k-Essence, Avoidance of the Weinberg's Cosmological Constant No-Go Theorem and Other Dark Energy Effects of Two Measures Field Theory

The dilaton-gravity sector of the Two Measures Field Theory (TMT) is explored in detail in the context of cosmology. The model possesses scale invariance which is spontaneously broken due to the intrinsic features of the TMT dynamics. The dilatondependence of the effective Lagrangian appears only as a result of the spontaneous breakdown of the scale invariance. If no fine tuning is made, the effective �-Lagrangian p(�,X) depends quadratically upon the kinetic energy X. Hence TMT may represent an explicit example of the effective k-essence resulting from first principles without any exotic term in the fundamental action intended for obtaining this result. Depending of the choice of regions in the parameter space, TMT exhibits different possible outputs for cosmological dynamics: a) Possibility of a power law inflation driven by the fieldwhich is followed by the late time evolution driven both by a small cosmological constant and the fieldwith a quintessence-like potential. TMT enables two ways for achieving small cosmological constant without fine tuning of dimensionfull parameters: either by a seesaw type mechanism or due to a correspondence principle between TMT and conventional field theories (i.e theories with only the measure of integration √ −g in the action). b) Possibility of resolution of the old cosmological constant problem. From the point of view of TMT, it becomes clear why the old cosmological constant problem cannot be solved (without fine tuning) in conventional field theories. c) The power law inflation without any fine tuning can end with damped oscillations ofaround the state with zero cosmological constant. d) There is a broad range of the parameters such that: the equation-of-state in the late time universe w = p/� < −1; w asymptotically (as t → ∞) approaches −1 from below; � approaches a cosmological constant, the smallness of which does not require fine tuning of dimensionfull parameters.