Gravitational waves (GWs) generated by axisymmetric rotating collapse, bounce, and early postbounce phases of a galactic core-collapse supernova will be detectable by current-generation gravitational wave observatories. Since these GWs are emitted from the quadrupole-deformed nuclear-density core, they may encode information on the uncertain nuclear equation of state (EOS). We examine the effects of the nuclear EOS on GWs from rotating core collapse and carry out 1824 axisymmetric general-relativistic hydrodynamic simulations that cover a parameter space of 98 different rotation profiles and 18 different EOS. We show that the bounce GW signal is largely independent of the EOS and sensitive primarily to the ratio of rotational to gravitational energy, and at high rotation rates, to the degree of differential rotation. The GW frequency of postbounce core oscillations shows stronger EOS dependence that can be parameterized by the cores EOS-dependent dynamical frequency $sqrt{Gbar{rho}_c}$. We find that the ratio of the peak frequency to the dynamical frequency follows a universal trend that is obeyed by all EOS and rotation profiles and that indicates that the nature of the core oscillations changes when the rotation rate exceeds the dynamical frequency. We find that differences in the treatments of low-density nonuniform nuclear matter, of the transition from nonuniform to uniform nuclear matter, and in the description of nuclear matter up to around twice saturation density can mildly affect the GW signal. We find that approximations and uncertainties in electron capture rates can lead to variations in the GW signal that are of comparable magnitude to those due to different nuclear EOS. This emphasizes the need for reliable nuclear electron capture rates and for self-consistent multi-dimensional neutrino radiation-hydrodynamic simulations of rotating core collapse.
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