Estimating Temperature Fluctuations in the Early Universe

A lagrangian for the $k-$ essence field is constructed for a constant scalar potential and its form determined when the scale factor was very small compared to the present epoch but very large 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 fluctuations of the temperature of the background radiation in these early stages (compared to the present epoch) of the universe. If the temperature at time $t_{a}$ is $T_{a}$ and at time $t_{b}$ the temperature is $T_{b}$ ($t_{b}>t_{a}$), then for small times, the probability for the logarithm of inverse temperature evolution can be estimated to be given by $$P(b,a)= |\langle ln~({1\over T_{b}}),t_{b}| ln~({1\over T_{a}}),t_{a}\rangle|^{2}$$ $$\approx\biggl({3m_{\mathrm Pl}^{2}\over \pi^{2} (t_{b}-t_{a})^{3}}\biggr) (ln~ T_{a})^{2}(ln~T_{b})^{2}\biggl(1 - 3\gamma (t_{a} + t_{b})\biggr)$$ where $0<\gamma<1$, $m_{\mathrm Pl}$ is the Planck mass and Planck's constant and the speed of light has been put equal to unity. There is the further possibility that a single scalar field may suffice for an inflationary scenario as well as the dark matter and dark energy realms.