Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions

Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are established. The solutions provide a very powerful method not only for examining the general properties of solutions and physical quantities when they are applied to, but also for numerical computations. The solutions are given in the expansion of a small parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole, which corresponds to Post-Minkowski expansion by $G$ and to post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. It is expected that these solutions will become a powerful weapon to construct the theoretical template towards LIGO and VIRGO projects.Comment: 24 pages, minor modification

Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion

Analytic solutions of the Regge-Wheeler equation are presented in the form of series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations between these solutions are established. The series solutions are given as the Post-Minkowskian expansion with respect to a parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole. This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. These solutions can also be useful for numerical computations.Comment: 22 page

A Bayesian construction of asymptotically unbiased estimators

A differential geometric framework to construct an asymptotically unbiased estimator of a function of a parameter is presented. The derived estimator asymptotically coincides with the uniformly minimum variance unbiased estimator, if a complete sufficient statistic exists. The framework is based on the maximum a posteriori estimation, where the prior is chosen such that the estimator is unbiased. The framework is demonstrated for the second-order asymptotic unbiasedness (unbiased up to $O(n^{-1})$ for a sample of size $n$). The condition of the asymptotic unbiasedness leads the choice of the prior such that the departure from a kind of harmonicity of the estimand is canceled out at each point of the model manifold. For a given estimand, the prior is given as an integral. On the other hand, for a given prior, we can address the bias of what estimator can be reduced by solving an elliptic partial differential equation. A family of invariant priors, which generalizes the Jeffreys prior, is mentioned as a specific example. Some illustrative examples of applications of the proposed framework are provided.Comment: 28 pages, 2 figure