Pseudospectrum and time-domain analysis of the EFT corrected black holes

TL;DR

This work investigates EFT gravity with dimension-6 corrections to Schwarzschild black holes, introducing a dimensionless parameter $oldsymbol{b5}$ that controls EFT effects. Using a hyperboloidal framework, the authors reformulate perturbations into a first-order operator $L$ and compute QNMs, time-domain waveforms, and pseudospectra, revealing that isospectrality between polar and axial perturbations is broken as $oldsymbol{b5}$ grows and that higher overtones are more sensitive to EFT corrections. The time-domain response shows small, continuous changes with $oldsymbol{b5}$, with waveform mismatch scaling as $oldsymbol{b5^2}$, indicating overall waveform stability. A physically motivated energy norm is introduced to define a pseudospectrum, uncovering complex, mode-dependent stability characterized by a critical perturbation size $oldsymbol{epsilon_c}$ that can vary nonmonotonically with $oldsymbol{b5}$. The results highlight nuanced implications of EFT corrections for QNM spectra and open avenues for extending to rotating BHs and tighter gravitational-wave constraints.

Abstract

We study the linear perturbations of a spherically symmetric black hole corrected by dimension-6 terms in the effective field theory (EFT) of gravity. The solution is asymptotically flat and characterized by two parameters -- a mass parameter $M$ and a dimensionless parameter $\varepsilon$ related to the EFT length scale $l$, and the perturbation equation incorporates a velocity factor which is not constant. The quasinormal modes (QNMs) and time-domain waveforms are studied within the hyperboloidal framework. This approach reproduces the breakdown of the isospectrality and reveals that higher overtones are more sensitive to $\varepsilon$. As for the time domain, the mismatch function is introduced and found to scale as $\varepsilon^2$, which demonstrates that the waveform is stable as $\varepsilon$ varies. Finally, a velocity-dependent energy norm is employed to compute the pseudospectrum and characterize the migration of the QNM spectrum. We further define a quantity $ε_c$ that describes the magnitude of the instability of a QNM spectrum. Our analysis reveals that the dependence of $ε_c$ on $\varepsilon$ is complicated -- it may increase, decrease or even be nonmonotonic.

Figures (9)

• Figure 1: $\frac{\mathrm{Re}(\omega (\varepsilon))}{\mathrm{Re}(\omega(0))}$ (a) and $\frac{\mathrm{Im}(\omega(\varepsilon))}{\mathrm{Im}(\omega(0))}$ (b) for $\ell=2$ QNM spectra in polar parity perturbations with the overtone number $n=0,1,2,3$. We vary the parameter $\varepsilon$ from $0$ to $0.05$ in increments of $\Delta\varepsilon=0.001$. For each color, the dashed line is tangent to the solid line at $\varepsilon=0$. The calculation is performed on the Chebyshev-Lobatto grid with $N=200$.
• Figure 2: $\frac{\mathrm{Re}(\omega (\varepsilon))}{\mathrm{Re}(\omega(0))}$ (a) and $\frac{\mathrm{Im}(\omega(\varepsilon))}{\mathrm{Im}(\omega(0))}$ (b) for $\ell=2$ QNM spectra in axial parity perturbations with the overtone number $n=0,1,2,3$. We vary the parameter $\varepsilon$ from $0$ to $0.05$ in increments of $\Delta\varepsilon=0.001$. For each color, the dashed line is tangent to the solid line at $\varepsilon=0$. The calculation is performed on the Chebyshev-Lobatto grid with $N=200$.
• Figure 3: Time-domain waveform at $\sigma=0$ for polar parity perturbation. (a) The absolute value of $\phi(\tau,0)$ for $\ell=2$ and $\varepsilon$ varying from $0$ to $0.05$ in increments of $\Delta\varepsilon=0.01$, together with $\tau$ varies from $0$ to $250$. (b) The absolute value of the difference between $\phi(\tau,0)$ in the case $\varepsilon=0.01\sim 0.05$ and $\varepsilon=0$. The calculation is performed on the Chebyshev-Lobatto grid with $N=100$.
• Figure 4: Time-domain waveform at $\sigma=0$ for axial parity perturbation. (a) The absolute value of $\phi(\tau,0)$ for $\ell=2$ and $\varepsilon$ varying from $0$ to $0.05$ in increments of $\Delta\varepsilon=0.01$, together with $\tau$ varying from $0$ to $250$. (b) The absolute value of the difference between $\phi(\tau,0)$ in the case $\varepsilon=0.01\sim 0.05$ and $\varepsilon=0$. The calculation is performed on the Chebyshev-Lobatto grid with $N=100$.
• Figure 5: The mismatch of the polar parity $(+)$ and axial parity $(-)$ time-domain waveform between $\varepsilon=0$ and $\varepsilon=0.01$, $0.02$, $0.03$, $0.04$, $0.05$. We set $\tau_{\text{min}}=0$ and $\tau_{\text{max}}=200$. The figure is plotted using double logarithmic coordinates, and the dashed lines are the fitting lines. The calculation is performed on the Chebyshev-Lobatto grid with $N=100$.