Spectroscopic analysis of stellar mass black-hole mergers in our local universe with ground-based gravitational wave detectors

Motivated by the recent discoveries of binary black-hole mergers by the Advanced Laser Interferometer Gravitational-Wave Observatory (Advanced LIGO), we investigate the prospects of ground-based detectors to perform a spectroscopic analysis of signals emitted during the ringdown of the final Kerr black hole formed by a stellar mass binary black-hole merger. Although it is unlikely that Advanced LIGO can measure multiple modes of the ringdown, assuming an optimistic rate of $240{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$, upgrades to the existing LIGO detectors could measure multiple ringdown modes in $\sim 6$ detections per year. New ground-based facilities such as Einstein Telescope or Cosmic Explorer could measure multiple ringdown modes in over 300 events per year. We perform Monte Carlo injections of $10^6$ binary black-hole mergers in a search volume defined by a sphere of radius 1500 Mpc centered at the detector, for various proposed ground-based detector models. We assume a uniform random distribution in component masses of the progenitor binaries, sky positions and orientations to investigate the fraction of the population that satisfies our criteria for detectability and resolvability of multiple ringdown modes. We investigate the detectability and resolvability of the subdominant modes $l=m=3$, $l=m=4$ and $l=2$, $m=1$. Our results indicate that the modes with $l=m=3$ and $l=2$, $m=1$ are the most promising candidates for subdominant mode measurability. We find that for stellar mass black-hole mergers, resolvability is not a limiting criteria for these modes. We emphasize that the measurability of the $l=2$, $m=1$ mode is not impeded by the resolvability criterion.

Figure 1

The following are sensitivity models for each detector [23] we consider in our study. The aLIGO curve corresponds to the design sensitivity of Advanced LIGO and the $\mathrm{A}+$ curve to the proposed upgrade to the Advanced LIGO detectors. The CE and the ET are two of the proposed next-generation ground-based detectors. We also perform the analysis with a flat noise curve at a strain per $\sqrt[2]{\mathrm{Hz}}$ of ${10}^{-25}$, to infer some conclusions which are independent of the shape of the noise curve. The shaded region shows the frequency band that corresponds to optimal tuning of the detectors for ringdown searches.

Figure 2

This figure presents the magnitude of mode amplitudes $\parallel A\parallel$ predicted by the fitting formulas given in [36] as a function of dimensionless symmetric mass ratio $\eta$. Comparing the amplitudes of different modes, we infer that the potential candidates for subdominant mode measurability correspond to $l=m=3$, $l=2$, $m=1$ and $l=m=4$.

Figure 3

We show the dimensionless central frequency of QNMs as a function of symmetric mass ratio $\eta$ as predicted by [35]. Note that modes with different $l$ have central frequencies that are well separated. One could naively expect that resolving modes with the same $l$ could be challenging. However, for stellar mass black-hole mergers this is not the case.

Figure 4

These contour plots show the differences in the central frequencies of the subdominant modes, $l=2$, $m=1$, $l=m=3$ and $l=m=4$, with the dominant mode. The color bar presents a measure of frequency difference in Hz. Notice that the central frequency of the $l=m=3$ and $l=m=4$ subdominant mode differs from the dominant mode by hundreds of Hz. It would be right to assume that resolvability of these modes is not challenging. However, it is very interesting to note that even for the $l=2$, $m=1$ subdominant mode the central frequency is separated by at least 20 Hz from the central frequency of the dominant mode. This is consistent with the fact that our results indicate that resolvability is not a limiting factor.

Figure 5

Scatter plots of all points that allow for measurability of subdominant modes in our analysis using a flat detector sensitivity curve at a strain of ${10}^{-25}{\sqrt[2]{\mathrm{Hz}}}^{-1}$. The x and y axes of these plots correspond to the central frequencies of $l=m=2$ and the measurable subdominant modes in Hz, respectively. From these plots, we can infer that if one were to perform detector detuning optimized towards a spectroscopic analysis of stellar mass black holes, a frequency band around 300 to 500 Hz would be the best choice for narrow banding.