The Volume Element of Space-Time and Scale Invariance

Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form S = ∫ L₁ Φ d⁴x + ∫ L₂ √ - g d⁴x where the volume element Φ d⁴x is independent of the metric. For global scale invariance, a "dilaton" φ has to be introduced, with non-trivial potentials V(φ) = f₁ e^αφ L₁ and U(φ) = f₂ e^2αφ in L₂. This leads to non-trivial mass generation and a potential for φ which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for "Quintessential models," which are scale invariant but formulated with the use of volume element Φ d⁴x alone. For closed strings and brunes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants.