1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
|
# 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, dMc)
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)
)
|
