Black-hole hair loss: Learning about binary progenitors from ringdown signals

Perturbed Kerr black holes emit gravitational radiation, which (for the practical purposes of gravitational-wave astronomy) consists of a superposition of damped sinusoids termed quasinormal modes. The frequencies and time constants of the modes depend only on the mass and spin of the black hole—a consequence of the no-hair theorem. It has been proposed that a measurement of two or more quasinormal modes could be used to confirm that the source is a black hole and to test if general relativity continues to hold in ultrastrong gravitational fields. In this paper, we propose a practical approach to testing general relativity with quasinormal modes. We will also argue that the relative amplitudes of the various quasinormal modes encode important information about the origin of the perturbation that caused them. This helps in inferring the nature of the perturbation from an observation of the emitted quasinormal modes. In particular, we will show that the relative amplitudes of the different quasinormal modes emitted in the process of the merger of a pair of nonspinning black holes can be used to measure the component masses of the progenitor binary.

Figure 1

This plot shows the relative luminosities, or radiated power, in modes $(2,2)$, $(3,3)$, (4, 4), and $(2,1)$. $L_{lm}$ represent the luminosities (in units $c=G=1$ in which luminosity is dimensionless) and $r_{lm}$ denote the ratios $r_{lm}=L_{lm}/L_{22}$. The different panels correspond to systems with different mass ratios as indicated in the panel. Note that as the mass ratio increases, the luminosity in each mode decreases but the amplitudes of all higher-order modes, relative to the (2, 2) mode, increase. We have omitted—both in the figure and in this work—the next most dominant modes, (5, 5), (3, 2), (4, 3), (6, 6), and (5, 4), as they are generally less than 1% as luminous as the (2, 2) mode (see, however, Pan et al. [51]).

Figure 2

Evolution of the first few dimensionless mode frequencies $f_{\ell m}=M{\omega }_{\ell m}$ as a function of the dimensionless time $t/M$, for different values of the mass ratio $q$ of the progenitor binary. Also shown in arbitrary units is the luminosity in the 22 mode. All mode frequencies, especially $f_{22}$ and $f_{33}$, stop evolving and stabilize soon after the binary merges to form a single black hole. The waveform is assumed to contain a superposition of only quasinormal modes a duration $10M$ after the luminosity in 22 mode reaches its peak.

Figure 3

This plot shows the amplitudes as a function of the mass ratio for different modes at the peak of the luminosity of the 22 mode (circles), an epoch $10M$ and $15M$ after the peak (respectively, squares and triangles). We have plotted the absolute amplitude ${\alpha }_{22}$ of the 22 mode and ratio of the subdominant mode amplitudes ${\alpha }_{\ell m}/{\alpha }_{22}$ relative to 22 (cf. Table ). The solid lines are the best fits [cf. Eqs. (31, 32, 33, 34)] to the amplitudes at $10M$ after the peak luminosity.

Figure 4

The signal-to-noise ratio integrand for LISA for a quasinormal mode signal that is composed of 22, 21, and 33 modes—the three most dominant ones. The source is assumed to be at a redshift of $z=1$ and the various angles are as in Table . The left panel corresponds to a black hole of mass $M=5\times {10}^6{M}_{\odot }$ and the right panel to a black hole of mass $M={10}^7{M}_{\odot }$. In both cases, the mass ratio of the progenitor binary is taken to be $q=10$.

Figure 5

SNR in Advanced LIGO (top set of four panels) and Einstein Telescope (bottom set of four panels) as a function of the black hole’s mass $M$ and progenitor binary’s mass ratio $q$ for different modes. Most of the contribution to the SNR comes from the 22 mode but other modes too have significant contributions, 33 being more important than 21. The source is assumed to be at a distance of 1 Gpc and various angles are as in Table .

Figure 6

Same as Fig. 5 but for LISA and the source is assumed to be at a redshift of $z=1$. Also, the mass ratio is allowed to vary over a larger range from 2 to 25 instead of 2 to 10. The “steps” that can be seen around $40\times {10}^6{M}_{\odot }$ and $80\times {10}^6{M}_{\odot }$ are mostly due to the LISA noise curve.

Figure 7

Dimensionless fractional errors ${\sigma }_k/{\lambda }_k$ in various parameters as a function of the black-hole mass and progenitor binary’s mass ratio. The black hole is assumed to form at a luminosity distance of $D_{\mathrm{L}}=1\mathrm{Gpc}$ and the various angles are assumed to be as in Table . The bottom-right panel plots the correlation coefficient between the luminosity distance and the orientation $\iota$ of the black hole’s spin with respect to the line of sight. The general trend for the errors is to increase with increasing mass ratio and decreasing mass, except for $D_{\mathrm{L}}$ for which there appears to be an “island” around $q=6$. We have used the ET-B sensitivity curve in computing the covariance matrix.

Figure 8

As in Fig. 7, but for supermassive black holes observed in LISA at a redshift of $z=1$. Note also that the mass ratio $q$ in this case is allowed to vary from 2 to 25.

Figure 9

Curves of constant mode frequencies and time constants in the $(M,j)$ plane obtained with Eq. (43) for the (2,2) mode and similar equations for other modes. Top panels correspond to the case when all measured values are exactly as predicted by black-hole perturbation theory. The bottom panels correspond to the case where the (3,3) and (2,1) mode frequencies and time constants differ from the GR value by 10%, but the (2,2) mode frequency and time constant are as in GR. An interesting thing to note is that for a large range of values of the black-hole spin, the curves of constant ${\tau }_{22}$ and ${\tau }_{33}$ almost overlap, providing the opportunity for a more accurate test of the no-hair theorem. (See the text for details.)