  Improvements in ground-based, advanced gravitational wave (GW) detectors may
allow in the near future to observe the GW signal of a nearby core-collapse
supernova. For the most common type of progenitors, likely with slowly rotating
cores, the dominant GW emission mechanisms are the post-bounce oscillations of
the proto-neutron star (PNS) before the explosion. We present a new procedure
to compute the eigenmodes of the system formed by the PNS and the stalled
accretion shock in general relativity including spacetime perturbations. The
new method improves on previous results by accounting for perturbations of both
the lapse function and the conformal factor. We apply our analysis to two
numerical core-collapse simulations and show that our improved method is able
to obtain eigenfrequencies that accurately match the features observed in the
GW signal and to predict the qualitative behaviour of quasi-radial
oscillations. Our analysis is possible thanks to a newly developed algorithm to
classify the computed eigenmodes in different classes (f-, p-, and g-modes),
improving previous results which suffered from misclassification issues. We
find that most of the GW energy is stored in the lowest order eigenmodes, in
particular in the $^2g_1$ mode and in the $^2f$ mode. Our results also suggest
that a low-frequency component of the GW signal attributed in previous works to
the characteristic frequency of the Standing Accretion Shock Instability should
be identified as the fundamental quadrupolar f-mode. We also develop a
formalism to estimate the contribution of quasi-radial ($l=0$) modes to the
quadrupolar component of the GW emission in the case of a deformed background,
with application to rapidly rotating cores. This work provides further support
for asteroseismology of core-collapse supernovae and the inference of PNS
properties based on GW observations.
