Dynamical excitation of space-time modes of compact objects

We discuss, in the perturbative regime, the scattering of Gaussian pulses of odd-parity gravitational radiation off a nonrotating relativistic star and a Schwarzschild black hole. We focus on the excitation of the $w$-modes of the star as a function of the width $b$ of the pulse and we contrast it with the outcome of a Schwarzschild black hole of the same mass. For sufficiently narrow values of $b$, the waveforms are dominated by characteristic space-time modes. On the other hand, for sufficiently large values of $b$ the backscattered signal is dominated by the tail of the Regge-Wheeler potential, the quasinormal modes are not excited and the nature of the central object cannot be established. We view this work as a useful contribution to the comparison between perturbative results and forthcoming $w$-mode 3D-nonlinear numerical simulation.

Figure 1

Dependence of the ring-down phase on the width $b$ of the Gaussian pulse: for $b=2$ (top panel) and $b=8$ (middle panel) the process of excitation of the space-time modes shows the same qualitative features for the black hole and for the star. The waveforms for $b=20$ (bottom panel) show there is basically no difference between the gravitational wave signal backscattered from a stars of Table  and from a Schwarzschild black hole with the same mass.

Figure 2

Energy spectra (from model A) for different values of $b$. The maximum frequency is consistent with ${\omega }_{\max }\simeq 3\sqrt{2}/b$. See text for discussion.

Figure 3

The ratio, as a function of $b$, of the energy released in gravitational waves by the black hole and different star models.

Figure 4

Fits of the ring-down part of the waveform with the fundamental ($n=0$) space-time mode for a black hole excited by a $b=2$ Gaussian pulse. We show the waveform ${\Psi }_2^{(\mathrm{o})}(t)$ and its absolute value on a logarithmic scale to highlight the differences with the fit.

Figure 5

Fits of the ring-down part of the waveform with the fundamental ($n=0$) space-time mode for the stellar model A exited by $b=2$ (upper panels) and a $b=8$ (bottom panels) Gaussian pulse; For each value of $b$ we show the waveform ${\Psi }_2^{(\mathrm{o})}(t)$ and its absolute value on a logarithmic scale to highlight the differences with the fit.

Figure 6

The residual $\mathcal{R}\equiv 1-\Theta$ [See Eq. (3.2)] of the fits of the waveform of the response to a $b=2$ Gaussian pulse of a black hole (upper panel) and stellar model A (bottom panel) as a function of the initial time ($u_{\mathrm{i}}$) of the fitting window around its best values that it is $u_{\mathrm{i}}=129$ for a black hole and $u_{\mathrm{i}}=131$ for model A.

Figure 7

Determination of the best window for the fit of the black hole waveform and model A ($b=2$). The initial time $u_{\mathrm{i}}$ is chosen so to minimize the residual $\mathcal{R}$ between the data and the fit. For the black hole we obtain $u_{\mathrm{i}}=129$ ($u_{\mathrm{f}}=254$), while for model A we have $u_{\mathrm{i}}=131$ ($u_{\mathrm{f}}=174$). The horizontal line indicates the values of the QNMs frequencies reported in Tables .