  In this work, we consider a generalised gravitational theory that contains
the Einstein term, a scalar field and the quadratic Gauss-Bonnet term. We focus
on the early-universe dynamics, and demonstrate that a simple choice of the
coupling function between the scalar field and the Gauss-Bonnet term and a
simplifying assumption regarding the role of the Ricci scalar can lead to new,
analytical, elegant solutions with interesting characteristics. We first argue,
and demonstrate in the context of two different models, that the presence of
the Ricci scalar in the theory at early times, when the curvature is strong,
does not affect the actual cosmological solutions. By considering therefore a
pure scalar-GB theory with a quadratic coupling function we derive a plethora
of interesting, analytic solutions: for a negative coupling parameter, we
obtain inflationary, de Sitter-type solutions or expanding solutions with a de
Sitter phase in their past and a natural exit mechanism at later times; for a
positive coupling function, we find instead singularity-free solutions with no
Big-Bang singularity. We show that the aforementioned solutions arise only for
this particular choice of coupling function, a result that may hint to some
fundamental role that this coupling function may hold in the context of an
ultimate theory.
