### abstract

- Realizations of scale invariance are studied in the context of a gravitational theory where the action (in the first-order formalism) is of the form S = ∫L1Φd4cursive Greek chi + ∫L2√-gd4cursive Greek chi where Φ is a density built out of degrees of freedom, the "measure fields" independent of gμν and matter fields appearing in L1, L2. If L1 contains the curvature, scalar potential V(φ) and kinetic term for φ, L2 another potential for φ, U(φ), then the true vacuum state has zero energy density, when theory is analyzed in the conformal Einstein frame (CEF), where the equations assume the Einstein form. Global scale invariance is realized when V(φ) = f1eαφ and U(φ) = f2e2αφ. In the CEF the scalar field potential energy Veff(φ) has, in addition to a minimum at zero, a flat region for αφ → ∞, with nonzero vacuum energy, which is suitable for either a new inflationary scenario for the early universe or for a slowly rolling decaying Λ-scenario for the late universe, where the smallness of the vacuum energy can be understood as a kind of seesaw mechanism.