  We gain tight rigorous bounds on the renormalisation fixed point function for
period doubling in families of unimodal maps with degree 2 critical point. By
writing the relevant eigenproblems in a modified nonlinear form, we use these
bounds, together with a contraction mapping argument, to gain tight bounds on
the essential eigenvalues and eigenfunctions of the linearised renormalisation
operator at the fixed point and also those of the operator encoding the
universal scaling of added uncorrelated noise. We gain bounds on the
corresponding power series coefficients and universal constants accurate to
over 400 significant figures, confirming and (in the case of noise) extending
the accuracy of previous numerical estimates, by using multi-precision interval
arithmetic with rigorous directed rounding to implement operations on a space
of analytic functions.
