  We study here the random fluctuations in the number of critical points with
values in an interval $I\subset \mathbb{R}$ for Gaussian spherical
eigenfunctions $\left\{f_{\ell }\right\} $, in the high energy regime where
$\ell \rightarrow \infty $. We show that these fluctuations are asymptotically
equivalent to the centred $L^{2}$-norm of $\left\{ f_{\ell }\right\} $ times
the integral of a (simple and fully explicit) function over the interval under
consideration. We discuss also the relationships between these results and the
asymptotic behaviour of other geometric functionals on the excursion sets of
random spherical harmonics.
