Estimating temperature fluctuations in the early universe

A lagrangian for a k-essence field is constructed for a constant scalar potential, and its form is determined when the scale factor is very small as compared to the present epoch but very large as compared to the inflationary epoch. This means that one is already in an expanding and flat universe. The form is similar to that of an oscillator with time-dependent frequency. Expansion is naturally built into the theory with the existence of growing classical solutions of the scale factor. The formalism allows one to estimate the temperature fluctuations of the background radiation at these early stages (as compared to the present epoch) of the Universe. If the temperature is T (a) at time t (a) and T (b) at time t (b) (t (b) > t (a) ), then, for small times, the probability evolution for the logarithm of the inverse temperature can be estimated as P(b, a) = vertical bar < ln(1/T-b), t(b)vertical bar ln (1/T-a), t(a)>vertical bar(2) approximate to (3m(Pl)(2)/pi(2)(t(b) - t(a))(3)) (ln T-a)(2)(lnT(b))(2)(1 - 3 gamma(t(a) + t(b))), where 0 < gamma < 1, m(Pl) is the Planck mass, and Planck's constant and the speed of light have been put equal to unity. There is a further possibility that a single scalar field may suffice for an inflationary scenario as well as the dark matter and dark energy realms.