  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.
