Maths utils

This module contains some fast utility functions that are useful in the same contexts as camb. They are entirely independent of the main camb code.

camb.mathutils.chi_squared(covinv, x)

Utility function to efficiently calculate x^T covinv x

Parameters:

• covinv – symmetric inverse covariance matrix

• x – vector

Returns:

covinv.dot(x).dot(x), but parallelized and using symmetry

camb.mathutils.pcl_coupling_matrix(P, lmax, pol=False)

Get Pseudo-Cl coupling matrix from power spectrum of mask. Uses multiple threads. See Eq A31 of astro-ph/0105302

Parameters:

• P – power spectrum of mask

• lmax – lmax for the matrix

• pol – whether to calculate TE, EE, BB couplings

Returns:

coupling matrix (square but not symmetric), or list of TT, TE, EE, BB if pol

camb.mathutils.scalar_coupling_matrix(P, lmax)

Get scalar Pseudo-Cl coupling matrix from power spectrum of mask, or array of power masks. Uses multiple threads. See Eq A31 of astro-ph/0105302

Parameters:

• P – power spectrum of mask, or list of mask power spectra

• lmax – lmax for the matrix (assumed square)

Returns:

coupling matrix (square but not symmetric), or list of couplings for different masks

camb.mathutils.threej(l2, l3, m2, m3)

Convenience wrapper around standard 3j function, returning array for all allowed l1 values

Parameters:

• l2 – L_2

• l3 – L_3

• m2 – M_2

• m3 – M_3

Returns:

array of 3j from max(abs(l2-l3),abs(m2+m3)) .. l2+l3

camb.mathutils.threej_coupling(W, lmax, pol=False)

Calculate symmetric coupling matrix :math`Xi` for given weights \(W_{\ell}\), where \(\langle\tilde{C}_\ell\rangle = \Xi_{\ell \ell'} (2\ell'+1) C_\ell\). The weights are related to the power spectrum of the mask P by \(W_\ell = (2 \ell + 1) P_\ell / 4 \pi\). See e.g. Eq D16 of arxiv:0801.0554.

If pol is False and W is an array of weights, produces array of temperature couplings, otherwise for pol is True produces set of TT, TE, EE, EB couplings (and weights must have one spectrum - for same masks - or three).

Use scalar_coupling_matrix() or pcl_coupling_matrix() to get the coupling matrix directly from the mask power spectrum.

Parameters:

• W – 1d array of Weights for each L, or list of arrays of weights (zero based)

• lmax – lmax for the output matrix (assumed symmetric, though not in principle)

• pol – if pol, produce TT, TE, EE, EB couplings for three input mask weights (or one if assuming same mask)

Returns:

symmetric coupling matrix or array of matrices