For which $n$ has the $S^1$-equivariant rational cohomology of $LBSO(n)$ been computed? Here, $SO(n)$ means the isometry group of the round sphere (preserving the orientation), $B$ stands for *classifying space* and $L$ for *free loop space*. The $S^1$-action on $LBSO(n)$ is, of course, via rotating the loops.


There are two fibrations $$SO(n) \to LBSO(n) \to BSO(n)$$ and $$LBSO(n) \to LBSO(n) \times\_{S^1} ES^1 \to BS^1$$
(both admitting a split on the right), and I am interested in knowing $$H^{\ast}\_{S^1}(LBSO(n);\mathbb Q) = H^{\ast}(LBSO(n) \times\_{S^1} ES^1;\mathbb Q),$$ including the ring structure. So one way to calculate this is to understand what happens in the Serre spectral sequences in the two fibrations above. I could do this (with some effort) for $n = 2,3$ but I do not have much hope figuring out all the differentials for bigger $n$. Is this something that can be found in the literature? 



Rational cohomology of $LBSO(n)$
1st method) If $X$ is a 1-connected then you can use the fact that you have an isomorphism of graded algebras (J.D.S. Jones "Cyclic homology and equivariant homology" Inventiones Math. 1987):


$$H^n\_{S^1}(LX;\mathbb{Q})\cong HC\_{-n}^{-}(S^\*(X;\mathbb{Q}))$$


where the right term is the negative cyclic homology of the cochain algebra of $X$.


Moreover if $X$ is formal you have an isomorphism of algebras:


$$H^n\_{S^1}(LX;\mathbb{Q})\cong HC\_{-n}^-(H^\*(X;\mathbb{Q})).$$


This is indeed the case for $X=BSO(n)$ which is formal. Now you use the fact $H^\*(BSO(n;\mathbb{Q}))$ is a polynomial algebra.


2nd method) Rationnaly $BSO(n)$ is formal and homotopy equivalent to a product of Eilenberg-MacLane spaces of type $K(\mathbb{Q},4k)$ and of a copy of $K(\mathbb{Q},2m)$ if $n=2m$, and now use the fact the for an $H$-space $Y$ we have $LY\cong Y\times \Omega Y$. Thus rationnaly you get that the cohomology of $LBSO(n)$ is isomorphic to 
$$H^\*(BSO(n);\mathbb{Q})\otimes H^\*(SO(n);\mathbb{Q}).$$


**Edit, Reference:** A good reference that will give you a cochain complex that computes $H^n\_{S^1}(LX;\mathbb{Q})$ is the paper ["A model for cyclic homology and algebraic K
-theory of 1-connected topological spaces"](http://projecteuclid.org/download/pdf_1/euclid.jdg/1214439821) by Micheline Vigué-Poirrier and Dan Burghelea (J. Differential Geom. Volume 22, Number 2 (1985), 243-253). In particular the theorem $A$ that gives an algebraic model from the minimal model of $X$.


