cosmology

cosmology.py is a Python routine that contains functions relevant for cosmological calculations, including a solver of Friedmann equations.

Adapted from the original cosmology in GW_turbulence (https://github.com/AlbertoRoper/GW_turbulence), created in Nov. 2022

Currently part of the cosmoGW code:

https://github.com/cosmoGW/cosmoGW/ https://github.com/cosmoGW/cosmoGW/blob/main/src/cosmoGW/cosmology.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: 27/11/2022 (GW_turbulence)

• Updated: 21/08/2025 (release cosmoGW 1.0: https://pypi.org/project/cosmoGW)

References

• [RoperPol:2018sap]: A. Roper Pol, A. Brandenburg, T. Kahniashvili, A. Kosowsky, S. Mandal, “The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence,”. Geophys. Astrophys. Fluid Dynamics 114, 1, 130 (2020), arXiv:1807.05479.

• [RoperPol:2019wvy]: A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili, A. Kosowsky, “Numerical simulations of gravitational waves from early-universe turbulence,” Phys. Rev. D 102, 083512 (2020), arXiv:1903.08585.

• [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.

• [He:2022qcs]: Y. He, A. Roper Pol, A. Brandenburg, “Modified propagation of gravitational waves from the early radiation era,” JCAP 06 (2023), 025, arXiv:2212.06082.

• [Planck:2018vyg]: [Planck], “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, Astron. Astrophys. 652 (2021) C4 (erratum), arXiv:1807.06209.

• [Rasanen]: Lecture notes on Cosmology by Syksy Räsänen (https://www.mv.helsinki.fi/home/syrasane/cosmo/)

cosmoGW.cosmology.Hs_fact()

Compute factor for Hubble rate at generation time (radiation era).

The factor is given in units of Hz/\({\rm MeV}^2\), such that the actual Hubble rate is given after multiplying fact x \(\sqrt{g}\) x \(T^2\), being g the number of dof and T the temperature scale (in MeV).

Returns

factastropy.units.Quantity

Factor in Hz/\({\rm MeV}^2\).

Reference

RoperPol:2022iel, eq. 29

cosmoGW.cosmology.Hs_from_a(a, dir0='', Neff=3)

Compute Hubble rate during RD era from scale factor, assuming adiabatic expansion:

\[a^3 g_S T^3 = {\rm \ constant}.\]

Parameters

andarray

Array of scale factors

dir0str, optional

Directory where the file of dof is stored (resources/cosmology/ default).

Nefffloat, optional

Effective number of neutrino species (default is 3)

Returns

Hsndarray

Array of Hubble rates during RD era Hs – Hubble rate during RD era

References

See values_0 and thermal_g.

cosmoGW.cosmology.Hs_val(g=100, T=<Quantity 100. GeV>)

Compute Hubble rate at generation time (radiation era).

Parameters

gfloat

Relativistic degrees of freedom (default 100).

Tastropy.units.Quantity

Temperature scale (default 100 GeV).

Returns

Hsastropy.units.Quantity

Hubble rate in frequency units (Hz).

Reference

RoperPol:2022iel, eq. 29

cosmoGW.cosmology.Omega_rad_dof(a, dir0='', Neff=3)

Compute radiation energy density ratio due to varying dofs.

Parameters

andarray

Array of scale factors

dir0str, optional

Directory where the file of dof is stored (resources/cosmology/ default)

Nefffloat, optional

Effective number of neutrino species (default is 3)

Returns

Om_rat_dofndarray

Ratio of radiation energy density due to accounting for T depending dofs

References

He:2022qcs, eq.A.2

cosmoGW.cosmology.Omega_vs_a(a, dir0='', a0=1, h0=0.6732, OmL0=0.6841, dofs=True, Neff=3)

Compute energy density ratios as a function of scale factor for a universe composed by matter, radiation, and dark energy:

\[\Omega (a) = \Omega_{R}^0 \times a^{-4} + \Omega_{M}^0 \times a^{-3} + \Omega_{L}^0\]

Parameters

andarray

Array of scale factors.

dir0str, optional

Directory for dof file (default resources/cosmology/).

a0float, optional

Present scale factor (default 1).

h0float, optional

Present Hubble rate scaling (default 0.6732).

OmL0float, optional

Present dark energy content (default 0.6841).

dofsbool, optional

If True, compensate radiation energy density using dofs.

Nefffloat, optional

Effective number of neutrino species (default 3).

Returns

Om_totndarray

Total energy density (normalized).

Om_radndarray

Radiation energy density (normalized).

Om_matndarray

Matter energy density (normalized).

References

He:2022qcs, eq.A.2

cosmoGW.cosmology.RD_dofs(dir0='', Neff=3)

Compute relativistic and adiabatic dofs during the RD era.

Parameters

dir0str, optional

Directory where the file of dof is stored (‘resources/cosmology/’ directory by default)

Nefffloat, optional

Effective number of neutrino species (default is 3)

Returns

Tndarray

Array of temperatures within RD era

as_Tndarray

Array of scale factors

gsndarray

Array of relativistic dofs

gSndarray

Array of adiabatic dofs

References

See values_0 and thermal_g.

cosmoGW.cosmology.as_a0_rat(g=100, T=<Quantity 100. GeV>, Neff=3)

Compute ratio of scale factors (generation/present) assuming adiabatic expansion of the universe.

Parameters

gfloat

Number of relativistic degrees of freedom (dof) at the time of generation (default is 100).

Tastropy.units.Quantity

Temperature scale at the time of generation (default is 100 GeV).

Nefffloat

Effective number of neutrino species (default is 3).

Returns

as_a0astropy.units.Quantity

Ratio of scale factors (a*/a0).

Reference

RoperPol:2022iel, eq. 28

cosmoGW.cosmology.as_fact(Neff=3)

Compute factor for scale factor ratio at generation time assuming adiabatic expansion of the universe.

The factor is in units of MeV and the ratio is obtained by multiplying fact x \(g^{-1/3}/T\), being \(g\) the number of dof and \(T\) the temperature scale (in MeV).

Parameters

Nefffloat

Effective number of neutrino species (default is 3).

Returns

factastropy.units.Quantity

Ratio to present-day scale factor, \(T_0 g_{0s}^{1/3}\).

Reference

RoperPol:2022iel, eq. 28

cosmoGW.cosmology.cut_var(x, x0, xf, inds, inds2)

Cut variable x using indices and add initial/final values. Used in norm_variables_cut function.

Parameters

xndarray

Input array.

x0float

Initial value to add.

xffloat

Final value to add.

indsndarray

Indices for first cut.

inds2ndarray

Indices for second cut.

Returns

yndarray

Cut and extended array.

cosmoGW.cosmology.dofs_vs_a(a, dir0='', Neff=3)

Compute relativistic and adiabatic dofs for scale factors.

Parameters

andarray

Array of scale factors

dir0str, optional

Directory where the file of dof is stored (‘/cosmology/’ directory by default)

Nefffloat, optional

Effective number of neutrino species (default is 3)

Returns

gsndarray

Array of relativistic dofs

gSndarray

Array of adiabatic dofs

References

See values_0 and thermal_g.

cosmoGW.cosmology.friedmann(a, dir0='', a0=1, h0=0.6732, OmL0=0.6841, dofs=True, Neff=3)

Use Friedmann equations to compute EOS and time derivatives.

Parameters

andarray

Array of scale factors.

dir0str, optional

Directory for dof file (default resources/cosmology/).

a0float, optional

Present scale factor (default 1).

h0float, optional

Present Hubble rate scaling (default 0.6732).

OmL0float, optional

Present dark energy content (default 0.6841).

dofsbool, optional

If True, compensate radiation energy density using dofs.

Nefffloat, optional

Effective number of neutrino species (default 3).

Returns

wfloat

Equation of state (p = w rho).

adfloat

Cosmic time derivative of the scale factor.

addfloat

Second cosmic time derivative of the scale factor.

apfloat

Conformal time derivative of the scale factor.

appfloat

Second conformal time derivative of the scale factor.

References

He:2022qcs, eqs. A.5-A.6

cosmoGW.cosmology.friedmann_solver(a, dir0='', a0=1.0, h0=0.6732, OmL0=0.6841, dofs=True, Neff=3, return_all=False, save=True, nm_fl='')

Numerically solve Friedmann equations for a(eta) and a(t).

A tutorial is available under cosmology/cosmology.ipynb

Parameters

andarray

Array of scale factors.

dir0str, optional

Directory for dof file (default resources/cosmology/).

a0float, optional

Present scale factor (default 1).

h0float, optional

Present Hubble rate scaling (default 0.6732).

OmL0float, optional

Present dark energy content (default 0.6841).

dofsbool, optional

If True, compensate radiation energy density using dofs.

Nefffloat, optional

Effective number of neutrino species (default 3).

return_allbool, optional

If True, return all variables used in the Friedmann solver.

savebool, optional

If True, save the solutions in the file ‘friedmann/solution#nm_fl.csv’ where #nm_fl is the input file name.

nm_flstr, optional

File name suffix.

Returns

tndarray

Cosmic time.

etandarray

Conformal time.

(other variables if return_all is True)

References

He:2022qcs, appendix A.

cosmoGW.cosmology.norm_variables_cut(eta_n, HH_n, a_n, Omega, Omega_mat, eta_n_0, dir0='', T=<Quantity 100. GeV>, OmM0=0.31589999999999996, h0=0.6732)

Cut normalized variables between initial time and present time.

Parameters

eta_nndarray

Normalized conformal time eta/eta_*.

HH_nndarray

Normalized conformal Hubble rate H/H_*.

a_nndarray

Normalized scale factor a/a_*.

Omegandarray

Ratio of total energy to present-time critical energy density.

Omega_matndarray

Matter energy density (normalized).

eta_n_0float

Normalized conformal present time.

Hsfloat

Hubble rate at the initial time.

astfloat

Scale factor at the initial time.

dir0str, optional

Directory for dof file (default resources/cosmology/).

Tfloat, optional

Temperature scale at the initial time in energy units (default is 100 GeV).

OmM0float, optional

Present-time content of matter (default is 0.3159).

h0float, optional

Present-time value of the Hubble rate H0 = h0 x 100 km/s/Mpc (default is 67.32 km/s/Mpc based on CMB observations).

Returns

eta_nnndarray

Normalized conformal time.

HH_nnndarray

Normalized conformal Hubble rate.

a_nnndarray

Normalized scale factor.

Omega_nnndarray

Ratio of total energy to present-time critical energy density.

Omega_mat_nnndarray

Matter energy density (normalized).

app_nnndarray

Second time derivative of the scale factor.

w_nnndarray

Equation of state p/rho.

cosmoGW.cosmology.normalized_variables(a, eta, ap_a, app_a, dir0='', T=<Quantity 100. GeV>, h0=0.6732)

Compute normalized variables for GW generation initial time, which are required to be used in the Pencil Code.

A tutorial is available under cosmology/cosmology_PC.ipynb

Parameters

andarray

Scale factors, normalized to present-time a_0 = 1.

etandarray

Conformal times, normalized to present-time a_0 = 1.

ap_andarray

Conformal Hubble time a’/a, normalized to present-time a_0 = 1.

app_andarray

a’’/a, normalized to present-time a_0 = 1.

dir0str, optional

Directory for dof file (default resources/cosmology/).

Tastropy.units.Quantity, optional

Temperature scale (default 100 GeV).

h0float, optional

Present Hubble rate scaling (default 0.6732).

Returns

a_nndarray

Normalized scale factor a/a_*.

eta_nndarray

Normalized conformal time eta/eta_*.

HH_nndarray

Normalized conformal Hubble rate H/H_*.

app_a_nndarray

Normalized second conformal time derivative of a.

Omegandarray

Ratio of total energy to present-time critical energy density.

wndarray

Equation of state.

eta_n_0float

Normalized conformal present time.

aEQ_nfloat

Normalized equipartition scale factor.

aL_nfloat

Normalized dark energy domination scale factor.

a_acc_nfloat

Normalized scale factor when acceleration starts.

eta_n_EQfloat

Normalized conformal time at equipartition.

eta_n_Lfloat

Normalized conformal time at dark energy domination.

eta_n_accfloat

Normalized conformal time when acceleration starts.

References

He:2022qcs, appendix A. RoperPol:2018sap.

cosmoGW.cosmology.ratio_app_a_n_factor(a, dir0='', a0=1, h0=0.6732, OmL0=0.6841, dofs=True, Neff=3)

Compute ratio of a’’/a (normalized) to conformal Hubble rate H (normalized) times a_*/a_0 during RD era.

Parameters

andarray

Array of scale factors.

dir0str, optional

Directory for dof file (default resources/cosmology/).

a0float, optional

Present scale factor (default 1).

h0float, optional

Present Hubble rate scaling (default 0.6732).

OmL0float, optional

Present dark energy content (default 0.6841).

dofsbool, optional

If True, compensate radiation energy density using dofs.

Nefffloat, optional

Effective number of neutrino species (default 3).

Returns

factorndarray

Ratio of a’’/a to conformal Hubble rate H times a_*/a_0.

References

He:2022qcs, eq. 3.11

cosmoGW.cosmology.rho_critical(H=<Quantity 3.24077929e-18 Hz>)

Compute critical energy density at a given epoch.

Parameters

Hastropy.units.Quantity

Hubble rate in units of frequency.

Returns

rho_castropy.units.Quantity

Critical energy density in GeV/m^3 units.

Reference

RoperPol:2019wvy, eq. 3

cosmoGW.cosmology.rho_radiation(g=100, T=<Quantity 100. GeV>)

Compute radiation energy density at a given epoch.

Parameters

gfloat

Relativistic degrees of freedom (default 100).

Tastropy.units.Quantity

Temperature scale (default 100 GeV).

Returns

rho_radastropy.units.Quantity

Radiation energy density in GeV/m^3 units.

Reference

RoperPol:2019wvy, eq. 3

cosmoGW.cosmology.thermal_g(dir0='', T=<Quantity 100. GeV>, s=0, file=True, Neff=3)

Return relativistic or adiabatic degrees of freedom as a function of T according to the thermal history of the Universe.

Note that for T < 0.5 MeV, after neutrino decoupling, entropic and relativistic g are approximately equal.

Parameters

dir0str, optional

Directory for dof file (default resources/cosmology/).

Tastropy.units.Quantity, optional

Temperature scale (default 100 GeV).

sint, optional

0 for relativistic, 1 for adiabatic dof.

filebool, optional

If True, read dof from file. It reads the file ‘dir0/cosmology/T_gs.csv’ and should be a pandas file with columns: [‘T [GeV]’, ‘g_*’, ‘gS’]

Nefffloat, optional

Effective number of neutrino species (default 3).

Returns

gfloat or ndarray

Relativistic or adiabatic degrees of freedom.

Reference

Rasanen - Lecture notes on Cosmology.

cosmoGW.cosmology.values_0(h0=1.0, neut=False, Neff=3, ret_rad=False)

Return cosmological parameters at present time.

Parameters

h0float

Hubble rate scaling (default 1).

neutbool

If True, include neutrinos in relativistic dofs.

Nefffloat

Effective number of neutrino species (default 3).

ret_radbool

If True, also return radiation energy density.

Returns

g0float

Relativistic degrees of freedom (2 if massive neutrinos are assumed).

g0sfloat

Entropic/adiabatic degrees of freedom (3.91 including neutrinos)

T0astropy.units.Quantity

Present temperature in MeV, corresponding to 2.72548 K

H0astropy.units.Quantity

Present Hubble rate in Hz, corresponding to H0 = 100 h0 km/s/Mpc.

rho_rad0astropy.units.Quantity, optional

Radiation energy density at present time.

Om_rad0float, optional

Radiation density ratio to critical at present time.

References

Rasanen - Lecture notes on Cosmology.