GW_models

GW_models.py is a Python routine that contains analytical and semi-analytical models of cosmological GW backgrounds.

Currently part of the cosmoGW code:

https://github.com/cosmoGW/cosmoGW/ https://github.com/cosmoGW/cosmoGW/blob/main/src/cosmoGW/GW_models.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: 29/08/2024
Updated: 21/08/2025 (release cosmoGW 1.0: https://pypi.org/project/cosmoGW)

Contributors

Antonino Midiri, Simona Procacci, Madeline Salomé, Isak Stomberg

References

[RoperPol:2022iel]: A. Roper Pol, C. Caprini, A. Neronov, D. Semikoz, “The gravitational wave signal from primordial magnetic fields in the Pulsar Timing Array frequency band,” Phys. Rev. D 105, 123502 (2022), arXiv:2201.05630.
[RoperPol:2023bqa]: A. Roper Pol, A. Neronov, C. Caprini, T. Boyer, D. Semikoz, “LISA and γ-ray telescopes as multi-messenger probes of a first-order cosmological phase transition,” arXiv:2307.10744 (2023).
[RoperPol:2023dzg]: A. Roper Pol, S. Procacci, C. Caprini, “Characterization of the gravitational wave spectrum from sound waves within the sound shell model,” Phys. Rev. D 109, 063531 (2024), arXiv:2308.12943.
[Hindmarsh:2019phv]: M. Hindmarsh, M. Hijazi, “Gravitational waves from first order cosmological phase transitions in the Sound Shell Model,” JCAP 12 (2019), 062, arXiv:1909.10040.
[Caprini:2024gyk]: A. Roper Pol, I. Stomberg, C. Caprini, R. Jinno, T. Konstandin, H. Rubira, “Gravitational waves from first-order phase transitions: from weak to strong,” JHEP 07 (2025), 217, arxiv:2409.03651.
[RoperPol:2025b]: A. Roper Pol, A. Midiri, M. Salomé, C. Caprini, “Modeling the gravitational wave spectrum from slowly decaying sources in the early Universe: constant-in-time and coherent-decay models,” in preparation
[RoperPol:2025a]: A. Roper Pol, S. Procacci, A. S. Midiri, C. Caprini, “Irrotational fluid perturbations from first-order phase transitions,” in preparation

Comments

RoperPol:2022iel/RoperPol:2023dzg and RoperPol:2025b/RoperPol:2025b consider spectral functions defined such that the average squared field corresponds to

\[ \begin{align}\begin{aligned}\langle v^2 \rangle \propto A k_\ast \int \zeta(K) dK, \quad {\rm \ in \ RoperPol:2022iel},\\\langle v^2 \rangle \propto A \int \zeta(K) d \ln K, \quad {\rm \ in \ RoperPol:2025b}.\end{aligned}\end{align} \]

The first convention can be chosen in the following functions if dlogK is set to False, while the second one is assumed when dlogK is True

cosmoGW.GW_models.Delta2_cit_aver(k, tini=1.0, tfin=10000.0, expansion=True)

Compute D^2(k, t) averaged over t0 for constant-in-time GW source.

Parameters

karray_like: Wave numbers.
tinifloat, optional: Initial time (default 1).
tfinfloat, optional: Final time.
expansionbool, optional: Include expansion of Universe (default is True).

Returns

D2ndarray: D^2(k, t) averaged over t0.

References

RoperPol:2022iel, RoperPol:2023dzg

cosmoGW.GW_models.Delta_cit(t, k, tini=1.0, tfin=10000.0, expansion=True)

Compute Delta(k, t) used in the analytical calculations of the GW spectrum when assuming a constant sourcing stress spectrum, i.e., Pi(k, t1, t2) = Pi(k)

Parameters

tarray_like: Times.
karray_like: Wave numbers.
tinifloat, optional: Initial time of sourcing (default 1).
tfinfloat, optional: Final time of sourcing (default 1e4).
expansionbool, optional: Include expansion of Universe (default is True).

Returns

Dndarray: Delta function values.

References

RoperPol:2022iel, RoperPol:2023dzg

cosmoGW.GW_models.EPi_correlators(k, a=5, b=0.6666666666666666, alp=2, tp='all', zeta=False, hel=False, norm=True, dlogk=True, model='dbpl', kk=None, EK_p=None)

Compute spectrum of projected or unprojected stresses from the two-point correlator of the source (Gaussian assumption).

Computes vortical, compressional, helical, and mixed components.

References

RoperPol:2025a, RoperPol:2022iel (eqs 11 and 20) for vortical fields, and RoperPol:2023dzg for compressional ones.

Parameters

karray_like: Wave numbers.
afloat: Low-k slope, \(k^a\) (default 5)
bfloat: High-k slope, \(k^{-b}\) (default 2/3).
alpfloat: Smoothness of transition, \(\alpha\) (default 2).
tpstr: Type of sourcing field (‘vort’, ‘comp’, ‘hel’ or ‘mix’ available)
zetabool: Integrate convolution over \(z \in (-1, 1)\).
helbool: Compute helical stresses.
normbool: Normalize spectrum to peak at kpeak with amplitude A.
dlogkbool: Use spectrum per unit log(k).
modelstr: Model for source spectrum (‘dbpl’ or ‘input’).
kkarray_like, optional: Wave numbers for input model.
EK_parray_like, optional: Spectrum for input model.

Returns

pindarray: Spectrum of anisotropic stresses.

cosmoGW.GW_models.K2int(dtfin, K0=1.0, dt0=11, b=0.0, expansion=False, beta=array([-20., -19.98666222, -19.97332444, ..., 19.97332444, 19.98666222, 20.], shape=(3000,)))

Compute integrated kinetic energy density squared for a power law,

\[K(\Delta t) = K_0 \left(\frac{\Delta t}{\Delta t_0}\right)^{-b},\]

where \(\Delta t\) and \(\Delta t_0\) represent time intervals, \(K_0\) is the amplitude at the end of the phase transition, and \(\Delta t_0 = 11\) is a numerical parameter of the fit.

Based on the results of Higgsless simulations of Caprini:2024gyk, see equation 2.27.

It determines the amplitude of GWs in the locally stationary UETC describing the decay of compressional sources, see equation 2.33,

\[K_{\rm int}^2 = \int K^2 (t) d t.\]

Parameters

dtfinfloat: Duration of sourcing (in units of \(1/\beta\) or \(1/H_\ast\) if expansion is False or True).
K0float, optional: Amplitude \(K_0\) at end of phase transition.
dt0float, optional: Numerical parameter \(\Delta t_0\) for fit.
bfloat, optional: Power law exponent.
expansionbool, optional: Consider expanding Universe.
betafloat, optional: Nucleation rate normalized to Hubble time, \(\beta/H_\ast\).

Returns

K2intfloat: Integrated squared kinetic energy density.

References

Caprini:2024gyk

cosmoGW.GW_models.OmGW_ssm_HH19(k, EK, Np=3000, Nk=1000, plot=False, cs2=0.3333333333333333)

Compute normalized GW spectrum \(\Omega(k)\) using Hindmarsh:2019phv sound-shell model under the delta assumption for the ‘stationary’ term (see appendix B of RoperPol:2023dzg for details).

The resulting GW spectrum is

\[\Omega_{\rm GW} (k) = \frac{3 \pi}{8 c_{\rm s}} \times \Gamma^2 \times (k/k_\ast)^2 \times (\Omega_\ast/{\cal A})^2 \times T_{\rm GW} \times \Omega (k).\]

where \(\Omega_\ast = \langle v^2 \rangle = v_{\rm rms}^2\) and \(\cal A = \int \zeta(K) d \ln k\) (using the normalization of RoperPol:2025a for \(\zeta\)).

Parameters

karray_like: Wave numbers.
EKarray_like: Kinetic spectrum.
Npint, optional: Number of p discretizations.
Nkint, optional: Number of k discretizations.
plotbool, optional: Plot interpolated spectrum.
cs2float, optional: Speed of sound squared.

Returns

kpndarray: Wave numbers.
Ommndarray: GW spectrum.

References

Hindmarsh:2019phv, RoperPol:2023dzg (appendix B, eq. B3)

cosmoGW.GW_models.TGW_func(s, Oms=0.1, lf=0.01, N=2, cs2=0.3333333333333333, expansion=True, tdecay='eddy', tp='magnetic')

Compute the logarithmic envelope of the GW template in the constant-in-time model for the unequal time correlator, \({\cal T}_{\rm GW}\).

It is obtained from integrating the Green’s function in the constant-in-time model, validated in RoperPol:2022iel and described for other sources in RoperPol:2025b

The function TGW_func determines two regimes, frequencies below and above an inverse duration,

\[f_{\rm br} = 1/\delta t_{\rm fin},\]

as used in RoperPol:2023bqa, EPTA:2023xxk, Caprini:2024hue. However, the original reference RoperPol:2022iel used

\[k_{\rm br} = f_{\rm br}/(2\pi) = 1/\delta t_{\rm fin},\]

such that \(\delta t_{\rm fin} = \delta t_{\rm fin}/(2\pi)\), and relates \(\delta t_{\rm fin}\) to the decay time (considered the eddy turnover time) as

\[\delta t_{\rm finm} = N \delta t_{\rm decay} = N/(v_A k_*) = N R_*/(2\pi v_A)\]

Hence, we will consider

\[f_{\rm br} = 1/\delta t_{\rm fin} = \frac{1}{2\pi \delta t_{\rm fin}} = \frac{v_{\rm A}}{N R_\ast} \delta t_{\rm fin}.\]

Main references are:

RoperPol:2022iel, equation 24
RoperPol:2023bqa, equation 15
EPTA:2023xxk, equation 21
Caprini:2024hue, equation 2.17
RoperPol:2025b

Parameters

sarray_like: Frequencies normalized by characteristic scale.
Omsfloat, optional: Source energy density.
lffloat, optional: Characteristic scale fraction.
Nint, optional: Source duration/eddy turnover ratio.
cs2float, optional: Speed of sound squared.
expansionbool, optional: Include expansion of Universe.
tdecaystr, optional: Decay time model (‘eddy’).
tpstr, optional: Source type (‘magnetic’, ‘kinetic’, ‘max’).

Returns

TGWndarray: Logarithmic function for spectral shape.

References

RoperPol:2022iel (eq. 24), RoperPol:2023bqa (eq. 15), Caprini:2024hue (eq. 2.17), EPTA:2023xxk (eq. 21).

cosmoGW.GW_models.compute_Delta_mn(t, k, p, ptilde, cs2=0.3333333333333333, m=1, n=1, tini=1.0, expansion=True)

Compute integrated Green’s functions and stationary UETC Delta_mn for the sound shell model.

Parameters

tfloat: Time.
kfloat: Wave number k.
pfloat: Wave number p.
ptildefloat: Second wave number \(\tilde{p} = |k - p|\).
cs2float, optional: Speed of sound squared (default 1/3).
m, nint, optional: Indices for Delta_mn.
tinifloat, optional: Initial time.
expansionbool, optional: Include expansion of Universe.

Returns

Delta_mnndarray: Integrated Green’s function.

References

RoperPol:2023dzg, eqs.56-59

cosmoGW.GW_models.compute_kin_spec_ssm(z, vws, fp, l=None, sp='sum', type_n='exp', cs2=0.3333333333333333, min_qbeta=-4, max_qbeta=5, Nqbeta=1000, min_TT=-1, max_TT=3, NTT=5000, corr=False, dens=True, normbeta=True)

Compute kinetic power spectral density for the sound-shell model.

Kinetic spectra computed for the sound-shell model from \(f' (z)\) and \(l(z)\) functions. \(f'\) and \(l\) functions need to be previously computed from the self-similar fluid perturbations induced by expanding bubbles using hydro_bubbles module.

Parameters

zarray_like: Values of z.
vwsarray_like: Wall velocities.
fparray_like: \(f'(z)\) from hydro_bubbles.
larray_like, optional: \(l(z)\) from hydro_bubbles (if lz is True)
spstr, optional: Type of function for kinetic spectrum.
type_nstr, optional: Nucleation history (‘exp’, ‘sim’).
cs2float, optional: Speed of sound squared (default 1/3).
min_qbeta, max_qbeta, Nqbetaint, optional: Wave number range and discretization.
min_TT, max_TT, NTTint, optional: Lifetime range and discretization.
corrbool, optional: Correct Rstar beta with max(vw, cs) (default False)
densbool, optional: Return power spectral density (True) or kinetic spectrum (False).
normbetabool, optional: Normalize k with beta (True) or Rstar (False).

Returns

qbetandarray: Wave number normalized.
Pvndarray: Power spectral density or kinetic spectrum if dens is True or False.

References

Equations (32)-(37) of RoperPol:2023dzg, RoperPol:2025a

cosmoGW.GW_models.effective_ET_correlator_stat(k, EK, tfin, Np=3000, Nk=1000, plot=False, expansion=True, kstar=1.0, extend=False, largek=3, smallk=-3, tini=1.0, cs2=0.3333333333333333, terms='all', inds_m=None, inds_n=None)

Compute normalized GW spectrum \(\zeta_{\rm GW}(k)\) from velocity field spectrum for compressional anisotropic stresses, assuming stationary UETC (e.g., sound waves under the sound-shell model).

Parameters

karray_like: Wave numbers.
EKarray_like: Kinetic spectrum.
tfinfloat: Final time.
Npint, optional: Number of p discretizations.
Nkint, optional: Number of k discretizations.
plotbool, optional: Plot interpolated spectrum.
expansionbool, optional: Include expansion of Universe.
kstarfloat, optional: Peak position of the kinetic spectrum.
extendbool, optional: Extend wave number array.
largek, smallkint, optional: Range for extended k (in log10).
tinifloat, optional: Initial time of GW production.
cs2float, optional: Speed of sound squared (default 1/3).
termsstr, optional: Terms to compute (‘all’).
inds_m, inds_nlist, optional: Indices for Delta_mn.
corr_Delta_0bool, optional: Correct Delta_0.

Returns

kpndarray: Wave numbers.
Ommndarray: Normalized GW spectrum, \(\zeta_{\rm GW}(k)\).

References

RoperPol:2023dzg, eq. 93