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# Copyright (c) 2017 Weitian LI <liweitianux@live.com>
# MIT license
"""
Simulate cluster formation (i.e., merging history) using the extended
Press-Schechter formalism.
References
----------
[1] Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
[2] Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
"""
import logging
import numpy as np
import scipy.integrate
import scipy.special
import scipy.optimize
from .cosmology import Cosmology
from .mergertree import MergerTree
logger = logging.getLogger(__name__)
class ClusterFormation:
"""
Simulate the cluster formation (i.e., merging history) using the extended
Press-Schechter formalism by Monte Carlo methods.
References
----------
[1] Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
[2] Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
Parameters
----------
M0 : float
Present-day (z=0) mass (unit: Msun) of the cluster.
configs : `ConfigManager`
A `ConfigManager` instance containing default and user configurations.
For more details, see the example configuration specifications.
Attributes
----------
cosmo : `~Cosmology`
Adopted cosmological model with custom utility functions.
mtree : `~MergerTree`
Merging history of this cluster.
"""
def __init__(self, M0, configs):
self.M0 = M0 # [Msun]
self.configs = configs
self._set_configs()
def _set_configs(self):
"""
Set up the necessary class attributes according to the configs.
"""
comp = "extragalactic/halos"
# Minimum mass change (unit: Msun) of the main-cluster for a merger
self.merger_mass_min = self.configs.getn(comp+"/merger_mass_min")
# Cosmology model
self.H0 = self.configs.getn("cosmology/H0")
self.OmegaM0 = self.configs.getn("cosmology/OmegaM0")
self.sigma8 = self.configs.getn("cosmology/sigma8")
self.cosmo = Cosmology(H0=self.H0, Om0=self.OmegaM0,
sigma8=self.sigma8)
logger.info("Loaded and set up configurations")
@property
def sigma_index(self):
"""
The power-law spectral index assumed for the following density
perturbations sigma(M).
References: Ref.[1],Eq.(2)
"""
n = -7/5
alpha = (n+3) / 6
return alpha
def f_sigma(self, mass):
"""
Current rms density fluctuations within a sphere of specified
mass (unit: Msun).
It is generally sufficient to consider a power-law spectrum of
density perturbations, which is consistent with the CDM models.
References: Ref.[1],Eq.(2)
"""
alpha = self.sigma_index
sigma = self.cosmo.sigma8 * (mass / self.cosmo.M8) ** (-alpha)
return sigma
def f_delta_c(self, z):
"""
w = delta_c(z) is the critical linear overdensity for a region
to collapse at redshift z.
This is a monotone decreasing function.
References: Ref.[1],App.A,Eq.(A1)
"""
return self.cosmo.overdensity_crit(z)
def f_dw_max(self, mass):
"""
Calculate the allowed maximum step size for tracing cluster
formation, therefore, the adopted step size is chosen to be half
of this maximum value.
dw^2 ~< abs(d(ln(sigma(M)^2)) / d(ln(M))) * (dMc / M) * sigma(M)^2
= 2 * alpha * sigma(M)^2 * dMc / M
References: Ref.[1],Sec.(3.1),Para.(1)
"""
alpha = self.sigma_index
dMc = self.merger_mass_min
return np.sqrt(2 * alpha * self.f_sigma(mass)**2 * dMc / mass)
def calc_z(self, delta_c):
"""
Solve the redshift from the specified delta_c (a.k.a. w).
"""
z = scipy.optimize.newton(
lambda x: self.f_delta_c(x) - delta_c,
x0=0, tol=1e-5)
return z
def calc_mass(self, S):
"""
Calculate the mass corresponding to the given S.
S = sigma(M)^2
References: Ref.[1],Sec.(3)
"""
alpha = self.sigma_index
mass = self.cosmo.M8 * (S / self.cosmo.sigma8**2)**(-1/(2*alpha))
return mass
@staticmethod
def cdf_K(dS, dw):
"""
The cumulative probability distribution function of sub-cluster
masses.
References: Ref.[1],Eq.(5)
"""
p = scipy.special.erfc(dw / np.sqrt(2*dS))
return p
@staticmethod
def cdf_K_inv(p, dw):
"""
Inverse function of the above ``cdf_K()``.
"""
dS = 0.5 * (dw / scipy.special.erfcinv(p))**2
return dS
def gen_dS(self, dw, size=None):
"""
Randomly generate values of dS by sampling the CDF ``cdf_K()``.
"""
r = np.random.uniform(size=size)
dS = self.cdf_K_inv(r, dw)
return dS
def simulate_mergertree(self):
"""
Simulate the merger tree of this cluster by tracing its formation
using the PS formalism.
References: Ref.[1],Sec.(3.1)
"""
self.mtree = self._trace_formation(self.M0, dMc=self.merger_mass_min)
return self.mtree
def _trace_formation(self, M, dMc, _z=None):
"""
Recursively trace the cluster formation and thus simulate its
merger tree.
"""
z = 0.0 if _z is None else _z
node_data = {"mass": M, "z": z, "age": self.cosmo.age(z)}
if M <= dMc:
# Stop the trace
return MergerTree(data=node_data)
# Trace the formation by simulate a merger/accretion event
# Notation: progenitor (*1) -> current (*2)
# Current properties
w2 = self.f_delta_c(z=z)
S2 = self.f_sigma(M) ** 2
dw = 0.5 * self.f_dw_max(M)
dS = self.gen_dS(dw)
# Progenitor properties
z1 = self.calc_z(w2 + dw)
S1 = S2 + dS
M1 = self.calc_mass(S1)
dM = M - M1
M_min = min(M1, dM)
if M_min <= dMc:
# Accretion
M_new = M - M_min
return MergerTree(
data=node_data,
main=self._trace_formation(M_new, dMc=dMc, _z=z1),
sub=None
)
else:
# Merger event
M_main = max(M1, dM)
M_sub = M_min
return MergerTree(
data=node_data,
main=self._trace_formation(M_main, dMc=dMc, _z=z1),
sub=self._trace_formation(M_sub, dMc=dMc, _z=z1)
)
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