An efficient finite element formulation for Newtonian noise analysis

Abstract

The Einstein Telescope is a third-generation underground gravitational wave observatory designed to achieve unprecedented sensitivity down to 3 Hz. Waves propagating in the soil due to anthropogenic or natural vibration sources generate density fluctuations which cause gravitational attraction, resulting in motion of the mirrors of the laser interferometer known as Newtonian noise. The latter is computed by integrating density fluctuations due to seismic wave fields over the soil domain surrounding the test mass. A finite element formulation is presented which evaluates the total Newtonian noise, as well as the bulk and surface contributions, from a seismic wave field defined on a finite element mesh using Gaussian quadrature. Linear and quadratic tetrahedral and brick finite elements are supported. The approach computes the total, bulk, and surface contributions, and expresses the corresponding volume and surface integrals in terms of finite element coupling matrices that depend only on geometry and material properties. This allows efficient evaluation of the Newtonian noise for different seismic wave fields without recomputing the integrals. The formulation is verified for plane P- and S-waves propagating in an elastic homogeneous full space with a mirror suspended in a spherical cavity, assuming the wavelength is much larger than the cavity radius, so that wave scattering can be ignored. Similar agreement is reported for the Newtonian noise on a test mass above the free surface of a homogeneous elastic halfspace in which a Rayleigh wave propagates. The methodology has been implemented in the ANNA Newtonian Noise analysis toolbox in MATLAB and is compatible with GNU Octave; a Python version is also available. The proposed finite element framework provides a physically consistent and computationally efficient approach for computing gravitational-seismic coupling in heterogeneous media.