GW Source Characterization¶

In GWDALI, a gravitational-wave source is described through a set of free parameters defining the binary intrinsic and extrinsic properties.

The framework supports multiple parameterizations for masses, distance, inclination, and spins. Independently of the chosen parameterization, a valid source description must contain:

exactly 2 mass parameters;
exactly 1 distance parameter;
exactly 1 inclination parameter;
exactly 6 spin parameters.

Additionally, detector-frame waveform generation requires:

RA
Dec
psi
t_coal
phi_coal

Mass Parameterizations¶

The following mass parameterizations are currently supported:

Parameter	Description
`m1`	Redshifted mass of the primary compact object, \((1+z)m_1\) [\(M_\odot\)].
`m2`	Redshifted mass of the secondary compact object, \((1+z)m_2\) [\(M_\odot\)].
`eta`	Symmetric mass ratio, \[\eta \equiv \frac{m_1m_2}{(m_1+m_2)^2}\]
`Mc`	Redshifted chirp mass, \[\mathcal{M}_c \equiv (1+z)\eta^{3/5}(m_1+m_2)\] [\(M_\odot\)].
`M`	Redshifted total mass, \[M \equiv (1+z)(m_1+m_2)\] [\(M_\odot\)].
`q`	Mass ratio, \[q \equiv \frac{m_2}{m_1}, \qquad m_1 > m_2\]
`inv_eta`	Inverse symmetric mass ratio, \(\eta^{-1}\).
`ln_Mc`	Logarithm of the redshifted chirp mass, \[\ln\left(\frac{(1+z)M_c}{M_\odot}\right)\]
`ln_eta`	Logarithm of the symmetric mass ratio, \(\ln(\eta)\).
`deltaM`	Dimensionless mass difference, \[\delta_M \equiv \frac{m_1-m_2}{m_1+m_2}\]

Distance Parameterizations¶

Exactly one distance parameter must be provided.

Parameter	Description
`dL`	Luminosity distance \(d_L\) [Gpc].
`inv_dL`	Inverse luminosity distance, \(d_L^{-1}\) [\(\mathrm{Gpc}^{-1}\)].
`inv_dL2`	Inverse squared luminosity distance, \(d_L^{-2}\) [\(\mathrm{Gpc}^{-2}\)].
`inv_sqrtdL`	Inverse square-root luminosity distance, \(d_L^{-1/2}\) [\(\mathrm{Gpc}^{-1/2}\)].
`lnDL`	Logarithm of the luminosity distance, \[\ln(d_L/\mathrm{Gpc})\]
`inv_lnDL`	Inverse logarithmic luminosity distance, \[\frac{1}{\ln(d_L/\mathrm{Gpc})}\]

Inclination Parameterizations¶

Exactly one inclination parameter must be provided.

Parameter	Description
`iota`	Inclination angle \(\iota\) [rad].
`cos_iota`	Cosine of the inclination angle, \(\cos(\iota)\).

Spin Parameterizations¶

Exactly six spin parameters must be provided.

Cartesian Spin Coordinates¶

Parameter	Description
`sx1`, `sy1`, `sz1`	Cartesian spin components of the primary compact object.
`sx2`, `sy2`, `sz2`	Cartesian spin components of the secondary compact object.

Spherical Spin Coordinates¶

Parameter	Description
`chi1`	Dimensionless spin magnitude of the primary compact object, \(\chi_1 \equiv \|\vec{S}_1\|\).
`theta1`	Polar angle of the primary spin [rad].
`phi1`	Azimuthal angle of the primary spin [rad].
`chi2`	Dimensionless spin magnitude of the secondary compact object, \(\chi_2 \equiv \|\vec{S}_2\|\).
`theta2`	Polar angle of the secondary spin [rad].
`phi2`	Azimuthal angle of the secondary spin [rad].

Symmetric/Antisymmetric Spin Coordinates¶

Parameter	Description
`chi_s`	Symmetric spin combination, \[\chi_s = \frac{\chi_1+\chi_2}{2}\]
`chi_a`	Antisymmetric spin combination, \[\chi_a = \frac{\chi_1-\chi_2}{2}\]

Extrinsic Parameters¶

Parameter	Description
`RA`	Right ascension [deg].
`Dec`	Declination [deg].
`psi`	Polarization angle [rad].
`phi_coal`	Coalescence phase [rad].
`t_coal`	Coalescence time [s].

Example: Source Dictionary Construction¶

Below we show a complete example of a gravitational-wave source dictionary compatible with get_strain().

This example uses:

(Mc, eta) as mass parameterization;
dL as distance parameterization;
iota as inclination parameterization;
Cartesian spin coordinates.

The resulting dictionary contains the required 15 parameters.

import numpy as np

GwPrms = {}

#========================================
# Distance sector (1 parameter)
#========================================

GwPrms['dL'] = 5.0          # Luminosity distance [Gpc]

#========================================
# Mass sector (2 parameters)
#========================================

GwPrms['Mc']  = 30.0        # Chirp mass [Msun]
GwPrms['eta'] = 0.24        # Symmetric mass ratio

#========================================
# Inclination sector (1 parameter)
#========================================

GwPrms['iota'] = 1.2        # Inclination [rad]

#========================================
# Extrinsic parameters
#========================================

GwPrms['RA']         = 120.0     # Right ascension [deg]
GwPrms['Dec']        = -30.0     # Declination [deg]
GwPrms['psi']        = 0.5       # Polarization angle [rad]
GwPrms['t_coal']     = 0.0       # Coalescence time [s]
GwPrms['phi_coal']   = 0.0       # Coalescence phase [rad]

#========================================
# Spin sector (6 parameters)
#========================================

GwPrms['sx1'] = 0.0
GwPrms['sy1'] = 0.0
GwPrms['sz1'] = 0.5

GwPrms['sx2'] = 0.0
GwPrms['sy2'] = 0.0
GwPrms['sz2'] = -0.3

print(GwPrms)

The source dictionary can then be passed directly to the waveform generation routines:

import GWDALI as gw

freq = np.arange(1.0, 1024.0, 0.1)

h = gw.get_strain(
    detectors,
    GwPrms,
    freq,
    approx="IMRPhenomD"
)

Sky Localization¶

In GWDALI, astronomical coordinates (RA, Dec) are aligned with the geocentric coordinates (longitude lon and latitude lat), such that:

\[\mathrm{RA} \parallel \mathrm{lon}, \qquad \mathrm{Dec} \parallel \mathrm{lat}\]