  We study the phase diagram of the Dicke model in terms of the excitation
energy and the radiation-matter coupling constant $\lambda$. Below a certain
critical value $\lambda_c$ of the coupling constant $\lambda$ all the energy
levels have a well-defined parity. For $\lambda > \lambda_c$ the energy
spectrum exhibits two different phases separated by a critical energy $E_c$
that proves to be independent of $\lambda$. In the upper phase, the energy
levels have also a well defined parity, but below $E_c$ the energy levels are
doubly degenerated. We show that the long-time behavior of appropriate
parity-breaking observables distinguishes between the different phases of the
energy spectrum of the Dicke model. Steady states reached from
symmetry-breaking initial conditions restore the symmetry only if their
expected energies are above the critical. This fact makes it possible to
experimentally explore the complete phase diagram of the excitation spectrum of
the Dicke model.
