GW_templates

GW_templates.py is a Python routine that contains analytical and semi-analytical templates of cosmological GW backgrounds, usually based on spectral fits, either from GW models (see GW_models) or from numerical simulations

Currently part of the cosmoGW code:

https://github.com/cosmoGW/cosmoGW/ https://github.com/cosmoGW/cosmoGW/blob/main/src/cosmoGW/GW_templates.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: 01/12/2022
Updated: 31/08/2024
Updated: 21/08/2025 (release cosmoGW 1.0: https://pypi.org/project/cosmoGW)

References

[Espinosa:2010hh]: J. R. Espinosa, T. Konstandin, J. M. No, G. Servant, “Energy Budget of Cosmological First-order Phase Transitions,” JCAP 06 (2010), 028, arXiv:1004.4187.
[Hindmarsh:2017gnf]: M. Hindmarsh, S. J. Huber, K. Rummukainen, D. J. Weir, “Shape of the acoustic gravitational wave power spectrum from a first order phase transition,” Phys.Rev.D 96 (2017) 10, 103520, Phys.Rev.D 101 (2020) 8, 089902 (erratum), arXiv:1704.05871.
[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:2019egz]: [LISA CosWG], “Detecting gravitational waves from cosmological phase transitions with LISA: an update,” JCAP 03 (2020) 024, arXiv:1910.13125.
[Hindmarsh:2020hop]: M. Hindmarsh, M. Lueben, J. Lumma, M. Pauly, “Phase transitions in the early universe,” SciPost Phys. Lect. Notes 24 (2021), 1, arXiv:2008.09136.
[Jinno:2022mie]: R. Jinno, T. Konstandin, H. Rubira, I. Stomberg, “Higgsless simulations of cosmological phase transitions and gravitational waves,” JCAP 02, 011 (2023), arXiv:2209.04369.
[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)
[EPTA:2023xxk]: [EPTA and InPTA Collaborations], “The second data release from the European Pulsar Timing Array - IV. Implications for massive black holes, dark matter, and the early Universe,” Astron. Astrophys. 685, A94 (2024), arXiv:2306.16227
[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.
[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.
[Caprini:2024hue]: E. Madge, C. Caprini, R. Jinno, M. Lewicki, M. Merchand, G. Nardini, M. Pieroni, A. Roper Pol, V. Vaskonen, “Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation,” JCAP 10 (2024) 020, arXiv:2403.03723.
[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

Reference values for turbulence template are based on RoperPol:2023bqa and RoperPol:2022iel. Template used in Caprini:2024hue for LISA and in EPTA:2023xxk

Note that the values used for a_turb, b_turb, bPi_vort, fPi are found assuming that the spectrum of the source is defined such that

\[\langle v^2 \rangle \propto 2 E_\ast \int \zeta(K) d \ln K, \quad {\rm as \ in \ RoperPol:2025b},\]

which is a different normalization than that in previous papers, where zeta is defined such that

\[\langle v^2 \rangle \propto 2 E_\ast k_\ast \int \zeta(K) dK, \quad {\rm in \ RoperPol:2022iel}.\]

Hence, the values are considered for the former \(\zeta\) (this choice yields different coefficients). However, the final result is not affected by this choice. See RoperPol:2025b for details

cosmoGW.GW_templates.OmGW_spec_sw(s, alphas, betas, vws=1.0, cs2=0.3333333333333333, quiet=True, a_sw=3, b_sw=1, c_sw=3, alp1_sw=0, alp2_sw=0, corrRs=True, expansion=True, Nsh=1.0, model_efficiency='fixed_value', OmGW_tilde=0.01, bs_HL_eff=20, model_K0='Espinosa', bs_k1HL=40, model_decay='sound_waves', interpolate_HL_decay=True, b=0, model_shape='sw_LISA', strength='weak', interpolate_HL_shape=False, interpolate_HL_n3=False, redshift=False, gstar=100, gS=0, T=<Quantity 100. GeV>, h0=1.0, Neff=3)

Compute the GW spectrum for sound waves, normalized to radiation energy density.

The GW spectrum is calculated as:

\[\Omega_{\rm GW} (f) = 3 \times {\rm ampl\_GWB} \times {\rm pref\_GWB} \times S(f)\]

where ampl_GWB is the efficiency, pref_GWB is the amplitude prefactor, and \(S(f)\) is the normalized spectral shape.

The GW spectrum as an observable at present time is then computed using

\[\Omega_{\rm GW}^0 (f) = \Omega_{\rm GW} (f) \times F_{\rm GW, 0},\]

where \(F_{\rm GW, 0}\) is the redshift from the time of generation to present time, computed in GW_back.py that depends on the degrees of freedom at the time of generation.

Parameters

sarray_like: Normalized wave number, divided by the mean bubble size Rstar, \(s = f R_\ast\).
alphasarray_like: Strength of the phase transition, \(\alpha\).
betasarray_like: Rate of nucleation of the phase transition, \(\beta/H_\ast\).
vwsarray_like, optional: Array of wall velocities, \(v_w\).
cs2float, optional: Square of the speed of sound (default 1/3).
quietbool, optional: Suppress debugging output.
a_sw, b_sw, c_swfloat, optional: Slopes of the sound wave template.
alp1_sw, alp2_swfloat, optional: Transition parameters of the sound wave template.
corrRsbool, optional: Correct Rstar beta with max(vw, cs).
expansionbool, optional: Include Universe expansion.
Nshfloat, optional: Number of shock formation times for source duration.
model_efficiencystr, optional: Model for GW efficiency (‘fixed_value’ or ‘higgsless’).
OmGW_tildefloat, optional: Value of GW efficiency, \(\tilde \Omega_{\rm GW}\) for fixed_value model.
bs_HL_effint, optional: Box size for Higgsless simulation interpolation.
model_K0str, optional: Model for kinetic energy ratio K0 (‘Espinosa’ or ‘higgsless’).
bs_k1HLint, optional: Box size for Higgsless simulation interpolation for k1.
model_decaystr, optional: Model for GW amplitude prefactor (‘sound_waves’ or ‘decay’).
interpolate_HL_decaybool, optional: Use Higgsless simulations to interpolate decay law.
bfloat, optional: Decay law exponent for kinetic energy density.
model_shapestr, optional: Model for spectral shape.
strengthstr, optional: Strength of the phase transition for sw_HLnew.
interpolate_HL_shapebool, optional: Use Higgsless simulations to interpolate k1 and k2.
interpolate_HL_n3bool, optional: Use Higgsless simulations to interpolate c_sw.
redshiftbool, optional: Redshift the GW spectrum and frequencies to present time.
gstar, gSint, optional: Degrees of freedom for redshift calculation.
Tastropy.units.Quantity: Temperature scale for redshift calculation.
h0float, optional: Hubble rate at present time.
Neffint, optional: Effective number of neutrino species.

Returns

freqsndarray: Array of frequencies (normalized or in Hz).
OmGWndarray: GW spectrum normalized to radiation energy density (or present critical density).

References

RoperPol:2023bqa, Caprini:2024gyk, Hindmarsh:2019phv

cosmoGW.GW_templates.OmGW_spec_turb(s, Oms, lfs, N=2, cs2=0.3333333333333333, a_turb=5, b_turb=0.6666666666666666, alp=0.35294117647058826, expansion=True, tdecay='eddy', tp='magnetic', alpPi=2.15, fPi=2.2, bPi=3, redshift=False, gstar=100, gS=0, T=<Quantity 100. GeV>, h0=1.0, Neff=3)

Compute the GW spectrum for turbulence, normalized to radiation energy density. The general shape of the GW spectrum is based on that of reference

\[\Omega_{\rm GW} (f) = 3 * ampl_GWB * pref_GWB * S(f),\]

where:

ampl_GWB is the efficiency of GW production by the specific source,
pref_GWB is the dependence of the GW amplitude on the source parameters (e.g. length scale and strength of the source),
S(f) is a normalized spectral shape, such that \(\int S(f) d \ln f = 1\).

The GW spectrum as an observable at present time is then computed using

\[\Omega_{\rm GW0} (f) = \Omega_{\rm GW} x F_{\rm GW, 0},\]

where \(F_{\rm GW, 0}\) is the redshift from the time of generation to present time, computed in GW_back.py that depends on the degrees of freedom at the time of generation.

Parameters

sarray_like: Normalized wave number, divided by the mean bubble size \(R_\ast\), \(s = f R_\ast\).
Omsfloat or array_like, optional: Energy density of the source, \(\Omega_\ast = {1 \over 2} v_{\rm rms}^2)\).
lfsfloat or array_like, optional: Characteristic scale of the turbulence as a fraction of the Hubble radius, \(l_\ast H_\ast\).
Nint, optional: Ratio between decay time and effective source duration.
cs2float, optional: Square of the speed of sound (default 1/3).
quietbool, optional: Suppress debugging output.
a_turb, b_turb, alpfloat, optional: Slopes and smoothness of the turbulent source spectrum.
expansionbool, optional: Include Universe expansion.
tdecaystr, optional: Determines finite duration in the cit model.
tpstr, optional: Type of source (‘magnetic’, ‘kinetic’, or ‘max’).
alpPi, fPi, bPifloat, optional: Parameters for the anisotropic stress fit.
redshiftbool, optional: Redshift the GW spectrum and frequencies to present time.
gstar, gSint, optional: Degrees of freedom for redshift calculation.
Tastropy.units.Quantity: Temperature scale for redshift calculation.
h0float, optional: Hubble rate at present time.
Neffint, optional: Effective number of neutrino species.

Returns

freqsndarray: Array of frequencies (normalized or in Hz).
OmGWndarray: GW spectrum normalized to radiation energy density (or present critical density).

Reference

RoperPol:2023bqa, equations 3 and 9.

cosmoGW.GW_templates.OmGW_spec_turb_alphabeta(s, alphas, betas, vws=1.0, eps_turb=1.0, model_K0='Espinosa', bs_HL_eff=20, N=2, cs2=0.3333333333333333, corrRs=True, quiet=True, a_turb=5, b_turb=0.6666666666666666, alp=0.35294117647058826, expansion=True, tdecay='eddy', tp='both', alpPi=2.15, fPi=2.2, bPi=3, redshift=False, gstar=100, gS=0, T=<Quantity 100. GeV>, h0=1.0, Neff=3)

Compute the GW spectrum for turbulence using \(\alpha\) and \(\beta\) (i.e., the parameters of the phase transition), following the description of RoperPol:2023bqa.

It is assumed that turbulence has two contributions (from velocity and magnetic fields), which are in equipartition,

\[\Omega_\ast = \Omega_v + \Omega_B \rightarrow \Omega_v = \Omega_B = 0.5 \Omega_\ast.\]

It takes the kinetic energy density from the PT, \(K = \rho_{\rm kin}/\rho_{\rm total}\), and assumes that the turbulence energy density is a fraction \(\epsilon_{\rm turb}\) of it,

\[\Omega_\ast = \epsilon_{\rm turb} K.\]

The duration of the GW production \(\delta t_{\rm fin}\) is assumed to be

\[\delta t_{\rm fin} = N_{\rm turb} \frac{l_\ast}{u_\ast},\]

where \(u_\ast\) is a characteristic velocity,

\[u_\ast = \sqrt{\max(\Omega_v, \frac{2}{1 + c_s^2} \Omega_B)}.\]

Parameters

sarray_like: Normalized wave number, divided by the mean bubble size \(R_\ast\), \(s = f R_\ast\).
alphasarray_like: Strength of the phase transition \(\alpha\).
betasarray_like: Rate of nucleation of the phase transition \(\beta/H_\ast\).
vwsarray_like, optional: Array of wall velocities.
eps_turbfloat, optional: Fraction of kinetic energy converted to turbulence.
model_K0str, optional: Model for kinetic energy ratio K0 (‘Espinosa’ or ‘higgsless’).
bs_HL_effint, optional: Box size for Higgsless simulation interpolation.
Nint, optional: Ratio between decay time and effective source duration.
cs2float, optional: Square of the speed of sound.
corrRsbool, optional: Correct Rstar beta with max(vw, cs).
quietbool, optional: Suppress debugging output.
a_turb, b_turb, alpfloat, optional: Slopes and smoothness of the turbulent source spectrum.
expansionbool, optional: Include Universe expansion.
tdecaystr, optional: Determines finite duration in the cit model.
tpstr, optional: Type of source (‘both’, ‘magnetic’, ‘kinetic’).
alpPi, fPi, bPifloat, optional: Parameters for the anisotropic stress fit.
redshiftbool, optional: Redshift the GW spectrum and frequencies to present time.
gstar, gSint, optional: Degrees of freedom for redshift calculation.
Tastropy.units.Quantity: Temperature scale for redshift calculation.
h0float, optional: Hubble rate at present time.
Neffint, optional: Effective number of neutrino species.

Returns

freqsndarray: Array of frequencies (normalized or in Hz).
OmGWndarray: GW spectrum normalized to radiation energy density (or present critical density).

cosmoGW.GW_templates.Sf_shape_sw(s, model='sw_LISA', Dw=1.0, a_sw=3, b_sw=1, c_sw=3, alp1_sw=0, alp2_sw=0, strength='weak', interpolate_HL=False, bs_k1HL=40, bs_k2HL=20, vws=None, alphas=None, quiet=False, interpolate_HL_n3=False, corrRs=True, cs2=0.3333333333333333)

Compute the GW spectral shape generated by sound waves based on different templates.

Several models are available, including analytic fits and interpolation from Higgsless simulations.

Parameters

sarray_like: Normalized wave number, divided by the mean bubble size, \(s = f R_\ast\).
modelstr, optional: Model for the sound-wave template (‘sw_SSM’, ‘sw_HL’, ‘sw_LISA’, ‘sw_LISAold’, ‘sw_HLnew’).
Dwfloat or array_like, optional: Ratio between peak frequencies \(\Delta_w\), determined by the fluid shell thickness.
a_sw, b_sw, c_swfloat, optional: Slopes for sound wave template (default 3, 1, 3).
alp1_sw, alp2_swfloat, optional: Transition smoothness parameters for sound wave template.
strengthstr, optional: Phase transition strength (‘weak’, ‘interm’, ‘strong’), used to determine peak2 in sw_HLnew.
interpolate_HLbool, optional: Use numerical data from Caprini:2024gyk for slope and peak positions.
bs_k1HL, bs_k2HLint, optional: Box size \(L/v_w\) of Higgsless simulations for peak interpolation.
vws, alphasarray_like, optional: Wall velocities and alpha values for parameter estimation.
quietbool, optional: Suppress warnings about interpolation range.
interpolate_HL_n3bool, optional: Use Higgsless simulations to estimate value of high-frequency slope c_sw.
corrRsbool, optional: Correct Rstar beta with max(vw, cs).
cs2float, optional: Square of the speed of sound (default 1/3).

Returns

Sndarray: Spectral shape of the GW spectrum as a function of \(s = f R_\ast\). If interpolate_HL is True, returns an array of shape (s, alphas, vws).

References

RoperPol:2023bqa, Hindmarsh:2019phv, Jinno:2022mie, Caprini:2024gyk

cosmoGW.GW_templates.Sf_shape_turb(s, Oms=0.1, lf=0.01, N=2, cs2=0.3333333333333333, expansion=True, tdecay='eddy', tp='magnetic', b_turb=0.6666666666666666, alpPi=2.15, fPi=2.2, bPi=3)

Compute the spectral shape for GWs generated by MHD turbulence.

Parameters

sarray_like: Normalized wave number, divided by the mean bubble size, \(s = f R_\ast\).
Omsfloat or array_like, optional: Energy density of the source, \(\Omega_\ast = {1 \over 2} v_{\rm rms}^2)\).
lffloat or array_like, optional: Characteristic scale of the turbulence as a fraction of the Hubble radius, \(l_\ast H_\ast\).
Nint, optional: Ratio between decay time and effective source duration.
cs2float, optional: Square of the speed of sound (default 1/3).
expansionbool, optional: Include Universe expansion.
tdecaystr, optional: Determines finite duration in the cit model.
tpstr, optional: Type of source (‘magnetic’, ‘kinetic’, or ‘max’).
b_turbfloat, optional: Slope of the velocity/magnetic field spectrum in the UV.
alpPi, fPi, bPifloat, optional: Parameters for the anisotropic stress fit.

Returns

Sndarray: Spectral shape for turbulence.

References

RoperPol:2023bqa, equation 9, based on RoperPol:2022iel, section II D. Also used in EPTA:2023xxk, Caprini:2024hue.

Also used in EPTA:2023xxk, Caprini:2024hue.

See further details in RoperPol:2025b.

cosmoGW.GW_templates.ampl_GWB_sw(model='fixed_value', OmGW_sw=0.01, vws=None, alphas=None, numerical=False, bs_HL=20, quiet=False)

Compute the GW efficiency parameter for sound waves.

This function returns the normalized amplitude (efficiency) of the GW spectrum from sound waves, based on either a fixed reference value or interpolation from Higgsless simulation results.

For model=’fixed_value’, the amplitude is set to a constant value (default: 1e-2), based on Hindmarsh:2019phv and Hindmarsh:2017gnf.
For model=’higgsless’, the amplitude is interpolated from simulation data (Caprini:2024gyk) for given wall velocities and alpha values. The simulation data is available for \(\alpha\) = 0.0046, 0.05, 0.5 and wall velocities in the range 0.32 to 0.8.

Parameters

modelstr, optional: Model for GW efficiency (‘fixed_value’ or ‘higgsless’).
OmGW_swfloat, optional: Reference value for GW efficiency, \(\tilde \Omega_{\rm GW}\) (default: 1e-2).
vwsarray_like, optional: Array of wall velocities for interpolation (required for ‘higgsless’).
alphasarray_like, optional: Array of alpha values for interpolation (required for ‘higgsless’).
numericalbool, optional: If True and model is ‘higgsless’, also return the full grid of simulation values and available alpha and v_wall values.
bs_HLint, optional: Box size (L/vw) used in the simulation (default: 20).
quietbool, optional: If False, print warnings about the validity range of the interpolation.

Returns

Omegasndarray: GW efficiency parameter for the requested values.
Omnumndarray, optional: Full grid of simulation values (only if numerical=True and model=’higgsless’).
val_alphasndarray, optional: Available alpha values in the simulation (only if numerical=True and model=’higgsless’).
val_vwsndarray, optional: Available wall velocity values in the simulation (only if numerical=True and model=’higgsless’).

References

Hindmarsh:2019phv, Hindmarsh:2017gnf, Caprini:2024gyk

cosmoGW.GW_templates.ampl_GWB_turb(a_turb=5, b_turb=0.6666666666666666, alp=0.35294117647058826)

Compute the GW efficiency parameter for turbulence.

Parameters

a_turbfloat, optional: Low-frequency slope of the turbulent source spectrum.
b_turbfloat, optional: High-frequency slope of the turbulent source spectrum.
alpfloat, optional: Smoothness parameter.

Returns

amplfloat: GW efficiency parameter for turbulence.

References

RoperPol:2023bqa, equation 9, based on the template of RoperPol:2022iel, section 3 D.

See footnote 3 of RoperPol:2023bqa for clarification (extra factor 1/2 for oscillation average).

cosmoGW.GW_templates.interpolate_HL_vals(df, vws, alphas, value='Omega_tilde_int_extrap', boxsize=40, numerical=False, quiet=False)

Interpolate Higgsless simulation results to arbitrary wall velocities (\(v_w\)) and alpha (\(\alpha\)).

This function uses numerical results from Caprini:2024gyk to interpolate quantities (such as GW efficiency or spectral parameters) for different values of wall velocity and phase transition strength. The data is read from:

resources/higgsless/parameters_fit_sims.csv

If numerical is True, the function also returns the full grid of simulation values and the available \(\alpha\) and \(v_w\) values.

Parameters

dfpandas.DataFrame: DataFrame containing the simulation results.
vwsarray_like: Array of wall velocities for interpolation.
alphasarray_like: Array of alpha values (phase transition strength) for interpolation.
valuestr, optional: Name of the column in the DataFrame to interpolate (default: ‘Omega_tilde_int_extrap’).
boxsizeint, optional: Box size (L/vw) used in the simulation (default: 40).
numericalbool, optional: If True, also return the full grid of simulation values and available alpha and v_wall values.
quietbool, optional: If False, print warnings about the validity range of the interpolation.

Returns

Omsssndarray: Interpolated values for the requested quantity.
Omegasndarray, optional: Full grid of simulation values (only if numerical=True).
val_alphasndarray, optional: Available alpha values in the simulation (only if numerical=True).
val_vwsndarray, optional: Available wall velocity values in the simulation (only if numerical=True).

References

Caprini:2024gyk

cosmoGW.GW_templates.pPi_fit(s, b=0.6666666666666666, alpPi=2.15, fPi=2.2, bPi=3)

Fit the spectrum of the anisotropic stresses for turbulence.

The spectrum can be computed numerically for a Gaussian source using EPi_correlators in GW_models. Default values are valid for a purely vortical velocity or magnetic field following a von Kárman spectrum, as indicated in RoperPol:2023bqa, equation 17.

The fit is:

\[p_\Pi = \left[1 + \left(\frac{f}{f_\Pi}\right)^{\alpha_\Pi} \right]^{-\frac{(b + b_\Pi)}{\alpha_\Pi}}.\]

Parameters

sarray_like: Array of frequencies, normalized by the characteristic scale, \(s = f R_\ast\).
bfloat, optional: High-frequency slope \(f^{-b}\).
alpPifloat, optional: Smoothness parameter of the fit, \(\alpha_\Pi\).
fPifloat, optional: Position of the fit break, \(f_\Pi\).
bPifloat, optional: Extra power law decay of the spectrum of the stresses compared to b, \(b_\Pi\).

Returns

Pindarray: Spectrum of the anisotropic stresses, \(p_\Pi(f)\).
fGWfloat: Frequency where \(s p_\Pi(s)\), which determines amplitude of GW spectrum for MHD turbulence, is maximum, \(f_{\rm GW}\).
pimaxfloat: Maximum value of \(p_\Pi\), i.e., at \(s = f_{\rm GW}\).

References

RoperPol:2023bqa, RoperPol:2025b

cosmoGW.GW_templates.pref_GWB_sw(Oms=0.1, lf=0.01, alpha=0, model='sound_waves', Nshock=1.0, b=0.0, expansion=True, beta=array([-20., -19.98666222, -19.97332444, ..., 19.97332444, 19.98666222, 20.], shape=(3000,)), cs2=0.3333333333333333)

Compute the prefactor for the GW spectrum from sound waves.

This function calculates the dependence of the GW spectrum amplitude on the kinetic energy density, \(\Omega_\ast\) and the mean bubble size \(l_\ast = R_\ast H_\ast\), for different physical models of the source duration and decay.

\[\Omega_\ast = v_{\rm rms}^2 = \frac{\langle w v^2 \rangle} {\langle w \rangle},\]

related to \(K = \kappa \alpha/(1 + \alpha)\) used in Caprini:2024gyk as

\[\Omega_\ast = \frac{K}{\Gamma} = \frac{\kappa \alpha}{(1 + c_s^2)}, \quad {\rm \ where \ } \Gamma = \frac{\langle w \rangle}{\langle \rho \rangle} = \frac{1 + c_{\rm s}^2}{1 + \alpha}\]

is the adiabatic index.

Note that RoperPol:2023dzg uses \(\Omega_K = {1 \over 2} v_{\rm rms}^2\) so an extra factor of 2 appears in its relation to \(K\).

For model=’sound_waves’, the prefactor is computed using the stationary sound wave assumption, including the effect of Universe expansion via the suppression factor Upsilon (Caprini:2024gyk eqs. 2.16 and 2.30). It corresponds to the linear growth with the source duration, which is set to

\[t_{\rm dur} = N_{\rm shock} t_{\rm shock} = N_{\rm shock} R_\ast/\sqrt{\Omega_\ast},\]

assuming sound waves do not decay and their UETC is stationary:

\[\Omega_{\rm GW} \sim K^2 R_\ast t_{\rm dur} = K^2 R_\ast N_{\rm shock} t_{\rm shock}.\]

When the Universe expansion is included, \(t_{\rm dur}\) is substituted by the suppression factor \(\Upsilon\) describing the growth with the source duration. For a radiation-dominated Universe (see RoperPol:2023dzg):

\[\Upsilon = \frac{t_{\rm dur}}{1 + t_{\rm dur}}\]

This choice includes Universe expansion and assumes sourcing occurs during the radiation-dominated era

\[\Omega_{\rm GW} \sim K^2 R_\ast \Upsilon(t_{\rm dur}) = K^2 R_\ast \frac{t_{\rm dur}}{1 + t_{\rm dur}}.\]

When \(N_{\rm shock} = 1, t_{\rm dur} = t_{\rm shock} = R_\ast/\sqrt{\Omega_{\rm s}}\) and it becomes equation 3 of RoperPol:2023bqa, based on Hindmarsh:2020hop, equation 8.24.
For model=’decay’, the prefactor includes the decay of the source and uses a power-law decay for the kinetic energy density. It uses Caprini:2024gyk, eq. 5.6, assuming a locally stationary UETC. For Minkowski space-time:

\[\Omega_{\rm GW} \sim K^2_{\rm int} R_\ast, \quad {\rm \ where \ } K^2_{\rm int} = \int K^2 dt,\]

while for a radiation-dominated expanding Universe:

\[\Omega_{\rm GW} \sim K^2_{\rm int, exp} R_\ast, \quad {\rm \ where \ } K^2_{\rm int, exp} = \int \frac{K^2}{t^2} dt.\]

Parameters

Omsfloat or array_like, optional: Kinetic energy density, defined as \(v_{\rm rms}^2\), such that \(\Omega_\ast = K/\Gamma = (1 + c_s^2) \kappa \alpha\).
lffloat or array_like, optional: Mean bubble size as a fraction of the Hubble radius.
alphafloat, optional: Ratio of vacuum to radiation energy densities.
modelstr, optional: Model for the prefactor (‘sound_waves’ or ‘decay’).
Nshockfloat, optional: Duration time in units of the shock time.
bfloat, optional: Power law decay exponent for the kinetic energy source. Default 0 recovers stationary sound waves.
expansionbool, optional: If True, include Universe expansion (radiation-dominated era).
betafloat, optional: Rate of nucleation of the phase transition.
cs2float, optional: Square of the speed of sound (default is 1/3 for radiation domination).

Returns

preffloat or ndarray: Prefactor for the GW spectrum, consisting of \(K^2 R_\ast\) or \(K^2_{\rm int, exp} R_\ast\), depending on the chosen model.

References

Caprini:2024gyk, RoperPol:2023bqa, RoperPol:2023dzg

cosmoGW.GW_templates.pref_GWB_turb(Oms=0.1, lf=0.01)

Compute the prefactor for the GW spectrum from turbulence.

The GW spectrum depends on the turbulence length scale (lf) and the fraction of turbulent to radiation energy density (Oms).

Parameters

Omsfloat or array_like, optional: Energy density of the source (i.e., 1/2 vrms^2).
lffloat or array_like, optional: Mean-size of the bubbles, as a fraction of the Hubble radius.

Returns

preffloat or ndarray: Prefactor for the GW spectrum from turbulence.

References

RoperPol:2023bqa, equation 9, based on RoperPol:2022iel, section II D.

Also used in EPTA:2023xxk, Caprini:2024hue.