In Hartle-Hawking "no boundary" proposal it is proposed that Riemannian spacetimes rather than Lorentzian dominated the path integral near the big bang.
Moments after the big bang however spacetimes with Lorentzian metrics started to dominate over the Riemannian. The dominance of the Riemannian spacetimes is characterized by positive definite metrics obtained by applying Wick rotation.
Now to compute the path integral some approximations are made and the process of analytic continuation is applied. What bothers me is analytic continuation of an approximate holomorphic function is not guaranteed to remain analytic in another region.
Why then this unreliable process applied—can one rely on this scheme which seems to be mathematically dubious?