# Detecting a stochastic gravitational wave signal

This note discusses trying to detect a generic gravitational wave with an unknown waveform emitted from a particular sky position in data from two separate gravitational wave detectors. We define two slightly different approaches to this problem.

## The signal

First we will define the gravitational wave signal at one timestamp, $$i$$, observed in one detector, $$L$$. We envisage two possible methods for this analysis with slightly different model definitions. The first uses

$$\label{eq:signal1}h_{i}^{L}=h_{0}\left({A_{+}}_{i}{F_{+}}_{i}^{L}(\psi_{i})+{A_{\times}}_{i}{F_{\times}}_{i}^{L}(\psi_{i})\right),\\$$

where $${A_{+}}_{i}$$ and $${A_{\times}}_{i}$$ are the signal amplitudes scale factors (which could be positive or negative) in the plus and cross polarisations at timestamp $$i$$ (which would be the same for different detectors), $${F_{+}}_{i}^{L}(\psi_{i})$$ and $${F_{\times}}_{i}^{L}(\psi_{i})$$ are the detector’s antenna response to the plus and cross polarisations for a given polarisation angle $$\psi_{i}$$11Note that $$\psi$$ could change between data points, so this is also indexed for the current timestamp., and $$h_{0}$$ is an overall underlying gravitational wave amplitude. The second uses

$$\label{eq:signal2}h_{i}^{L}={A_{+}}_{i}{F_{+}}_{i}^{L}(\psi_{i})+{A_{\times}}_{i}{F_{\times}}_{i}^{L}(\psi_{i}),\\$$

where, in this case, the $${A_{+}}_{i}$$ and $${A_{\times}}_{i}$$ are the actual signal amplitudes in the plus and cross polarisations at timestamp $$i$$.