  In this article we develop new methods for exhibiting convex semialgebraic
sets that are not spectrahedral shadows. We characterize when the set of
nonnegative polynomials with a given support is a spectrahedral shadow in terms
of sums of squares. As an application of this result we prove that the cone of
copositive matrices of size $n\geq5$ is not a spectrahedral shadow, answering a
question of Scheiderer. Our arguments are based on the model theoretic
observation that any formula defining a spectrahedral shadow must be preserved
by every unital $\mathbb{R}$-linear completely positive map $R\to R$ on a real
closed field extension $R$ of $\mathbb{R}$.
