Quasi-normal modes ratios as agnostic test of general relativity

Abstract

In this letter, we provide a novel test of general relativity based on ringdown analysis. The test is performed on agnostic models, where the postmerger signal is fitted with a superposition of damped sinusoids. If at least two modes are detected, one has to compute the ratio of the frequencies and of the damping times and compare them against the predictions of general relativity. By considering ratios, the dependency on the black hole's mass is scaled away. Most notably, we find that the ratios vary very little with the spin, the real part depends mostly on the angular momentum of the mode $\ell$ and the imaginary part depends mostly on the overtone number $n$: different combinations create specific mode islands. We provide a qualitative explanation of these islands through a semi-analytical argument. We discuss the application of the method to future detectors. Finally, we show that ratios in alternative theories of gravity or between different field content drastically differ from those of general relativity.

Figures (4)

• Figure 1: Frequency ratio $\mathcal{F}_{j j'}$ versus damping time ratio $\mathcal{T}_{j j'}$ for various combination of $j, j'$. The black dotted lines correspond to $j'=(2,2,0)$ and selected values of $j=(2,2,1),(2,1,0),(2,0,0),(3,3,0),(3,2,0),(4,4,0)$, for values of the spin $a=[0,0.99]$; an open triangle marker is drawn at $a=0$, and a circular one at $a=0.9$. The solid lines represent correspond to $j'=(2,2,0)$ and $j=(\ell,m,n)$, with $\ell=[2,4]$, $m=[-\ell,\ell]$, $n=[0,1]$ for values of the spin $a=[0,0.9]$; different colours correspond to mode combinations of same $\ell/\ell'$. The cyan circle highlights the ratio between the quadratic mode $j=(2,2,0)\times (2,2,0)$ and the fundamental mode $j'=(2,2,0)$. The dashed-opaque lines correspond to $j=(\ell, m, n)$ and $j'=(\ell',m',0)$, with $\ell=[\ell',4]$, $m,n$ varying as before and $\ell'= [2,4]$, $m'=[-\ell',\ell']$, for values of the spin $a=[0,0.9]$. The gridlines are obtained from WKB estimates, see equations \ref{['eq:F_WKB']}--\ref{['eq:T_WKB']}.
• Figure 2: Detectability forecast for QNM ratios of $(\ell,m)=(2,2),(2,1),(2,0),(3,3),(3,2),(4,4)$ modes for a GW150914-like binary with $q=2$ and ringdown SNR=$85$ as if detected by ET. The upper plot shows distributions for $f_{\ell m}$ and $\tau_{\ell m}$, while the lower plot the ratios $\mathcal{F}_{\ell m 2 2}$ and $\mathcal{T}_{\ell m 2 2}$. The light (dark) contours show the $1\sigma$ ($2\sigma$) confidence level of a bi-normal distribution of $2\times10^3$ randomly sampled points. Red dots show the injected values. In the top panel we show with black empty markers GR values of the other $m$ for same mass and spin of the injected values. In the lower panel for $\mathcal{F}_{j 22}$ and $\mathcal{T}_{j 22}$ contours and samples, the colours for each $j$ correspond to those of the upper panel. We also show the GR predictions of ratios between other mode combinations at the spin of the injection $a=0.62$, split in different classes of relevance as described in the text.
• Figure 3: Comparison of $\mathcal{F}_{j j'}$ versus $\mathcal{T}_{j j'}$ in GR and HDG. For GR, we plot the same mode combinations as in figure \ref{['Fig:lmn-220']}, whereas for HDG we selected $j=[(2,2,1),(2,1,0),(3,3,0)]$, $j'=(2,2,0)$ for even/odd-parity perturbations displayed in the left/right panel. For each HDG set of frequencies, we span $a=[0.5,0.7]$ with steps of $\delta a = 0.01$ corresponding to different colors (refer to the bar legend) and we span the coupling constant of the theory as $\lambda=[0,0.3]$ and each step in $\delta\lambda=0.05$ is marked by a dot in each curve.
• Figure 4: Comparison of frequency and damping time ratios for scalar modes $s=0$ over the tensor $(-2,2,2,0)$ mode (solid lines). We also report combinations of $\ell=[2,3]$ modes over the $(-2,2,2,0)$ mode (dotted lines) with the same color scheme of figure \ref{['Fig:lmn-220']}. Open triangles mark the ratios at $a=0$, circular ones at $a=0.9$.