Testing the no-hair theorem with black hole ringdowns using TIGER

The Einstein Telescope, a proposed third-generation gravitational-wave observatory, would enable tests of the no-hair theorem by looking at the characteristic frequencies and damping times of black hole ringdown signals. In previous work it was shown that with a single $500–1000M_{\odot }$ black hole at a distance $\lesssim 6$ Gpc (or redshift $z\lesssim 1$), deviations of a few percent in the frequencies and damping times of dominant and subdominant modes would be within the range of detectability. Given that such sources may be relatively rare, it is of interest to see how well the no-hair theorem can be tested with events at much larger distances and with smaller signal-to-noise ratios, thus accessing a far bigger volume of space and a larger number of sources. We employ a model-selection scheme called TIGER (Test Infrastructure for GEneral Relativity), which was originally developed to test general relativity with weak binary coalescence signals that will be seen in second-generation detectors, such as Advanced LIGO and Advanced Virgo. TIGER is well suited for the regime of low signal-to-noise ratios, and information from a population of sources can be combined so as to arrive at a stronger test. By performing a range of simulations using the expected noise power spectral density of the Einstein Telescope, we show that with TIGER, similar deviations from the no-hair theorem (such as those considered in previous works) will be detectable with great confidence using $\mathcal{O}(10)$ sources distributed uniformly in a comoving volume out to $50\mathrm{Gpc}(z\lesssim 5)$.

Figure 1

The strain sensitivity of the Einstein Telescope as envisaged in Ref. [18], labeled ET-D, compared with the older ET-B curve.

Figure 2

Single-source GR background distributions (dark gray) and foreground distributions (light gray) for a 10% deviation in ${\omega }_{22}$ (left), a 10% deviation in ${\omega }_{33}$ (middle), and a 10% deviation in ${\tau }_{22}$ (right). In all three cases there is significant overlap between background and foreground; for a maximum tolerable false-alarm probability of $\beta =0.05$, the efficiencies are, respectively, 47%, 46%, and 5%.

Figure 3

GR background distributions (dark gray) and foreground distributions (light gray) for a 10% deviation in ${\omega }_{22}$ (left), a 10% deviation in ${\omega }_{33}$ (middle), and a 10% deviation in ${\tau }_{22}$ (right). This time we considered catalogs of ten sources each. Again with $\beta =0.05$, this time efficiencies of 98% are attained for the two mode frequencies. On the other hand, the deviation in ${\tau }_{22}$ remains hard to detect, with an efficiency of only 14%.

Figure 4

Growth of the efficiency $\zeta$ with the number of sources per catalog, for a 10% deviation in ${\omega }_{22}$ (left), a 10% deviation in ${\omega }_{33}$ (middle), and a 10% deviation in ${\tau }_{22}$ (right), for maximum tolerable false-alarm probabilities $\beta =0.05$ and $\beta =0.01$, respectively. In order to understand uncertainties in $\zeta$ due to having a finite number of catalogs, the available simulated sources were randomly combined into catalogs to obtain 1000 different realizations. Shown are the median efficiencies (solid and dashed lines) and 95% confidence intervals.

Figure 5

Growth of the efficiency with the number of sources per catalog, this time for a 25% deviation in ${\tau }_{22}$, again for maximum tolerable false-alarm probabilities of $\beta =0.05$ and $\beta =0.01$. As in Fig. 4, medians and 95% confidence intervals for $\zeta$ are shown, obtained from combining simulated sources into catalogs in many different ways.

Figure 6

Left: The cumulative number of times that a given hypothesis (any of the $H_{i_1{i}_2\dots i_k}$, or ${\mathcal{H}}_{\mathrm{GR}}$) has the largest Bayes factor against noise ($B_{\text{noise}}^{i_1{i}_2\dots i_k}$, or $B_{\text{noise}}^{\mathrm{GR}}$), for single sources up to an SNR of 30. Right: The same, but focusing on SNRs up to 12.

Figure 7

The progression of combined log Bayes factors within an example catalog of 20 sources. Shown are $\ln {\mathcal{B}}_{\mathrm{GR}}^{i_1{i}_2\dots i_k}$, as well as the log odds ratio $\ln {\mathcal{O}}_{\mathrm{GR}}^{\text{modGR}}$, with an increasing number of sources (sorted by SNR), for the case where the injections have $\delta {\widehat{\omega }}_{22}=0.1$.

Figure 8

Top panels: Posterior density functions for $\delta {\widehat{\omega }}_{22}$ (left), $\delta {\widehat{\omega }}_{33}$ (middle), and $\delta {\widehat{\tau }}_{22}$ (right), both for a single source at a distance of 20.69 Gpc ($z=2.47$) with an SNR of 19.14, and for a catalog of 20 sources. Bottom: Evolution of medians and 95% confidence intervals of PDFs as more and more sources are included.