  Calcium silicate perovskite (CaSiO$_3$) is one of the major mineral
components of the lower mantle, but has been the subject of relatively little
work compared to the more abundant Mg-based materials. One of the major
problems related to CaSiO$_3$ that is still the subject of research is its
crystal structure under lower mantle conditions - a cubic Pm$\bar{3}$m
structure is accepted in general, but some have suggested that lower-symmetry
structures may be relevant. In this paper, we use a fully first-principles
vibrational self-consistent field method to perform high accuracy anharmonic
vibrational calculations on several candidate structures at a variety of points
along the geotherm near the base of the lower mantle to investigate the
stability of the cubic structure and related distorted structures. Our results
show that the cubic structure is the most stable throughout the lower mantle,
and that this result is robust against the effects of thermal expansion.
