When black holes collide: Probing the interior composition by the spectrum of ringdown modes and emitted gravitational waves

The merger of colliding black holes (BHs) should lead to the production of ringdown or quasinormal modes (QNMs), which may very well be sensitive to the state of the interior. We put this idea to the test with a recent proposal that the interior of a BH consists of a bound state of highly excited, long, closed, interacting strings; figuratively, a collapsed polymer. We show, using scalar perturbations for simplicity, that such BHs do indeed have a distinct signature in their QNM spectrum: A new class of modes whose frequencies are parametrically lower than the lowest-frequency mode of a classical BH and whose damping times are parametrically longer. The reason for the appearance of the new modes is that our model contains another scale, the string length, which is parametrically larger than the Planck length. This distinction between the collapsed-polymer model and general-relativistic BHs could be made with gravitational-wave observations and offers a means for potentially measuring the strength of the coupling in string theory. For example, GW150914 already allows us to probe the strength of the string coupling near the regime which is predicted by the unification of the gravitational and gauge-theory couplings. We also derive bounds on the amplitude of the collapsed-polymer QNMs that can be placed by current and future gravitational-wave observations.

Figure 1

QNM spectrum of putative polymer modes for GW150914 with various $g_s^2$, as well as noise spectral density against frequency. For $g_s^2=1$, the QNM amplitude, frequency and damping time for the polymer modes are the same as those of a classical BH. The ratio between the signal and noise roughly corresponds to the SNR. The spectrum is detectable if this ratio is above the threshold $(\sim 5)$.

Figure 2

Top: SNR of the putative QNM of a collapsed polymer for GW150914 as a function of $g_s^2$ (red, solid). The SNR scales with $g_s^3$ (blue, dashed) for $g_s^2\sim 1$ as explained in Sec. 2c. The SNR threshold of 5 (black, dot-dashed) allows us to constrain $g_s^2$ as $g_s^2\lesssim 0.65$. As the detector sensitivity increases, one will be able to probe $g_s^2$ for the unification of the gravitational and gauge theory couplings (green dot). Bottom: Same as the top panel but for GW151226.

Figure 3

Projected upper bound on $g_s^2$ as a function of the total mass of an equal-mass BH binary at various distances apart using aLIGO’s design sensitivity. 410 Mpc corresponds to the distance for GW150914 [28]. The horizontal line represents $g_s^2=4\pi /25$. The upper bound on $g_s^2$ scales with $r^{2/3}$.

Figure 4

Upper bound on the relative amplitude $\gamma$ of the polymer QNMs with respect to the BH QNMs as a function of $g_s^2$. The thinner red, solid curve is the current bound from GW150914 with aLIGO’s O1 run, while the thicker red, solid curve is the projected bound for a GW150914-like event with aLIGO’s design sensitivity. Blue, dashed lines are the analytic prediction for the upper bound on $\gamma$ valid around $g_s^2=1$, while green dots are the bounds at $g_s^2=4\pi /25$. The black, dot-dashed line is the predicted relative amplitude proportional to the $g_s^4$ scaling (see Sec. 2c).