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The dilaton-gravity sector of the Two Measures Field Theory (TMT)is exploredin detail in the context of cosmology. The model possesses scale invariancewhich is spontaneously broken due to the intrinsic features of the TMTdynamics. The effective model represents an explicit example of the effectivek-essence resulting from first principles without any exotic term in thefundamental action. Depending of the choice of regions in the parameter space,TMT exhibits different possible outputs for cosmological dynamics: a) Absenceof initial singularity of the curvature while its time derivative is singular.This is a sort of "sudden" singularities studied by Barrow on purely kinematicgrounds. b) Power law inflation in the subsequent stage of evolution. Dependingon the region in the parameter space (but without fine tuning) the inflationends with a graceful exit either into the state with zero cosmological constant(CC) or into the state driven by both a small CC and the field phi with aquintessence-like potential. c) Possibility of resolution of the old CCproblem. From the point of view of TMT, it becomes clear why the old CC problemcannot be solved (without fine tuning) in conventional field theories. d) TMTenables two ways for achieving small CC without fine tuning of dimensionfullparameters: either by a seesaw type mechanism or due to a correspondenceprinciple between TMT and conventional field theories (i.e theories with onlythe measure of integration sqrt{-g} in the action. e) There is a wide range ofthe parameters such that in the late time universe: the equation-of-statew=p/\rho <-1; w asymptotically (as t\to\infty) approaches -1 from below; \rhoapproaches a constant, the smallness of which does not require fine tuning ofdimensionfull parameters.Comment: 37 pages, 20 figures. Minor misprints corrected, reference added. The  final version published in Phys. Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Fine-tuning free paradigm of two-measures theory:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>k</mml:mi></mml:math>-essence, absence of initial singularity of the curvature, and inflation with graceful exit to the zero cosmological constant state

Fine-tuning free paradigm of two-measures theory:k-essence, absence of initial singularity of the curvature, and inflation with graceful exit to the zero cosmological constant state