Homology of smooth loop spaces of spheres

Question

I'm looking for a reference on the homology of $C^\infty(S^1,S^n)$. So far I've only been able to find references for spaces of maps which are $C^k$ or in a Sobolev class. I also have references which show that $C^0(S^1,S^n)$ and $C^k(S^1,S^n)$ are homotopy equivalent, and I'm wondering if this is still true for $k=\infty$.