ET sensitivity to the anisotropic Stochastic Gravitational Wave Background

We study the sensitivity of a pair of Einstein Telescopes (ET) (hypothetically located at the two sites currently under consideration for ET) to the anisotropies of the Stochastic Gravitational Wave Background (SGWB). We focus on the ℓ =0,2,4 multipoles of an expansion of the SGWB in spherical harmonics, since the sensitivity to other multipoles is suppressed due to the fact that this pair of detector operates in a regime for which the product between the observed frequency and the distance between the two sites is much smaller than one. In this regime, the interferometer overlap functions for the anisotropic signal acquire very simple analytic expressions. These expressions can also be applied to any other pairs of interferometers (each one of arbitrary opening angle between its two arms) operating in this regime. Once the measurements at the vertices of the two sites are optimally combined, the sensitivity to the multipoles of the SGWB depends only on the latitude of the two sites, on the difference of their longitude, but not on the orientation of their arms.