# Gravitational-wave utilities¶

The gw_utils module contains functions that are useful in connecting numerical relativity simulation with gravitational-waves observations.

Reference on kuibit.gw_utils

## Detectors¶

gw_utils defines a new data type, Detectors that can contain quantities that are specific to the each of operating gravitational-wave detector (Hanford, Livingston, Virgo). A Detectors object det has three attributes: det.hanford, det.livingston, and det.virgo. You can access the fields in det as shown, or you can use the index notation, e.g., det[0] (which is det.hanford). The fields are (by convention) in alphabetical order. Using Detectors is very convinent because it allows us to forget about the order of the fields, while guaranteeing that an order exist.

Technically, Detectors is a namedtuple. You can think of it as an ordered and immutable dictionary.

## luminosity_distance_to_redshift¶

Compute the redshift starting from a given luminosity distance in Megaparsec. The assumed cosmology is LCDM with no radiation (only matter and dark energy) and the default values are the ones provided by the Planck mission.

The computation is based on a numerical inversion, which requires an initial guess. The default one (0.1) should work in most scenarios, but it should be changed in case of failure.

## sYlm¶

Compute the spin-weighted spherical harmonics using recursion relationships. The angles are defined as $$\theta$$ the meridional angle and $$\phi$$ the azimulathal one.

## antenna_responses¶

Compute the antenna pattern $$F$$ for Hanford, Livingston, and the Virgo interferometers for a given source localization (as right ascension and declination in degrees, and the UTC time). The antenna pattern is used to compute the strain measured by a detector with the formula. The output of this antenna_responses_from_sky_localization() is a Detectors, a namedtuple with attributes hanford, livingston, and virgo, each containing a standard tuple with the responses of that detector for the cross and plus polarizations.

# GW150914
antenna = antenna_responses_from_sky_localization(8, -70, "2015-09-14 09:50:45")
(Fc_H, Fp_H) = antenna_LIGO.hanford  # or antenna_LIGO['hanford']

# Alternatively, unpack everything
((Fc_H, Fp_H), (Fc_L, Fp_L), (Fc_V, Fp_V)) = antenna


If you are working with a single generic detector, you can use antenna_responses() which takes the spherical angles with respect to a detector on the $$z=0$$ plane and with arms on the two other directions.

## signal_to_noise_ratio_from_strain¶

signal_to_noise_ratio_from_strain() takes a strain, a noise curve, and two boundary frequencies and return the signal-to-noise ratio as

$\rho^2 = 4 \int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\frac{\|\tilde{h}\|^2}{S_n(f)}df$

## effective_amplitude_spectral_density¶

effective_amplitude_spectral_density() computes an effective amplitude spectral density given a gravitational strain. The strain is copied and optionally windowed prior to being converted into frequency space. The effective amplitude spectral density is defined as

$h_{\rm eff}(f) = f \sqrt{(|\tilde{h}_{+}(f)|^2 + |\tilde{h}_{\times}(f)|^2) / 2}$

The effective amplitude spectral density is an useful quantity to estimate detectability of signals given the sensitivity curve of a detector.

Note

By default, kuibit works with r_ext h, where r_ext is the extraction radius and h the complex strain. Be careful with this!