A battle of designs: triangular vs. L-shaped detectors and parity violation in the gravitational-wave background

We investigate the prospects for detecting a parity-violating gravitational-wave background (GWB) with third-generation ground-based detector networks. We focus on a network consisting of one Einstein Telescope (ET) and two Cosmic Explorer (CE) detectors. In our analysis we vary the ET design, detector orientations, and arm lengths to assess the impact of geometry and scale on detection capabilities. We find that, among the configurations studied, networks with a $15$ km triangular ET design provide the strongest parity-violation constraining power, followed by networks with an L-shaped ET design, while networks with $10$ km triangular ET configurations are the least sensitive. Under current bounds, ET alone cannot confidently detect parity violation.