Echoes from the abyss: Tentative evidence for Planck-scale structure at black hole horizons

In classical general relativity (GR), an observer falling into an astrophysical black hole is not expected to experience anything dramatic as she crosses the event horizon. However, tentative resolutions to problems in quantum gravity, such as the cosmological constant problem, or the black hole information paradox, invoke significant departures from classicality in the vicinity of the horizon. It was recently pointed out that such near-horizon structures can lead to late-time echoes in the black hole merger gravitational wave signals that are otherwise indistinguishable from GR. We search for observational signatures of these echoes in the gravitational wave data released by the advanced Laser Interferometer Gravitational-Wave Observatory (LIGO), following the three black hole merger events GW150914, GW151226, and LVT151012. In particular, we look for repeating damped echoes with time delays of $8M\log M$ ($+\text{spin}$ corrections, in Planck units), corresponding to Planck-scale departures from GR near their respective horizons. Accounting for the “look elsewhere” effect due to uncertainty in the echo template, we find tentative evidence for Planck-scale structure near black hole horizons at false detection probability of 1% (corresponding to $2.5\sigma$. significance level). Future observations from interferometric detectors at higher sensitivity, along with more physical echo templates, will be able to confirm (or rule out) this finding, providing possible empirical evidence for alternatives to classical black holes, such as in “firewall” or “fuzzball” paradigms.

In this paper, we use 2-tailed Gaussian probability to assign a significance to a p-value, e.g., $1-\mathrm{p}\text{-value}=68\%$ and 95% correspond to $1\sigma$ and $2\sigma$, respectively.

Figure 1

Spacetime depiction of gravitational wave echoes from a membrane/firewall on the stretched horizon, following a black hole merger event.

Figure 2

LIGO original template for GW150914, along with our best-fit template for the echoes.

Figure 3

Amplitude spectral densities (ASD’s) of our best-fit echo template [Eq. (9)] and the main event, for GW150914. Since we have a quasi-periodic model, there are resonances in the spectrum. The ASDs are the square root of the power spectral densities, which are averages of the square of the fast Fourier transforms of the data. The noise spectra from Hanford and Livingston detectors are also shown.

Figure 4

Best-fit (or maximum) ${\mathrm{SNR}}^2$ near the expected time of merger echoes (Eq’s. (1) and (6), for the combined (top) and GW150914 (bottom) events. The significance of the peaks is quantified by the p-value of their ${\mathrm{SNR}}_{\max }$ within the gray rectangle (see Appendix pp5 for detail of calculation).

Figure 5

Average number of noise peaks higher than a particular SNR-value within a time interval $2\%\times {\overline{\mathrm{\Delta }t}}_{\text{echo}}$ for combined (left) and GW150914 (right) events. The red dots show the observed SNR peak at $t_{\text{echo}}=1.0054\mathrm{\Delta }{t}_{\text{echo}}$ (Fig. 4). The horizontal bar shows the correspondence between SNR values and their significance.

Figure 6

Best-fit (or maximum) ${\mathrm{SNR}}^2$ near the expected time of merger echoes (similar to the Fig. 4 in the main text), for all the three events.

Figure 7

(a)  p-value for our maximum SNR, as a function of $\gamma$, computed within a time interval $(-1\%,1\%)\times {\overline{\mathrm{\Delta }t}}_{\text{echo}}$ for GW150914 (top) and combined events (bottom). (b)  ${\mathrm{SNR}}^2$ near the expected time of best-fit merger echoes [Eq. (6)] for GW150914 in Hanford (red) and Livingston (green) detectors. Not only do we see that the two detectors see coincident SNR peaks, but also their ratio $2.42/3.49=0.69$ is comparable to the SNR ratio for the main events $13.3/18.6=0.72$ seen in the two detectors. Note that, unlike Fig. 4 in the main text, here we have fixed the echo parameters to their best-fit values for combined detectors [36].

Figure 8

Same as Fig. 4 in the main text, but over an extended range of $x=\frac{t_{\text{echo}}-t_{\text{merger}}}{\mathrm{\Delta }{t}_{\text{echo}}}$. The SNR peaks at the predicted value of $x=1$ have false detection probability of 0.11 (0.011) and significance of $1.6\sigma$ ($2.5\sigma$), for GW150914 (combined events) (See also [36]).

Figure 9

Best-fit templates for LIGO main events and echoes (using the joint best fit described in the main text), in Fourier space (similar to Fig. 3 in the main text). The amplitude spectral distribution (ASD) for each detector is shown for comparison.

Figure 10

Resulting prior distribution on $x=\frac{t_{\text{echo}}-t_{\text{merger}}}{\mathrm{\Delta }{t}_{\text{echo}}}$, assuming a random phase for the echo template.

Figure 11

An alternative false detection probability (p-value) as a function of uncertainty in $t_{\text{echo}}$ defined in Eq. (c1).

Figure 12

Distribution of p-values for combined events for different stretches of data within 1 minute of the main events. Surprisingly, the blue line which is closest to the main event, and is used to define p-value in the main text (Fig. 5), happens to give the smallest p-value. The shaded region depicts the Poisson error range for blue histogram, showing that the variation in p-values is clearly much larger. We interpret this as non-Gaussianity and/or nonstationarity in the LIGO noise properties. Here the y-axis on the left (right) shows p-value (number of higher peaks) within the mentioned range of data. The total number of “peaks” considered in each histogram is $(38-9)/0.02=1450$.

Figure 13

SNR peaks and their significance: We maximize SNR along the red lines for fixed values of $x=\frac{t_{\text{echo}}-t_{\text{merger}}}{\mathrm{\Delta }{t}_{\text{echo}}}$, which is used to search for best-fit echo parameters, within $1\sigma$ region for $\mathrm{\Delta }{t}_{\text{echo}}$ (vertical lines). In particular, the grey trapezoid ($x=1\pm 0.05$) corresponds to Fig. 4 in the main text. After we find an SNR peak at some value of $x\simeq 1$, we quantify its significance by how often a comparable ${\mathrm{SNR}}_{\max }$ can be found along the blue lines with unit slope for $x\gg 1$.