We present results from a new set of three-dimensional general-relativistic hydrodynamic simulations of rotating iron core collapse. We assume octant symmetry and focus on axisymmetric collapse, bounce, the early postbounce evolution, and the associated gravitational wave (GW) and neutrino signals. We employ a finite-temperature nuclear equation of state, parametrized electron capture in the collapse phase, and a multispecies neutrino leakage scheme after bounce. The latter captures the important effects of deleptonization, neutrino cooling and heating and enables approximate predictions for the neutrino luminosities in the early evolution after core bounce. We consider $12\ensuremath{-}{M}_{\ensuremath{\bigodot}}$ and $40\ensuremath{-}{M}_{\ensuremath{\bigodot}}$ presupernova models and systematically study the effects of (i) rotation, (ii) progenitor structure, and (iii) postbounce neutrino leakage on dynamics, GW, and neutrino signals. We demonstrate that the GW signal of rapidly rotating core collapse is practically independent of progenitor mass and precollapse structure. Moreover, we show that the effects of neutrino leakage on the GW signal are strong only in nonrotating or slowly rotating models in which GW emission is not dominated by inner core dynamics. In rapidly rotating cores, core bounce of the centrifugally deformed inner core excites the fundamental quadrupole pulsation mode of the nascent protoneutron star. The ensuing global oscillations ($f\ensuremath{\sim}700--800\text{ }\text{ }\mathrm{Hz}$) lead to pronounced oscillations in the GW signal and correlated strong variations in the rising luminosities of antineutrino and heavy-lepton neutrinos. We find these features in cores that collapse to protoneutron stars with spin periods $\ensuremath{\lesssim}2.5\text{ }\text{ }\mathrm{ms}$ and rotational energies sufficient to drive hyperenergetic core-collapse supernova explosions. Hence, GW or neutrino observations of a core-collapse event could deliver strong evidence for or against rapid core rotation. Joint $\mathrm{GW}+\mathrm{\text{neutrino}}$ observations would allow one to make statements with high confidence. Our estimates suggest that the GW signal should be detectable throughout the Milky Way by advanced laser-interferometer GW observatories, but a water-Cherenkov neutrino detector would have to be of near-megaton size to observe the variations in the early neutrino luminosities from a core collapse event at 1 kpc.

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