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 $S = \int L_{1} \Phi d^4x$+$\int L_{2}\sqrt{-g}d^4x$wherethevolumeelement$\Phi d^4x$ is independent of the metric. For global scale invariance, a “dilaton” $\varphi$ has to be introduced, with non-trivial potentials $V(\varphi )$ = $f_1{e}^{\alpha \varphi }$ in $L_1$ and $U(\varphi )$ = $f_2{e}^{2\alpha \varphi }$ in $L_2$. This leads to non-trivial mass generation and a potential for $\varphi$ 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 $\mathrm{\Phi }{d}^{4}x$ alone. For closed strings and branes (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.