Figure 1

Fundamental mode in the spherically symmetric polytrope BU0 ($\Gamma =2.0$, $K=100$, central density ${\rho }_c=1.28\times {10}^{-3}$): The mode is excited with a nodeless spherically symmetric density perturbation. This plot shows a discrete Fourier transform of the central density time evolution ${\rho }_c(t)$, for both a model in the Cowling approximation and in full general relativity. The resulting major peak is the fundamental mode $F$, which can be compared to published estimates. Since we do not impose an exact eigenfunction, the initial perturbation (and also the discretization error) also have some overlap with the first overtone H1, although the perturbation is nodeless. The differenced plots refer to central finite differences of the Fourier transforms, which allow to locate the maxima of the peaks more accurately. (The difference graph values have been rescaled for illustration.)Reuse & Permissions

Figure 2

Example of a signal obtained by the mode extraction technique described in the text. The background model is a member of the sequence C (number C1, see Table ), with $K=1000$, $\Gamma =2.5$, and ${\rho }_c=5\times {10}^{-3}$. A nonaxisymmetric $\ell =2$, $m=2$ density perturbation excites the corresponding two $f$-modes, and the resulting oscillations are extracted by a discrete Fourier analysis of the density on discrete rings. The plot shows the resulting signal of the mode $A_2$ (for $m=2$) for an evolution in full general relativity over 13 ms. The signal exhibits oscillations which are dominated by two frequencies. Also visible is a decay in the signal strength, which is a numerical artifact caused by the finite-volume scheme and the inaccurate discretization near the stellar surface.Reuse & Permissions

Figure 3

Fourier transform of the signal from Fig. 2 obtained in postprocessing. The initial $l=2$, $m=2$ perturbation excites both the co- and counterrotating $f$-modes. To determine the actual mode frequency, we do not use the location of the peak, but the root of the derivates obtained from second order central differencing, as shown in the inset.Reuse & Permissions

Figure 4

Resolution test for the rapidly rotating model S3 (in full GR). Amplitude extraction results from the Fourier transform in both a simulation with 90 grid cells per coordinate direction and 120 cells are compared. The finer resolved simulation has also been run for longer time (up to 20 ms) to reduce peak width and the resulting size of the error bars in the Fourier transform. The graph shows the Fourier transform, its derivatives, and the null line to visually identify the location of the roots. It is apparent that the discrete peak maximum locations disagree slightly, but the locations of the root agree very well, better than the error bar estimate obtained from the half-width of the peak.Reuse & Permissions

Figure 5

Frequencies of $\ell =|m|=2$ $f$-modes in the models of sequence S ($\Gamma =2.0$). The models are characterized by their ratio of rotational kinetic over gravitational binding energy $T/|W|$, where $T/|W|=0.08$ is very close to the mass-shedding limit. Both results from simulations in the Cowling approximation (blue dashed line) and those from full simulations (black solid line) are displayed for comparison. The approximate location of the neutral point is indicated for each graph.Reuse & Permissions

Figure 6

Comparison between the Cowling approximation and full general relativity for the $\ell =|m|=3$ $f$-mode frequencies along sequence S.Reuse & Permissions

Figure 7

Comparison between $\ell =|m|=2$ and $\ell =|m|=3$ $f$-mode frequencies along sequence S. The neutral point of the $m=3$ mode is shifted with respect to the $m=2$ mode.Reuse & Permissions

Figure 8

Influence of the equation of state on the $\ell =|m|=2$ $f$-mode frequencies. The results are from simulations of sequences S and C (cf. Tables , also Fig. 5). While the corotating modes directly show the effect of the lower compressibility of the $\Gamma =2.5$ models in a shift in the frequency band, the counterrotating modes have comparable frequencies in the admissible range of stellar modes, and consequently a similar location of the neutral point in terms of $T/|W|$.Reuse & Permissions

Figure 9

Same as Fig. 8, but for the $\ell =|m|=3$ modes.Reuse & Permissions

Figure 10

Frequency band of the CFS instability based on the extraction of $m=2$ and $m=3$ $f$-mode frequencies in two sequences of uniformly rotating polytropes. Each band is constrained by the neutral point (low frequencies on the left end) and the frequency of the counterrotating mode in the most rapidly rotating model. For the $m=3$ mode in the $\Gamma =2.5$ sequence of polytropes, the counterrotating mode frequency is extrapolated from Fig. 9 to around $\approx 2.5\mathrm{kHz}$ (a lower limit is given by 1.1 kHz). To compare with the frequency variation of the detector sensitivity, the noise curves (in $1/\sqrt{\mathrm{Hz}}$) of Advanced LIGO and the proposed Einstein Telescope are also displayed. The displayed ordering of the frequency bands has no relation to the relative expected amplitudes of the associated instabilities.Reuse & Permissions