Detector response¶

Bilby’s Interferometer.get_detector_response — which GWForge used everywhere until now — makes two approximations that are harmless for LIGO/Virgo and wrong for XG detectors:

The long-wavelength approximation. The detector tensor is treated as frequency-independent, i.e. the wave is assumed not to change appreciably while light traverses an arm. The correction scales with $fL/c$ and becomes important near the free spectral range $c/2L$ — 3.75 kHz for CE’s 40 km arms, which is inside the analysis band.
A static antenna pattern. $F_{+,\times}$ and the geocentre-to-vertex delay are frozen at geocent_time. ET and CE see BNS inspirals lasting hours, over which the Earth rotates through a large angle, so both are really functions of frequency through the time-frequency relation $t(f)$.

GWForge.ifo.antenna drops both. It is on by default.

Using it¶

Both corrections are controlled by two flags, available wherever GWForge projects a signal onto a detector.

For injections, in the [Injections] section of the config file:

[Injections]
injection-method = bilby
earth-rotation = True
finite-size = True

For gwforge_optimal_snr, on the command line:

gwforge_optimal_snr --injection-file injections.h5 --output-file snrs.h5 \
                    --no-earth-rotation --no-finite-size   # restores the old behaviour

From Python:

from GWForge.ifo import antenna

signal = antenna.detector_response(
    interferometer, polarizations, parameters,
    earth_rotation=True, finite_size=True,
)

Warning

Because these default to True, injections and optimal SNRs computed with this version differ from those produced by earlier versions. Set both to False to reproduce the old numbers exactly — that combination is asserted to reproduce bilby to a relative $2.5\times10^{-16}$.

Note

The pycbc injection method is unaffected: it uses LAL’s own projection via Detector.project_wave, which is a separate code path.

The physics¶

The implementation follows Baral et al. (arXiv:2304.09889), with polarisation tensors as defined in Nishizawa et al. (arXiv:0903.0528) — the same convention bilby uses.

Finite arm length¶

Each arm gets its own transfer function, from Rakhmanov, Romano & Whelan (arXiv:0808.3805):

$$ D(x, y) = \tfrac{1}{2}\left[ e^{-i\pi x(1+y)},\mathrm{sinc}\big(x(1-y)\big)

e^{ i\pi x(1-y)},\mathrm{sinc}\big(x(1+y)\big)\right] $$

with $x = fL/c$ and $y = -\hat\Omega\cdot\hat{a}$. The two half-outer-products $\tfrac{1}{2}\hat{x}\otimes\hat{x}$ and $\tfrac{1}{2}\hat{y}\otimes\hat{y}$ — whose difference is exactly bilby’s detector_tensor — are contracted separately so each carries its own $D$.

Two things worth knowing about the size of this effect:

$D(0, y) = 1$ exactly, so switching finite_size off is a genuine limit and not an approximation.
To first order $D \approx 1 - i\pi x y$. The leading correction is a phase, not an amplitude change — it is the extra light travel to the arm midpoint. That is why it grows linearly in $f$ rather than quadratically, and why much of it is degenerate with the coalescence time. The genuinely new content appears at $\mathcal{O}(x^2)$.
At $y = 0$ the response is $\mathrm{sinc}(x)\cos(\pi x)$, which vanishes at $f = c/2L$.

Earth rotation¶

The frequency bin at $f$ was emitted $\tau(f)$ seconds before merger, so the Earth’s orientation there is its orientation at geocent_time - tau(f). GWForge uses the 3.5PN time-to-coalescence of arXiv:0907.0700 eq. (3.8b) — deliberately the same expression GWFast uses, so the cross-validation below compares geometry rather than two different time-frequency relations. Mode $m$ reaches frequency $f$ when the quadrupole is at $2f/m$, i.e. $\tau_m(f) = \tau_2(2f/m)$.

The sidereal angle is advanced at the sidereal rate, $2\pi/86164.09,\mathrm{s}$.

Validation¶

Two reference implementations of this physics exist and they disagree on conventions, so tests/test_antenna_response.py layers its checks outward from claims that cannot be wrong.

Check	Result
Both corrections off $\equiv$ bilby’s `get_detector_response`, CE40 and all three ET arms	$2.5\times10^{-16}$ relative
$D(x,y)$ vs brute-force integration of the round-trip light path (shares no code)	$<10^{-9}$
$D(0,y) = 1$; null at $f = c/2L$	exact
First-order term is $-i\pi xy$, residual scales as $x^2$	confirmed
$\tau(f)$ vs GWFast `IMRPhenomD.tau_star`	$10^{-13}$ relative
Time-varying $F_{+,\times}$ vs GWFast `_PatternFunction`, CE40	$<10^{-8}$ absolute

The first row is the important one: it anchors every sign, the handedness of $\psi$, and the delay convention against code GWForge already relies on, so a convention error surfaces there rather than being absorbed into the cross-code comparison.

Notes on the GWFast comparison¶

GWFast is an independent code with its own conventions; matching it required establishing these, and each was verified rather than assumed:

det_xax = xarm_azimuth + 45. GWFast orients a detector by the bisector of its arms, bilby by the x-arm. Verified against H1, L1 and Virgo, where GWFast’s tabulated xax minus bilby’s xarm_azimuth is exactly 45.0 in all three cases.
theta = pi/2 - dec, phi = ra.
Only the real $F_{+,\times}$ amplitudes are compared. GWFast writes $h \sim e^{+i\Psi}$ where bilby writes $e^{-i\ldots}$, so the complex responses differ by a conjugation that says nothing about geometry.
The two codes are placed at identical Earth orientations. GWFast advances its pattern by $2\pi t/86400$ — a solar day — which over a one-hour inspiral is ~$7\times10^{-4}$ rad adrift of true sidereal rotation. GWFast’s t enters its pattern functions only as $2\pi t$, so passing it $\mathrm{GMST}/2\pi$ removes that approximation and leaves a pure geometry comparison. GWForge uses the sidereal rate; a separate test asserts this.

GWFast is also not used for ET: it models a spherical Earth and ignores detector elevation, and it places a triangle’s three arms at a single vertex, whereas bilby walks them apart. ET’s three arms are instead checked against bilby directly, and asserted to have genuinely distinct responses (otherwise the null stream would be wrong).

Check	Result
Both corrections off \(\equiv\) bilby’s `get_detector_response`, CE40 and all three ET arms	\(2.5\times10^{-16}\) relative
\(D(x,y)\) vs brute-force integration of the round-trip light path (shares no code)	\(<10^{-9}\)
\(D(0,y) = 1\); null at \(f = c/2L\)	exact
First-order term is \(-i\pi xy\), residual scales as \(x^2\)	confirmed
\(\tau(f)\) vs GWFast `IMRPhenomD.tau_star`	\(10^{-13}\) relative
Time-varying \(F_{+,\times}\) vs GWFast `_PatternFunction`, CE40	\(<10^{-8}\) absolute