# Paper No. 12-09

**D Chakrabarty**

**Detection of Bistable Phase Space of a Real Galaxy, using a new non-parametric Bayesian Test of Hypothesis**

**Abstract:** In lieu of direct detection of dark matter, inverse learning of the distribution of the gravitational mass in distant galaxies is of crucial importance in Cosmology. Such inverse learning is typically invoked within under-abundant information domains, which in turn motivates the need for undertaking simplifying model assumptions. Assuming the topology of the phase space structure of a galaxy allows for the learning of the density of the gravitational mass in galaxies, using partial and noisy data on velocity and location vectors of individual particles that reside in such a galactic system. The simplest assumption about the phase space topology is that of an isotropic phase space. This is equivalent to the assumption that the phase space pdf that the particle data are drawn from, is an isotropic function of the particle phase space coordinates.

We present a new distribution-free test of hypothesis that tests for relative support in two or more measured particle data sets, for such simplified isotropic phase space pdfs that the respective data sets are drawn from. This test is designed to work in parameter space, for disparate sample sizes, in situations characterised by little and/or differential information about the prior for the null, i.e. the prior probability that a given particle data set is drawn from an isotropic phase space. The problem of differential sensitivity to a choice of the prior, given disparate sample sizes, is circumvented by this test. It works by computing the fraction of the inferred state space vectors, the posterior probability for which exceeds the maximal posterior achieved under the null. We illustrate applications of this test to two independent particle data sets in simulated as well as a real galactic system. The dynamical implications of the results of application to the real galaxy is indicated to be the residence of the observed particle samples in disjoint volumes of the galactic phase space. This result is used to suggest the serious risk borne in attempts at learning of gravitational mass density of galaxies, using particle data.