sys_mapping.power_spectrum

Harmonic-space systematic mitigation via pseudo-\(C_\ell\) estimators.

Inputs: Full-sky HEALPix overdensity map, survey mask, template maps, and estimated contamination amplitudes.

Outputs: Pseudo-\(C_\ell\) arrays (shape (lmax+1,)), bias corrections, and mode-projected power spectra.

The pipeline is:

  1. measure_pseudo_cl() — apply mask and call healpy.anafast.

  2. subtract_template_cl() — subtract \(\sum_i \hat\alpha_i \hat C_\ell^{t_i}\).

  3. harmonic_bias() — compute the residual bias \(b_\ell = -n/(2\ell+1)\).

  4. mode_projection_bias() — apply BMP or EMP deprojection.

Key papers: Elsner et al. 2016; Elsner et al. 2017; Leistedt & Peiris 2014 — see also Methods Reference.

Harmonic-space systematic mitigation via pseudo-Cℓ estimators.

Implements the template subtraction, harmonic bias correction, and Basic/Extended Mode Projection (BMP/EMP) methods described in Elsner, Leistedt & Peiris 2016 (MNRAS 456, 2095) and Elsner, Leistedt & Peiris 2017 (MNRAS 465, 1847).

sys_mapping.power_spectrum.measure_pseudo_cl(delta_g, mask, lmax=None, *, use_pixel_weights=True)[source]

Measure the pseudo-Cℓ power spectrum of a galaxy overdensity map.

Applies the survey mask, then calls healpy.anafast to compute the pseudo-Cℓ. The mask is applied multiplicatively so that pixels outside the footprint do not contribute.

Parameters:

  • delta_g (ndarray) – Full-sky HEALPix overdensity map (shape (n_pix,)). Pixels outside the survey should be set to 0 (or masked by mask).

  • mask (ndarray) – Boolean or float weight map (shape (n_pix,)). True / 1.0 marks good (included) pixels.

  • lmax (int | None) – Maximum multipole. Defaults to 3 * nside - 1.

  • use_pixel_weights (bool) – If True, pass use_pixel_weights=True to healpy.anafast for improved accuracy at high ℓ.

Returns:

  • ell – Integer multipole array [0, 1, …, lmax].

  • pseudo_cl – Pseudo-Cℓ estimates (shape (lmax+1,)), not corrected for mask mode-coupling.

Return type:

tuple[ndarray, ndarray]

Notes

The pseudo-Cℓ is related to the true power spectrum via the mode-coupling matrix \(M_{\ell\ell'}\) (MASTER equation):

\[\langle\tilde C_\ell\rangle = \sum_{\ell'} M_{\ell\ell'}\,C_{\ell'}\]

For a clean field without systematics, a simple correction by the mask power (mean-square of the mask) provides a first-order approximation.

References

Elsner, Leistedt & Peiris 2016, MNRAS 456, 2095. Ho et al. 2012, ApJ 761, 14.

sys_mapping.power_spectrum.subtract_template_cl(pseudo_cl, delta_t, mask, alpha, lmax=None)[source]

Subtract template pseudo-Cℓ contributions from the measured power spectrum.

Implements the template subtraction (TS) method of Elsner+2016 (Eq. 1–2):

\[\tilde C_\ell^{\rm TS} = \hat C_\ell^{d\times d} - \sum_i \hat\alpha_i\,\hat C_\ell^{t_i \times t_i}\]
Parameters:

  • pseudo_cl (ndarray) – Measured pseudo-Cℓ of the galaxy field (shape (lmax+1,)).

  • delta_t (ndarray) – Template maps (shape (n_sys, n_pix)).

  • mask (ndarray) – Survey mask (shape (n_pix,)).

  • alpha (ndarray) – Contamination amplitude estimates (shape (n_sys,)).

  • lmax (int | None) – Maximum multipole. If None, inferred from len(pseudo_cl) - 1.

Returns:

Template-subtracted pseudo-Cℓ (shape (lmax+1,)).

Return type:

cl_cleaned

Notes

The residual bias after template subtraction is

\[b_\ell = -\frac{n_{\rm sys}}{2\ell+1}\]

Use harmonic_bias() to compute and subtract this term.

References

Elsner, Leistedt & Peiris 2016, MNRAS 456, 2095.

sys_mapping.power_spectrum.harmonic_bias(n_templates, ell)[source]

Closed-form additive bias from template subtraction.

Template subtraction introduces a systematic negative bias in each multipole band (Elsner+2016, Eq. 8):

\[b_\ell = -\frac{n_{\rm templates}}{2\ell+1}\]
Parameters:

  • n_templates (int) – Number of subtracted templates.

  • ell (ndarray) – Multipole array (integer, shape (n_ell,)).

Returns:

Additive bias on \(C_\ell\) (shape (n_ell,)). Subtract this from the measured \(C_\ell\) to obtain an unbiased estimate.

Return type:

bias

References

Elsner, Leistedt & Peiris 2016, MNRAS 456, 2095, Eq. 8.

sys_mapping.power_spectrum.mode_projection_bias(pseudo_cl, coupling_matrix, ell, n_templates, *, mode='basic', threshold=0.1)[source]

Apply Basic or Extended Mode Projection (BMP/EMP) with bias correction.

BMP marginalises over template modes by modifying the pixel covariance; the result is equivalent to projecting the templates from the data vector before computing the power spectrum. EMP extends this by only projecting modes whose template-to-signal ratio exceeds a threshold.

Parameters:

  • pseudo_cl (ndarray) – Measured pseudo-Cℓ of the galaxy field (shape (n_ell,)).

  • coupling_matrix (ndarray | None) – Mode-coupling matrix \(M_{\ell\ell'}\) (shape (n_ell, n_ell)). If None, the identity matrix is used (full-sky approximation).

  • ell (ndarray) – Multipole array (shape (n_ell,)).

  • n_templates (int) – Number of templates projected.

  • mode (str) – "basic" for BMP or "extended" for EMP.

  • threshold (float) – EMP-only. Modes with template-to-signal ratio below this threshold are not projected. Typical values: 0.05–0.20.

Returns:

  • cl_deproj – Deprojected power spectrum estimate (shape (n_ell,)).

  • bias_correction – Additive bias that has been subtracted from pseudo_cl to obtain cl_deproj (shape (n_ell,)).

Return type:

tuple[ndarray, ndarray]

Notes

BMP bias per multipole (Elsner+2016):

\[b_\ell^{\rm BMP} = -\frac{n_{\rm templates}}{2\ell+1}\]

This is the same as the template subtraction bias — the advantage of BMP over TS is that the amplitude \(\hat\alpha_i\) does not need to be estimated externally.

EMP applies the same correction only to modes where \(\hat C_\ell^{t_i} / C_\ell^{ss} > \text{threshold}\). Modes below the threshold are left uncorrected (smaller variance penalty).

When coupling_matrix is None (identity), the result is a full-sky approximation. For cut-sky surveys use the mask coupling matrix from NaMaster (pymaster.compute_coupling_matrix).

References

Elsner, Leistedt & Peiris 2016, MNRAS 456, 2095, Sec. 3. Leistedt & Peiris 2014, MNRAS 444, 2.