Fast and accurate parameter estimation of high-redshift sources with the Einstein Telescope

TL;DR

This paper tackles the challenge of parameter estimation for high-redshift gravitational-wave sources expected to be detected by the Einstein Telescope. It demonstrates that neural posterior estimation via Dingo-IS can produce fast, accurate posteriors, validated against Bilby and contrasted with Fisher-matrix predictions, and reveals eight sky-mode degeneracies intrinsic to the ET triangular design. The study shows that FIM significantly underestimates sky localization uncertainties and can misrepresent distance errors at high redshift, while NPE achieves comparable results with substantially lower computational and energy cost. The findings have important implications for cosmology with dark sirens and for the design and analysis strategies of 3G detectors, though they rely on certain simplifications (e.g., a single ET triangle, fixed low-frequency cut) that motivate future extensions to more realistic detector networks and waveform models.

Abstract

The Einstein Telescope (ET), along with other third-generation gravitational wave (GW) detectors, will be a key instrument for detecting GWs in the coming decades. However, analyzing the data and estimating source parameters will be challenging, especially given the large number of expected detections-on the order of $10^5$ per year-which makes current methods based on stochastic sampling impractical. In this work, we use Dingo-IS to perform neural posterior estimation (NPE) of high-redshift events detectable with ET in its triangular configuration. NPE is a likelihood-free inference technique that leverages normalizing flows to approximate posterior distributions. After training, inference is fast, requiring only a few minutes per source, and accurate, as corrected through importance sampling and validated against standard Bayesian inference methods. To confirm previous findings on the ability to estimate parameters for high-redshift sources with ET, we compare NPE results with predictions from the Fisher information matrix (FIM) approximation. We find that NPE correctly recovers the eight degenerate sky modes induced by the triangular detector geometry, missed by the FIM analysis, resulting in an underestimation of sky localization uncertainties for most sources. FIM also overestimates the uncertainty in luminosity distance by a factor of $\sim 3$ on average when the injected luminosity distance is $d^{\mathrm{inj}}_{\mathrm{L}} > 10^5~$Mpc, further confirming that ET will be particularly well suited for studying the early Universe.

Figures (16)

• Figure 1: Locations and orientations of the three independent detectors (ET-1, ET-2, and ET-3) constituting ET-$\Delta$. See Sec. \ref{['sec:detectors']} for further details.
• Figure 2: The injected signal $h(\theta_i)$ of Event 1 projected on ET-1 in dark blue, the corresponding data $d_i$ after adding noise $n_i$ in light blue, and the adopted amplitude spectral density (ASD) in orange. See Sec \ref{['sec:detectors']} and Sec. \ref{['sec:single']} for further details.
• Figure 3: The $\log$ loss as a function of training epochs, shown as a solid line for the training set and a dashed line for the test set. See Sec. \ref{['sec:dingo']} for details.
• Figure 4: For each posterior sample drawn for Event 1, we compare the inferred density $\log q(\theta | d)$ with the normalized posterior $\log \mathcal{L}(d | \theta) + \log \pi(\theta) - \log \mathcal{Z}(d)$. The difference between these log densities determines the importance weights (Eq. \ref{['eq:weights']}), which are color-coded. The sample efficiency is $\sim 16\%$. See Sec. \ref{['sec:dingo']} for details.
• Figure 5: Marginalized one- and two-dimensional posterior distributions for Event 1 over all sets of parameters, comparing Dingo-IS (light blue), Bilby (orange), and GWFish+Priors (black). Vertical and horizontal lines mark the true injected values (see Table \ref{['tab:jsd']}). Contours represent 68% and 95% credible regions. Sky modes are discussed in Sec. \ref{['sec:skyloc']}. See Sec. \ref{['sec:single']} for other details.