# Source code for camb.correlations

r"""
Functions to transform CMB angular power spectra into correlation functions (cl2corr)
and vice versa (corr2cl), and calculate lensed power spectra from unlensed ones.

The lensed power spectrum functions are not intended to replace those calculated by default when getting CAMB results,
but may be useful for tests, e.g. using different lensing potential power spectra, partially-delensed
lensing power spectra, etc.

These functions are all pure python/scipy, and operate and return cls including
factors :math:\ell(\ell+1)/2\pi (for CMB) and :math:[L(L+1)]^2/2\pi (for lensing).

A. Lewis December 2016
"""

import numpy as np
import os

try:
from .mathutils import gauss_legendre
except:
# use np.polynomial.legendre if can't load fast native (so can use module without compiling camb)
# Fortran version is much faster than current np.polynomial
gauss_legendre = None

from scipy.special import lpn as legendreP
else:
np.pi = 3.1415927  # needed to get docs right for np.pi/32 default argument

_gauss_legendre_cache = {}

def _cached_gauss_legendre(npoints, cache=True):
if cache and npoints in _gauss_legendre_cache:
return _gauss_legendre_cache[npoints]
else:
if gauss_legendre is not None:
xvals = np.empty(npoints)
weights = np.empty(npoints)
gauss_legendre(xvals, weights, npoints)
xvals.flags.writeable = False
weights.flags.writeable = False
else:
xvals, weights = np.polynomial.legendre.leggauss(npoints)
if cache:
_gauss_legendre_cache[npoints] = xvals, weights
return xvals, weights

[docs]def legendre_funcs(lmax, x, m=(0, 2), lfacs=None, lfacs2=None, lrootfacs=None):
r"""
Utility function to return array of Legendre and :math:d_{mn} functions for all :math:\ell up to lmax.
Note that :math:d_{mn} arrays start at :math:\ell_{\rm min} = \max(m,n), so returned arrays are different sizes

:param lmax: maximum :math:\ell
:param x: scalar value of :math:\cos(\theta) at which to evaluate
:param m: m values to calculate :math:d_{m,n}, etc as relevant
:param lfacs: optional pre-computed :math:\ell(\ell+1) float array
:param lfacs2: optional pre-computed :math:(\ell+2)*(\ell-1) float array
:param lrootfacs: optional pre-computed sqrt(lfacs*lfacs2) array
:return: :math:(P,P'),(d_{11},d_{-1,1}), (d_{20}, d_{22}, d_{2,-2}) as requested, where P starts
at :math:\ell=0, but spin functions start at :math:\ell=\ell_{\rm min}
"""
allP, alldP = legendreP(lmax, x)
# Polarization functions all start at L=2
fac1 = 1 - x
fac2 = 1 + x
res = []
if 0 in m:
res.append((allP, alldP))

if 1 in m:
lfacs1 = np.arange(1, lmax + 1, dtype=np.float64)
lfacs1 *= (1 + lfacs1)
d11 = fac1 * alldP[1:] / lfacs1 + allP[1:]
dm11 = fac2 * alldP[1:] / lfacs1 - allP[1:]
res.append((d11, dm11))

if 2 in m:
if lfacs is None:
ls = np.arange(2, lmax + 1, dtype=np.float64)
lfacs = ls * (ls + 1)
lfacs2 = (ls + 2) * (ls - 1)
lrootfacs = np.sqrt(lfacs * lfacs2)
P = allP[2:]
dP = alldP[2:]

fac = fac1 / fac2
d22 = (((4 * x - 8) / fac2 + lfacs) * P
+ 4 * fac * (fac2 + (x - 2) / lfacs) * dP) / lfacs2
if x > 0.998:
# for stability use series at small angles (thanks Pavel Motloch)
d2m2 = np.empty(lmax - 1)
indser = int(np.sqrt((400.0 + 3 / (1 - x ** 2)) / 150)) - 1
d2m2[indser:] = ((lfacs[indser:] - (4 * x + 8) / fac1) * P[indser:]
+ 4 / fac * (-fac1 + (x + 2) / lfacs[indser:]) * dP[indser:]) / lfacs2[indser:]
sin2 = 1 - x ** 2
d2m2[:indser] = lfacs[:indser] * lfacs2[:indser] * sin2 ** 2 / 7680 * (20 + sin2 * (16 - lfacs[:indser]))
else:
d2m2 = ((lfacs - (4 * x + 8) / fac1) * P
+ 4 / fac * (-fac1 + (x + 2) / lfacs) * dP) / lfacs2
d20 = (2 * x * dP - lfacs * P) / lrootfacs
res.append((d20, d22, d2m2))

return res

[docs]def cl2corr(cls, xvals, lmax=None):
r"""
Get the correlation function from the power spectra, evaluated at points cos(theta) = xvals.
Use roots of Legendre polynomials (np.polynomial.legendre.leggauss) for accurate back integration with corr2cl.
Note currently does not work at xvals=1 (can easily calculate that as special case!).

:param cls: 2D array cls(L,ix), with L (:math:\equiv \ell) starting at zero and ix-0,1,2,3 in
order TT, EE, BB, TE. cls should include :math:\ell(\ell+1)/2\pi factors.
:param xvals: array of :math:\cos(\theta) values at which to calculate correlation function.
:param lmax: optional maximum L to use from the cls arrays
:return: 2D array of corrs[i, ix], where ix=0,1,2,3 are T, Q+U, Q-U and cross
"""

if lmax is None:
lmax = cls.shape - 1
xvals = np.asarray(xvals)
ls = np.arange(0, lmax + 1, dtype=np.float64)
corrs = np.zeros((len(xvals), 4))
lfacs = ls * (ls + 1)
lfacs = 1
facs = (2 * ls + 1) / (4 * np.pi) * 2 * np.pi / lfacs

ct = facs * cls[:lmax + 1, 0]
# For polarization, all arrays start at 2
cp = facs[2:] * (cls[2:lmax + 1, 1] + cls[2:lmax + 1, 2])
cm = facs[2:] * (cls[2:lmax + 1, 1] - cls[2:lmax + 1, 2])
cc = facs[2:] * cls[2:lmax + 1, 3]
ls = ls[2:]
lfacs = lfacs[2:]
lfacs2 = (ls + 2) * (ls - 1)
lrootfacs = np.sqrt(lfacs * lfacs2)
for i, x in enumerate(xvals):
(P, _), (d20, d22, d2m2) = legendre_funcs(lmax, x, [0, 2], lfacs, lfacs2, lrootfacs)
corrs[i, 0] = np.dot(ct, P)  # T
corrs[i, 1] = np.dot(cp, d22)  # Q+U
corrs[i, 2] = np.dot(cm, d2m2)  # Q-U
corrs[i, 3] = np.dot(cc, d20)  # cross

return corrs

[docs]def gauss_legendre_correlation(cls, lmax=None, sampling_factor=1):
r"""
Transform power specturm cls into correlation functions evaluated at the
roots of the Legendre polynomials for Gauss-Legendre quadrature. Returns correlation function array,
evaluation points and weights.
Result can be passed to corr2cl for accurate back transform.

:param cls: 2D array cls(L,ix), with L (:math:\equiv \ell) starting at zero and ix=0,1,2,3 in
order TT, EE, BB, TE. Should include :math:\ell(\ell+1)/2\pi factors.
:param lmax: optional maximum L to use
:param sampling_factor: uses Gauss-Legendre with degree lmax*sampling_factor+1
:return: corrs, xvals, weights; corrs[i, ix] is 2D array where ix=0,1,2,3 are T, Q+U, Q-U and cross
"""

if lmax is None: lmax = cls.shape - 1
xvals, weights = _cached_gauss_legendre(int(sampling_factor * lmax) + 1)
return cl2corr(cls, xvals, lmax), xvals, weights

[docs]def corr2cl(corrs, xvals, weights, lmax):
r"""
Transform from correlation functions to power spectra.
Note that using cl2corr followed by corr2cl is generally very accurate (< 1e-5 relative error) if
xvals, weights = np.polynomial.legendre.leggauss(lmax+1)

:param corrs: 2D array, corrs[i, ix], where ix=0,1,2,3 are T, Q+U, Q-U and cross
:param xvals: values of :math:\cos(\theta) at which corrs stores values
:param weights: weights for integrating each point in xvals. Typically from np.polynomial.legendre.leggauss
:param lmax: maximum :math:\ell to calculate :math:C_\ell
:return: array of power spectra, cl[L, ix], where L starts at zero and ix=0,1,2,3 in order TT, EE, BB, TE.
They include :math:\ell(\ell+1)/2\pi factors.
"""
# For polarization, all arrays start at 2
ls = np.arange(2, lmax + 1, dtype=np.float64)
lfacs = ls * (ls + 1)
lfacs2 = (ls + 2) * (ls - 1)
lrootfacs = np.sqrt(lfacs * lfacs2)
cls = np.zeros((lmax + 1, 4))
for i, (x, weight) in enumerate(zip(xvals, weights)):
(P, _), (d20, d22, d2m2) = legendre_funcs(lmax, x, [0, 2], lfacs, lfacs2, lrootfacs)
cls[:, 0] += (weight * corrs[i, 0]) * P
T2 = (corrs[i, 1] * weight / 2) * d22
T4 = (corrs[i, 2] * weight / 2) * d2m2
cls[2:, 1] += T2 + T4
cls[2:, 2] += T2 - T4
cls[2:, 3] += (weight * corrs[i, 3]) * d20

cls[1, :] *= 2
cls[2:, :] = (cls[2:, :].T * lfacs).T

return cls

[docs]def lensing_correlations(clpp, xvals, lmax=None):
r"""
Get the :math:\sigma^2(x) and :math:C_{{\rm gl},2}(x) functions from the lensing power spectrum

:param clpp: array of :math:[L(L+1)]^2 C_L^{\phi\phi}/2\pi lensing potential power spectrum (zero based)
:param xvals: array of :math:\cos(\theta) values at which to calculate correlation function.
:param lmax: optional maximum L to use from the clpp array
:return: array of :math:\sigma^2(x), array of :math:C_{{\rm gl},2}(x)
"""
if lmax is None: lmax = clpp.shape - 1
ls = np.arange(1, lmax + 1, dtype=np.float64)
cldd = clpp[1:] / (ls * (ls + 1))
cphil3 = (2 * ls + 1) * cldd / 2  # (2*l+1)l(l+1)/4pi C_phi_phi
sigmasq = np.zeros(xvals.shape)
Cg2 = np.zeros(xvals.shape)
llp1 = ls * (ls + 1)
for i, x in enumerate(xvals):
P, dP = legendreP(lmax, x)
fac1 = 1 - x
fac2 = 1 + x
d_11 = fac1 * dP[1:] / llp1 + P[1:]
d_m11 = fac2 * dP[1:] / llp1 - P[1:]
sigmasq[i] = np.dot(1 - d_11, cphil3)
Cg2[i] = np.dot(d_m11, cphil3)
return sigmasq, Cg2

[docs]def lensing_R(clpp, lmax=None):
r"""
Get :math:R \equiv \frac{1}{2} \langle |\nabla \phi|^2\rangle

:param clpp: array of :math:[L(L+1)]^2 C_L^{\phi\phi}/2\pi lensing potential power spectrum
:param lmax: optional maximum L to use from the cls arrays
:return: R
"""
if lmax is None: lmax = clpp.shape - 1
ls = np.arange(1, lmax + 1, dtype=np.float64)
cldd = clpp[1:] / (ls * (ls + 1))
cphil3 = (2 * ls + 1) * cldd / 4
return np.sum(cphil3)

[docs]def lensed_correlations(cls, clpp, xvals, weights=None, lmax=None, delta=False, theta_max=None,
apodize_point_width=10):
r"""
Get the lensed correlation function from the unlensed power spectra, evaluated at points :math:\cos(\theta) = xvals.
Use roots of Legendre polynomials (np.polynomial.legendre.leggauss) for accurate back integration with corr2cl.
Note currently does not work at xvals=1 (can easily calculate that as special case!).

To get the lensed cls efficiently, set weights to the integral weights for each x value, then function returns
lensed correlations and lensed cls.

Uses the non-perturbative curved-sky results from Eqs 9.12 and 9.16-9.18 of
astro-ph/0601594 <https://arxiv.org/abs/astro-ph/0601594>_, to second order in :math:C_{{\rm gl},2}

:param cls: 2D array of unlensed cls(L,ix), with L (:math:\equiv\ell) starting at zero and ix=0,1,2,3 in
order TT, EE, BB, TE. cls should include :math:\ell(\ell+1)/2\pi factors.
:param clpp: array of :math:[L(L+1)]^2 C_L^{\phi\phi}/2\pi lensing potential power spectrum (zero based)
:param xvals: array of :math:\cos(\theta) values at which to calculate correlation function.
:param weights: if given also return lensed :math:C_\ell, otherwise just lensed correlations
:param lmax: optional maximum L to use from the cls arrays
:param delta: if true, calculate the difference between lensed and unlensed (default False)
:param theta_max: maximum angle (in radians) to keep in the correlation functions
:param apodize_point_width: smoothing scale for apodization at truncation of correlation function
:return: 2D array of corrs[i, ix], where ix=0,1,2,3 are T, Q+U, Q-U and cross;
if weights is not None, then return corrs, lensed_cls
"""

if lmax is None: lmax = cls.shape - 1
xvals = np.asarray(xvals)
ls = np.arange(0, lmax + 1, dtype=np.float64)
lfacs = ls * (ls + 1)
lfacsall = lfacs.copy()
lfacs = 1
cldd = clpp[1:lmax + 1] / lfacs[1:]
cphil3 = (2 * ls[1:] + 1) * cldd / 2  # (2*l+1)l(l+1)/4pi C_phi_phi
facs = (2 * ls + 1) / (4 * np.pi) * 2 * np.pi / lfacs

ct = facs * cls[:lmax + 1, 0]
# For polarization, all arrays start at 2
cp = facs[2:] * (cls[2:lmax + 1, 1] + cls[2:lmax + 1, 2])
cm = facs[2:] * (cls[2:lmax + 1, 1] - cls[2:lmax + 1, 2])
cc = facs[2:] * cls[2:lmax + 1, 3]
ls = ls[2:]
lfacs = lfacs[2:]
lfacs2 = (ls + 2) * (ls - 1)
lrootfacs = np.sqrt(lfacs * lfacs2)
rootfac1 = np.sqrt(lfacs2)
rootfac2 = np.sqrt((ls[1:] + 3) * (ls[1:] - 2))
rootrat = lfacs2[1:] / rootfac1[1:] / rootfac2
rootfac3 = np.sqrt((ls[2:] - 3) * (ls[2:] + 4))
if weights is not None:
lensedcls = np.zeros((lmax + 1, 4))
if delta:
delta_diff = 1
else:
delta_diff = 0

if theta_max is not None:
xmin = np.cos(theta_max)
imin = np.searchsorted(xvals, xmin)  # assume xvals sorted
else:
imin = 0

corrs = np.zeros((len(xvals[imin:]), 4))

for i, x in enumerate(xvals[imin:]):
(P, dP), (d11, dm11), (d20, d22, d2m2) = legendre_funcs(lmax, x, [0, 1, 2], lfacs, lfacs2,
lrootfacs)
sigma2 = np.dot(1 - d11, cphil3)
Cg2 = np.dot(dm11, cphil3)

c2fac = lfacsall[1:] * Cg2 / 2
c2fac2 = c2fac[1:] ** 2
fac = np.exp(-lfacsall * sigma2 / 2)
difffac = fac - delta_diff
f = ct * fac
# T (don't really need the term second order Cg2 here, but include for consistency)
corrs[i, 0] = np.dot(ct * difffac, P) + np.dot(f[1:], c2fac * (dm11 + c2fac * P[1:] / 4)) \
+ np.dot(f[2:], c2fac2 * d2m2) / 4
sinth = np.sqrt(1 - x ** 2)
sinfac = 4 / sinth
fac1 = 1 - x
fac2 = 1 + x
d1m2 = sinth / rootfac1 * (dP[2:] - 2 / fac1 * dm11[1:])
d12 = sinth / rootfac1 * (dP[2:] - 2 / fac2 * d11[1:])
d1m3 = (-(x + 0.5) * sinfac * d1m2[1:] / rootfac2 - rootrat * dm11[2:])
d2m3 = (-fac2 * d2m2[1:] * sinfac - rootfac1[1:] * d1m2[1:]) / rootfac2
d3m3 = (-(x + 1.5) * d2m3 * sinfac - rootfac1[1:] * d1m3) / rootfac2
d13 = ((x - 0.5) * sinfac * d12[1:] / rootfac2 - rootrat * d11[2:])
d04 = ((-lfacs[2:] + (18 * x ** 2 + 6) / sinth ** 2) * d20[2:] -
6 * x * lfacs2[2:] * dP[4:] / lrootfacs[2:]) / (rootfac2[1:] * rootfac3)
d2m4 = (-(6 * x + 4) / sinth * d2m3[1:] - rootfac2[1:] * d2m2[2:]) / rootfac3
d4m4 = (-7 / 5.0 * (lfacs2[2:] - 6) * d2m2[2:] +
12 / 5.0 * (-lfacs2[2:] + (9 * x + 26) / fac1) * d3m3[1:]) / (lfacs2[2:] - 12)
# + (second order Cg2 terms are needed for <1% accuracy on BB)
f = cp * fac[2:]
corrs[i, 1] = np.dot(cp * difffac[2:], d22) + np.dot(f[1:], c2fac[2:] * d13) \
+ (np.dot(f, c2fac2 * d22) + np.dot(f[2:], c2fac2[2:] * d04)) / 4
# -
f = cm * fac[2:]
corrs[i, 2] = np.dot(cm * difffac[2:], d2m2) + (np.dot(f, c2fac[1:] * dm11[1:])
+ np.dot(f[1:], c2fac[2:] * d3m3)) / 2 \
+ (np.dot(f, c2fac2 * (2 * d2m2 + P[2:])) + np.dot(f[2:], c2fac2[2:] * d4m4)) / 8

# cross
f = cc * fac[2:]
corrs[i, 3] = np.dot(cc * difffac[2:], d20) + (np.dot(f, c2fac[1:] * d11[1:])
+ np.dot(f[1:], c2fac[2:] * d1m3)) / 2 \
+ (3 * np.dot(f, c2fac2 * d20) + np.dot(f[2:], c2fac2[2:] * d2m4)) / 8
if weights is not None:
weight = weights[i + imin]
if theta_max is not None and i < apodize_point_width * 4:
weight *= 1 - np.exp(-((i + 1.) / apodize_point_width) ** 2 / 2)

lensedcls[:, 0] += (weight * corrs[i, 0]) * P
T2 = (corrs[i, 1] * weight / 2) * d22
T4 = (corrs[i, 2] * weight / 2) * d2m2
lensedcls[2:, 1] += T2 + T4
lensedcls[2:, 2] += T2 - T4
lensedcls[2:, 3] += (weight * corrs[i, 3]) * d20

if weights is not None:
lensedcls[1, :] *= 2
lensedcls[2:, :] = (lensedcls[2:, :].T * lfacs).T
return corrs, lensedcls
else:
return corrs

[docs]def lensed_cls(cls, clpp, lmax=None, lmax_lensed=None, sampling_factor=1.4, delta_cls=False,
theta_max=np.pi / 32, apodize_point_width=10, leggaus=True, cache=True):
r"""
Get the lensed power spectra from the unlensed power spectra and the lensing potential power.
Uses the non-perturbative curved-sky results from Eqs 9.12 and 9.16-9.18 of
astro-ph/0601594 <https://arxiv.org/abs/astro-ph/0601594>_, to second order in :math:C_{{\rm gl},2}.

Correlations are calculated for Gauss-Legendre integration if leggaus=True; this slows it by several seconds,
but will be must faster on subsequent calls with the same lmax*sampling_factor.
If Gauss-Legendre is not used, sampling_factor needs to be about 2 times larger for same accuracy.

For a reference implementation with the full integral range and no apodization set theta_max=None.

Note that this function does not pad high :math:\ell with a smooth fit (like CAMB's main functions); for
accurate results should be called with lmax high enough that input cls are effectively band limited
(lmax >= 2500, or higher for accurate BB to small scales).
Usually lmax truncation errors are far larger than other numerical errors for lmax<4000.

:param cls: 2D array of unlensed cls(L,ix), with L starting at zero and ix=0,1,2,3 in order TT, EE, BB, TE.
cls should include :math:\ell(\ell+1)/2\pi factors.
:param clpp: array of :math:[L(L+1)]^2 C_L^{\phi\phi}/2\pi lensing potential power spectrum (zero based)
:param lmax: optional maximum L to use from the cls arrays
:param lmax_lensed: optional maximum L for the returned cl array (lmax_lensed <= lmax)
:param sampling_factor: npoints = int(sampling_factor*lmax)+1
:param delta_cls: if true, return the difference between lensed and unlensed (optional, default False)
:param theta_max: maximum angle (in radians) to keep in the correlation functions; default: pi/32
:param apodize_point_width: if theta_max is set, apodize around the cut using half Gaussian of approx
width apodize_point_width/lmax*pi
:param leggaus: whether to use Gauss-Legendre integration (default True)
:param cache: if leggaus = True, set cache to save the x values and weights between calls (most of the time)
:return: 2D array of cls[L, ix], with L starting at zero and ix=0,1,2,3 in order TT, EE, BB, TE.
cls include :math:\ell(\ell+1)/2\pi factors.
"""
if lmax is None:
lmax = cls.shape - 1
npoints = int(sampling_factor * lmax) + 1
if leggaus:
xvals, weights = _cached_gauss_legendre(npoints, cache)
else:
theta = np.arange(1, npoints + 1) * np.pi / (npoints + 1)
xvals = np.cos(theta[::-1])
weights = np.pi / npoints * np.sin(theta)
_, lensedcls = lensed_correlations(cls, clpp, xvals, weights, lmax, delta=True,
theta_max=theta_max,
apodize_point_width=int(apodize_point_width * sampling_factor))
if not delta_cls:
lensedcls += cls[:lmax + 1, :]
if lmax_lensed is not None:
return lensedcls[:lmax_lensed + 1, :]
else:
return lensedcls

[docs]def lensed_cl_derivatives(cls, clpp, lmax=None, theta_max=np.pi / 32,
apodize_point_width=10, sampling_factor=1.4):
r"""
Get derivative dcl of lensed :math:D_\ell\equiv \ell(\ell+1)C_\ell/2\pi with respect to :math:\log(C^{\phi}_L).
To leading order (and hence not actually accurate), the lensed correction to power spectrum ix
is given by dcl[ix,:,:].dot(np.ones(clpp.shape)).

Uses the non-perturbative curved-sky results from Eqs 9.12 and 9.16-9.18 of
astro-ph/0601594 <https://arxiv.org/abs/astro-ph/0601594>_, to second order in :math:C_{{\rm gl},2}

:param cls: 2D array of unlensed cls(L,ix), with L starting at zero and ix=0,1,2,3 in order TT, EE, BB, TE.
cls should include :math:\ell(\ell+1)/2\pi factors.
:param clpp: array of :math:[L(L+1)]^2 C_L^{\phi\phi}/2\pi lensing potential power spectrum (zero based)
:param lmax: optional maximum L to use from the cls arrays
:param theta_max: maximum angle (in radians) to keep in the correlation functions
:param apodize_point_width: if theta_max is set, apodize around the cut using half Gaussian of approx
width apodize_point_width/lmax*pi
:param sampling_factor: npoints = int(sampling_factor*lmax)+1
:return: array dCL[ix, ell, L], where ix=0,1,2,3 are T, EE, BB, TE and r
esult is :math:d[D^{\rm ix}_\ell]/ d (\log C^{\phi}_L)

"""

if lmax is None: lmax = cls.shape - 1
npoints = int(sampling_factor * lmax) + 1
xvals, weights = _cached_gauss_legendre(npoints)

ls = np.arange(0, lmax + 1, dtype=np.float64)
lfacs = ls * (ls + 1)
lfacsall = lfacs.copy()
lfacs = 1
cldd = clpp[1:lmax + 1] / lfacs[1:]
cphil3 = (2 * ls[1:] + 1) * cldd / 2  # (2*l+1)l(l+1)/4pi C_phi_phi
facs = (2 * ls + 1) / (4 * np.pi) * 2 * np.pi / lfacs

ct = facs * cls[:lmax + 1, 0]
# For polarization, all arrays start at 2
cp = facs[2:] * (cls[2:lmax + 1, 1] + cls[2:lmax + 1, 2])
cm = facs[2:] * (cls[2:lmax + 1, 1] - cls[2:lmax + 1, 2])
cc = facs[2:] * cls[2:lmax + 1, 3]
ls = ls[2:]
lfacs = lfacs[2:]
lfacs2 = (ls + 2) * (ls - 1)
lrootfacs = np.sqrt(lfacs * lfacs2)
rootfac1 = np.sqrt(lfacs2)
rootfac2 = np.sqrt((ls[1:] + 3) * (ls[1:] - 2))
rootrat = lfacs2[1:] / rootfac1[1:] / rootfac2
rootfac3 = np.sqrt((ls[2:] - 3) * (ls[2:] + 4))
dcl = np.zeros((4, lmax + 1, lmax + 1))

if theta_max is not None:
xmin = np.cos(theta_max)
imin = np.searchsorted(xvals, xmin)  # assume xvals sorted
else:
imin = 0

corrs = np.zeros((4, lmax + 1))

for i, x in enumerate(xvals[imin:]):
(P, dP), (d11, dm11), (d20, d22, d2m2) = legendre_funcs(lmax, x, [0, 1, 2], lfacs, lfacs2,
lrootfacs)
sigma2 = np.dot(1 - d11, cphil3)
dsigma2 = 1 - d11
Cg2 = np.dot(dm11, cphil3)
dCg2 = dm11

c2fac = lfacsall[1:] * Cg2 / 2
c2fac2 = c2fac[1:] ** 2
fac = np.exp(-lfacsall * sigma2 / 2)
f = -lfacsall / 2 * ct * fac
orderfac = 1  # set to zero to neglect second order in cg2 (doesn't make much difference)
# T (don't really need the term second order Cg2 here, but include for consistency)
corr = np.dot(f[1:], P[1:] + c2fac * (dm11 + orderfac * c2fac * P[1:] / 4)) \
+ orderfac * np.dot(f[2:], c2fac2 * d2m2) / 4
f = -f
corr2 = np.dot(f[1:], (dm11 + orderfac * 2 * c2fac * P[1:] / 4)) + orderfac * 2 * np.dot(f[2:],
c2fac[1:] * d2m2) / 4
corrs[0, 1:] = (dsigma2 * corr + dCg2 * corr2) * cphil3
sinth = np.sqrt(1 - x ** 2)
sinfac = 4 / sinth
fac1 = 1 - x
fac2 = 1 + x
d1m2 = sinth / rootfac1 * (dP[2:] - 2 / fac1 * dm11[1:])
d12 = sinth / rootfac1 * (dP[2:] - 2 / fac2 * d11[1:])
d1m3 = (-(x + 0.5) * sinfac * d1m2[1:] / rootfac2 - rootrat * dm11[2:])
d2m3 = (-fac2 * d2m2[1:] * sinfac - rootfac1[1:] * d1m2[1:]) / rootfac2
d3m3 = (-(x + 1.5) * d2m3 * sinfac - rootfac1[1:] * d1m3) / rootfac2
d13 = ((x - 0.5) * sinfac * d12[1:] / rootfac2 - rootrat * d11[2:])
d04 = ((-lfacs[2:] + (18 * x ** 2 + 6) / sinth ** 2) * d20[2:] -
6 * x * lfacs2[2:] * dP[4:] / lrootfacs[2:]) / (rootfac2[1:] * rootfac3)
d2m4 = (-(6 * x + 4) / sinth * d2m3[1:] - rootfac2[1:] * d2m2[2:]) / rootfac3
d4m4 = (-7 / 5.0 * (lfacs2[2:] - 6) * d2m2[2:] +
12 / 5.0 * (-lfacs2[2:] + (9 * x + 26) / fac1) * d3m3[1:]) / (lfacs2[2:] - 12)
# + (second order Cg2 terms are needed for <1% accuracy on BB)
f = -lfacsall[2:] / 2 * cp * fac[2:]
corr = np.dot(f, d22) + np.dot(f[1:], c2fac[2:] * d13) \
+ orderfac * (np.dot(f, c2fac2 * d22) + np.dot(f[2:], c2fac2[2:] * d04)) / 4
f = lfacsall[2:] / 2 * cp * fac[2:]
corr2 = np.dot(f[1:], d13) + orderfac * 2 * (np.dot(f, c2fac[1:] * d22) + np.dot(f[2:], c2fac[3:] * d04)) / 4
corrs[1, 1:] = (dsigma2 * corr + dCg2 * corr2) * cphil3
# -
f = -lfacsall[2:] / 2 * cm * fac[2:]
corr = np.dot(f, d2m2) + (np.dot(f, c2fac[1:] * dm11[1:])
+ np.dot(f[1:], c2fac[2:] * d3m3)) / 2 \
+ orderfac * (np.dot(f, c2fac2 * (2 * d2m2 + P[2:])) + np.dot(f[2:], c2fac2[2:] * d4m4)) / 8
f = -f
corr2 = (np.dot(f, dm11[1:]) + np.dot(f[1:], d3m3)) / 2 \
+ orderfac * 2 * (np.dot(f, c2fac[1:] * (2 * d2m2 + P[2:])) + np.dot(f[2:], c2fac[3:] * d4m4)) / 8

corrs[2, 1:] = (dsigma2 * corr + dCg2 * corr2) * cphil3

# cross
f = -lfacsall[2:] / 2 * cc * fac[2:]
corr = np.dot(f, d20) + (np.dot(f, c2fac[1:] * d11[1:])
+ np.dot(f[1:], c2fac[2:] * d1m3)) / 2 \
+ orderfac * (3 * np.dot(f, c2fac2 * d20) + np.dot(f[2:], c2fac2[2:] * d2m4)) / 8
f = -f
corr2 = (np.dot(f, d11[1:]) + np.dot(f[1:], d1m3)) / 2 \
+ orderfac * 2 * (3 * np.dot(f, c2fac[1:] * d20) + np.dot(f[2:], c2fac[3:] * d2m4)) / 8

corrs[3, 1:] = (dsigma2 * corr + dCg2 * corr2) * cphil3

weight = weights[i + imin]
if theta_max is not None and i < apodize_point_width * 4:
weight *= 1 - np.exp(-((i + 1.) / apodize_point_width) ** 2 / 2)

dcl[0, :, :] += np.outer(P, (weight * corrs[0, :]))
T2 = np.outer(d22, (corrs[1, :] * weight / 2))
T4 = np.outer(d2m2, (corrs[2, :] * weight / 2))
dcl[1, 2:, :] += T2 + T4
dcl[2, 2:, :] += T2 - T4
dcl[3, 2:, :] += np.outer(d20, (weight * corrs[3, :]))

# put into ell(ell+1)C_ell/2pi units [two pi cancels from correlation integral]
dcl[:, 1, :] *= 2
for i in range(dcl.shape):
dcl[i, 2:, :] = (dcl[i, 2:, :].T * lfacs).T
return dcl

[docs]def lensed_cl_derivative_unlensed(clpp, lmax=None, theta_max=np.pi / 32,
apodize_point_width=10, sampling_factor=1.4):
r"""
Get derivative dcl of lensed minus unlensed power :math:D_\ell \equiv \ell(\ell+1)\Delta C_\ell/2\pi with respect
to :math:\ell(\ell+1)C^{\rm unlens}_\ell/2\pi

The difference in power in the lensed spectrum is given by dCL[ix, :, :].dot(cl),
where cl is the appropriate :math:\ell(\ell+1)C^{\rm unlens}_\ell/2\pi.

Uses the non-perturbative curved-sky results from Eqs 9.12 and 9.16-9.18 of
astro-ph/0601594 <https://arxiv.org/abs/astro-ph/0601594>_, to second order in :math:C_{{\rm gl},2}

:param clpp: array of :math:[L(L+1)]^2 C_L^{\phi\phi}/2\pi lensing potential power spectrum (zero based)
:param lmax: optional maximum L to use from the clpp array
:param theta_max: maximum angle (in radians) to keep in the correlation functions
:param apodize_point_width: if theta_max is set, apodize around the cut using half Gaussian of approx
width apodize_point_width/lmax*pi
:param sampling_factor: npoints = int(sampling_factor*lmax)+1
:return: array dCL[ix, ell, L], where ix=0,1,2,3 are TT, EE, BB, TE and result is
:math:d\left(\Delta D^{\rm ix}_\ell\right) / d D^{{\rm unlens},j}_L where j[ix] are TT, EE, EE, TE

"""

if lmax is None:
lmax = clpp.shape - 1
npoints = int(sampling_factor * lmax) + 1
xvals, weights = _cached_gauss_legendre(npoints)

ls = np.arange(0, lmax + 1, dtype=np.float64)
lfacs = ls * (ls + 1)
lfacsall = lfacs.copy()
lfacs = 1
cldd = clpp[1:lmax + 1] / lfacs[1:]
cphil3 = (2 * ls[1:] + 1) * cldd / 2  # (2*l+1)l(l+1)/4pi C_phi_phi
facs = (2 * ls + 1) / (4 * np.pi) * 2 * np.pi / lfacs

ct = facs
# For polarization, all arrays start at 2
cp = facs[2:]
cm = facs[2:]
cc = facs[2:]
ls = ls[2:]
lfacs = lfacs[2:]
lfacs2 = (ls + 2) * (ls - 1)
lrootfacs = np.sqrt(lfacs * lfacs2)
rootfac1 = np.sqrt(lfacs2)
rootfac2 = np.sqrt((ls[1:] + 3) * (ls[1:] - 2))
rootrat = lfacs2[1:] / rootfac1[1:] / rootfac2
rootfac3 = np.sqrt((ls[2:] - 3) * (ls[2:] + 4))
dcl = np.zeros((4, lmax + 1, lmax + 1))

if theta_max is not None:
xmin = np.cos(theta_max)
imin = np.searchsorted(xvals, xmin)  # assume xvals sorted
else:
imin = 0

corr = np.zeros((4, lmax + 1))

for i, x in enumerate(xvals[imin:]):
(P, dP), (d11, dm11), (d20, d22, d2m2) = legendre_funcs(lmax, x, [0, 1, 2], lfacs, lfacs2,
lrootfacs)
sigma2 = np.dot(1 - d11, cphil3)
Cg2 = np.dot(dm11, cphil3)

c2fac = lfacsall[1:] * Cg2 / 2
c2fac2 = c2fac[1:] ** 2
fac = np.exp(-lfacsall * sigma2 / 2)
difffac = fac - 1
f = ct * fac
# T (don't really need the term second order Cg2 here, but include for consistency)
corr[0, :] = ct * difffac * P
corr[0, 1:] += f[1:] * c2fac * (dm11 + c2fac * P[1:] / 4)
corr[0, 2:] += f[2:] * c2fac2 * d2m2 / 4

sinth = np.sqrt(1 - x ** 2)
sinfac = 4 / sinth
fac1 = 1 - x
fac2 = 1 + x
d1m2 = sinth / rootfac1 * (dP[2:] - 2 / fac1 * dm11[1:])
d12 = sinth / rootfac1 * (dP[2:] - 2 / fac2 * d11[1:])
d1m3 = (-(x + 0.5) * sinfac * d1m2[1:] / rootfac2 - rootrat * dm11[2:])
d2m3 = (-fac2 * d2m2[1:] * sinfac - rootfac1[1:] * d1m2[1:]) / rootfac2
d3m3 = (-(x + 1.5) * d2m3 * sinfac - rootfac1[1:] * d1m3) / rootfac2
d13 = ((x - 0.5) * sinfac * d12[1:] / rootfac2 - rootrat * d11[2:])
d04 = ((-lfacs[2:] + (18 * x ** 2 + 6) / sinth ** 2) * d20[2:] -
6 * x * lfacs2[2:] * dP[4:] / lrootfacs[2:]) / (rootfac2[1:] * rootfac3)
d2m4 = (-(6 * x + 4) / sinth * d2m3[1:] - rootfac2[1:] * d2m2[2:]) / rootfac3
d4m4 = (-7 / 5.0 * (lfacs2[2:] - 6) * d2m2[2:] +
12 / 5.0 * (-lfacs2[2:] + (9 * x + 26) / fac1) * d3m3[1:]) / (lfacs2[2:] - 12)
# + (second order Cg2 terms are needed for <1% accuracy on BB)

f = cp * fac[2:]
corr[1, 2:] = cp * difffac[2:] * d22 + f * c2fac2 * d22 / 4
corr[1, 3:] += f[1:] * c2fac[2:] * d13
corr[1, 4:] += f[2:] * c2fac2[2:] * d04 / 4

# -
f = cm * fac[2:]
corr[2, 2:] = cm * difffac[2:] * d2m2 + f * c2fac[1:] * dm11[1:] / 2 + f * c2fac2 * (2 * d2m2 + P[2:]) / 8
corr[2, 3:] += f[1:] * c2fac[2:] * d3m3 / 2
corr[2, 4:] += f[2:] * c2fac2[2:] * d4m4 / 8

# cross
f = cc * fac[2:]
corr[3, 2:] = cc * difffac[2:] * d20 + f * c2fac[1:] * d11[1:] / 2 + 3 / 8. * f * c2fac2 * d20
corr[3, 3:] += f[1:] * c2fac[2:] * d1m3 / 2
corr[3, 4:] += f[2:] * c2fac2[2:] * d2m4 / 8

weight = weights[i + imin]
if theta_max is not None and i < apodize_point_width * 4:
weight *= 1 - np.exp(-((i + 1.) / apodize_point_width) ** 2 / 2)

dcl[0, :, :] += np.outer(P, (weight * corr[0, :]))
T2 = np.outer(d22, (corr[1, :] * weight / 2))
T4 = np.outer(d2m2, (corr[2, :] * weight / 2))
dcl[1, 2:, :] += T2 + T4
dcl[2, 2:, :] += T2 - T4
dcl[3, 2:, :] += np.outer(d20, (weight * corr[3, :]))

# put into ell(ell+1)C_ell/2pi units [two pi cancels from correlation integral]
dcl[:, 1, :] *= 2
for i in range(dcl.shape):
dcl[i, 2:, :] = (dcl[i, 2:, :].T * lfacs).T
return dcl