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.airy_ai_fast(x)

Fast Airy \(Ai(x)\) approximation.

Uses a fitted Fortran implementation optimized for < 1e-7 absolute accuracy.

Parameters:

x – scalar or array-like input values

Returns:

Airy \(Ai(x)\), with scalar or array shape matching x

camb.mathutils.airy_fast(x)

Fast Airy \(Ai(x)\) and derivative \(Ai'(x)\) approximation.

Uses a fitted Fortran implementation optimized for < 1e-7 absolute accuracy.

Parameters:

x – scalar or array-like input values

Returns:

tuple (ai, aip) for Airy \(Ai(x)\) and \(Ai'(x)\), with scalar or array shapes matching x

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.legendre_polynomials(x, lmax)

Legendre polynomials \(P_\ell(x)\) and derivatives \(dP_\ell/dx\) for all \(0\le \ell \le\) lmax (requires \(|x| < 1\)).

Parameters:

• x – scalar or 1D array of x values

• lmax – maximum \(\ell\)

Returns:

P, dP arrays; shape (lmax+1,) for scalar x, else (len(x), lmax+1)

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.phi_derivative(L, K, nu, chi)

Evaluate d phi_L^nu(K, chi) / d chi using the adjacent-order recurrence.

camb.mathutils.phi_first_peak_amplitude(L, K, nu)

Return abs(phi_recurs(L, K, nu, phi_first_peak_chi(L, K, nu))).

If phi_first_peak_chi(..., return_status=True) reports no peak found, this is the amplitude at the search boundary rather than at a stationary point.

camb.mathutils.phi_first_peak_chi(L, K, nu, return_status=False)

Return the first peak position at or after the hyperspherical Bessel turning point.

If return_status is true, also return whether no stationary peak was found before the search boundary, in which case the returned position is that boundary.

camb.mathutils.phi_olver(L, K, nu, chi)

Evaluate the regular hyperspherical Bessel function \(\phi_L^\nu(K,\chi)\) using the leading-order Olver map to a flat spherical Bessel function.

Fast with peak-relative accuracy around 1e-4; use phi_recurs() for a slower high-accuracy reference. For K=0 this returns the spherical Bessel function \(j_L(\nu\chi)\). Falls back to the recursive result where the Olver approximation may be unreliable.

Parameters:

• L – multipole index

• K – dimensionless curvature sign, one of -1, 0, 1

• nu – dimensionless radial eigenvalue; for closed models (K=1), an integer mode with nu >= 3 and nu > L

• chi – non-negative scalar dimensionless radial distance or 1D array of non-negative values

Returns:

scalar value or 1D array matching chi

camb.mathutils.phi_recurs(L, K, nu, chi)

Evaluate the regular hyperspherical Bessel function \(\phi_L^\nu(K,\chi)\) by recurrence.

Uses upward recurrence in the safe oscillatory region and Miller backward recurrence elsewhere, normalized by the exact low-order solution. The recurrence follows Abbott and Schaefer (1986) for the seed and recurrence formulae. Miller starts use the continued-fraction construction of Tram (2017) and Lesgourgues and Tram (2014) for K=0,-1; for K=1 they use either the finite closed-spectrum endpoint or the closed-space Gegenbauer Miller start.

Parameters:

• L – multipole index

• K – dimensionless curvature sign, one of -1, 0, 1

• nu – dimensionless radial eigenvalue; for closed models (K=1), an integer mode with nu >= 3 and nu > L

• chi – non-negative scalar dimensionless radial distance or 1D array of non-negative values

Returns:

scalar value or 1D array matching chi

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

camb.mathutils.threej_pt(l1, l2, l3, m1, m2, m3)

Convenience testing function to get 3j for specific arguments. Normally use threej to get an array at once for same cost.

Parameters:

• l1 – L_1

• l2 – L_2

• l3 – L_3

• m1 – M_1

• m2 – M_2

• m3 – M_3

Returns:

Wigner 3j (integer zero if outside triangle constraints)