2L MisA Configuration in Gravitational-wave Astronomy

• 2L MisA configuration is a network topology featuring two 15 km L-shaped interferometers misaligned by 45°, which improves localization of gravitational wave signals.
• It employs advanced neural posterior estimation (Dingo-IS) to compress strain data and efficiently infer parameters for high-mass, high-redshift binary black holes.
• Performance metrics show enhanced sky and volume localization compared to triangular designs, although with some trade-offs in distance precision.

The term 2L MisA Configuration designates a network topology for next-generation gravitational-wave (GW) observatories, specifically concerning the Einstein Telescope (ET). In this configuration, two L-shaped interferometers—each with 15 km arms—are sited in Sardinia (ET-S) and the Meuse–Rhine Euroregion (ET-EMR), with their arms oriented at a 45° offset relative to one another. This geometry diverges from the original triangular ET scheme and enables improved sky localization and inference capability for massive, high-redshift binary black hole (BBH) mergers via sophisticated neural posterior estimation (NPE) methods. The following sections detail the configuration’s architectural, algorithmic, and performance properties, along with its scientific implications and trade-offs relative to alternative detector networks (Santoliquido et al., 23 Dec 2025).

1. Geometry and Sensitivity Characteristics

The 2L MisA configuration consists of two ET interferometers:

• Sites and Arm Orientation: ET-S is located at 40.517° N, 9.417° E (Sardinia) and ET-EMR at 50.723° N, 5.921° E (Meuse–Rhine). The chord distance between sites is 1166 km. The ET-EMR interferometer’s arms are rotated by 45° about the local vertical with respect to ET-S.
• Interferometer Specification: Each detector is L-shaped with 15 km arms—significantly longer than existing detectors.
• Sensitivity: Both detectors adopt the “HFLF-cryogenic” design, integrating high-frequency room-temperature and low-frequency cryogenic instruments. The amplitude spectral density (ASD) ensures sensitivity down to 6 Hz, which is essential for detecting short-duration, massive, high-redshift GW sources.

This spatial and orientation configuration constitutes a non-colocated, misaligned L-style baseline designed to break degeneracies inherent in purely triangular or aligned-topologies.

2. Network Response and Bayesian Inference Formalism

Upon GW arrival, each ET detector measures strain $d_i(t)$ according to:

$$d_i(t) = F^+_i(\theta, \phi, \psi) h_+(t) + F^\times_i(\theta, \phi, \psi) h_\times(t) + n_i(t)$$

where $F^+$ and $F^\times$ are the antenna-pattern functions:

$$F^+(\theta,\phi,\psi) = \frac{1}{2}(1+\cos^2 \theta)\cos2\phi\cos2\psi-\cos\theta\sin2\phi\sin2\psi$$

$$F^\times(\theta,\phi,\psi) = \frac{1}{2}(1+\cos^2 \theta)\cos2\phi\sin2\psi+\cos\theta\sin2\phi\cos2\psi$$

Assuming Gaussian stationary noise, the network likelihood is

$$\mathcal{L}(d|\theta)\propto\exp\Biggl[\sum_i\Bigl(d_i\big|h_i(\theta)\Bigr)-\tfrac12\bigl(h_i(\theta)\big|h_i(\theta)\bigr)\Biggr]$$

with inner product

$$(a|b)=4\,\mathrm{Re}\!\int_{f_{\rm min}}^{f_{\rm max}}\frac{a^*(f)\,b(f)}{S(f)}\,df$$

The optimal SNR for signal $\theta_{\text{inj}}$ is computed network-wise,

$$\rho = \sqrt{\sum_i\bigl(h_i(\theta_{\rm inj})\big|h_i(\theta_{\rm inj})\bigr)}$$

This formalism underpins all inference and detection calculations for the 2L MisA network.

3. Neural Posterior Estimation: Dingo-IS Implementation

The computational challenge of analyzing $\mathcal{O}(10^5)$ BBH GW events annually renders standard Bayesian sampling approaches computationally infeasible. The configuration leverages neural posterior estimation (NPE)—specifically, Dingo-IS, which utilizes conditional normalizing flows $q_\phi(\theta|d)$ refined through importance sampling.

Key properties:

• Input Compression: The embedding network reduces the raw strain time series (8 s, 0.125 Hz bins, 6–256 Hz) to 256 features.
• Normalizing-Flow Backbone: $\sim 4\times10^8$ parameters, mapping base Gaussian distributions into complex posteriors.
• Importance Reweighting: For $n=10^5$ samples, weights $w_i$ are assigned via

$$w_i = \frac{\mathcal{L}(d|\theta_i) \pi(\theta_i)}{q_\phi(\theta_i|d)}$$

The effective sample number $n_\text{eff}$ and sample-efficiency $\epsilon = n_\text{eff}/n > 1\%$ are required for proper posterior reliability.

• Validation: Comparative tests using Jensen–Shannon divergence and sky area comparisons ($\Delta\Omega_{90}$) confirm high fidelity to conventional methods.

This approach defines the state-of-the-art for rapid, precise GW parameter inference within the 2L MisA network context.

4. Performance Metrics: High-redshift, High-mass BBH Localization

Restricting attention to sources with $D_L \lesssim 500$ Gpc and $M_d > 100\,M_\odot$, analysis reveals:

• Luminosity Distance Uncertainty: 2L MisA yields broader, often multimodal posteriors in $D_L$ relative to the triangular ET ($\Delta$), mainly due to partial degeneracy between arrival time and sky direction for short merger durations.

| Configuration | Median $\Delta D_L/D_L$ | 90% CI | |-------------------|------------------------|-----------------| | $\Delta$ (triangular) | 0.25 | [0.10–0.60] | | 2L MisA | 0.35 | [0.15–0.75] | | 2L MisA + CE | 0.20 | [0.08–0.45] |

• Sky Localization: 2L MisA greatly reduces ambiguous sky islands compared to $\Delta$:

| Configuration | $<10\,\mathrm{deg}^2$ | $<100\,\mathrm{deg}^2$ | $<1000\,\mathrm{deg}^2$ | |-------------------|----------------------|------------------------|-------------------------| | $\Delta$ (triangular) | 0.7% | 8.5% | 26.2% | | 2L A (aligned) | 0.3% | 4.4% | 24.3% | | 2L MisA | 3.6% | 15.0% | 34.8% | | 2L MisA + CE | 36.7% | 71.5% | 94.2% |

• Multimodalities: 2L MisA typically produces 2–6 sky modes for most events, whereas $\Delta$ yields up to 8 disconnected islands in $\gtrsim 80\%$ of cases. With the addition of CE, $\gtrsim 90\%$ of events are localized to a single sky island.
• Volume Localization: The 90% comoving volume $\Delta V_{90}$ for 2L MisA shrinks by $\sim 30$–$50\%$ compared to the triangular topology.

This suite of metrics demonstrates 2L MisA’s superiority in spatial and volume inference for high-$z$, high-mass BBH signals, with trade-offs in distance precision.

5. Comparative Advantages and Scientific Trade-offs

The 2L MisA configuration’s distinctive misaligned, non-colocated design yields:

• Sky Degeneracy Mitigation: Breaks the eight-fold degeneracy endemic to the triangular ET, leading to more precise sky– and volume–localization.
• Polarization Coverage: Slightly less than optimal polarization sampling compared to the triangular scheme.
• Distance Inference: Luminosity distance posteriors are degraded in $\sim 20\%$ of cases, with multimodal structures; incorporating Cosmic Explorer (CE) largely restores unimodality and precision.
• Information Gain: Quantified via KL divergence, 2L MisA outperforms triangular ET but does not reach the hybrid networks’ performance when CE or additional LIGO sites are included.

A plausible implication is that the 2L MisA approach provides a cost-effective and operationally tractable pathway to improve GW source localization, especially when synergistically deployed alongside widely spaced detectors such as CE.

6. Operational Guidelines and Future Prospects

Deployment and optimization of the 2L MisA configuration is governed by:

• Baseline Optimization: 1166 km separation achieves sufficient arrival time differentiation for breaking spatial degeneracies while minimizing infrastructure complexity.
• Instrument Sensitivity: HFLF-cryogenic design curves are necessary for maximizing detection rates below 10 Hz.
• Algorithm Selection: Dingo-IS NPE enables tractable scaling to $\sim 10^5$ events/year; importance sampling ensures robust posteriors even when multimodal support arises.

The scientific potential of the 2L MisA configuration is expected to be fully realized through its integration within a global XG GW network that includes the Cosmic Explorer, promising precision cosmology and population studies of early-Universe and intermediate-mass BBHs (Santoliquido et al., 23 Dec 2025).