GW_back

GW_back.py is a Python routine that contains functions relevant for cosmological stochastic gravitational wave backgrounds (SGWB).

Adapted from the original cosmoGW in GW_turbulence (https://github.com/AlbertoRoper/GW_turbulence), created in Dec. 2021

Currently part of the cosmoGW code:

https://github.com/cosmoGW/cosmoGW/ https://github.com/cosmoGW/cosmoGW/blob/main/src/cosmoGW/GW_back.py

Note

For full documentation, visit Read the Docs.

To use it, first install cosmoGW:

pip install cosmoGW

Author

Alberto Roper Pol (alberto.roperpol@unige.ch)

Dates

Created: 01/12/2021 (GW_turbulence)
Updated: 21/08/2025 (release cosmoGW 1.0: https://pypi.org/project/cosmoGW)

References

[Maggiore:1999vm]: M. Maggiore, “Gravitational wave experiments andearly universe cosmology,” Phys.Rept. 331 (2000) 283-367,`arXiv:gr-qc/9909001 <https://arxiv.org/abs/gr-qc/9909001>`_.
[RoperPol:2018sap]: A. Roper Pol, A. Brandenburg, T. Kahniashvili,A. Kosowsky, S. Mandal, “The timestep constraint in solving thegravitational wave equations sourced by hydromagnetic turbulence,”Geophys. Astrophys. Fluid Dynamics 114, 1, 130 (2020),`arXiv:1807.05479 <https://arxiv.org/abs/1807.05479>`_.
[RoperPol:2021xnd]: A. Roper Pol, S. Mandal, A. Brandenburg,T. Kahniashvili, “Polarization of gravitational waves from helicalMHD turbulent sources,” JCAP 04 (2022), 019,`arXiv:2107.05356 <https://arxiv.org/abs/2107.05356>`_.
[RoperPol:2022iel]: A. Roper Pol, C. Caprini, A. Neronov, D. Semikoz,”The gravitational wave signal from primordial magnetic fields in thePulsar Timing Array frequency band,” Phys. Rev. D 105, 123502 (2022),`arXiv:2201.05630 <https://arxiv.org/abs/2201.05630>`_.

cosmoGW.GW_back.Omega_A(A=1.0, fref=0, beta=0, h0=1.0)

Returns the amplitude of the SGWB energy density spectrum, expressed as a power law (PL), given the amplitude A of the characteristic strain, also expressed as a PL.

A is always given for the reference frequency of \(f_{\rm yr} = 1/(1 {\rm yr})\) and is used in the common process reported by PTA collaborations.

The GW energy density and characteristic amplitude can be expressed as:

\[\Omega_{\rm GW} = \Omega_{\rm ref} * (f/f_{\rm ref})^\beta h_c = A * (f/f_{\rm yr})^\alpha\]

Parameters

Afloat: Amplitude of the characteristic strain PL using 1yr as the reference frequency.
freffloat: Reference frequency used for the PL expression of the GW background given in units of frequency (default 1 yr^(-1)).
betafloat: Slope of the PL.
h0float: Parameterizes the uncertainties (Hubble tension) in the value of the Hubble rate (default 1).

Returns

Omreffloat: Amplitude of the GW energy density PL.

Reference

RoperPol:2022iel, eq. 44

cosmoGW.GW_back.fac_hc_OmGW(d=1, h0=1.0)

Returns the factor to transform the strain function \(h_c (f)\) to the GW energy density \(\Omega_{\rm GW}(f)\) away from the source.

Parameters

dint: Option to give the factor to convert from energy density to strain if set to -1 (default 1).
h0float: Parameterizes the uncertainties (Hubble tension) in the value of the Hubble rate (default 1).

Returns

facfloat: Factor to convert from the strain function hc(f) to the GW energy density OmGW(f) in frequency units (Hz).

Reference

Maggiore:1999vm, eq. 17

cosmoGW.GW_back.hc_OmGW(f, OmGW, d=1, h0=1.0)

Transforms the GW energy density \(\Omega_{\rm GW}(f)\) to the characteristic strain spectrum function \(h_c(f)\) away from the source.

Note

Careful with the different notations (can be confusing!!), see below.

\[\Omega_{\rm GW}(f) = \frac{2 \pi^2}{3 H_0^2} f^2 h_c^2(f)\]

Parameters

farray_like: Frequency array (in units of frequency, e.g. Hz).
OmGWarray_like: GW energy density spectrum OmGW(f).
dint: Option to convert from energy density to strain if set to -1 (default 1).
h0float: Parameterizes the uncertainties (Hubble tension) in the value of the Hubble rate (default 1).

Returns

hcarray_like: Strain spectrum.

References

Maggiore:1999vm, eq. 17

Maggiore defines \(S_h(f)\) in eq. B12 such that

\[h_c^2(f) = 2 f S_h^{\mathrm{Mag}}(f)\]

Note that this is different than the notation in RoperPol:2021xnd, eq. B.18, used in interferometry.py, where

\[h_c^2(f) = f S_h^{\pm}(f)\]

such that

\[S_h^{\pm}(f) = 2 S_h^{\mathrm{Mag}}(f)\]

Hence, from \(h_c^2(f)\) we can compute

\[S_h^{\mathrm{Mag}}(f) = \frac{h_c^2(f)}{2f} \qquad S_h^{\pm}(f) = \frac{h_c^2(f)}{f}\]

cosmoGW.GW_back.hc_Sf(f, Sf, d=1)

Transforms the power spectral density \(S_f(f)\) to the characteristic strain spectrum function \(h_c(f)\).

Parameters

farray_like: Frequency array (in units of frequency, e.g. Hz).
Sfarray_like: Power spectral density \(S_f(f)\) (in units of 1/Hz^3).
dint: Option to convert from strain to power spectral density if set to -1 (default 1).

Returns

hcarray_like: Strain spectrum.

Reference

RoperPol:2022iel, eq. 42

cosmoGW.GW_back.shift_OmGW_today(k, OmGW, g=100, gS=0.0, T=<Quantity 100. GeV>, d=1, h0=1.0, kk=True, Neff=3)

Shifts the GW energy density spectrum from the time of generation to the present time. Assumes the time of generation is within the radiation dominated era. Test.

Parameters

karray_like: Array of wave numbers (normalized by the Hubble scale).
OmGWarray_like: GW energy density spectrum per logarithmic interval (normalized by the radiation energy density).
gfloat: Number of relativistic degrees of freedom (dof) at the time of generation (default is 100).
gSfloat: Number of adiabatic dof (default is gS = g).
Tfloat: Temperature scale at the time of generation in energy units (convertible to MeV) (default is 100 GeV).
dint: Option to reverse the transformation if set to -1 (default 1).
h0float: Parameterizes the uncertainties (Hubble tension) in the value of the Hubble rate (default 1).
kkbool: If True, kf corresponds to k_* HH_*, otherwise refers to the length in terms of the Hubble size \({\cal H}_\ast l_\ast = 2 \pi {\cal H}_\ast/k_\ast\).
Nefffloat: Effective number of neutrino species (default is 3).

Returns

farray_like: Shifted wave number to frequency as a present time observable (in Hz).
OmGW0array_like: Shifted spectrum OmGW to present time.

Reference

See functions shift_onlyOmGW_today and shift_frequency_today.

cosmoGW.GW_back.shift_frequency_today(k, g=100, gS=0.0, T=<Quantity 100. GeV>, d=1, kk=True, Neff=3)

Transforms the normalized wave number at the time of generation by the Hubble rate \(H_\ast\) to the present time frequency.

Parameters

karray_like: Array of wave numbers (normalized by the Hubble scale).
gfloat: Number of relativistic degrees of freedom (dof) at the time of generation (default is 100).
gSfloat: Number of adiabatic dof (default is gS = g).
Tfloat: Temperature scale at the time of generation in energy units (convertible to MeV) (default is 100 GeV).
dint: Option to reverse the transformation if set to -1 (default 1).
kkbool: If True, kf corresponds to \(k_\ast {\cal H}_\ast\), otherwise refers to the length in terms of the Hubble size \({\cal H}_\ast l_\ast = 2 \pi {\cal H}_\ast/k_\ast\).
Nefffloat: Effective number of neutrino species (default is 3).

Returns

farray_like: Shifted wave number to frequency as a present time observable (in Hz).

Reference

RoperPol:2022iel, eq. 32

cosmoGW.GW_back.shift_hc_today(k, hc, g=100, gS=0.0, T=<Quantity 100. GeV>, d=1, kk=True, Neff=3)

Shifts the characteristic amplitude spectrum from the time of generation to the present time.

Assumes the time of generation is within the radiation dominated era.

Parameters

karray_like: Array of wave numbers (normalized by the Hubble scale).
hcarray_like: Spectrum of GW characteristic amplitude per logarithmic interval.
gfloat: Number of relativistic degrees of freedom (dof) at the time of generation (default is 100).
gSfloat: Number of adiabatic dof (default is gS = g).
Tfloat: Temperature scale at the time of generation in energy units (convertible to MeV) (default is 100 GeV).
dint: Option to reverse the transformation if set to -1 (default 1).
kkbool: If True, kf corresponds to \(k_* \mathcal{H}_*\). Otherwise, refers to the length in terms of the Hubble size, i.e. \(\mathcal{H}_* l_* = 2\pi {\cal H}_\ast/k_*\).
Nefffloat: Effective number of neutrino species (default is 3).

Returns

farray_like: Shifted wave number to frequency as a present time observable (in Hz).
hc0array_like: Shifted \(h_c(f)\) spectrum to present time.

Reference

RoperPol:2018sap, eq. B.12

cosmoGW.GW_back.shift_onlyOmGW_today(OmGW, g=100, gS=0.0, d=1, h0=1.0, Neff=3)

Shifts the GW energy density spectrum from the time of generation to the present time (assumed to be within the RD era).

Parameters

OmGWarray_like: GW energy density spectrum per logarithmic interval (normalized by the radiation energy density).
gfloat: Number of relativistic degrees of freedom (dof) at the time of generation (default is 100).
gSfloat: Number of adiabatic dof (default is gS = g).
dint: Option to reverse the transformation if set to -1 (default 1).
h0float: Parameterizes the uncertainties (Hubble tension) in the value of the Hubble rate (default 1).
Nefffloat: Effective number of neutrino species (default is 3).

Returns

OmGW0array_like: Shifted spectrum OmGW to present time.

Reference

RoperPol:2022iel, eq. 27