Taller in the saddle: constraining CMB physics using saddle points

The statistics of extremal points in the cosmic microwave background (CMB) temperature (hot and cold spots) have been well explored in the literature, and have been used to constrain models of the early Universe. Here, we extend the study of critical points in the CMB to the set that remains after removing extrema, namely the saddle points. We perform stacks of temperature and polarization about temperature saddle points in simulations of the CMB, as well as in data from the Planck satellite. We then compute the theoretical profile of saddle-point stacks, given the underlying power spectra of the CMB. As an example of the utility of such stacks, we constrain models of cosmic birefringence, and compare the constraining power of the saddle points with that of extremal points. We find that, in the specific example of birefringence, we can place tighter constraints using saddle points in our analysis than using extrema. In fact, we find our saddle-point analysis yields close to optimal constraints, as seen by comparing to a power spectrum analysis. We, therefore, suggest that stacking on saddle points may, in general, be a useful way of testing for non-standard physics effects that change the CMB power spectra.