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 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} .