project_2d

Project 3D power spectra to 2D tomographic bins using the Limber approximation

File	structure/projection/project_2d.py
Attribution	CosmoSIS Team
URL

The Limber approximation integrates a 3D power spectrum over the radial direction to get a 2D angular power spectrum. It is an approximation which is only valid on smaller scales.

\(C_\ell = A \int_0^{\chi_1} W_1(\chi) W_2(\chi) P(k=(l+0.5)/\chi, z(\chi)) / \chi^2 d\chi\)

The full integral must integrate over k(ell) also. This module is a python interface to a Limber integrator written in C, and can generate a range of different spectra depending on what options you set.

It can generate spectra for any pair of (Shear, Position, Intrinsic, Magnification, Cmbkappa) though beware that for certain combinations and scales the Limber approximation will be inaccurate.

Depending which spectra you ask for, it will look for different input P(k) value:

Option name	Input-3d-spectrum	Default output name
shear-shear	matter_power_nl	shear_cl
weyl-weyl	weyl_curvature_spectrum_nl	shear_cl
shear-intrinsic	matter_intrinsic_power	shear_cl_gi
intrinsic-intrinsic	intrinsic_power	shear_cl_ii
intrinsicb-intrinsicb	intrinsic_power_bb	shear_cl_bb
position-position	matter_power_nl	galaxy_cl
magnification-density	matter_power_nl	magnification_density_cl
magnification-magnification	matter_power_nl	magnification_cl
position-shear	matter_power_nl	galaxy_shear_cl
density-intrinsic	matter_intrinsic_power	galaxy_intrinsic_cl
magnification-intrinsic	matter_intrinsic_power	magnification_intrinsic_cl
magnification-shear	matter_power_nl	magnification_shear_cl
shear-cmbkappa	matter_power_nl	shear_cmbkappa_cl
cmbkappa-cmbkappa	matter_power_nl	cmbkappa_cl
intrinsic-cmbkappa	matter_intrinsic_power	intrinsic_cmbkappa_cl
density-cmbkappa	matter_power_nl	galaxy_cmbkappa_cl
fast-shear-shear-ia	matter_power_nl	shear_cl
fast-lingal-shear-ia	matter_power_nl	galaxy_shear_cl
fast-position-shear-ia	matter_power_nl	galaxy_shear_cl
lingal-lingal	matter_power_nl	galaxy_cl
lingal-shear	matter_power_nl	galaxy_shear_cl
lingal-magnification	matter_power_nl	galaxy_magnification_cl
lingal-intrinsic	matter_intrinsic_power	galaxy_intrinsic_cl
nlgal-nlgal	matter_power_nl	galaxy_cl
nlgal-shear	matter_power_nl	galaxy_shear_cl
nlgal-magnification	matter_power_nl	galaxy_magnification_cl

For each of the spectra listed above you can set a parameter in the parameter file to describe whether that term should be calculated and what input n(z) and output names should be used for it.

You can set either:

shear-shear = T ; to use the default wl_number_density n(z) section and save to default shear_cl

shear-shear = euclid-ska ; to cross-correlate n(z) from nz_euclid and nz_ska sections, and save to shear_cl

shear-shear = red-red:shear_cl_red ; to auto-correlate n(z) from the nz_red section and then save to shear_cl_red

If no spectra are chosen at all then only “shear-shear=T” is assumed.

The same forms can be used for all the other spectra, though note that the magnification spectra also require information on the luminosity function.

Lingal refers to clustering spectra for a linearly-biased sample. Nlgal is for non-linearly biased samples.

Parts of this code and the underlying implementation of limber are based on cosmocalc: https://bitbucket.org/beckermr/cosmocalc-public

Assumptions

• The Limber approximation is reasonable for the fields being integrated

• Flat sky approximation

• GR is assumed in various ways

Setup Parameters

Name	Type	Default	Description
verbose	bool	False	Print more output.
fatal_errors	bool	False	Raise an error instead of returning non-zero on error loading splines. Handy for debugging.
save_kernels	bool	False	Save the kernels n(chi) and w(chi) to the block
do_exact	str		Spectra for which to do exact (non-limber) calculation at low ell (space-separated)
auto_only	str		Spectra for which to only compute auto-correlations, not inter-bin correlations (space-separated)
clip_chi_kernels	float	1e-06	Fraction of the integration kernel peaks below which to set the kernel to zero
sig_over_dchi	float	50.0	Ratio of the kernel width to sampling. Sets the sampling of the kernels. Larger is more precise.
shear_kernel_dchi	float	5.0	Sample spacing for shear kernels
limber_ell_start	int	300	For spectra listed in do_exact, the minimum ell to switch to Limber
ell_min_logspaced	real	-1	Minimum ell value for log-spaced values (usually higher than linear)
ell_max_logspaced	real	-1	Maximum ell value for log-spaced values (usually higher than linear)
n_ell_logspaced	real	-1	Number of log-spaced C_ell values produced
ell_min_linspaced	real	-1	Minimum ell value for linearly-spaced values (usually lower than log-spaced)
ell_max_linspaced	real	-1	Maximum ell value for linearly-spaced values (usually higher than log-spaced)
n_ell_linspaced	real		Number of linearly-spaced C_ell values produced
dlogchi	int	-1	spacing in log-chi for exact non-limber calculation (or -1 to auto-set)
chi_pad_upper	float	2.0	Lower padding fraction in chi for non-limber calculation
shear-shear	str or bool		See note in the explanation above for this and related parameters
limber_abs_tol	real	0.0	Absolute tolerance for the Limber integral
limber_rel_tol	real	0.001	Relative tolerance for the Limber integral
lin_bias_prefix	str	b	Parameter name to use for linear bias values, e.g. b for b1, b2, b3, etc.
do_rsd	bool	False	Whether to compute RSD in non-limber calculations

Input values

Section	Name	Type	Default	Description
matter_power_nl	k_h	real 1d		Sample values of nonlinear spectrum in Mpc/h. Many inputs P(k) sections may be required depending on what C_ell spectra are requested - see above.
	z	real 1d		Redshift of nonlinear spectrum samples
	P_k	real 2d		Nonlinear spectrum in (Mpc/h)^{-3}
distances	z	real 1d		Redshift samples of d_m
	a	real 1d		Scale factor samples of d_m
	d_m	real 1d		Comoving distance to sample points.
cosmological_parameters	h0	real		The Hubble parameter H0/100 km/s/Mpc
	chi_star	real		CMB distance. Only needed if doing CMB Kappa spectra.
wl_number_density	z	real 1d		Redshift samples of binned n(z). A different section name to wl_number_density will be used depending on the options chosen (see above)
	nbin	int		Number of tomographic bins used
	bin_{i}	real 1d		Number density n(z) samples for each bin i=1..nbin.

Output values

Output values

Section	Name	Type	Description
kernel_{sample}	n_of_chi_chi_{i}	real 1d	The chi values for the n(chi) calculation. Only if save_kernels=T.
	n_of_chi_n_{i}	real 1d	The n values for the n(chi) calculation. Only if save_kernels=T.
	w_of_chi_chi_{i}	real 1d	The chi values for the w(chi) calculation. Only if save_kernels=T and shear spectrum calculated.
	w_of_chi_n_{i}	real 1d	The n values for the w(chi) calculation. Only if save_kernels=T and shear spectrum calculated.
	ww_of_chi_chi_{i}	real 1d	The chi values for the ww(chi) calculation. Only if save_kernels=T and Weyl spectrum calculated.
	ww_of_chi_n_{i}	real 1d	The n values for the ww(chi) calculation. Only if save_kernels=T and Weyl spectrum calculated.
shear_cl	nbin_a	int	Number of tomographic bins for first of the two quantities correlated. Various sections will have these outputs depending on the options chosen
	nbin_b	int	Number of tomographic bins for second of the two quantities correlated
	nbin	int	Only if auto-correlation is calculated. Number of tomographic bins for the quantity (nbin=nbin_a=nbin_b)
	ell	int 1d	Values at which c_ell are calculated
	bin_{i}_{j}	real 1d	S for relevant i and j combinations. C_ell calculated at corresponding ell.
	chi_peak_{i}_{j}	real	Only if get_kernel_peaks=T. Peak of the n(z) or w(z) for this bin combination
	arcmin_per_Mpch_{i}_{j}	real	Only if get_kernel_peaks=T. Conversion factor from mpc/h to arcmin for this bin