We study the scalar sector of the Two Measures Field Theory ( TMT ) model in the context of cosmological dynamics .
The scalar sector includes the dilaton \phi and the Higgs \upsilon fields .
The model possesses gauge and scale invariance .
The latter is spontaneously broken due to intrinsic features of the TMT dynamics .
The scalar sector dynamics represents an explicit example of k -essence resulting from first principles where \phi plays the role of the inflaton .
In the model with the inflaton \phi alone , in different regions of the parameter space the following different effects can take place without fine tuning of the parameters and initial conditions : a ) Possibility of a power law inflation driven by the scalar field \phi which is followed by the late time evolution driven both by a small cosmological constant and the scalar field \phi with a quintessence-like potential ; smallness of the cosmological constant can be achieved without fine tuning of dimensionfull parameters .
b ) Possibility of resolution of the old cosmological constant problem : this is done in a consistent way hinted by S. Weinberg in his comment concerning the question of how one can avoid his no-go theorem .
c ) The power law inflation without any fine tuning may end with damped oscillations of \phi around the state with zero cosmological constant .
d ) There are regions of the parameters where the equation-of-state w = p / \rho in the late time universe is w < -1 and w asymptotically ( as t \rightarrow \infty ) approaches -1 from below .
This effect is achieved without any exotic term in the action .
In a model with both \phi and \upsilon fields , a scenario which resembles the hybrid inflation is realized but there are essential differences , for example : the Higgs field undergos transition to a gauge symmetry broken phase < \upsilon > \neq 0 soon after the end of a power law inflation ; there are two oscillatory regimes of \upsilon , one around \upsilon = 0 at 50 e-folding before the end of inflation , another - during transition to a gauge symmetry broken phase where the scalar dark energy density approaches zero without fine tuning ; the gauge symmetry breakdown is achieved without tachyonic mass term in the action .