wale.RateFunction module

wale.RateFunction.get_phi_projec_2cell(theta1, theta2, zarr, chis, dchis, w, y, recal, variance, **kwargs)

Compute projected 2-cell φ(y) by solving saddle-point equations.

Parameters:

• theta1 (float) – Angular scales of the two cells.

• theta2 (float) – Angular scales of the two cells.

• zarr (ndarray) – Redshifts at which to evaluate.

• chis (ndarray) – Comoving distances corresponding to redshifts.

• dchis (float) – Integration step size in χ.

• w (ndarray) – Lensing weight function W(χ).

• y (ndarray) – Slope values for SCGF (y-grid).

• recal (float) – Empirical recalibration factor.

• variance (object) – Provides nonlinear_sigma2(χ) interface.

• deld (float, optional) – Step size for finite differences (default: 1e-8).

Returns:

phi_proj – Projected φ(y) values computed via saddle-point approximation.

Return type:

ndarray

wale.RateFunction.get_psi_2cell(variance, chi, recal, z, delta1, delta2, theta1, theta2)

Compute the 2-cell action ψ(δ₁, δ₂) using large-deviation theory.

Parameters:

• variance (object) – Object providing nonlinear_sigma2 method for computing variances.

• chi (float or ndarray) – Comoving distance(s).

• recal (float) – Empirical recalibration factor.

• z (float) – Redshift.

• delta1 (float) – Density contrasts in each cell.

• delta2 (float) – Density contrasts in each cell.

• theta1 (float) – Angular scales of the two cells.

• theta2 (float) – Angular scales of the two cells.

Returns:

psi – Value of the large-deviation action ψ(δ₁, δ₂).

Return type:

float

wale.RateFunction.get_psi_2nd_derivative_delta1(deld, variance, chi, recal, z, delta1, delta2, theta1, theta2)

Compute ∂²ψ/∂δ₁² using second-order central finite differences.

Returns:

second_derivative – Second partial derivative of ψ with respect to δ₁.

Return type:

float

wale.RateFunction.get_psi_2nd_derivative_delta2(deld, variance, chi, recal, z, delta1, delta2, theta1, theta2)

Compute ∂²ψ/∂δ₂² using second-order central finite differences.

Returns:

second_derivative – Second partial derivative of ψ with respect to δ₂.

Return type:

float

wale.RateFunction.get_psi_derivative_delta1(deld, variance, chi, recal, z, delta1, delta2, theta1, theta2)

Compute ∂ψ/∂δ₁ using central finite differences.

Returns:

derivative – Numerical partial derivative of ψ with respect to δ₁.

Return type:

float

wale.RateFunction.get_psi_derivative_delta2(deld, variance, chi, recal, z, delta1, delta2, theta1, theta2)

Compute ∂ψ/∂δ₂ using central finite differences.

Returns:

derivative – Numerical partial derivative of ψ with respect to δ₂.

Return type:

float

wale.RateFunction.get_psi_mixed_derivative_delta1_delta2(deld, variance, chi, recal, z, delta1, delta2, theta1, theta2)

Compute the mixed partial derivative ∂²ψ/∂δ₁∂δ₂ using central differences.

Returns:

mixed_derivative – Mixed second-order partial derivative of ψ.

Return type:

float

wale.RateFunction.get_scaled_cgf(theta1, theta2, zarr, chis, dchis, lensing_weight, y, recal, variance)

Wrapper to compute the scaled cumulant generating function (SCGF).

Parameters:

• theta1 (float) – Angular scales.

• theta2 (float) – Angular scales.

• zarr (ndarray) – Redshift values.

• chis (ndarray) – Comoving distances.

• dchis (float) – Integration step size.

• lensing_weight (ndarray) – Lensing kernel W(χ).

• y (ndarray) – SCGF slope values.

• recal (float) – Recalibration factor.

• variance (object) – Provides nonlinear sigma²(R₁, R₂, z).

Returns:

scgf – The scaled cumulant generating function φ(y).

Return type:

ndarray

wale.RateFunction.get_tau(rho)

Compute the large-deviation theory mapping τ(ρ) for a given density contrast ρ.

Parameters:

rho (float or array_like) – Local density ρ = 1 + δ.

Returns:

tau – The corresponding τ(ρ) value based on the ν parameter.

Return type:

float or ndarray

wale.RateFunction.psi_derivative_determinant(deld, delta1, delta2, z, variance, chi, recal, theta1, theta2)

Compute the determinant of the Hessian matrix of ψ(δ₁, δ₂):

det(H) = ∂²ψ/∂δ₁² * ∂²ψ/∂δ₂² − (∂²ψ/∂δ₁∂δ₂)²

This determinant is used for computing the normalization in saddle-point approximations.

Returns:

result – Determinant of the ψ Hessian matrix.

Return type:

float