  We compare two approaches to refine the "linear model" of cavity
optomechanics, in order to describe radiation pressure effects that are beyond
first order in the coupling constant. We compare corrections derived from (I) a
widely used phenomenological Hamiltonian that conserves the photon number and
(II) a two-mode truncation of C. K. Law's microscopic model, which we take as
the "true" Hamiltonian of the system. While these approaches agree at first
order, the latter model does not conserve the photon number, resulting in
challenging computations from second order onwards. Our numerics suggest that
the phenomenological Hamiltonian significantly improves the linear model, yet
it does not fully capture all second-order corrections arising from the C. K.
Law model. We conclude that, even when the mechanical frequency is much lower
than the cavity one, photon number conservation must be eventually given up to
model cavity optomechanics with high accuracy.
