Observational tests of the black hole area increase law

The black hole area theorem implies that when two black holes merge, the area of the final black hole should be greater than the sum of the areas of the two original black holes. We examine how this prediction can be tested with gravitational-wave observations of binary black holes. By separately fitting the early inspiral and final ringdown stages, we calculate the posterior distributions for the masses and spins of the two initial and the final black holes. This yields posterior distributions for the change in the area and thus a statistical test of the validity of the area increase law. We illustrate this method with a GW150914-like binary black hole waveform calculated using numerical relativity, and detector sensitivities representative of both the first observing run and the design configuration of Advanced LIGO. We obtain a $\sim 74.6\%$ probability that the simulated signal is consistent with the area theorem with current sensitivity, improving to $\sim 99.9\%$ when Advanced LIGO reaches design sensitivity. An important ingredient in our test is a method of estimating when the postmerger signal is well fit by a damped sinusoid ringdown waveform.

Figure 1

Posterior distribution for the sky location (right ascension $\alpha$ and declination $\delta$) obtained from the full simulated signal. The colorbar shows the signal-to-noise ratio (SNR), which is a function of the likelihood, and the red lines indicate the values that were injected. The center and outer dashed lines in the histograms represent the median value and the 90% credible interval, respectively. These correspond to the values and the errors given on top of the histograms.

Figure 2

(Top) Posterior distribution of the chirp mass and the effective spin obtained from three different runs. The red cross indicates the injected values. (Bottom) Jensen-Shannon divergence between consecutive grid runs. The $x$ axis indicates the corresponding time $t_{\mathrm{grid}}$ and the JS divergence is calculated between the posterior distributions at $t_{\mathrm{grid}}$ and $t_{\mathrm{grid}}+\mathrm{\Delta }t$, where $\mathrm{\Delta }t=2.5\mathrm{ms}$. The colored circles indicate the times corresponding to the posterior distributions in the top figure. The chosen transition time is $t_{\mathrm{grid}}-t_{\mathrm{ref}}=-27.5\mathrm{ms}$.

Figure 3

(Top) Posterior distribution of the final mass $M_f$ and the final spin ${\chi }_f$ obtained from three different runs. The red cross indicates the injected values. (Bottom) Jensen-Shannon divergence between consecutive grid runs. The $x$ axis indicates the corresponding time $t_{\mathrm{grid}}$ and the JS divergence is calculated between the posterior distributions at $t_{\mathrm{grid}}$ and $t_{\mathrm{grid}}+\mathrm{\Delta }t$, where $\mathrm{\Delta }t=0.5\mathrm{ms}$. The colored circles indicate the times corresponding to the posterior distributions in the top figure. The chosen transition time is $t_{\mathrm{grid}}-t_{\mathrm{ref}}=0.5\mathrm{ms}$.

Figure 4

Whitened strain in each detector with the maximum posterior (MAP) waveform from the inspiral analysis (blue) and the ringdown analysis (orange).

Figure 5

(Top) Posterior distribution on the ratio of the final to initial areas, $A_f/(A_1+A_2)$, for two different Advanced LIGO sensitivities, O1 and ZDHP. The shaded region $A_f/A_i<1$ indicates violation of the area theorem, and $A_f/A_i>4$ indicates violation of the conservation of energy. The vertical red line is the expected area increase. (Bottom) Distribution of the ratio $\mathrm{\Delta }{A}_{\text{measured}}/\mathrm{\Delta }{A}_{\text{expected}}$. The measured area change corresponds to the distribution shown in the top figure. The expected area change is given by the initial parameters obtained in the inspiral analysis and the corresponding expected final parameters from the fitting formulas in [47]. The vertical red line indicates agreement between the measured and the expected values, i.e., $\mathrm{\Delta }{A}_{\text{measured}}/\mathrm{\Delta }{A}_{\text{expected}}=1$.

Figure 6

Jensen-Shannon divergence between consecutive grid times of the inspiral (top) and ringdown (bottom) analyses using the ZDHP PSD. Insets show the resulting posterior distribution for the chosen transition times. The light grey posteriors correspond to the results obtained with O1 sensitivity. For the inspiral (ringdown) analysis we find that the JS divergence settles at $t_{\mathrm{grid}}-t_{\mathrm{ref}}=-27.5\mathrm{ms}$ ($t_{\mathrm{grid}}-t_{\mathrm{ref}}=0.5\mathrm{ms}$), consistent with the O1 PSD results shown in Fig. 2 (Fig. 3).

Figure 7

(Top) Posterior distribution on the energy radiated during the coalescence. The red line indicates the expected value for the injected parameters, $E_i-E_f\simeq 3.4M_{\odot }$. The shaded region shows the theoretical limit of 29% in the energy emitted. (Bottom) Distribution of the ratio $\mathrm{\Delta }{E}_{\text{measured}}/\mathrm{\Delta }{E}_{\text{expected}}$. The expected radiated energy is given by the initial parameters obtained in the inspiral analysis and the corresponding expected final parameters from the fitting formulas in [47]. The vertical red line indicates agreement between the measured and the expected values, i.e., $\mathrm{\Delta }{E}_{\text{measured}}/\mathrm{\Delta }{E}_{\text{expected}}=1$.

Figure 8

Lines of constant area as a function of the final mass and spin. The solid line shows the expected value, while dashed lines indicate areas $\pm 25\%,50\%,75\%$. For comparison, the 90% credible interval from the ringdown analysis with O1 sensitivity is shown.

Figure 9

Posterior distribution on the ratio of the final to initial areas, $A_f/(A_1+A_2)$, with ZDHP sensitivity. The shaded region $A_f/A_i<1$ indicates violation of the area theorem. The shaded region $A_f/A_i>4$ indicates violation of the conservation of energy. Vertical lines indicate the expected value for each case. The dashed posterior distribution is the result shown in the previous section. The solid posterior distribution is a signal that slightly violates the area theorem, and the dotted posterior distribution is a signal that clearly violates the area theorem.