Sensitivity of third-generation interferometers to extra polarizations
  in the Stochastic Gravitational Wave Background
  When modified theories of gravity are considered, at most six gravitational
wave polarization modes are allowed and classified in tensor modes, the only
ones predicted by General Relativity (GR), along with additional vector and
scalar modes. Therefore, gravitational waves represent a powerful tool to test
alternative theories of gravitation. In this paper, we forecast the sensitivity
of third-generation ground-based interferometers, Einstein Telescope and Cosmic
Explorer, to non-GR polarization modes focusing on the stochastic gravitational
wave background. We consider the latest technical specifications of the two
independent detectors and the full network in order to estimate both the
optimal signal-to-noise ratio and the detectable energy density limits relative
to all polarization modes in the stochastic background for several locations on
Earth and orientations of the two observatories. By considering optimal
detector configurations, we find that in 5 years of observation the detection
limit for tensor and extra polarization modes could reach
$h_0^2\Omega^{T,V,S}_{GW} \approx 10^{-12}-10^{-11}$, depending on the network
configuration and the stochastic background (i.e., if only one among vector and
scalar modes exists or both are present). This means that the network
sensitivity to different polarization modes can be approximately improved by a
factor $10^3$ with respect to second-generation interferometers. We finally
discuss the possibility of breaking the scalar modes degeneracy by considering
both detectors angular responses to sufficiently high gravitational wave
frequencies.
