  In this paper we prove new classification results for nonnegatively curved
gradient expanding and steady Ricci solitons in dimension three and above,
under suitable integral assumptions on the scalar curvature of the underlying
Riemannian manifold. In particular we show that the only complete expanding
solitons with nonnegative sectional curvature and integrable scalar curvature
are quotients of the Gaussian soliton, while in the steady case we prove
rigidity results under sharp integral scalar curvature decay. As a corollary,
we obtain that the only three dimensional steady solitons with less than
quadratic volume growth are quotients of $\mathbb{R}\times\Sigma^{2}$, where
$\Sigma^{2}$ is Hamilton's cigar.
