Behroozi¶

class hmf.core.fits.Behroozi(*, A_200=0.1858659, A_300=0.1995973, A_400=0.2115659, A_600=0.2184113, A_800=0.2480968, A_1200=0.2546053, A_1600=0.26, A_2400=0.26, A_3200=0.26, a_200=1.466904, a_300=1.521782, a_400=1.559186, a_600=1.614585, a_800=1.869936, a_1200=2.128056, a_1600=2.301275, a_2400=2.529241, a_3200=2.661983, b_200=2.571104, b_300=2.254217, b_400=2.048674, b_600=1.869559, b_800=1.588649, b_1200=1.507134, b_1600=1.464374, b_2400=1.436827, b_3200=1.40521, c_200=1.193958, c_300=1.270316, c_400=1.335191, c_600=1.446266, c_800=1.581345, c_1200=1.79505, c_1600=1.965613, c_2400=2.237466, c_3200=2.439729, A_exp=0.14, a_exp=0.06)[source]¶

The Behroozi, Wechsler & Conroy (2013) mass function.

\(f(\sigma)\) is that of Tinker08 (evaluated at the virial overdensity), and modify_dndm_kernel() applies the empirical high-redshift correction of App. G to the mass function:

\[n(>M) = \theta(M, z)\, n_{\rm T08}(>M), \qquad \theta = 10^{\alpha(z)(M/M_\star)^{\gamma(z)}},\]

with \(M_\star = 10^{11.5}\,M_\odot\) (eqs. G2-G3).

Parameters:

A_200 (float, default 0.1858659) – The amplitude A at Delta = 200.
A_300 (float, default 0.1995973) – The amplitude A at Delta = 300.
A_400 (float, default 0.2115659) – The amplitude A at Delta = 400.
A_600 (float, default 0.2184113) – The amplitude A at Delta = 600.
A_800 (float, default 0.2480968) – The amplitude A at Delta = 800.
A_1200 (float, default 0.2546053) – The amplitude A at Delta = 1200.
A_1600 (float, default 0.26) – The amplitude A at Delta = 1600.
A_2400 (float, default 0.26) – The amplitude A at Delta = 2400.
A_3200 (float, default 0.26) – The amplitude A at Delta = 3200.
a_200 (float, default 1.466904) – The slope a at Delta = 200.
a_300 (float, default 1.521782) – The slope a at Delta = 300.
a_400 (float, default 1.559186) – The slope a at Delta = 400.
a_600 (float, default 1.614585) – The slope a at Delta = 600.
a_800 (float, default 1.869936) – The slope a at Delta = 800.
a_1200 (float, default 2.128056) – The slope a at Delta = 1200.
a_1600 (float, default 2.301275) – The slope a at Delta = 1600.
a_2400 (float, default 2.529241) – The slope a at Delta = 2400.
a_3200 (float, default 2.661983) – The slope a at Delta = 3200.
b_200 (float, default 2.571104) – The scale b at Delta = 200.
b_300 (float, default 2.254217) – The scale b at Delta = 300.
b_400 (float, default 2.048674) – The scale b at Delta = 400.
b_600 (float, default 1.869559) – The scale b at Delta = 600.
b_800 (float, default 1.588649) – The scale b at Delta = 800.
b_1200 (float, default 1.507134) – The scale b at Delta = 1200.
b_1600 (float, default 1.464374) – The scale b at Delta = 1600.
b_2400 (float, default 1.436827) – The scale b at Delta = 2400.
b_3200 (float, default 1.40521) – The scale b at Delta = 3200.
c_200 (float, default 1.193958) – The cut-off c at Delta = 200.
c_300 (float, default 1.270316) – The cut-off c at Delta = 300.
c_400 (float, default 1.335191) – The cut-off c at Delta = 400.
c_600 (float, default 1.446266) – The cut-off c at Delta = 600.
c_800 (float, default 1.581345) – The cut-off c at Delta = 800.
c_1200 (float, default 1.79505) – The cut-off c at Delta = 1200.
c_1600 (float, default 1.965613) – The cut-off c at Delta = 1600.
c_2400 (float, default 2.237466) – The cut-off c at Delta = 2400.
c_3200 (float, default 2.439729) – The cut-off c at Delta = 3200.
A_exp (float, default 0.14) – A(z) = A (1+z)^-A_exp.
a_exp (float, default 0.06) – a(z) = a (1+z)^-a_exp.

Model metadata

alias: ClassVar[str | None] = 'Behroozi'¶: The short name the model is registered under, if any. Set by the alias class keyword; never inherited.

calibration_domain: ClassVar[Domain | None] = Domain(bounds=(('z', Interval(lower=0.0, upper=9.0, unit=None, lower_open=False, upper_open=False)),), source='Behroozi et al. 2013, App. G: the correction is fit to the Consuelo simulation (Omega_m = 0.25, sigma_8 = 0.8, n_s = 1; Sec. 4) over a = 0.1-1, i.e. z ~ 0-9 (Fig. 23), with n(>M) at M = 10^11.5-10^13 Msun (Fig. 23, right). The range of sigma, and of masses for dn/dM: not stated. Tinker08 is applied at the virial overdensity, outside its calibrated Delta range at z >~ 1, deliberately.')¶: Where the fit was calibrated against simulations, with its source; None for a fit that was not calibrated (e.g. an analytic one).

measured_mass_definition: ClassVar[MeasuredMassDefinition] = MeasuredMassDefinition(kind='so_virial', linking_length=None, overdensity=None, preferred=None, note='Behroozi et al. 2013, Sec. 4: Bryan & Norman (1998) virial SO, ROCKSTAR.')¶: The halo mass definition the fit was measured in.

modifies_dndm: ClassVar[bool] = True¶: Whether modify_dndm_kernel() changes the mass function (see “Post-processing the mass function” in the module documentation); if False, it is the identity.

normalized¶: Whether \(f\) integrates to 1 over \(\ln\sigma^{-1}\) (all mass in haloes).

parameter_source: ClassVar[str] = 'Behroozi et al. 2013 (arXiv v2), App. G, eqs. G2-G3 (the correction); f(sigma) parameters from Tinker et al. 2008, Table 2.'¶: The exact source (paper, version, table or equation) of the parameter defaults, e.g. “Tinker et al. 2008, ApJ 688, 709 (published), Table 2”.

references: ClassVar[tuple[str, ...]] = ('Behroozi, P. S., Wechsler, R. H., Conroy, C., 2013. ApJ 770, 57. https://doi.org/10.1088/0004-637X/770/1/57', 'Tinker, J., et al., 2008. ApJ 688, 709. https://doi.org/10.1086/591439')¶: References to cite when the model is used, each a formatted citation.

requires: ClassVar[frozenset[str]] = frozenset({'delta_halo', 'z'})¶: The inputs (from INPUTS) that fsigma() needs, besides sigma.

simulations: ClassVar[SimulationDetails | None] = SimulationDetails(box_size=<Quantity [420.] Mpc / littleh>, n_particles=(2744000000,), omega_m=(0.25,), omega_b=None, sigma_8=(0.8,), h=(0.7,), n_s=(1.0,), softening=<Quantity [0.008] Mpc / littleh>, transfer=None, z_start=None, initial_conditions=('2LPT',), halo_finder='ROCKSTAR', n_min=None, z_range=(0.0, 9.0), names=('Consuelo',), notes="The correction is fitted to Consuelo only, after an incompleteness correction (App. G1). Bolshoi (250 Mpc/h, 2048^3, Omega_m = 0.27, sigma_8 = 0.82) is shown for comparison. Softening: 'eight times worse' than Bolshoi's 1 kpc/h.", source='Behroozi et al. 2013 (arXiv v2), Sec. 4 and App. G.')¶: The simulations the fit was calibrated on (None for a fit to no simulations).

valid_domain: ClassVar[Domain] = Domain(bounds=(('delta_halo', Interval(lower=75.0, upper=30000.0, unit=None, lower_open=True, upper_open=False)), ('sigma', Interval(lower=0.0, upper=inf, unit=None, lower_open=True, upper_open=True)), ('z', Interval(lower=0.0, upper=inf, unit=None, lower_open=False, upper_open=True))), source='sigma > 0, z >= 0, and 75 < Delta <= 3e4: eq. 8 needs Delta > 75, and the spline of the default parameters keeps all of them > 0 only for 42.7 < Delta < 3.07e4. Parameters that are not > 0 raise in any case.')¶

fsigma() raises outside it.

Type:: Where the formula is defined or sensible

Methods

check_required(x: FitInputs) → None¶

Raise if an input needed to evaluate the fit is missing from x.

Raises:: ValueError – If an input in requires (or needed to check valid_domain) is missing.

check_valid(x: FitInputs) → None¶

Raise if any input is outside valid_domain.

Raises:: DomainError – Always, if any value is outside the valid domain: this does not depend on any policy.

classmethod coerce(value: object) → Self¶

Convert a value to a model of this kind: the converter of a model field.

Parameters:

value – An instance of cls, which is returned as it is; or a name, import path or class that get() accepts, which is looked up and instantiated with its default parameters.

Returns:

An instance of cls.

Return type:

Model

Raises:

ModelNotFoundError – If value is a name that no model of this kind is known by.
TypeError – If value is of any other type, or names a class that is not a concrete subclass of cls.

Examples

>>> from hmf.core.filters import Filter, TopHat
>>> Filter.coerce("TopHat") == TopHat()
True

classmethod domain_inputs() → frozenset[str]¶: The inputs (besides sigma) needed to evaluate the fit and check both domains.

classmethod fields_info() → tuple[FieldInfo, ...]¶

Describe this class’s fields, without creating an instance.

Returns:: The name, type, default and documentation of each field.
Return type:: tuple of FieldInfo

The multiplicity function, \(f(\sigma) = \nu f(\nu)\).

m must be a Quantity (see hmf.core.units); every other input is dimensionless. Code that already has plain arrays in canonical units calls fsigma_kernel() or evaluate_fsigma() instead (see “Evaluating a fit” in the module documentation). It applies no calibration policy (see evaluate_fsigma() for that), but always checks valid_domain.

Parameters:

sigma – \(\sigma(m, z)\).
z – Redshift.
omega_m_z – \(\Omega_m(z)\).
delta_halo – The halo overdensity relative to the mean density.
delta_c – The critical overdensity for collapse.
n_eff – The effective spectral index.
m – The halo mass, a Quantity (e.g. in Msun_h).

Returns:

\(f(\sigma)\) (dimensionless), broadcast over the inputs.

Return type:

numpy.ndarray

Raises:

ValueError – If an input the fit requires is not given.
UnitBoundaryError – If m is not a Quantity.
DomainError – If any input is outside valid_domain, or makes the fit’s parameters unphysical.

fsigma_kernel(x: FitInputs) → ndarray[tuple[Any, ...], dtype[float64]]¶

fsigma() at kernel level, from inputs that are plain arrays.

The inputs are in canonical units (the mass in Msun/h); see hmf.core._kernels. It checks the valid domain, as fsigma() does, and applies no calibration policy (see evaluate_fsigma()).

Parameters:

x – The inputs.

Return type:

numpy.ndarray

Raises:

ValueError – If an input the fit requires is not in x.
DomainError – As for fsigma().

classmethod get(name: str | type[Model]) → type[Self]¶

Look up a model of this kind.

Parameters:

name – An alias ("Tinker08"), a qualified name ("hmf.core.fits:Tinker08"), an import path to a class not yet imported ("pkg.mod:Cls" or "pkg.mod.Cls"), or a class.

Returns:

The model class (not an instance). It is always a subclass of cls.

Return type:

type

Raises:

ModelNotFoundError – If no model of this kind is known by name, even after loading the hmf.models entry points and trying name as an import path.
TypeError – If name resolves to something that is not a concrete subclass of cls.

classmethod get_aliases() → Mapping[str, str]¶: Return a read-only mapping from alias to qualified name, for this kind.

classmethod get_models() → Mapping[str, type[Self]]¶

Return the registered models of this kind, by qualified name.

Plugins advertised in the hmf.models entry-point group are loaded first.

Returns:: A read-only mapping from qualified name to class. Called on a kind, it is a live view of the registry. Called on a non-kind subclass, it holds only the models that subclass it.
Return type:: Mapping

in_calibration_domain(x: FitInputs) → bool | ndarray[tuple[Any, ...], dtype[bool]]¶

Whether each input is inside calibration_domain.

Parameters:: x – The inputs, from inputs().
Returns:: A bool if every input is a scalar, else the broadcast array. True if the fit has no calibration domain (it is None).
Return type:: bool or numpy.ndarray of bool
Raises:: ValueError – If an input needed to check the domain was not given.

Collect the inputs, checking that the required ones are given.

Parameters are as for fsigma() (m is a Quantity). Code that already has plain arrays in canonical units builds FitInputs directly (see “Evaluating a fit” in the module documentation).

Returns:: The inputs as plain arrays, in canonical units.
Return type:: FitInputs
Raises:: ValueError – If an input in requires (or needed to check valid_domain) is missing.

modify_dndm(m: Any, dndm: Any, *, z: ArrayLike, ngtm: Any, H0: Any, omega_m0: ArrayLike) → Any¶

Map the mass function computed from fsigma() to the fit’s.

The identity unless modifies_dndm (see “Post-processing the mass function” in the module documentation). Every fit takes the same arguments, whether it uses them or not. This method converts them and calls modify_dndm_kernel(), which a mass-function Stage calls directly.

Parameters:

m – Halo masses, a Quantity: in h-units, or physical units (converted with H0).
dndm – The mass function from fsigma(), a Quantity (number density per mass).
z – Redshift.
ngtm – The cumulative mass function n(>m) from fsigma(), a Quantity.
H0 – The Hubble constant, a scalar Quantity (e.g. 70 * H0_unit). It converts physical units, and gives h to fits whose formula is in physical units.
omega_m0 – The density parameter of CDM + baryons today.

Returns:

The fit’s mass function, in dndm_unit.

Return type:

Quantity

modify_dndm_kernel(m: ArrayLike, dndm: ArrayLike, *, z: ArrayLike, ngtm: ArrayLike, h: ArrayLike, omega_m0: ArrayLike) → ndarray[tuple[Any, ...], dtype[float64]]¶

modify_dndm() at kernel level, on plain arrays.

A pure kernel in canonical units (see hmf.core._kernels): the identity, unless the fit sets modifies_dndm (its class documentation describes the correction). All arguments broadcast. It checks the arguments that say where the mass function is evaluated (m, z, h, omega_m0), from their extent; dndm and ngtm are values it maps, and may be NaN (e.g. masked by a domain policy).

Parameters:

m – Halo masses, in Msun/h.
dndm – The mass function from fsigma(), in h^4 / (Msun Mpc^3).
z – Redshift.
ngtm – The cumulative mass function n(>m) from fsigma(), in h^3 / Mpc^3.
h – The dimensionless Hubble parameter, H0 / (100 km/s/Mpc).
omega_m0 – The density parameter of CDM + baryons today.

Returns:

The fit’s mass function, in h^4 / (Msun Mpc^3).

Return type:

numpy.ndarray

Raises:

DomainError – If an m, h or omega_m0 is not finite and > 0, or a z is not finite and >= 0.

parameters(delta_halo: ArrayLike, z: ArrayLike) → tuple[ndarray[tuple[Any, ...], dtype[float64]], ndarray[tuple[Any, ...], dtype[float64]], ndarray[tuple[Any, ...], dtype[float64]], ndarray[tuple[Any, ...], dtype[float64]]]¶

The parameters \((A, a, b, c)\) at overdensity delta_halo and z.

Parameters:

delta_halo – The halo overdensity relative to the mean density (> 75).
z – Redshift.

Returns:

A, a, b, c, broadcast over the inputs.

Return type:

tuple of numpy.ndarray

Raises:

DomainError – If any parameter is not finite and > 0 (the spline extrapolates them far outside the tabulated overdensities).

classmethod qualified_name() → str¶: Return this class’s qualified name, package.module:Class.