def local_term(ells, chi, delta, flat=False):
out = []
chi = float(chi)
delta = float(delta)
for ell in ells:
ell = int(ell)
p_k_interp = matter_power.p_k_interp(chi)
if flat:
if (ell/chi) > matter_power.K_MAX:
out.append(0.)
continue
# Maximum k_par.
k_max = np.sqrt(max(matter_power.K_MAX**2 - (ell/chi)**2, 0))
# Reduce x_max if envelope is significant.
if delta == 0:
envelop_width = 1000 * k_max
else:
envelop_width = 5 * (2 * np.pi / delta)
k_max = min(k_max, 5 * envelop_width)
nk = sph_bessel.pow_2_gt(k_max * delta * 5) + 1
k = np.linspace(0, k_max, nk, endpoint=True)
delta_k = k_max / (nk - 1)
# Envelope of a Gaussian with width of several oscillations. This
# controls the oscillations out to high k.
envelope = np.exp(-0.5*k**2 / envelop_width**2)
p_k_local = p_k_interp(np.sqrt((ell/chi)**2 + k**2))
p_k_local *= envelope
# Fourier factor.
p_k_local *= np.cos(k * delta)
I1 = integrate.romb(p_k_local, dx=delta_k)
I1 /= np.pi * chi**2
else:
p_k = lambda k: k**2 * p_k_interp(k)
I1 = sph_bessel.integrate_f_jnjn(p_k, ell, chi, delta,
matter_power.K_MAX) * (2 / np.pi)
out.append(I1)
return out
def __init__(self, ell, chi_max, limber=False):
ell = int(ell)
chi_max = float(chi_max)
self._ell = ell
self._chi_max = chi_max
nchi = 20
#delta_chi = chi_max / nchi
#chi_list = np.arange(1, nchi + 1, dtype=float) / nchi * chi_max
CHI_MIN = 500.
chi_list = np.linspace(CHI_MIN, chi_max, nchi, endpoint=True)
delta_chi = chi_list[1] - chi_list[0]
data = []
n_total = 0
for chi in chi_list:
this_chi_data = {}
p_k_interp = matter_power.p_k_interp(chi)
# Put in Gaussian cut-off to control oscillations.
if limber:
# Local term.
delta_k = 2 * np.pi / (2 * chi)
k_max = np.sqrt(max(matter_power.K_MAX**2 - (ell/chi)**2, 0))
k = np.arange(0, k_max, delta_k)
k = np.concatenate((k, -k[:0:-1]))
nk = len(k) # Guaranteed to be odd.
if nk > 10:
p_k_local = p_k_interp(np.sqrt((ell/chi)**2 + k**2))
#window = fftpack.ifftshift(signal.kaiser(nk, beta=14, sym=True))
window = fftpack.ifftshift(signal.hann(nk, sym=True))
p_k_local *= window
fft_norm = len(k) * delta_k / 2 / np.pi / chi**2
I1 = fftpack.ifft(p_k_local).real
I1 = fftpack.fftshift(I1) * fft_norm
deltas = np.linspace(-chi, chi, len(I1), endpoint=True)
else:
I1 = np.zeros(10, dtype=float)
deltas = np.linspace(-chi, chi, len(I1), endpoint=True)
# Limber term.
if ell/chi > matter_power.K_MAX:
I2 = 0
else:
I2 = p_k_interp(ell/chi) / chi**2
else:
p_k = lambda k: k**2 * p_k_interp(k)
scale = ell / chi
nscales = min(5, 2 * ell - 1)
# These deltas for the integral.
deltas, delta_u = exp_sample(scale, nscales)
ndelta_int = len(deltas)
# Extra deltas for interpolation.
delta_max = min(1.99 * chi, 1.99 * (chi_max - chi))
factor = 1.5
deltas = list(deltas)
while factor * deltas[-1] < delta_max:
deltas.append(factor * deltas[-1])
deltas.append(delta_max)
deltas = np.array(deltas)
I1 = np.empty_like(deltas)
for ii in range(len(deltas)):
I1[ii] = sph_bessel.integrate_f_jnjn(p_k, ell, chi, deltas[ii],
matter_power.K_MAX) * (2 / np.pi)
I2 = integrate.romb(I1[:ndelta_int]
* np.exp(scale * deltas[:ndelta_int]),
dx=delta_u) * 2
this_chi_data['chi'] = chi
this_chi_data['deltas'] = deltas
this_chi_data['I1'] = I1
this_chi_data['I2'] = I2
data.append(this_chi_data)
n_total += len(deltas)
I2 = [td["I2"] for td in data]
I2 = np.array(I2)
I3 = integrate.cumtrapz(I2, dx=delta_chi, initial=0.)
for ii in range(nchi):
data[ii]["I3"] = I3[ii]
chi_delta = np.empty((n_total, 2), dtype=float)
I1 = np.empty(n_total, dtype=float)
ii = 0
for d in data:
ndelta = len(d["deltas"])
chi_delta[ii:ii + ndelta,0] = d["chi"]
chi_delta[ii:ii + ndelta,1] = d["deltas"]
I1[ii:ii + ndelta] = d["I1"]
ii += ndelta
self.i1 = interpolate.LinearNDInterpolator(chi_delta, I1, fill_value=0.)
self.i2 = interpolate.interp1d(chi_list, I2, kind="cubic")
self.i3 = interpolate.interp1d(chi_list, I3, kind="cubic")
self.data = data