def p_z(ell_min, ell_max, s=-2):
"""Generator for p^z matrix elements (for use with matrix_expectation_value)
This function is specific to the case where waveforms have s=-2.
p^z = \\cos\theta = 2 \\sqrt{\\pi/3} Y_{1,0}
This is what Ruiz+ (2008) [0707.4654] calls "l^z", which is a bad name.
The matrix elements yielded are
< s, ellp, mp | \\cos\theta | s, ell, m > =
\\sqrt{ \frac{2*ell+1}{2*ellp+1} } *
< ell, m, 1, 0 | ellp, m > < ell, -s, 1, 0 | ellp, -s >
where the terms on the last line are the ordinary Clebsch-Gordan coefficients.
Because of the magnetic selection rules, we only have mp == m
We could have used `_swsh_Y_mat_el` but I am just preemptively
combining the prefactors.
"""
import numpy as np
from spherical_functions import clebsch_gordan as CG
for ell in range(ell_min, ell_max + 1):
ellp_min = max(ell_min, ell - 1)
ellp_max = min(ell_max, ell + 1)
for ellp in range(ellp_min, ellp_max + 1):
for m in range(-ell, ell + 1):
if (m < -ellp) or (m > ellp):
continue
cg1 = CG(ell, m, 1, 0, ellp, m)
cg2 = CG(ell, -s, 1, 0, ellp, -s)
prefac = np.sqrt((2.0 * ell + 1.0) / (2.0 * ellp + 1.0))
yield ellp, m, ell, m, (prefac * cg1 * cg2)
def p_z(ell_min, ell_max, s=-2):
"""Generator for pᶻ matrix elements (for use with `matrix_expectation_value`)
This function is specific to the case where waveforms have s=-2.
pᶻ = cosθ = 2 √(π/3) Y₁₀
This is what Ruiz et al. (2008) [0707.4654] call "lᶻ".
The matrix elements yielded are
⟨s,j,n|cosθ|s,l,m⟩ = √[(2l+1)/(2j+1)] ⟨l,m,1,0|j,m⟩ ⟨l,-s,1,0|j,-s⟩
where the terms on the last line are the ordinary Clebsch-Gordan coefficients.
Because of the magnetic selection rules, we only have nonzero elements for
n==m.
We could have used `_swsh_Y_mat_el` but I am just preemptively combining the
prefactors.
"""
for ell in range(ell_min, ell_max + 1):
ellp_min = max(ell_min, ell - 1)
ellp_max = min(ell_max, ell + 1)
for ellp in range(ellp_min, ellp_max + 1):
for m in range(-ell, ell + 1):
if (m < -ellp) or (m > ellp):
continue
cg1 = CG(ell, m, 1, 0, ellp, m)
cg2 = CG(ell, -s, 1, 0, ellp, -s)
prefac = np.sqrt((2.0 * ell + 1.0) / (2.0 * ellp + 1.0))
yield ellp, m, ell, m, (prefac * cg1 * cg2)
def swsh_Y_mat_el(s, l3, m3, l1, m1, l2, m2):
"""Compute a matrix element treating Y_{\ell, m} as a linear operator
From the rules for the Wigner D matrices, we get the result that
<s, l3, m3 | Y_{l1, m1} | s, l2, m2 > =
\sqrt{ \frac{(2*l1+1)(2*l2+1)}{4*\pi*(2*l3+1)} } *
< l1, m1, l2, m2 | l3, m3 > < l1, 0, l2, −s | l3, −s >
where the terms on the last line are the ordinary Clebsch-Gordan coefficients.
See e.g. Campbell and Morgan (1971).
"""
from spherical_functions import clebsch_gordan as CG
cg1 = CG(l1, m1, l2, m2, l3, m3)
cg2 = CG(l1, 0., l2, -s, l3, -s)
return np.sqrt( (2.*l1 + 1.) * (2.*l2 + 1.) / (4. * np.pi * (2.*l3 + 1)) ) * cg1 * cg2
def swsh_Y_mat_el(s, l3, m3, l1, m1, l2, m2):
"""Compute a matrix element treating Yₗₘ as a linear operator
From the rules for the Wigner D matrices, we get the result that
⟨s,l₃,m₃|Yₗ₁ₘ₁|s,l₂,m₂⟩ =
√[(2l₁+1)(2l₂+1)/(4π(2l₃+1))] ⟨l₁,m₁,l₂,m₂|l₃,m₃⟩ ⟨l₁,0,l₂,−s|l₃,−s⟩
where the terms on the last line are the ordinary Clebsch-Gordan
coefficients. See, e.g., Campbell and Morgan (1971).
"""
cg1 = CG(l1, m1, l2, m2, l3, m3)
cg2 = CG(l1, 0.0, l2, -s, l3, -s)
return np.sqrt((2.0 * l1 + 1.0) * (2.0 * l2 + 1.0) /
(4.0 * np.pi * (2.0 * l3 + 1))) * cg1 * cg2
def ethbar_chi_z(ell_min, ell_max, s=-2):
"""Generator for the matrix element ⟨h|ð̄χ|ðN⟩ in the z direction
For the z direction χ is an m = 0 function. The matrix element is
√[(6/4π) ((2l+1)/(2j+1))] ⟨l,-s-1,1,1|j,-s⟩ ⟨l,m,1,0|j,m⟩
"""
for ell in range(ell_min, ell_max + 1):
ellp_min = max(ell_min, ell - 1)
ellp_max = min(ell_max, ell + 1)
for ellp in range(ellp_min, ellp_max + 1):
cg2 = CG(ell, -s - 1, 1, 1, ellp, -s)
prefac = np.sqrt((2.0 * ell + 1.0) / (2.0 * ellp + 1.0))
for m in range(-ell, ell + 1):
if (m < -ellp) or (m > ellp):
continue
cg1 = np.sqrt(2) * CG(ell, m, 1, 0, ellp, m)
yield ellp, m, ell, m, (prefac * cg1 * cg2)
def mat_el_ethbar_chi(s, l3, m3, l1, m1, l2, m2):
cg1 = np.sqrt(2) * CG(l1, m1, l2, m2, l3, m3)
cg2 = CG(l1, 1.0, l2, -1 - s, l3, -s)
return np.sqrt((2.0 * l1 + 1.0) * (2.0 * l2 + 1.0) /
(4.0 * np.pi * (2.0 * l3 + 1))) * cg1 * cg2