Spectroscopy of Kerr Black Holes with Earth- and Space-Based Interferometers

We estimate the potential of present and future interferometric gravitational-wave detectors to test the Kerr nature of black holes through “gravitational spectroscopy,” i.e., the measurement of multiple quasinormal mode frequencies from the remnant of a black hole merger. Using population synthesis models of the formation and evolution of stellar-mass black hole binaries, we find that Voyager-class interferometers will be necessary to perform these tests. Gravitational spectroscopy in the local Universe may become routine with the Einstein Telescope, but a 40-km facility like Cosmic Explorer is necessary to go beyond $z\sim 3$. In contrast, detectors like eLISA (evolved Laser Interferometer Space Antenna) should carry out a few—or even hundreds—of these tests every year, depending on uncertainties in massive black hole formation models. Many space-based spectroscopical measurements will occur at high redshift, testing the strong gravity dynamics of Kerr black holes in domains where cosmological corrections to general relativity (if they occur in nature) must be significant.

Figure 1

Noise PSDs for various space-based and advanced Earth-based detector designs. “$\mathrm{N}i\mathrm{A}k$” refers to non-sky-averaged eLISA PSDs with pessimistic (N1) and optimistic (N2) acceleration noise and armlength $L=k\mathrm{Gm}$ (cf. [20]). In the high-frequency regime, we show noise PSDs for (top to bottom): the first AdLIGO observing run (O1); the expected sensitivity for the second observing run (O2) and the advanced LIGO (AdLIGO) design sensitivity [21]; the pessimistic and optimistic ranges of AdLIGO designs with squeezing ($\mathrm{A}+$, $\mathrm{A}++$) [22]; Vrt and Voyager [23, 24]; Cosmic Explorer (CE1), basically $\mathrm{A}+$ in a 40-km facility [25]; CE2 wide and CE2 narrow, i.e., 40-km detectors with Voyager-type technology but different signal extraction tuning [24, 26]; and two possible Einstein Telescope designs, namely, ET-B [27] and ET-D in the “xylophone” configuration [28].

Figure 2

Rates of binary BH mergers that yield detectable ringdown signals (filled symbols) and allow for spectroscopical tests (hollow symbols). Left panel: Rates per year for Earth-based detectors of increasing sensitivity. Right panel: Rates per year for six-link (solid) and four-link (dashed) eLISA configurations with varying armlength and acceleration noise.

Figure 3

Left: Redshift distribution of events with $\rho >8$ (top) and $\rho >{\rho }_{\mathrm{GLRT}}$ (bottom) for model M1 and Earth-based detectors. In the bottom-left panel, the estimated AdLIGO rate ($\approx 2.6\times {10}^{-2}\text{\hspace{0.17em}events}/\text{year}$) is too low to display. Right: Same for models Q3nod, Q3d, and Pop III. Different eLISA design choices have an almost irrelevant impact on the distributions.