  A new class of gravity-matter models defined in terms of two independent
non-Riemannian volume forms (alternative generally covariant integration
measure densities) on the space-time manifold are studied in some detail. These
models involve an additional $R^2$ (square of the scalar curvature) term as
well as scalar matter field potentials of appropriate form so that the
pertinent action is invariant under global Weyl-scale symmetry. Scale
invariance is spontaneously broken upon integration of the equations of motion
for the auxiliary volume-form degrees of freedom. After performing transition
to the physical Einstein frame we obtain: (i) An effective potential for the
scalar field with two flat regions which allows for a unified description of
both early universe inflation as well as of present dark energy epoch; (ii) For
a definite parameter range the model possesses a non-singular "emergent
universe" solution which describes an initial phase of evolution that precedes
the inflationary phase; (iii) For a reasonable choice of the parameters the
present model conforms to the Planck Collaboration data.
