Probing neutron star structure via $f$-mode oscillations and damping in dynamical spacetime models

Gravitational wave and electromagnetic observations can provide new insights into the nature of matter at supranuclear densities inside neutron stars. Improvements in electromagnetic and gravitational wave sensing instruments continue to enhance the accuracy with which they can measure the masses, radii, and tidal deformability of neutron stars. These better measurements place tighter constraints on the equation of state of cold matter above nuclear density. In this article, we discuss a complementary approach to get insights into the structure of neutron stars by providing a model prediction for nonlinear fundamental eigenmodes ($f$ modes) and their decay over time, which are thought to be induced by time-dependent tides in neutron star binaries. Building on pioneering studies that relate the properties of $f$ modes to the structure of neutron stars, we systematically study this link in the nonperturbative regime using models that utilize numerical relativity. Using a suite of fully relativistic numerical relativity simulations of oscillating Tolman-Oppenheimer-Volkof stars, we establish blueprints for the numerical accuracy needed to accurately compute the frequency and damping times of $f$-mode oscillations, which we expect to be a good guide for the requirements in the binary case. We show that the resulting $f$-mode frequencies match established results from linear perturbation theory, but the damping times within numerical errors depart from linear predictions. This work lays the foundation for upcoming studies aimed at a comparison of theoretical models of $f$-mode signatures in gravitational waves, and their uncertainties with actual gravitational wave data, searching for neutron star binaries on highly eccentric orbits, and probing neutron star structure at high densities.

Figure 1

Plot of density contours $\rho$ (in code units) of a strongly perturbed TOV solution. The star was excited using a pressure perturbation that was derived in the Cowling approximation to induce $f$-mode oscillations. The perturbation amplitude is chosen as $\alpha =0.141$ for the star shown in the figure.

Figure 2

The panel shows the real part of ${\psi }_4$ from the PPM run with a resolution of 0.08 M. We observe that ${\psi }_4$ takes the form of a decaying exponential. Furthermore, since the gravitational waves are linearly polarized, we have confirmed that the imaginary part of ${\psi }_4$ vanishes.

Figure 3

Convergence of the extrema of ${\psi }_4$. Here low, med, high refer to runs with fine resolutions of 0.125 M, 0.08 M. and 0.0625 M respectively for the WENO case. This information can be inferred by noticing that the resolutions satisfy the relation $(\mathrm{high}-\mathrm{med})=c_n(\mathrm{med}-\mathrm{low})$; see Eq. (13). We found $c_n\sim 0.197$ by taking $c_n$ to be the best fit of our data to Eq. (13). Using Eq. (14) for $c_{2.84}\sim 0.197$, we found that $n=2.84$ which is consistent with the assumed order of convergence. This result is close to the order of our lowest order contributions to our numerical method, which demonstrates both the convergence and internal consistency of our numerical methods. We acknowledge that the value in this plot differs from that quoted in Table 2 as this convergence test applies to a longer portion of the waveform. The value quoted in Table 2 only uses the portion of the waveform that was used to compute the damping times. See Sec. 3f for more details.

Figure 4

Estimate for the relative error in ${\psi }_4$. Here low, med, high refer to runs with fine resolutions of 0.125 M, 0.08 M, and 0.0625 M respectively for the WENO case. This error estimate was obtained using Eq. (15) and dividing by the value of ${\psi }_4$ at each instant in time. Unlike Fig. 3 where we used a constant value of $c_n$ for a best fit to Eq. (13), our value of $c_n$ and $n$ here are solutions to Eqs. (13) and (14) at each point in time.

Figure 5

Fast Fourier transform (FFT) of the ${\psi }_4$ data from WENO HLLE simulation with a fine resolution of 0.0625 M. The gravitational waves here were extracted from 10 M. The 10M data best illuminate the excitations in the frequency spectrum because it occurs before any AMR boundary introduces high frequency features (we also present the frequency spectrum extracted at 100 M in Fig. 10). We note the primary is near the expected $f$-mode excitation frequency of 1.579 kHz. We also observe secondary excitations at 3.710, 5.684, and 7.658 kHz which are consistent with the first overtones ($p$ modes) of the $f$ mode. We note that the spikes at the higher frequencies are suppressed after applying the filtering. See the Appendix for more details.

Figure 6

Frequency evolution of the waveform as time progressed of the PPM and PPM big ${\rho }_{\mathrm{atm}}$ runs at a fine resolution of 0.125 M. This frequency was computed by examining the roots of the waveform time series and finding the time interval between them. This allows us to track the frequency evolution throughout the run. For the high atmosphere plot in red, the frequency increases after about 10 ms. The low atmosphere run in blue remained constant throughout the entirety of the run.

Figure 7

Linear fit of the extrema of ${\psi }_4$ that was used to obtain damping times of the gravitational waves for the PPM CCZ4 simulation with a fine resolution of 0.16 M. We note in particular that the extrema of ${\psi }_4$ differed by an offset that may skew the results. Therefore, we examined the slopes computed using the maxima and minima individually to provide an error estimate for the damping time of each simulation. The quoted value for the damping times was computed using the full dataset of extrema.

Figure 8

Damping times for simulations of neutron star $f$-mode oscillations as a function of fine resolution. The black dash-dot line represents the expected damping time for a static TOV of this mass computed using linear theory. The star value at $\mathrm{\Delta }x\to 0\mathrm{M}$ is the value obtain for the frequencies by extrapolating to infinite resolution. We notice that the runs using PPM reconstruction behaved similarly to one another. However, the high atmosphere PPM ${\rho }_{\mathrm{atm}}$ and PPM CCZ4 runs diverge from the PPM run at finer resolutions. This causes the Richardson extrapolated value of those high atmosphere sequences to significantly overshoot the expected damping time from linear theory. We described in Sec. 3d that the high atmosphere runs experienced increased frequency as the run progressed. We believe the effects of this evolving frequency propagate into the damping time calculation, leading to the divergence of these sequences of damping times. The WENO reconstruction runs also exhibited similar characteristics. In addition, the WENO runs are significantly more accurate than the PPM runs at a given resolution. PPM models required twice the resolution than the WENO models for similar accuracy. The high perturbation run exhibited entirely different damping time characteristics from the others, indicating that it was likely not in the linear perturbation regime.

Figure 9

Plot of the damping time $\tau$ vs analysis time with the statistical fitting error at each data point for the WENO HLLE and PPM datasets with $\mathrm{\Delta }x=0.0625\mathrm{M}$. The analysis time is defined as the cutoff time for waveform data to be used in the analysis. Data beyond the analysis time were discarded for the purpose of the analysis. The first 2 ms of data which are included in the analysis were also discarded to allow for any transient effects to decay. We notice that statistical error clearly decreases as the analysis time increases. However, we observe that $\tau$ decreases away from the expected value of 298 ms with increased analysis time. We noticed that in both curves in the plot there appears to be a local minimum in the statistical uncertainty at an analysis time of 15. As such, we selected an analysis time of 15 ms for the damping time analysis in this paper.

Figure 10

Plot of the ${\psi }_4$ data extracted at 10 M. The left plot depicts the unfiltered data. We note that the unfiltered data lack any sort of high frequency component as they have yet to pass through a refinement boundary. When we filter the data as in the right plot, we observe that the waveform is now centered at the origin rather than offset as it had been previously. The waveform is also hardly distorted, with most of the distortion occurring at the end of the waveform.

Figure 11

Plot of the ${\psi }_4$ data extracted at 100 M. The left plot depicts the unfiltered data. We note that the unfiltered data are much noisier than the data extracted at 10 M in Fig. 10, having passed through multiple refinement layers. These refinement layers are believed to induce high frequency noise in the waveform. In the plot, we show the postfiltered data. Compared to Fig. 10 the data have been significantly changed by the filtering. In turn, the postfiltered data appear to match the data in Fig. 10 much more closely. This supports the notion that the unfiltered 100 M data were contaminated by high frequency noise passing through refinement boundaries. We also note that the end of the waveform is distorted in a similar manner as Fig. 10. For this reason, we removed the end of the data from the computation of the damping times.

Figure 12

Plots of the FFTs of the WENO HLLE simulation with a fine resolution of 0.0625 M before and after the filtering. The gravitational waves here were extracted from 100 M, the portion that was analyzed to extract the frequencies and damping times. The left plot illustrates the unfiltered FFT, which contains the four distinct peaks seen in Fig. 5. However, there is a significant amount of noise after the second peak. On the right we show the FFT after the filtering has been applied. We observed that the noise is no longer present. However, the second peak is significantly suppressed, while the third and fourth peaks are no longer visible. Thus we did not consider effects from these modes in our analysis of the frequency or damping times.