Higgs-inflaton symbiosis, cosmological constant problem and superacceleration phase of the universe in two measures field theory with spontaneously broken scale invariance

We study the scalar sector of the Two Measures Field Theory (TMT) model in the context of cosmological dynamics. The scalar sector includes the dilatonand the Higgsfields. 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 whereplays the role of the inflaton . In the model with the inflatonalone, 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 fieldwhich is followed by the late time evolution driven both by a small cosmological constant and the scalar fieldwith 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 � around the state with zero cosmological constant. d) There are regions of the parameters where the equation-of-state w = p/� in the late time universe is w 6= 0 soon after the end of a power law inflation; there are two oscillatory regimes of �, one around � = 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.