbers: 98.80.Cq, 98.65. Dx,98.70.Vc

The role of the quintessence field as a probable candidate for the repulsive dark energy, the conditions for tracking and the requisites for tracker fields are examined. The concept of 'integrated tracking' is introduced and a new criterion for the existence of tracker potentials is derived assuming monotonic increase in the scalar energy density parameter Ω φ with the evolution of the universe as suggested by the astrophysical constraints. It provides a technique to investigate generic potentials of the tracker fields. The general properties of the tracker fields are discussed and their behaviour with respect to tracking parameter ǫ is analyzed. It is shown that the tracker fields around the limiting value ǫ ≃ 2 3 give the best fit with the observational constraints.