  Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$
admitting Riemannian metrics of positive scalar curvature. Denote by
$\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that
$\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(S^d)$, where
$S^d$ denotes the $d$-dimensional sphere with standard smooth structure.
  We also show a similar result for simply connected non-spin manifolds $M^d$
with $d\geq 5$ and $d\neq 8$. In this case let $W^d$ be the total space of the
non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then
$\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(W^d)$.
