Measuring the maximally allowed polarization states of the isotropic stochastic gravitational wave background with the ground-based detectors

We discuss the polarizational study of isotropic gravitational wave backgrounds with the second generation detector network, paying special attention to the impacts of adding LIGO-India. The backgrounds can be characterized by at most five spectral components (three parity-even ones and two parity-odd ones). They can be algebraically decomposed through the difference of the corresponding overlap reduction functions defined for the individual spectra. We newly identify two interesting relations between the overlap reduction functions, and these relations generally hamper the algebraic decomposition in the low frequency regime $f\lesssim 30$Hz. We also find that LIGO-India can significantly improve the network sensitives to the odd spectral components.