C_\ell(\kappa_I)/C_\ell(\delta) should be scale independent for NLA

[NiallJeffrey]

In the NLA Intrinsic Alignment model, the ratio of the kappa power spectrum with pure IA (no lensing shear) to the matter power spectrum should be scale invariant. However, I find a low \ell discrepancy. Code snippet to reproduce:

import matplotlib.pyplot as plt
import numpy as np
import pyccl as ccl

# Cosmology
z = np.linspace(0, 2.5, 10000)
z_fiducial = 0.9
sigma_z = 0.01
nz = np.exp(-0.5*((z-z_fiducial)/sigma_z)**2)
theory_fnla = -0.003018

cosmo_truth = ccl.Cosmology(Omega_c=0.286-0.046, Omega_b=0.046 , h=0.7, A_s=1.703983e-9, n_s=0.96)

# Calculate cl for IA shear with NLA
ell = np.unique(np.geomspace(2,1024*4,1000).astype(int)).astype(float)
t_i_nla = ccl.WeakLensingTracer(cosmo_truth, dndz=(z, nz), has_shear=False,
                                ia_bias=(z, np.ones_like(z)*0.49), use_A_ia=True)
cl_II_nla = ccl.angular_cl(cosmo_truth, t_i_nla, t_i_nla, ell)

# Convert cl to IA kappa 
prefactor_ell = []
for ell_val in ell:
    prefactor_ell.append((((ell_val-1.)*(ell_val+2.))/((ell_val)*(ell_val+1.))))

prefactor_ell = np.array(prefactor_ell)
cl_II_nla_kappa = cl_II_nla/prefactor_ell

# Calculate cl for density
clu1 = ccl.NumberCountsTracer(cosmo_truth, has_rsd=False, dndz=(z,nz), bias=(z,z*0.+1.))
cls_clu = ccl.angular_cl(cosmo_truth, clu1, clu1, ell) 

# Calculate empirical f_nla: -sqrt(C_ell(IA)/C_ell(\delta))
f_nla_empiral = -np.sqrt(cl_II_nla_kappa/cls_clu)

# Plot result
_ = plt.figure(figsize=(5,4))
_ = plt.plot(ell, f_nla_empiral, ls=':',lw=3,
         label='$-\sqrt{C_\ell(\kappa_I)/C_\ell(\delta)}$')
_ = plt.plot(ell,ell*0+theory_fnla, alpha=0.5, lw=3,label='Theory prediction')
_ = plt.xlabel(r'$\ell$', fontsize=12)
_ = plt.ylabel(r'Empirical $f_{NLA}$', fontsize=12)
_ = plt.xscale('log')
_ = plt.legend()
_ = plt.grid()
_ = plt.title(r'$C_\ell(\kappa_I)/C_\ell(\delta)\ $' + 'should be scale independent for NLA', x=0.48)
_ = plt.tight_layout()
_ = plt.savefig('f_NLA_comparison_error.png',dpi=200)
_ = plt.close()

[damonge]

Could this be due to the spin-2 nature of IAs? I.e. remember there's a prefactor of the form sqrt((ell+2)!/(ell-2)!)/(ell+1/2)^2 due to this.

[damonge]

In the code I see you're correcting to kappa, so in that case the correction factor would be ell*(ell+1)/(ell+0.5)^2

[NiallJeffrey]

@damonge Ah - that's not the number I get. For example using this eq. 11 - https://arxiv.org/pdf/2210.13260.pdf - I get the power spectra from shear to kappa changes by ((ell-1)(ell+2))/(ell (ell+1)). Do you see where this difference comes from?

[NiallJeffrey]

@damonge I actually realise you mean there is an additional correction factor. This works! The error is now sub-percent.

I need to convince myself that this makes sense for NLA in this context before I close the issue.

[damonge]

Kappa takes a factor ell*(ell+1)/(k*chi)^2 from translating between the transverse and full Laplacian (this is so you can express it as an integral over the matter overdensity). In the Limber approximation k*chi=(ell+1/2), hence the correction factor of ell*(ell+1)/(ell+0.5)^2 for each factor of kappa going into the power spectrum.

[damonge]

@NiallJeffrey let me know if the above was not convincing! Will close for now.