The proposed third-generation gravitational-wave detectors Einstein Telescope will have a triangular design that consists of three colocated interferometers. Summing the strain outputs from the three interferometers will cancel any gravitational-wave
signal and the resultant signal-free stream is known as null stream. The null stream is in a fixed subspace of the observation space of Einstein telescope where no gravitational-wave signal can exist. In this paper, we establish the decomposition of the observation space of Einstein Telescope into the signal space that contains all possible gravitational-wave signals and the null space that contains the null stream. We show that the results of Bayesian parameter estimation and model selection using the strain data in the signal space are identical to that using the full set of strain data. This implies that one could use a fraction of the strain data to extract all information of the source which reduces the memory usage and speeds up the likelihood evaluation. We also discuss the existence of a fixed null space in Einstein Telescope allows the unbiased estimation of the noise properties in the signal-free subspace.
We present a null-stream-based Bayesian unmodeled framework to probe generic gravitational-wave polarizations. Generic metric theories allow six gravitational-wave polarization states, but general relativity only permits the existence of two of them
namely the tensorial polarizations. The strain signal measured by an interferometer is a linear combination of the polarization modes and such a linear combination depends on the geometry of the detector and the source location. The detector network of Advanced LIGO and Advanced Virgo allows us to measure different linear combinations of the polarization modes and therefore we can constrain the polarization content by analyzing how the polarization modes are linearly combined. We propose the basis formulation to construct a null stream along the polarization basis modes without requiring modeling the basis explicitly. We conduct a mock data study and we show that the framework is capable of probing pure and mixed polarizations in the Advanced LIGO-Advanced Virgo 3-detector network without knowing the sky location of the source from electromagnetic counterparts. We also discuss the effect of the presence of the uncaptured orthogonal polarization component in the framework, and we propose using the plug-in method to test the existence of the orthogonal polarizations.
The detection of gravitational wave signals by Advanced LIGO and Advanced Virgo enables us to probe the polarization content of gravitational waves. In general relativity, only tensor modes are present, while in a variety of alternative theories one
can also have vector or scalar modes. Recently test were performed which compared Bayesian evidences for the hypotheses that either purely tensor, purely vector, or purely scalar polarizations were present. Indeed, with only three detectors in a network and allowing for mixtures of tensor polarizations and alternative polarization states, it is not possible to identify precisely which non-standard polarizations might be in the signal and by what amounts. However, we demonstrate that one can still infer whether, in addition to tensor polarizations, alternative polarizations are present in the first place, irrespective of the detailed polarization content. We develop two methods to do this for sources with electromagnetic counterparts, both based on the so-called null stream. Apart from being able to detect mixtures of tensor and alternative polarizations, these have the added advantage that no waveform models are needed, and signals from any kind of transient source with known sky position can be used. Both formalisms allow us to combine information from multiple sources so as to arrive at increasingly more stringent bounds. For now we apply these on the binary neutron star signal GW170817, showing consistency with the tensor-only hypothesis with p-values of 0.315 and 0.790 for the two methods.
