A numerical framework for Newtonian-noise estimation at the Einstein Telescope: 2-D simulations beyond the plane-wave approximation

This paper presents a numerical framework using spectral-element simulations to estimate Newtonian noise for the Einstein Telescope, demonstrating its accuracy against analytical models and revealing that a lower P-wave fraction in homogeneous media suggests enhanced prospects for noise mitigation compared to previous assumptions.

The Gist

Imagine you are trying to listen to a whisper in a crowded room. That whisper is a gravitational wave—a ripple in space-time caused by colliding black holes or neutron stars. The Einstein Telescope (ET) is a super-sensitive "ear" being built deep underground to hear these whispers.

But there's a problem: the Earth itself is noisy. Even when the ground seems still, it's actually vibrating with tiny tremors from wind, ocean waves, and distant traffic. These vibrations create a "static" noise that drowns out the cosmic whispers. This is called Newtonian Noise.

Here is the simple breakdown of what this paper does, using some everyday analogies:

1. The Problem: The "Ghost" in the Machine

Newtonian noise isn't caused by the ground shaking the detector directly (like a car bumping a table). Instead, it's a gravity ghost.

• The Analogy: Imagine you are floating in a pool. If a huge wave passes through the water, the water's density changes slightly. Even though you aren't touching the wave, the changing weight of the water around you pulls on you slightly.
• In the Telescope: Seismic waves (vibrations in the rock) change the density of the rock around the telescope's mirrors. This changing density creates tiny, fluctuating gravitational pulls that trick the telescope into thinking a gravitational wave has arrived.

2. The Old Way vs. The New Way

For years, scientists tried to predict this noise using simple math formulas.

• The Old Way (The Flat Map): They assumed the Earth was a perfectly flat, uniform layer of cake. They treated the vibrations like a single, perfect wave rolling across a calm ocean. This is like trying to predict the weather in a mountain range by only looking at a flat beach. It works okay for a quick guess, but it misses the bumps, the valleys, and the complex bounces.
• The New Way (The 3D Movie): This paper introduces a numerical framework. Instead of simple math, the authors built a computer simulation—a "movie" of how seismic waves actually move through rock. They used a super-powerful calculator (called a spectral-element solver) to watch how waves bounce, scatter, and interact with the ground, just like light bouncing off a disco ball.

3. The Experiment: From One Rock to a Crowd

The authors tested their new "movie" in two stages:

• Stage 1: The Soloist. They dropped a single "rock" (a seismic source) on the surface and watched the waves travel to the underground detector.
  • The Result: The computer simulation matched the old, simple math perfectly. This proved their new "movie" engine was working correctly.
• Stage 2: The Crowd. Real life isn't just one rock; it's a chaotic mix of 30 different sources (like a crowd of people stomping randomly). They simulated 30 random sources to mimic real-world background noise.

4. The Big Discovery: The "P-Wave" Surprise

This is the most exciting part of the paper. Seismic waves come in two main flavors:

• P-Waves (The Squeezers): These squeeze the rock like a sponge. They are the main culprits for creating Newtonian noise.
• S-Waves (The Shakers): These shake the rock side-to-side. They are less effective at creating this specific type of noise.

The Old Assumption: Scientists used to guess that about 33% (1/3) of the noise came from the dangerous P-waves.
The New Finding: The computer simulation showed that in this specific setup, the P-wave fraction was actually much lower—only about 14%.

Why does this matter?
Think of Newtonian noise mitigation like trying to cancel out noise-canceling headphones. If you know exactly what kind of noise you are fighting (P-waves vs. S-waves), you can build better "noise-canceling" systems.

• If the noise is mostly P-waves, it's harder to cancel.
• If the noise is mostly S-waves (as this paper suggests might be the case), it is easier to cancel.

The Bottom Line

This paper is a "proof of concept." It says: "We built a super-accurate computer simulator for Earth vibrations. When we tested it, it worked. And when we used it to look at the noise, we found it might be less scary than we thought."

The Future:
Right now, the simulation is like a 2D cartoon. The next step is to make it a full 3D movie that includes real mountains, caves, and complex geology. If they can do that, they can pick the perfect spot for the Einstein Telescope and design the best possible system to silence the Earth's noise, allowing us to finally hear the universe's deepest secrets.

Technical Summary

1. Problem Statement

The Einstein Telescope (ET), a planned third-generation underground gravitational-wave observatory, aims to detect signals down to a few Hertz. However, its low-frequency sensitivity (specifically 1.7–6 Hz) is expected to be limited by Newtonian noise (NN). NN arises from gravitational force fluctuations on the detector's test masses caused by seismic-induced density fluctuations in the surrounding rock.

Current NN estimation methods rely heavily on analytical or semi-analytical models that assume:

• Homogeneous or horizontally layered media.
• Plane-wave approximations with infinite coherence.
• Neglect of geological heterogeneity, scattering, mode conversions, and complex wave interactions.

These simplifications fail to capture realistic subsurface complexities, such as scattering from geological interfaces and the specific correlations between P-waves (longitudinal) and S-waves (transversal). Furthermore, acquiring the dense seismic array data required for cancellation schemes based on single-point predictions is practically challenging for underground installations. There is a critical need for a fully numerical framework capable of resolving geological heterogeneity and complex wave fields to provide site-specific NN estimates.

2. Methodology

The authors developed a numerical framework based on spectral-element simulations to model seismic wave fields and subsequent Newtonian noise.

• Numerical Solver: The study utilizes Salvus, a highly parallelized, GPU-accelerated spectral-element solver. This method combines the geometric adaptability of finite elements with the high-order accuracy of spectral methods, making it suitable for heterogeneous media.
• Physical Model:
  • Wave Equation: The seismic wave field $\vec{\xi}(\vec{x}, t)$ is solved using the momentum conservation equation in a continuum medium.
  • Density Fluctuations: Density fluctuations $\delta\rho$ are derived from the displacement field using the continuity equation: $\delta\rho = -\vec{\nabla} \cdot (\rho \vec{\xi})$. This is decomposed into a bulk term (density changes within the medium) and a surface term (density jumps at interfaces, e.g., cavern walls).
  • Newtonian Force Calculation: The gravitational force fluctuation $\delta \vec{F}_M$ on a test mass is calculated by integrating the density fluctuations over the volume (Eq. 3). The authors also employ a "dipole equation" (Eq. 8) which unifies bulk and surface integrals, accounting for a virtual cavern around the test mass.
• Simulation Scenarios (2-D Proof of Concept):
  1. Single Source Benchmark: A single upward-pointing force source (Ricker wavelet, 2 Hz) on the surface of a homogeneous medium to validate the numerical results against analytical plane-wave predictions.
  2. Ambient Noise Array: An array of 30 stochastic surface sources with independent time-domain inputs (band-limited to 10 Hz) to approximate stationary ambient seismic noise.
• Parameters: The simulations use a homogeneous upper-crustal reference medium ($c_P = 3000$ m/s, $c_S \approx 1764$ m/s, $\rho \approx 2224$ kg/m³) in a 20 km $\times$ 8 km domain with absorbing boundaries.

3. Key Contributions

• Development of a Numerical Framework: Established a workflow to simulate seismic wave fields and compute Newtonian noise directly from the displacement field, moving beyond analytical approximations.
• Validation against Analytical Models: Successfully benchmarked the numerical results against analytical plane-wave predictions in a homogeneous medium, demonstrating excellent agreement for both bulk and cavern contributions.
• Quantification of P-wave Fraction: Introduced a method to estimate the P-wave fraction ($p$) in a complex, multi-source wave field by analyzing the ratio of the power spectral densities of the divergence (P-wave) and curl (S-wave) of the displacement field.
• Demonstration of Enhanced Mitigation Potential: Provided evidence that the P-wave content in ambient seismic noise may be significantly lower than previously assumed, suggesting better prospects for NN mitigation strategies.

4. Key Results

• Benchmarking (Single Source):
  • The numerical simulation of a single source perfectly reproduced analytical predictions.
  • The bulk integral (blue curve) matched the analytical P-wave prediction, while the dipole integral (orange curve), which includes the virtual cavern effect, correctly captured the combined P-wave and S-wave contributions.
  • The S-wave contribution was shown to arise solely from the surface (cavern wall) term, as S-waves do not cause bulk density fluctuations in a homogeneous medium.
• Ambient Noise Simulation (30 Sources):
  • The Amplitude Spectral Density (ASD) of the calculated Newtonian noise fell within the theoretical upper (pure P-wave) and lower (pure S-wave) bounds but fluctuated significantly, indicating complex correlations between wave types not captured by simple analytical models.
  • P-wave Fraction ($p$): The study estimated the P-wave fraction in the wave field near the test mass. The frequency-averaged median was found to be $p \approx 0.137 \pm 0.02$.
  • This value is significantly lower than the commonly assumed $p = 1/3$ in analytical estimates.

5. Significance and Outlook

• Implications for Mitigation: Since P-waves are generally harder to mitigate than S-waves (due to their ability to cause bulk density fluctuations), a lower P-wave fraction ($p \approx 0.14$ vs. $0.33$) implies that the Newtonian noise level may be lower than previously estimated, and the potential for successful mitigation (e.g., via seismic sensor arrays) is higher.
• Future Directions:
  • The current 2-D homogeneous study serves as a proof of concept. The framework is designed to be extended to 3-D simulations.
  • Future work will integrate detailed local geological models, topography, and inhomogeneities to account for scattering and mode conversions.
  • The ultimate goal is to generate site-specific NN estimates for the Einstein Telescope to optimize seismic array designs for noise cancellation.

Conclusion: This paper successfully demonstrates that numerical spectral-element simulations can accurately model Newtonian noise, validating them against analytical theory while revealing that realistic multi-source wave fields may contain less P-wave energy than standard models assume. This offers a promising new pathway for optimizing the low-frequency sensitivity of the Einstein Telescope.