def get_resp_legs(source, lmax):
r"""Defines the responses terms for a CMB map anisotropy source.
Args:
source (str): anisotropy source (e.g. 'p', 'f', ...).
lmax (int): responses are given up to lmax.
Returns:
4-tuple (r, rR, -rR, cL): source spin response *r* (positive or zero),
the harmonic responses for +r and -r (2 1d-arrays), and the scaling between the G/C modes
and the potentials of interest. (for lensing, cL is given by :math:`L\sqrt{L (L + 1)}`).
"""
if source in ['p', 'x']:
# lensing (gradient and curl): _sX -> _sX - 1/2 alpha_1 \eth _sX - 1/2 \alpha_{-1} \bar \eth _sX
return {s : (1, -0.5 * uspin.get_spin_lower(s, lmax), -0.5 * uspin.get_spin_raise(s, lmax),
lambda ell : uspin.get_spin_raise(0, np.max(ell))[ell]) for s in [0, -2, 2]}
if source == 'f': # Modulation: _sX -> _sX + f _sX.
return {s : (0, 0.5 * np.ones(lmax + 1, dtype=float), 0.5 * np.ones(lmax + 1, dtype=float),
lambda ell: np.ones(len(ell), dtype=float)) for s in [0, -2, 2]}
if source in ['a', 'a_p']: # Polarisation rotation _\pm 2 X -> _\pm 2 X + \mp 2 i a _\pm 2 X
ret = {s: (0, -np.sign(s) * 1j * np.ones(lmax + 1, dtype=float),
-np.sign(s) * 1j * np.ones(lmax + 1, dtype=float),
lambda ell: np.ones(len(ell), dtype=float)) for s in [-2, 2]}
ret[0]=(0, np.zeros(lmax + 1, dtype=float),
np.zeros(lmax + 1, dtype=float),
lambda ell: np.ones(len(ell), dtype=float))
return ret
assert 0, source + ' response legs not implemented'
def get_mf_resp(qe_key, cls_cmb, cls_ivfs, lmax_qe, lmax_out):
"""Deflection-induced mean-field response calculation.
See Carron & Lewis 2019 in prep.
"""
# This version looks stable enough
assert qe_key in ['p_p', 'ptt'], qe_key
GL = np.zeros(lmax_out + 1, dtype=float)
CL = np.zeros(lmax_out + 1, dtype=float)
if qe_key == 'ptt':
lmax_cmb = len(cls_cmb['tt']) - 1
spins = [0]
elif qe_key == 'p_p':
lmax_cmb = min(len(cls_cmb['ee']) - 1, len(cls_cmb['bb'] - 1))
spins = [-2, 2]
elif qe_key == 'p':
lmax_cmb = min(len(cls_cmb['ee']) - 1, len(cls_cmb['bb']) - 1, len(cls_cmb['tt']) - 1, len(cls_cmb['te']) - 1)
spins = [0, -2, 2]
else:
assert 0, qe_key + ' not implemented'
assert lmax_qe <= lmax_cmb
if qe_key == 'ptt':
cl_cmbtoticmb = {'tt': cls_cmb['tt'][:lmax_qe + 1] ** 2 * cls_ivfs['tt'][:lmax_qe + 1]}
cl_cmbtoti = {'tt': cls_cmb['tt'][:lmax_qe + 1] * cls_ivfs['tt'][:lmax_qe + 1]}
elif qe_key == 'p_p':
cl_cmbtoticmb = {'ee': cls_cmb['ee'][:lmax_qe + 1] ** 2 * cls_ivfs['ee'][:lmax_qe + 1],
'bb': cls_cmb['bb'][:lmax_qe + 1] ** 2 * cls_ivfs['bb'][:lmax_qe + 1]}
cl_cmbtoti = {'ee': cls_cmb['ee'][:lmax_qe + 1] * cls_ivfs['ee'][:lmax_qe + 1],
'bb': cls_cmb['bb'][:lmax_qe + 1] * cls_ivfs['bb'][:lmax_qe + 1]}
else:
assert 0, 'not implemented'
# Build remaining fisher term II:
FisherGII = np.zeros(lmax_out + 1, dtype=float)
FisherCII = np.zeros(lmax_out + 1, dtype=float)
for s1 in spins:
for s2 in spins:
cl1 = uspin.spin_cls(s1, s2, cls_ivfs)[:lmax_qe + 1] * (0.5 ** (s1 != 0) * 0.5 ** (s2 != 0))
# These 1/2 factor from the factor 1/2 in each B of B Covi B^dagger, where B maps spin-fields to T E B.
cl2 = np.copy(uspin.spin_cls(s2, s1, cls_cmb)[:lmax_cmb + 1])
cl2[:lmax_qe + 1] -= uspin.spin_cls(s2, s1, cl_cmbtoticmb)[:lmax_qe + 1]
if np.any(cl1) and np.any(cl2):
for a in [-1, 1]:
ai = uspin.get_spin_lower(s2, lmax_cmb) if a == - 1 else uspin.get_spin_raise(s2, lmax_cmb)
for b in [1]: # a, b symmetry
aj = uspin.get_spin_lower(-s1, lmax_cmb) if b == 1 else uspin.get_spin_raise(-s1, lmax_cmb)
hL = 2 * (-1) ** (s1 + s2) * uspin.wignerc(cl1, cl2 * ai * aj, s2, s1, -s2 - a, -s1 - b, lmax_out=lmax_out)
GL += (- a * b) * hL
CL += (-1) * hL
# Build remaining Fisher term II:
for s1 in spins:
for s2 in spins:
cl1 = uspin.spin_cls(s2, s1, cl_cmbtoti)[:lmax_qe + 1] * (0.5 ** (s1 != 0))
cl2 = uspin.spin_cls(s1, s2, cl_cmbtoti)[:lmax_qe + 1] * (0.5 ** (s2 != 0))
if np.any(cl1) and np.any(cl2):
for a in [-1, 1]:
ai = uspin.get_spin_lower(s2, lmax_qe) if a == -1 else uspin.get_spin_raise(s2, lmax_qe)
for b in [1]:
aj = uspin.get_spin_lower(s1, lmax_qe) if b == 1 else uspin.get_spin_raise(s1, lmax_qe)
hL = 2 * (-1) ** (s1 + s2) * uspin.wignerc(cl1 * ai, cl2 * aj, -s2 - a, -s1, s2, s1 - b, lmax_out=lmax_out)
FisherGII += (- a * b) * hL
FisherCII += (-1) * hL
GL -= FisherGII
CL -= FisherCII
print("CL[1] ",CL[1])
print("GL[1] (before subtraction) ", GL[1])
print("GL[1] (after subtraction) ", GL[1] - CL[1])
GL -= CL[1]
CL -= CL[1]
GL *= 0.25 * np.arange(lmax_out + 1) * np.arange(1, lmax_out + 2)
CL *= 0.25 * np.arange(lmax_out + 1) * np.arange(1, lmax_out + 2)
return GL, CL