GWForge.fisher.antenna_derivatives

Closed-form derivatives of the detector response geometry.

ra, dec and psi reach the strain only through the detector projection, never through the waveform, so their derivatives can be written down exactly. This module differentiates the same expressions GWForge.ifo.antenna.antenna_response() evaluates – the u, v, m and n triad, the propagation direction, the finite-size arm transfer function and the vertex delay – with respect to the internal angles \((\phi, \theta, \psi)\), and chain-rules them onto the source parameters.

That holds under the frequency-dependent, Earth-rotating response too, which changes only how \(F_{+,\times}\) and \(\tau\) are built. Freezing the polarizations and differencing the response alone reproduces the full derivative bit-identically; tests/test_fisher.py asserts exactly that.

The work is done on scalars rather than on \(3\times3\times n\) polarization tensors: contracting an arm with the polarization basis reduces to \((\hat a\cdot m)\) and \((\hat a\cdot n)\), so only those and their derivatives are ever needed.

Classes

AntennaDerivatives

Exact derivatives of \(F_+\), \(F_\times\) and \(\tau\).

Module Contents

class GWForge.fisher.antenna_derivatives.AntennaDerivatives(interferometer, ra, dec, geocent_time, psi, frequencies, start_time, times_to_coalescence=None, earth_rotation=True, finite_size=True)

Exact derivatives of \(F_+\), \(F_\times\) and \(\tau\).

Differentiates the same expressions GWForge.ifo.antenna.antenna_response() evaluates, with respect to the internal angles \((\phi, \theta, \psi)\), then chain-rules those onto the source parameters ra, dec, psi and geocent_time:

\[\frac{\partial}{\partial\alpha} = \frac{\partial}{\partial\phi}, \qquad \frac{\partial}{\partial\delta} = -\frac{\partial}{\partial\theta}, \qquad \frac{\partial}{\partial t_c} \supset -\dot{\mathrm{gmst}}\,\frac{\partial}{\partial\phi},\]

the last because \(\phi = \alpha - \mathrm{gmst}(t_c)\). So the Earth-rotation part of the geocent_time derivative comes free from the sky partials. geocent_time additionally enters \(\tau\) explicitly with unit derivative, which is the dominant \(-2\pi i f h\) term.

Variables:

• cross (plus,) – \(F_+\) and \(F_\times\) and their derivatives, keyed by source parameter name ("ra") or sorted pair (("dec", "ra")).

• delay (dict) – Same for \(\tau\).

sidereal_rate