zeus21.sfrd

Bulk of the Zeus21 calculation. Compute SFRD from cosmology.

Authors: zeus21 v2 collaboration - June 2026: Emily Bregou ; Hector Afonso G. Cruz ; Sarah Libanore ; Julian B. Muñoz ; Yonny Sklansky ; Emilie Thélie ; Alessandra Venditti

arXiv:2302.08506, arXiv:2306.09403, arXiv:2407.18294, Sklansky et al. (in prep)

Classes

`Z_init`	Initial redshift matrices for the calculation
`SFRD_class`	Compute all quantities and methods associated with the star formation rate density and the astrophysical model
`PopIII_relvel`	Pre-compute velocity–dependent Pop III star formation suppression (see sec 4A in arXiv:2407.18294)

Module Contents

class zeus21.sfrd.Z_init(UserParams, CosmoParams)[source]

Initial redshift matrices for the calculation

Parameters:

UserParams (UserParams class)
CosmoParams (CosmoParams class)

dlogzint[source]

Set the log step for the redshift binning, based on the required input

Type:: array

zintegral[source]

Redshift array over which will be performed all integration and for which the output will be computed

Type:: array

zGreaterMatrix[source]

Redshift associated with the distance at radius R. Dimension (z, R)

Type:: matrix

zGreaterMatrix_nonan[source]

Redshift associated with the distance at radius R; when z > zmax_AstroBreak (50 by default in constants), we set z > 100 to prevent computing things where we don’t trust the astrophysical model. Dimension (z, R)

Type:: matrix

dlogzint[source]

zintegral[source]

zGreaterMatrix[source]

zGreaterMatrix_nonan[source]

class zeus21.sfrd.SFRD_class(UserParams, CosmoParams, AstroParams, HMFinterp, z_Init=None)[source]

Compute all quantities and methods associated with the star formation rate density and the astrophysical model

Parameters:

UserParams (UserParams class)
CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
z_Init (Z_init class, optional) – Initial redshift matrices for the calculation (see sfrd.py for details). Default is None.

SFRD_II_interp[source]

Average SFRD for popII stars, interpolated over redshift.

Type:: interpolator

J_21_LW_II[source]

Lyman-Werner flux from popII stars, units of erg/s/cm^2/Hz/s, interpolated over redshift

Type:: interpolator

J21LW_interp_conv_avg

Lyman-Werner flux iteratively computed to account for popIII contribution, interpolated over redshift

Type:: interpolator

SFRD_III_cnvg_interp

Average SFRD for popIII stars, determines part-of and is affected by the LW flux; interpolated over redshift

Type:: : interpolator

J_21_LW_III

Lyman-Werner flux, units of erg/s/cm^2/Hz/s from popIII stars, interpolated over redshift

Type:: interpolator

SFRD_II_avg[source]

Average SFRD for popII stars, units Msun/yr

Type:: array

SFRD_III_avg[source]

Average SFRD for popIII stars, units Msun/yr

Type:: array

SFRD_avg[source]

Total average SFRD, units Msun/yr

Type:: array

SFRDbar2D_II[source]

Average SFRD for popII computed at z corresponding to each shell

Type:: matrix

SFRDbar2D_III[source]

Average SFRD for popIII computed at z corresponding to each shell

Type:: matrix

fesctab_II[source]

Escape fraction for popII, z-independent, as function of the halo mass

Type:: array

fesctab_III[source]

Escape fraction for popIII, z-independent, as function of the halo mass

Type:: array

reio_integrand_II_interp[source]

Number of ionizing photons produced by popII, interpolated in redshift

Type:: integrand

reio_integrand_II_interp[source]

Number of ionizing photons produced by popIII, interpolated in redshift

Type:: integrand

niondot_avg_II[source]

Number of ionizing photons produced by popII computed at the redshifts of the analysis

Type:: array

niondot_avg_III[source]

Number of ionizing photons produced by popIII computed at the redshifts of the analysis

Type:: array

niondot_avg[source]

Number of ionizing photons produced at the redshifts of the analysis

Type:: array

sigmaofRtab

Variance of the matter field on scales associated with the shells and at the observed rerdshift

Type:: matrix

Matom[source]

Minimum mass for atomic-cooling halos at given redshift

Type:: method

Mmol_0[source]

Minimum mass for molecular halos without LW or VCB feedback

Type:: method

Mmol_vcb[source]

Minimum mass for molecular halos without LW feedback

Type:: method

Mmol_LW[source]

Minimum mass for molecular halos without VCB feedback

Type:: method

Mmol[source]

Minimum mass for molecular halos with LW and VCB feedback

Type:: method

dMh_dt[source]

Mass accretion rate, in units of M_sun/yr

Type:: method

fstar_ofz[source]

Star formation efficiency generative function

Type:: method

fduty[source]

Duty cycle to damp star formation in low- or high-mass halos or both

Type:: method

SFE_II[source]

Star formation efficiency for popII stars

Type:: method

SFE_III[source]

Star formation efficiency for popIII stars

Type:: method

SFE[source]

Total star formation efficiency

Type:: method

SFR[source]

Star formation rate

Type:: method

SRFD_integrand

Integrand to compute the star formation rate density

Type:: method

J_LW_21[source]

Mean cosmological LW background specific intensity

Type:: method

J_LW_Discrete[source]

Radial kernel of the LW specific intensity before R integration

Type:: method

dSFRDIII_dJ[source]

Response of the popIII star formation rate density to the LW background

Type:: method

fesc_II[source]

Escape fraction of ionizing photons in halos hosting popII stars

Type:: method

fesc_III[source]

Escape fraction of ionizing photons in halos hosting popIII stars

Type:: method

compute_sigmaR_nu[source]

Compute the local mass function conditioned over the environment

Type:: method

compute_gamma[source]

Compute linear and quadratic gamma exponents for the SFRD-delta (popII+popIII) and niondot-delta (only popII) lognormal approximations

Type:: method

gamma_II_index2D

Linear gamma exponent for SFRD in popII

Type:: array

gamma2_II_index2D

Quadratic gamma exponent for SFRD in popII

Type:: array

gamma_niondot_II_index2D

Linear gamma exponent for niondot in popII

Type:: array

gamma2_niondot_II_index2D

Quadratic gamma exponent for niondot in popII

Type:: array

gamma_III_index2D

Linear gamma exponent for SFRD in popIII

Type:: array

gamma2_III_index2D

Quadratic gamma exponent for SFRD in popIII

Type:: array

compute_numerical_der_gamma[source]

Compute first and second numerical derivatives of an array wrt the other (used for SFRD and niondot wrt delta)

Type:: method

classmethod light_init()[source]

SFRD_II_interp[source]

J_21_LW_II[source]

SFRD_II_avg[source]

SFRD_III_avg[source]

SFRD_avg[source]

SFRDbar2D_II[source]

SFRDbar2D_III[source]

fesctab_II[source]

fesctab_III[source]

reio_integrand_II_interp[source]

reio_integrand_III_interp[source]

niondot_avg_II[source]

niondot_avg_III[source]

niondot_avg[source]

Matom(z)[source]

Compute minimum mass for atomic-cooling halos

Parameters:: z (float) – Redshift
Returns:: Matom – Minimum halo mass, in Msun
Return type:: float

Mmol_0(z)[source]

Compute minimum mass for molecular-cooling halos without LW or VCB feedback

Parameters:: z (float) – Redshift
Returns:: Mmol_0 – Minimum halo mass, in Msun
Return type:: float

Mmol_vcb(CosmoParams, AstroParams, z, vCB)[source]

Compute minimum mass for molecular-cooling halos without LW feedback

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
z (float) – Redshift
vCB (float) – Baryon-DM relative velocity

Returns:

Mmol_vcb – Minimum halo mass, in Msun

Return type:

float

Mmol_LW(AstroParams, J21LW_interp, z)[source]

Compute minimum mass for molecular halos without VCB feedback

Parameters:

AstroParams (AstroParams class)
J21LWinterp (interpolator) – Interpolator of the LW flux, function of z
z (float) – Redshift

Returns:

Mmol_LW – Minimum halo mass, in Msun

Return type:

float

Mmol(CosmoParams, AstroParams, J21LW_interp, z, vCB)[source]

Compute minimum mass for molecular halos with LW and VCB feedback

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
J21LWinterp (interpolator or False) – Interpolator of the LW flux, function of z. If False, no LW feedback.
z (float) – Redshift
vCB (float or False) – Baryon-DM relative velocity. If False, no feedback from streaming velocitites.

Returns:

Mmol – Minimum halo mass, in Msun

Return type:

float

dMh_dt(CosmoParams, AstroParams, HMFinterp, massVector, z)[source]

Compute halo mass accretion rate, in units of M_sun/yr

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
massVector (array) – Halo masses
z (array) – Redshift

Returns:

Mhdot – Halo mass accretion rate

Return type:

array

fstar_ofz(CosmoParams, z, massVector, eps, dlog10eps, zpiv, Mc, alphastar, betastar, fstarmax)[source]

Compute star formation efficiency as function of z – by changing the parameters, the user can run both popII and popIII

Parameters:

CosmoParams (CosmoParams class)
z (float) – Redshift
massVector (array) – Halo masses
eps (float) – Star formation efficiency at pivot redshift and mass
dlog10eps (float) – Logaritmic redshift evolution
zpiv (float) – Pivot reference redshift
Mc (float) – Pivot reference halo mass
alphastar (float) – Power-law coefficient
betastar (float) – Second power-law coefficient
fstarmax (float) – Cap

Returns:

fstar – Star formation efficiency

Return type:

array

fduty(CosmoParams, AstroParams, massVector, z, lower_cutoff=None, upper_cutoff=None, is_sharp_cutoff=False, vCB=None, J21LW_interp=None)[source]

Compute duty fraction to damp star formation

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
massVector (array) – Halo masses
z (float) – Redshift
lower_cutoff (str or float or None) – Apply cutoff at the low-mass end; if None, no cutoff; if str == “Mmol” or “Matom”, cutoff at the molecular/atomic-cooling limit respectively; if float, custom cutoff at user-provided value
upper_cutoff (str or float or None) – Apply cutoff at the high-mass end; if None, no cutoff; if str == “Matom”, cutoff at the atomic-cooling limit; if float, custom cutoff at user-provided value
is_sharp_cutoff (bool) – If true, apply sharp cutoff; exponential cutoff otherwise
vCB (float or None) – Baryon-DM relative velocity (None by default: only matters for molecular-cooling limit computation for Pop IIIs)
J21LW_interp (interpolator or None) – Interpolator of the LW flux, function of z (None by default: only matters for molecular-cooling limit computation for Pop IIIs)

Returns:

fduty – Duty cycle

Return type:

array

SFE_II(CosmoParams, AstroParams, massVector, z)[source]

Star formation efficiency for popII stars

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
massVector (array) – Halo masses
z (float) – Redshift

Returns:

SFE – Star formation efficiency

Return type:

array

SFE_III(CosmoParams, AstroParams, massVector, z, vCB, J21LW_interp)[source]

Star formation efficiency for popIII stars.

By default, this computes a single main Pop III component with a low-mass cutoff at the molecular-cooling limit and a user-controlled high-mass cutoff Mup_III. Setting Mup_III=”Matom” (default) in the AstroParams recovers standard minihalo-only behavior, with high-mass cutoff at the atomic-cooling limit.

If AstroParams.DETACH_III_ACH is True, main component is forced to stop at the atomic-cooling limit and a detached atomic-cooling-halo component with independent parameters is added between Matom and Mup_III.

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
massVector (array) – Halo masses
z (float) – Redshift
vCB (float or None) – Baryon-DM relative velocity
J21LW_interp (interpolator or None) – Interpolator of the LW flux, function of z

Returns:

SFE_tot – Star formation efficiency

Return type:

array

SFE(CosmoParams, AstroParams, massVector, z, pop, vCB=None, J21LW_interp=None)[source]

Total star formation efficiency

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
massVector (array) – Halo masses
z (float) – Redshift
pop (int) – Which population (2 for popII or 3 for popIII)
vCB (float or None) – Baryon-DM relative velocity (None by default: only matters for molecular-cooling limit computation for Pop IIIs)
J21LW_interp (interpolator or None) – Interpolator of the LW flux, function of z (None by default: only matters for molecular-cooling limit computation for Pop IIIs)

Returns:

SFE – Star formation efficiency for the input population

Return type:

array

SFR(CosmoParams, AstroParams, HMFinterp, massVector, z, pop, vCB=None, J21LW_interp=None)[source]

Star formation rate in Msun/yr for given population

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
massVector (array) – Halo masses
z (float) – Redshift
pop (int) – Which population (2 for popII or 3 for popIII)
vCB (float or None) – Baryon-DM relative velocity (None by default: only matters for molecular-cooling limit computation for Pop IIIs)
J21LW_interp (interpolator or None) – Interpolator of the LW flux, function of z (None by default: only matters for molecular-cooling limit computation for Pop IIIs)

Returns:

SFR – Star formation rate

Return type:

array

SFRD_integrand(CosmoParams, AstroParams, HMFinterp, massVector, z, pop, vCB=None, J21LW_interp=None)[source]

Integrand for the star formation rate density for a given population

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
massVector (array) – Halo masses
z (float) – Redshift
pop (int) – Which population (2 for popII or 3 for popIII)
vCB (float or None) – Baryon-DM relative velocity (None by default: only matters for molecular-cooling limit computation for Pop IIIs)
J21LW_interp (interpolator or None) – Interpolator of the LW flux, function of z (None by default: only matters for molecular-cooling limit computation for Pop IIIs)

Returns:

integrand – Integrand to be used in the main class

Return type:

array

J_LW_21(CosmoParams, AstroParams, sfrdIter, z, pop)[source]

Mean background specific intensity, units of erg/s/cm^2/Hz/sr

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
sfrdIter (array) – Star formation rate density
z (float) – Redshift
pop (int) – Which population (2 for popII or 3 for popIII)

Returns:

JW – LW specific intensity

Return type:

array

J_LW_Discrete(CosmoParams, AstroParams, z, pop, rGreater, SFRD_interp_input)[source]

Radial kernel of the LW specific intensity before R integration, units of erg/s/cm^2/Hz/sr

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
z (float or array) – Redshift
pop (int) – Which population (2 for popII or 3 for popIII)
rGreater (matrix) – Radii
SFRD_interp_input (interpolator) – Interpolator for the star formation rate density in redshift

Returns:

RK – Radial kernel of the LW specific intensity

Return type:

array

dSFRDIII_dJ(CosmoParams, AstroParams, HMFinterp, z, vCB, J21LW_interp)[source]

Response of the popIII star formation rate density to the LW background

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
z (float) – Redshift
vCB (bool) – Include contribution from baryon-CDM relative velocity (popIII) or not (popII)
J21LW_interp (bool) – Include contribution from LW feedback (popIII) or not (popII)

Returns:

integral – Response integrated over the mass array

Return type:

array

fesc_II(AstroParams, Mh)[source]

Escape fraction of ionizing photons in halos hosting popII stars

Parameters:

AstroParams (AstroParams class)
Mh (array) – Halo masses

Returns:

fesc – Escape fraction

Return type:

array

fesc_III(AstroParams, Mh)[source]

Escape fraction of ionizing photons in halos hosting popIII stars

Parameters:

AstroParams (AstroParams class)
Mh (array) – Halo masses

Returns:

fesc – Escape fraction

Return type:

array

compute_sigmaR_nu(CosmoParams, HMFinterp, z_array, R_array, Mh_array, dorv_array, dorv)[source]

Compute the local mass function conditioned over the environment

Parameters:

CosmoParams (CosmoParams class)
HMFinterp (HMFinterp class)
z_array (array) – Redshifts
R_array (array) – Shell radii
Mh_array (array) – Halo masses
dorv_array (array) – Input values of either the denisty or velocity field
dorv (str) – Compute the output wrt the density field (“delta”) or the velocity field (“vel”)

Returns:

HMF_corr (array) – Local HMF in Eulerian space
mArray (array) – Halo masses, dimension (z,R,Mh,delta or v)
zGreaterArray (array) – Redshifts of the sources, dimension (z,R,Mh,delta or v)
out (array) – Either delta_R (if dorv == delta) or velocity, dimension (z,R,Mh,delta or v)

compute_gamma(CosmoParams, AstroParams, HMFinterp, z_array, R_array, Mh_array, input_sigmaofRtab, fesctab_II)[source]

Compute linear and quadratic gamma exponents for the SFRD-delta (popII+popIII) and niondot-delta (onlypopII) lognormal approximations

Parameters:

CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
z_array (array) – Redshifts
R_array (array r) – Shell radii
Mh_array (array) – Halo masses
input_sigmaofRtab (array) – Variance of the matter field smoothed over R
fesctab_II – Escape fraction for popII stars

Return type:

Empty

compute_numerical_der_gamma(arr1, arr2, order)[source]

Compute first and second numerical derivatives of arr1 wrt arr2

Parameters:

arr1 (array) – Quantity to be derived
arr2 (array) – Quantity to perform the numerical derivative
order (int) – Compute either first (1) or second (2) order derivative

Returns:

darr1_darr2 – Numerical derivative

Return type:

array

class zeus21.sfrd.PopIII_relvel(UserParams, CosmoParams, AstroParams, HMFinterp, z_Init=None, SFRD_Init=None)[source]

Pre-compute velocity–dependent Pop III star formation suppression (see sec 4A in arXiv:2407.18294)

Parameters:

UserParams (UserParams class)
CosmoParams (CosmoParams class)
AstroParams (AstroParams class)
HMFinterp (HMFinterp class)
z_Init (Z_init class, optional) – Initial redshift matrices for the calculation (see sfrd.py for details). Default is None.
SFRD_Init (SFRD_class class, optional) – Initial star formation rate density for the calculation (see sfrd.py for details). Default is None.

vcb_expFitParams

Coefficients to approximate SFRD(vCB) / SFRD

Type:: float