Parameter estimation of gravitational wave echoes from exotic compact objects

Relativistic ultracompact objects without an event horizon may be able to form in nature and merge as binary systems, mimicking the coalescence of ordinary black holes. The postmerger phase of such processes presents characteristic signatures, which appear as repeated pulses within the emitted gravitational waveform, i.e., echoes with variable amplitudes and frequencies. Future detections of these signals can shed new light on the existence of horizonless geometries and provide new information on the nature of gravity in a genuine strong-field regime. In this work we analyze phenomenological templates used to characterize echolike structures produced by exotic compact objects, and we investigate for the first time the ability of current and future interferometers to constrain their parameters. Using different models with an increasing level of accuracy, we determine the features that can be measured with the largest precision, and we span the parameter space to find the most favorable configurations to be detected. Our analysis shows that current detectors may already be able to extract all the parameters of the echoes with good accuracy, and that multiple interferometers can measure frequencies and damping factors of the signals at the level of percent.

Figure 1

Sketch of some phenomenological waveforms used in this work. The center panel also shows the meaning of the two parameters $t_{\text{echo}}$ and $\mathrm{\Delta }t$, which identify the time shift between different pulses in the template.

Figure 2

Relative (percentage) errors on the parameters of the template echoI as a function of the number of echoes. All the results are derived for Advance LIGO, assuming $\delta ={10}^{-10}$ and ${\beta }_1=0.003$. The bottom right panel shows the change of the signal-to-noise ratio due to the increasing numbers of echoes.

Figure 3

Relative (percentage) errors on the parameters of the template echoI computed for Advanced LIGO, as a function of the width of the Gaussian width ${\beta }_1$. Both panels refer to $t_{\text{echo}}\approx \mathrm{\Delta }t\approx 2.95\times {10}^{-2}$ ($\delta ={10}^{-10}$).

Figure 4

Contour plots in the ${\beta }_1-t_{\text{echo}}$ parameter space for the relative errors on the echoI parameters. White and black dashed curves represent configurations with fixed accuracy. The data refer to Advanced LIGO at design sensitivity.

Figure 5

Errors on the echoI template for different GW interferometers. The data refer to the best case scenario with ${\beta }_1=0.006$ (and different values of $t_{\text{echo}}$).

Figure 6

(Left and center panels) Same as Fig. 3 but for the parameters of the echoIIa model with phase shift $\varphi =0$ and $\varphi =-\pi /2$. (Right panel) Comparison between the SNR of echoI and echoIIa as a function of ${\beta }_1$. All the results refer to Advanced LIGO, assuming $t_{\text{echo}}\approx \mathrm{\Delta }t$, with $\delta ={10}^{-10}$.

Figure 7

Correlation coefficients for the echoIIa template between $f_1$ and $({\beta }_1,f_2)$, assuming $\varphi =0$ (empty dots) and $\varphi =-\pi /2$ (empty dots), for LIGO with $\delta ={10}^{-10}$ and $t_{\text{echo}}\approx \mathrm{\Delta }t$.

Figure 8

Same as Fig. 4 but for the parameters of echoIIb in the ${\beta }_1\times {\beta }_2$ plane. The phase offset is fixed to $\varphi =-\pi /2$.

Figure 9

Each dot specifies a configurations in the ${\beta }_1\times {\beta }_2$ plane with $\varphi =0$ for which the relative error ${\epsilon }_{\alpha }>1$.

Figure 10

Relative errors of the echoIIb model for the best case scenario with $\varphi =-\pi /2$, computed for different interferometers.