Parametrised moduli spaces of surfaces as infinite loop spaces

We study the $E_2$-algebra $\mathrm{\Lambda }{\mathfrak{M}}_{\ast ,1}={\coprod }_{g⩾0}\mathrm{\Lambda }{\mathfrak{M}}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\mathrm{\Omega }B\mathrm{\Lambda }{\mathfrak{M}}_{\ast ,1}$: it is the product of ${\mathrm{\Omega }}^{\mathrm{\infty }}\mathbf{M}\mathbf{T}\mathbf{S}\mathbf{O}(2)$ with a certain free ${\mathrm{\Omega }}^{\mathrm{\infty }}$-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups ${\mathrm{\Gamma }}_{g,n}$ with $g⩾0$ and $n⩾1$.