Observing binary black hole ringdowns by advanced gravitational wave detectors

The direct discovery of gravitational waves from compact binary systems leads for the first time to explore the possibility of black hole spectroscopy. Newly formed black holes produced by coalescing events are copious emitters of gravitational radiation, in the form of damped sinusoids, the quasinormal modes. The latter provides a precious source of information on the nature of gravity in the strong field regime, as they represent a powerful tool to investigate the validity of the no-hair theorem. In this work we perform a systematic study on the accuracy with which current and future interferometers will measure the fundamental parameters of ringdown events, such as frequencies and damping times. We analyze how these errors affect the estimate of the mass and the angular momentum of the final black hole, constraining the parameter space which will lead to the most precise measurements. We explore both single and multimode events, showing how the uncertainties evolve when multiple detectors are available. We also prove that, for the second generation of interferometers, a network of instruments is a crucial and necessary ingredient to perform strong-gravity tests of the no-hair theorem. Finally, we analyze the constraints that a third generation of detectors may be able to set on the mode’s parameters, comparing the projected bounds against those obtained for current facilities.

Figure 1

Noise spectral densities of the terrestrial interferometers considered in this paper. From top to bottom we show AdLIGO, AdVirgo, and KAGRA at design sensitivity, LIGO A+, Voyager, the Einstein Telescope, and the Cosmic Explorer.

Figure 2

Contour plots for $\mathrm{\Delta }{f}_{22}$ in the $M-j$ plane for the frequency (left panel) and the quality factor (right panel) of the fundamental (2, 2) mode. The plots refer to a BH at $d=400\mathrm{Mpc}$, with radiation efficiency ${\epsilon }_{\mathrm{QNM}}={\epsilon }_{\mathrm{QNM}}^{(2,2)}=0.03$. The white dashed curves identify lines of constant SNR, computed assuming a single detector with the AdLIGO sensitivity. The orange dot identifies a BH configuration with the same mass and spin of GW150914.

Figure 3

Same as Fig. 2, but for contour lines of fixed relative uncertainty for the frequency (left) and the quality factor (right) of the dominant mode with $l=m=2$.

Figure 4

Same as Figs. 2 and 3 but for the relative errors on the BH mass $M$ and spin parameter $j$, derived from $\mathrm{\Delta }{f}_{22}$ and $\mathrm{\Delta }{Q}_{22}$.

Figure 5

Contour plots for fixed relative errors on the frequency/quality factor (left), spin/mass (right), as a function of the number of interferometers, for the optimistic scenario ${\epsilon }_{\mathrm{QNM}}=0.03$.

Figure 6

(Top) Contour plots of relative uncertainty for the frequency of the (3,3) and (4,4) modes, with ${\epsilon }_{\mathrm{QNM}}^{(3,3)}={\epsilon }_{\mathrm{QNM}}^{(4,4)}=0.003$. (Bottom) Curves of fixed accuracy $\mathrm{\Delta }{Q}_{33}=\mathrm{\Delta }{Q}_{44}=(50,100)\%$ for a network of interferometers with an increasing number of detectors.

Figure 7

Contour plots for the critical values ${\mathcal{R}}_s$ and ${\mathcal{R}}_b$ required to distinguish frequencies and quality factors of the (2, 2) and the (3, 3) modes.

Figure 8

Change in the SNR for the secondary mode ${\rho }^{(3,3)}$, as a function of the number of terrestrial interferometers. The white dashed curve corresponds to the threshold ${\rho }^{(3,3)}=5$.

Figure 9

Same as Fig. 7 but for the case of four interferometers observing the ringdown event.

Figure 10

Left and center panels show the relative errors on frequency and quality factor of the (2, 2) and the (3, 3) modes, for second and third generations of interferometers. Uncertainties on the mass and spin computed from the dominant (2, 2) component are drawn in the right plot. We consider three mass configurations $M=(20,50,70)M_{\odot }$, assuming BHs with spin parameter $j=0.9$.

Figure 11

Contour plots for the critical value ${\mathcal{R}}_b$ need to distinguish both the frequency and the quality factor of the (2, 2) and the (3, 3) modes.