ANNA: a toolbox for Newtonian Noise Analysis

Abstract

The Einstein Telescope (ET) is a third-generation underground gravitational wave observatory designed to achieve an 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. ANNA Newtonian Noise Analysis is a toolbox that computes Newtonian Noise from a seismic wave field defined on a finite element mesh, using Gaussian quadrature. 3D finite element meshes composed of linear and quadratic tetrahedral (4-node and 10-node) and brick (8-node and 20-node) elements are supported. The user computes (or interpolates) a seismic wave field on a finite element mesh and the toolbox computes the total Newtonian noise, as well as the bulk and surface contributions. ANNA runs in the MATLAB programming and numeric computing platform and is compatible with the open-source GNU Octave Scientific Programming Language; a Python version is also available. The toolbox is verified for plane P- and S-waves propagating in an elastic homogeneous full space with a mirror suspended in a spherical cavity, assuming that the wavelength is much larger than the radius of the cavity, so that wave scattering can be ignored. Excellent agreement with analytical solutions is obtained. Similar good agreement is reported for the Newtonian noise on a test mass suspended at a finite distance above the free surface of a homogeneous elastic halfspace in which a Rayleigh wave propagates. The proposed finite element framework provides a physically consistent and computationally efficient approach for computing gravitational-seismic coupling in heterogeneous media.

Figures (12)

• Figure 1: Point mass $m$ at a position ${\bf x}_{0}$ and volume $\Omega({\bf x},t)$ with density $\rho({\bf x},t)$.
• Figure 2: (a) Finite element mesh of a sphere with center at ${\bf x}_{0} = \{ 0,0,0 \}^{\mathrm{T}}$ and radius $R=2000\,\hbox{m}$ and a spherical cavity with radius $r_{0}=20\,\hbox{m}$, used to compute the Newtonian noise due to a plane harmonic P-wave in a frequency range around 5 Hz. (b) Displacement $\hat{u}_{x}({\bf x},\omega)$ due to a plane harmonic P-wave with unit amplitude and frequency of 5 Hz, propagating in the direction ${\bf e}_{\mathrm{k}}=\{\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\}^{\mathrm{T}}$.
• Figure 3: (a) Real part of the total Newtonian noise $\delta\hat{a}_{\mathrm{t}x}({\bf x}_{0},\omega)$ computed analytically () and with the numerical model (), due to a plane harmonic P-wave in a linear elastic full space with a cavity with radius $r_{0}=20\,\hbox{m}$; the P-wave has unit amplitude and propagates in the direction ${\bf e}_{\mathrm{k}}=\{\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\}^{\mathrm{T}}$. The bulk contribution $\delta\hat{a}_{\mathrm{b}x}({\bf x}_{0},\omega)$ ( and ) and the surface contribution $\delta\hat{a}_{\mathrm{s}x}({\bf x}_{0},\omega)$ ( and ) are also shown. (b) Relative error $\varepsilon_{\mathrm{t}x}(\omega)$ on the total Newtonian noise.
• Figure 4: (a) Finite element mesh of a sphere with center at ${\bf x}_{0} = \{ 0,0,0 \}^{\mathrm{T}}$ and radius $R=2000\,\hbox{m}$ and a spherical cavity with radius $r_{0}=20\,\hbox{m}$, used to compute the Newtonian noise due to a plane harmonic S-wave in a frequency range around 5 Hz. (b) Displacement $\hat{u}_{z}({\bf x},\omega)$ due to a plane harmonic S-wave with unit amplitude and frequency of 5 Hz propagating in the direction ${\bf e}_{\mathrm{k}}=\{1,0,0\}^{\mathrm{T}}$ with polarization vector ${\bf e}_{\mathrm{s}}=\{0,0,1\}^{\mathrm{T}}$.
• Figure 5: (a) Real part of the total Newtonian noise $\delta\hat{a}_{\mathrm{t}z}({\bf x}_{0})$ computed analytically () and with the numerical model (), due to a plane harmonic S-wave in a linear elastic full space with a cavity with radius $r_{0}=20\,\hbox{m}$; the S-wave has unit amplitude and propagates in the direction ${\bf e}_{\mathrm{k}}=\{1,0,0\}^{\mathrm{T}}$ with polarization vector ${\bf e}_{\mathrm{s}}=\{0,0,1\}^{\mathrm{T}}$. (b) Relative error $\varepsilon_{\mathrm{t}z}(\omega)$ on the total Newtonian noise.