-
Notifications
You must be signed in to change notification settings - Fork 22
/
WaveformTools.py
347 lines (236 loc) · 11.8 KB
/
WaveformTools.py
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
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
import numpy as np
import matplotlib.pyplot as plt
from scipy import optimize
from scipy import interpolate
import PhenomA as pa
import LISA as li
""" Constants """
C = 299792458. # m/s
YEAR = 3.15581497632e7 # sec
TSUN = 4.92549232189886339689643862e-6 # mass of sun in seconds (G=C=1)
MPC = 3.08568025e22/C # mega-Parsec in seconds
TOBS_MAX = 4*YEAR # Maximum observation period (LISA's nominal mission lifetime)
""" Cosmological values """
H0 = 69.6 # Hubble parameter today
Omega_m = 0.286 # density parameter of matter
def get_Dl(z, Omega_m, H0):
""" calculate luminosity distance in geometric units """
# see http://arxiv.org/pdf/1111.6396v1.pdf
x0 = (1. - Omega_m)/Omega_m
xZ = x0/(1. + z)**3
Phi0 = (1. + 1.320*x0 + 0.4415*x0**2 + 0.02656*x0**3)
Phi0 /= (1. + 1.392*x0 + 0.5121*x0**2 + 0.03944*x0**3)
PhiZ = (1. + 1.320*xZ + 0.4415*xZ**2 + 0.02656*xZ**3)
PhiZ /= (1. + 1.392*xZ + 0.5121*xZ**2 + 0.03944*xZ**3)
return 2.*C/H0*(1.0e-3*MPC)*(1. + z)/np.sqrt(Omega_m)*(Phi0 - PhiZ/np.sqrt(1. + z))
def get_z(z, Dl, Omega_m, H0):
""" calculate redishift uisng root finder """
return get_Dl(z, Omega_m, H0) - Dl
def SetFreqBounds(self, lisa):
""" """
Mc = self.M*self.eta**(3./5) # chirp mass
# Determine start frequency of binary
if (self.f_start == None): # T_merge was specified
self.f_start = (5.*Mc/self.T_merge)**(3./8.)/(8.*np.pi*Mc)
else:
self.T_merge = 5.*Mc/(8.*np.pi*self.f_start*Mc)**(8./3.)
# Determine the end frequency
if (self.T_merge > lisa.Tobs):
self.f_end = (5.*Mc/(np.abs(lisa.Tobs-self.T_merge)))**(3./8.)/(8.*np.pi*Mc)
else:
self.f_end = pa.get_freq(self.M, self.eta, "cut") # PhenomA cut-off frequency i.e. frequency upper bound
return
def calc_k(theta, phi):
""" Calculate the unit-direction vector pointing towards the source """
sth = np.sin(theta)
k = -np.array([sth*np.cos(phi), sth*np.sin(phi), np.cos(theta)])
return k
def calc_k_dot_r(k, rij):
""" Dot product between unit-direction vector and the S/C unit-separation vectors """
k_dot_r = k[0]*rij[0,:,:,:] + k[1]*rij[1,:,:,:] + k[2]*rij[2,:,:,:]
return k_dot_r
def get_XX_TDI(OBJ, lisa, f, Aeff, theta, phi, iota):
""" Construct cos(\iota) and \psi averaged Michelson-equivalent TDI response """
N = len(f)
# stationary time of SPA
tStar = pa.dPsieff_df(f, OBJ.M, OBJ.eta, 0.0)/(2*np.pi)
# Direction vectors
k = calc_k(theta, phi)
x = lisa.SC_Orbits(tStar)
rij = lisa.SC_Seps(tStar, x)
rij_OUTER_rij = rij.reshape((3,1, 3,3, N))*rij.reshape((1,3, 3,3, N))
# GW basis tesnrors
u = np.array([np.cos(theta)*np.cos(phi), np.cos(theta)*np.sin(phi), -np.sin(theta)])
v = np.array([np.sin(phi), -np.cos(phi), 0.0])
ep = np.outer(u,u) - np.outer(v,v)
ec = np.outer(u,v) + np.outer(v,u)
# construct the detector tensor for LISA
dp12 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,0,1,:], ep)
dc12 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,0,1,:], ec)
dp21 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,1,0,:], ep)
dc21 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,1,0,:], ec)
dp13 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,0,2,:], ep)
dc13 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,0,2,:], ec)
dp31 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,2,0,:], ep)
dc31 = np.einsum('nmk,nm->k', rij_OUTER_rij[:,:,2,0,:], ec)
# Piece together the transfer function
kDOTrij = calc_k_dot_r(k, rij)
kDOTr12 = kDOTrij[0,1,:]
kDOTr21 = kDOTrij[1,0,:]
kDOTr13 = kDOTrij[0,2,:]
kDOTr31 = kDOTrij[2,0,:]
TransArg12 = f/(2*lisa.fstar)*(1. - kDOTr12)
TransArg21 = f/(2*lisa.fstar)*(1. - kDOTr21)
TransArg13 = f/(2*lisa.fstar)*(1. - kDOTr13)
TransArg31 = f/(2*lisa.fstar)*(1. - kDOTr31)
Trans12 = 0.5*np.sinc(TransArg12/np.pi)*np.exp(1j*TransArg12)*np.exp(1j*2*np.pi*f*np.dot(k,x[:,0,:])/C)
Trans21 = 0.5*np.sinc(TransArg21/np.pi)*np.exp(1j*TransArg21)*np.exp(1j*2*np.pi*f*np.dot(k,x[:,1,:])/C)
Trans13 = 0.5*np.sinc(TransArg13/np.pi)*np.exp(1j*TransArg13)*np.exp(1j*2*np.pi*f*np.dot(k,x[:,0,:])/C)
Trans31 = 0.5*np.sinc(TransArg31/np.pi)*np.exp(1j*TransArg31)*np.exp(1j*2*np.pi*f*np.dot(k,x[:,2,:])/C)
if (iota == None):
# iota = pi/2, psi = 0
y12_a = Trans12*Aeff*0.5*dp12/2
y21_a = Trans21*Aeff*0.5*dp21/2
y13_a = Trans13*Aeff*0.5*dp13/2
y31_a = Trans31*Aeff*0.5*dp31/2
else:
y12_a = 0.5*Trans12*Aeff*(0.5*(1 + np.cos(iota)**2)*dp12 + 1j*np.cos(iota)*dc12)
y21_a = 0.5*Trans21*Aeff*(0.5*(1 + np.cos(iota)**2)*dp21 + 1j*np.cos(iota)*dc21)
y13_a = 0.5*Trans13*Aeff*(0.5*(1 + np.cos(iota)**2)*dp13 + 1j*np.cos(iota)*dc13)
y31_a = 0.5*Trans31*Aeff*(0.5*(1 + np.cos(iota)**2)*dp31 + 1j*np.cos(iota)*dc31)
X_TDI = (y12_a - y13_a)*np.exp(-1j*f/lisa.fstar) + (y12_a - y13_a)
if (iota == None):
XX_TDI = 8./5*np.abs(X_TDI)**2
else:
XX_TDI = 1./2*np.abs(X_TDI)**2
if (iota == None):
# iota = pi/2, psi = pi/4
y12_a = Trans12*Aeff*0.5*dc12/2
y21_a = Trans21*Aeff*0.5*dc21/2
y13_a = Trans13*Aeff*0.5*dc13/2
y31_a = Trans31*Aeff*0.5*dc31/2
else:
y12_a = 0.5*Trans12*Aeff*(-0.5*(1 + np.cos(iota)**2)*dc12 + 1j*np.cos(iota)*dp12)
y21_a = 0.5*Trans21*Aeff*(-0.5*(1 + np.cos(iota)**2)*dc21 + 1j*np.cos(iota)*dp21)
y13_a = 0.5*Trans13*Aeff*(-0.5*(1 + np.cos(iota)**2)*dc13 + 1j*np.cos(iota)*dp13)
y31_a = 0.5*Trans31*Aeff*(-0.5*(1 + np.cos(iota)**2)*dc31 + 1j*np.cos(iota)*dp31)
X_TDI = (y12_a - y13_a)*np.exp(-1j*f/lisa.fstar) + (y12_a - y13_a)
if (iota == None):
XX_TDI += 8./5*np.abs(X_TDI)**2
else:
XX_TDI += 1./2*np.abs(X_TDI)**2
return XX_TDI
def CalcStrain(self, lisa, theta=None, phi=None, iota=None):
""" Calculate the characteristic GW strain """
Delta_logf = np.log(self.f_end) - np.log(self.f_start)
if (Delta_logf > 0.00005): # Generate a track
N = 500 # number of points
f = np.logspace(np.log10(self.f_start), np.log10(self.f_end), N)
Aeff = pa.Aeff(f, self.M, self.eta, self.Dl)
if (theta == None and phi == None): # generate sky averaged response
self.Figure_Type = 'track'
X_char = np.sqrt(16./5*f)*Aeff
else: # Generate X Michelson channel
self.Figure_Type = 'track_sky_dependent'
XX_TDI = get_XX_TDI(self, lisa, f, Aeff, theta, phi, iota)
X_char = np.sqrt(4*f*XX_TDI)
else:
N = 1
f = np.array([self.f_start])
Aeff = pa.Aeff(f, self.M, self.eta, self.Dl)
if (theta == None and phi == None): # generate sky averaged response
self.Figure_Type = 'point'
X_char = np.sqrt(16./5*Aeff**2*np.sqrt(f)*(self.f_end - f))
else:
self.Figure_Type = 'point_sky_dependent'
XX_TDI = get_XX_TDI(self, lisa, f, Aeff, theta, phi, iota)
X_char = np.sqrt(4*XX_TDI*np.sqrt(f)*(self.f_end - f))
return f, X_char
def CalcSNR(self, f, X_char, lisa):
""" Calculate the signal to noise ratio for the source """
if (self.Figure_Type == 'track'):
N = len(f) # number of frequency samples
d_logf = np.log(f[1:]) - np.log(f[:N-1])
term_i = X_char[1:]**2/lisa.Sn(f[1:])
term_im1 = X_char[:N-1]**2/lisa.Sn(f[:N-1])
snrSQ = np.sum(0.5*(term_i + term_im1)*d_logf)
elif (self.Figure_Type == 'track_sky_dependent'):
N = len(f) # number of frequency samples
d_logf = np.log(f[1:]) - np.log(f[:N-1])
term_i = X_char[1:]**2/lisa.Pn_WC(f[1:])*lisa.NC
term_im1 = X_char[:N-1]**2/lisa.Pn_WC(f[:N-1])*lisa.NC
snrSQ = np.sum(0.5*(term_i + term_im1)*d_logf)
elif (self.Figure_Type == 'point'):
snrSQ = (X_char**2/np.sqrt(f)/lisa.Sn(f))[0]
elif (self.Figure_Type == 'point_sky_dependent'):
snrSQ = (X_char**2/np.sqrt(f)/lisa.Pn_WC(f)*lisa.NC)[0]
return np.sqrt(snrSQ)
def PlotStrain(self, freqs, X_char, lisa):
""" Plot the characteristic strain curves """
fig, ax = plt.subplots(1, figsize=(8,6))
plt.tight_layout()
ax.set_xlabel(r'f [Hz]', fontsize=20, labelpad=10)
ax.set_ylabel(r'Characteristic Strain', fontsize=20, labelpad=10)
ax.tick_params(axis='both', which='major', labelsize=20)
ax.set_xlim(1.0e-5, 1.0e0)
ax.set_ylim(3.0e-22, 1.0e-15)
f = np.logspace(np.log10(1.0e-5), np.log10(1.0e0), 1000)
if (self.Figure_Type == 'track'):
ax.loglog(freqs, np.sqrt(freqs)*X_char)
ax.loglog(f, np.sqrt(f*lisa.Sn(f)))
elif (self.Figure_Type == 'track_sky_dependent'):
ax.loglog(freqs, np.sqrt(freqs)*X_char)
ax.loglog(f, np.sqrt(f*lisa.Pn_WC(f)))
elif (self.Figure_Type == 'point_sky_dependent'):
ax.loglog(freqs, np.sqrt(freqs)*X_char, 'r.')
ax.loglog(f, np.sqrt(f*lisa.Pn_WC(f)))
elif (self.Figure_Type == 'point'):
ax.loglog(freqs, np.sqrt(freqs)*X_char, 'r.')
ax.loglog(f, np.sqrt(f*lisa.Sn(f)))
return
class Binary():
"""
Binary Class
-------------------------------------------
Inputs:
Specify source-frame masses: m1, m2
Specify a distance parameter: z, Dl (redshift, luminosity distance IN SECONDS)
Specify an initial condition parameter: T_merge, f_start
(note that an upper limit of 4 years will be set on the
observation period)
Methods:
CalcStrain: Calculate the characteristic strain of the binary. If (the optional
arguments) sky angles are provided use the stataionary phase approximation
signal generator, else use PhenomA amplitude exclusively
CalcSNR: Calculate the SNR averaged over polarization, inclination,
and sky angles. Theta, phi (spherical polar) are optional arguments
allowing the user to calculate the SNR at a specific sky location
averaged over only polarization and inclination angles
PlotStrain: Plot the characteristic strain
"""
def __init__(self, m1, m2, z=None, Dl=None):
# source-frame component masses
self.m1 = m1
self.m2 = m2
# Store distance parameters
if (Dl == None): # convert redshift into luminosity distance
self.z = z # TODO: check that one of these is provided
self.Dl = get_Dl(self.z, Omega_m, H0) # Dl returned in seconds (i.e. G=c=1, geometric units)
print("Redshift provided. \n\tLuminosity Distance........... {} Mpc".format(self.Dl/MPC))
else: # convert luminosity distance to redshift
self.Dl = Dl # TODO: check that one of these is provided
self.z = optimize.root(get_z, 1., args=(self.Dl, Omega_m, H0)).x[0]
print("Luminosity Distance provided. \n\tredshift........... {}".format(self.z))
# adjust source-frame masses to detector-frame masses
self.m1 *= 1. + self.z
self.m2 *= 1. + self.z
# calculate relevant mass parameters
self.M = self.m1 + self.m2 # total mass
self.eta = self.m1*self.m2/self.M**2 # symmetric mass ratio
self.f_start = None
self.f_end = None
# Methods
SetFreqBounds = SetFreqBounds
CalcStrain = CalcStrain
CalcSNR = CalcSNR
PlotStrain = PlotStrain