Savage-Dickey ratio in CW tests of GR

Our problem

Signal

In the context of testing Einstein’s General Theory of Relativity (GR) using continuous gravitational waves (GWs), we can define all possible signal models as subsets of the most generic conceivable signal (Isi 2015): \[s (t) = \frac{1}{2} \sum_p a_p A_p(t),\] for the antenna patterns \(A_p\), a complex coefficient \(a_p\) and with the sum over all possible polarizations: plus (\(+\)), cross (\(\times\)), vector x (x), vector y (y) breathing (b) and longitudinal (l). Note longitudinal mode is fully degenerate with breathing and can be safely excluded. The GR hypothesis is obtained by setting: \[\label{eqn:a_plus} a_+ = h_0 (1+\cos^2 \iota)/2,\] \[a_\times = h_0 \cos \iota~ e^{-i\pi/2},\] \[\label{eqn:a_others} a_{\rm x} = a_{\rm y} = a_{\rm b} = a_{\rm l} = 0,\] where \(\iota\) is the inclination angle and \(h_0\) an overall amplitude determined by the properties of the source.

Note that the polarizations can be categorized in terms of their associated graviton spin: tensor (\(+\), \(\times\)), vector (x, y) and scalar (b, l). These groups are not separable: a signal model that includes one element, must also include the other (e.g. it is not possible to have a model that allows plus \(+\) but not \(\times\)). The reason for this is that the distinction between modes of a same spin is contingent on the relative orientation of source and detector (i.e. the difference is not intrinsic to the signal).

In order for our notation to agree with that used in lalapps_pulsar_parameter_estimation_nested , for each polarization \(p\), parametrize \(a_p\) by real amplitude \(h_p\) and relative phase \(\phi_p\) such that \(a_p=h_p e^{i\phi_p}\). Furthermore, define an angle \(\phi_s\) for each spin \(s\) and an internal offset \(\psi_s\) between components of a given spin; that way: \[\phi_+ = \phi_{\rm t}\] \[\phi_\times = \phi_{\rm t} + \psi_{\rm t}\] \[\phi_{\rm x} = \phi_{\rm v}\] \[\phi_{\rm y} = \phi_{\rm v} + \psi_{\rm v}\] \[\phi_{\rm b} = \phi_{\rm s}\] \[\phi_{\rm l} = \phi_{\rm s} + \psi_{\rm s}\]

Hypotheses

Given some data, the three relevant hypotheses we must consider are: Gaussian noise (\({\cal H}_{\rm n}\)), GR signal with Gaussian noise (\({\cal H}_{\rm GR}\)), non–GR signal with Gaussian noise (\({\cal H}_{\rm nGR}\)). The first two are computed in LALSuite directly; the last one is not so straightforward: the non–GR signal hypothesis is a composite hypothesis made up of the “or” junction of the hypotheses corresponding to the presence of all the possible signals that are not allowed by GR (Li 2012).

Broadly speaking, there are two ways in which GR can be violated: there may be some vector or scalar component in addition to the GR signal or the signal may be composed by any other arbitrary combination of the six polarizations that is different from eqs. (\ref{eqn:a_plus}-\ref{eqn:a_others}). Let us denote the former by \({\cal H}_{\rm GR+v}\), \({\cal H}_{\rm GR+s}\) and \({\cal H}_{\rm GR+sv}\), while the latter can be indexed by their component spins, e.g. \({\cal H}_{\rm sv}\) representing the scalar–vector model. In this notation, \({\cal H}_{\rm nGR}\) can be obtained from the eight non–GR sub–hypotheses by: \[\label{eq:hypotheses} {\cal H}_{\rm nGR} = {\cal H}_{\rm GR+s} \lor {\cal H}_{\rm GR+v} \lor {\cal H}_{\rm GR+sv} \lor {\cal H}_{\rm s} \lor {\cal H}_{\rm t} \lor {\cal H}_{\rm v} \lor {\cal H}_{\rm st} \lor {\cal H}_{\rm sv} \lor {\cal H}_{\rm tv} \lor {\cal H}_{\rm stv}\] Note that \({\cal H}_{\rm t}\) is logically different from \({\cal H}_{\rm GR}\), since the latter is more restrictive. [MI: even then, we might decide, based on physics/geometry, that the only possible relation between tensor modes is as given in GR (i.e. phase difference of \(\pi/2\)); in that case, we could discard \({\cal H}_{\rm t}\).]