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.
