On Ultrapowers and Cohesive Ultrafilters

We characterize the Tukey order, the Galvin property/ Cohesive ultrafilters from \cite{Kanamori1978} in terms of ultrapowers. We use this characterization to measure the distance between the Tukey order and other well-known orders of ultrafilters. Secondly, we improve two theorems of Kanamori \cite{Kanamori1978} from the 70's. We then study the point spectrum and the depth spectrum of an ultrafilter, and give a simple positive answer to Kanamori's question \cite[Question 2]{Kanamori1978} starting from a supercompact cardinal. We also prove that a positive answer requires more than $o(\kappa)=\kappa^{++}$. Finally, we prove several consistency results regarding the point and depth spectrum.