Initial Singularity, Lambda-Problem and Crossing the Phantom Divide in Scale Invariant TMT Model

In the framework of the scale invariant model of the Two Measures Field Theory (TMT), we study the dilaton-gravity sector in the context of spatially flat FRW cosmology. The scale invariance is spontaneously broken due to the intrinsic features of the TMT dynamics. If no fine tuning is made, the effective $\phi$-Lagrangian $p(\phi,X)$ depends quadratically upon the kinetic term $X$. Hence TMT represents an explicit example of the effective k-essence resulting from first principles without any exotic term in the underlying action intended for obtaining this result. Depending of the choice of regions in the parameter space (but without fine tuning), TMT exhibits interesting outputs for cosmological dynamics, for example: a) Absence of initial singularity of the curvature while its time derivative is singular. This is a sort of"sudden"singularities studied by Barrow on purely kinematic grounds. b) Power law inflation in the subsequent stage of evolution which ends with a graceful exit into the state with zero cosmological constant (CC). c) Possibility of resolution of the old CC problem. From the point of view of TMT, it becomes clear why the old CC problem cannot be solved (without fine tuning) in conventional field theories; d) There is a wide range of the parameters such that in the late time universe: the equation-of-state w=p/\rho<-1; w asymptotically (as t\to\infty) approaches -1 from below; $\rho$ approaches a constant, the smallness of which does not require fine tuning of dimensionfull parameters.