A k-essence scalar field model having (non canonical) Lagrangian of the form L = −V (φ)F (X) where X = 1 2 g µν ∇µφ∇νφ with constant V (φ) is shown to be consistent with luminosity distanceredshift data observed for type Ia Supernova. For constant V (φ), F (X) satisfies a scaling relation which is used to set up a differential equation involving the Hubble parameter H, the scale factor a and the k-essence field φ. H and a are extracted from SNe Ia data and using the differential equation the time dependence of the field φ is found to be: φ(t) ∼ λ0 + λ1t + λ2t 2 . The constants λi have been determined. The time dependence is similar to that of the quintessence scalar field (having canonical kinetic energy) responsible for homogeneous inflation. Furthermore, the scaling relation and the obtained time dependence of the field φ is used to determine the X-dependence of the function F (X).

A phenomenological model is described for Cosmic Microwave Background Radiation (CMBR) evolution with dark energy an essential ingredient in the form of a k−essence scalar field. The following features of this evolution can be successfully obtained from this model: (a) the observed variation of the rate of change of scale factor a(t), i.e.ȧ, with time and (b) the observed value of the epoch when the universe went from a decelerating phase to an accelerated phase. These two features have been matched with graphical transcriptions of SNe Ia data. The model also indicates that the evolution is sensitive to the presence of inhomogeneity and this sensitivity increases as one goes further into the past. Further, the value of the inhomogeneity parameter determines the epoch of switch over to an accelerated phase. A positive value of inhomogeneity parameter leads to switch over at earlier epochs, while a negative value leads to switch over at later epochs. If the value of the inhomogeneity parameter is a bit negative then the crossover point from deceleration to acceleration gives better agreement with the observed value.

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