\texttt{GWBird}: a toolkit for the characterization of the Stochastic Gravitational Wave Background for Ground, Space, and Pulsar Timing Array detectors

The detection of the Stochastic Gravitational Wave Background (SGWB) is one of the most challenging tasks for both current and next-generation detectors. Successfully distinguishing the SGWB from instrumental noise and environmental effects requires accurate and flexible analysis tools capable of detecting the signal and determining its origin. In this paper, we introduce a unified framework and a user-friendly tool for SGWB characterization: \texttt{GWBird} (Gravitational Wave Background Inventory of Response functions for Detectors). This code enables the computation of overlap reduction functions (ORFs), power-law integrated sensitivity curves (PLS), angular response functions, and angular PLS (APLS). It supports the full range of gravitational wave polarization modes (tensor, scalar, and vector), allowing for the characterization of both isotropic and anisotropic SGWB components for all the polarizations. Additionally, the code includes functions for circular polarization characterization, which is particularly relevant for probing parity-violating signals. The framework integrates analyses for ground-based, space-based, and Pulsar Timing Array (PTA) detectors, offering a versatile framework for SGWB analysis. The \texttt{GWBird} code is publicly available at:~\github{https://github.com/ilariacaporali/GWBird}