echo-IA

1.1 Simulation boxes

1.1.1 Projected correlation functions

Measure position–shape alignments along a chosen line-of-sight (often also get clustering for free).

• Estimators: position–position: (w_{gg}, \tilde{\xi}{gg,0}); position–shape: (w{g+}, \tilde{\xi}_{g+,2}).
• Inputs: ((x,y,z)) positions for both position and shape samples; projected shapes (major-axis direction from projected inertia-tensor eigenvector, axis ratio (b/a)); projection axis (x/y/z) as line-of-sight; box size (units must match positions); jackknife regions for covariance: use (x^3) sub-boxes with (x\in\mathbb{Z}), require (x^3 \gtrsim N_r^{3/2}) and sub-box size (\gg) max (r) or (r_p).
• Codes: MeasureIA, Halotools.
• Figures/conventions: log–log; optionally plot (r_p\,w_{g+}) or (r^2\,\tilde{\xi}_{g+,2}).

1.1.2 3D correlation functions

• Estimators: (\xi_{g+}(r)), (\xi_{++}(r)), (\xi_{gg}(r)).
• Inputs: full 3D positions; 3D shape tensors/major axes; box size; binning in (r).
• Codes: Halotools, Corrfunc-style custom pair-counting, MeasureIA modules.
• Figures/conventions: (\xi(r)) vs (r) (log–log); highlight 1–2 halo-like features if relevant.

1.1.3 Power spectra

• Estimators: 3D Fourier modes; (P_{g+}(k)), (P_{++}(k)).
• Inputs: density field on grid; shape field on grid; window handling.
• Codes: custom FFT pipelines, Pylians; use CCL for theory comparison.
• Figures/conventions: (P_{g+}(k)), (P_{++}(k)) vs (k).

1.1.4 Misalignment angles

• Estimators: angle between galaxy/halo major axis and LSS/tidal directions.
• Inputs: inertia-tensor eigenvectors; filament/sheet directions or tidal field.
• Codes: custom routines; simulation-toolkit alignment modules.
• Figures/conventions: histograms of (\cos\theta); compare to randoms.

1.1.5 Three-point (3pt) correlation functions

• Estimators: bispectra (B) or 3pt (\zeta) with shape legs (e.g., (gg+), (g++)).
• Inputs: positions, shapes, triangle configs or (k)-triangle bins.
• Codes: custom 3pt estimators; fast bispectrum libraries.
• Figures/conventions: (B(k_1,k_2,k_3)) slices; reduced statistics where helpful.

1.2.1 Projected correlation functions

• Estimators: (w_{g+}, w_{gg}) in RA/Dec/(z).
• Inputs: RA, Dec, (z); galaxy shapes; survey masks/weights; random catalogs.
• Codes: TreeCorr.
• Figures/conventions: as in sims, but mask-aware errors (JK or mocks).

1.2.2 Power spectra

• Estimators: pseudo-(C_\ell) (mask-aware).
• Inputs: shear/ellipticity maps, masks, weights; (\ell)-binning.
• Codes: NaMaster (PyMaster).
• Figures/conventions: (C_\ell^{g+}, C_\ell^{++}); E/B decomposition; deconvolved bandpowers.

1.2.3 Misalignment angles (optional)

• Estimators: alignments between galaxy PAs and density-field directions.
• Inputs: galaxy PAs from shapes; density tracers (photo-(z)/spec-(z)).
• Codes: custom catalog routines; cross-matching scripts.
• Figures/conventions: angle distributions; alignment vs redshift.

2.1 Angular Power Spectra (C_\ell)

Goal: Predict IA (C_\ell) for cosmic shear and galaxy samples using pyCCL.

• Estimators: (C_\ell^{II}, C_\ell^{GI}, C_\ell^{GG}); cross: (C_\ell^{gI}, C_\ell^{gG}).
• Inputs:
  • Cosmology: ccl.Cosmology(Ω_m, σ_8, h, n_s, …)
  • Redshift distributions: 1D (n(z)) for sources/lenses on chosen (z)-grid
  • Multipoles: array of (\ell)
  • IA model: NLA / z-NLA / LF-z-NLA / TATT

    # NLA-type example (sketch)
    source = ccl.WeakLensingTracer(cosmo, dndz=(z, nz_s), IA_bias=(z, IA))
    cls = ccl.angular_cl(cosmo, source, source, ells)
    

  • Galaxy bias: (b(z))

    lens = ccl.NumberCountsTracer(cosmo, dndz=(z, nz_l), bias=(z, b))
    

• Real-space 2pt (example):

  xi_plus  = ccl.correlation(cosmo, ell=ells, C_ell=cls,
                             theta=theta_deg, type="GG+", method="FFTLog")
  xi_minus = ccl.correlation(cosmo, ell=ells, C_ell=cls,
                             theta=theta_deg, type="GG-", method="FFTLog")
  

• Figures/conventions: plot (C_\ell) or (\xi_\pm(\theta)) on log–log; optionally split IA vs lensing or show fractional IA.

2.3 Advanced models

• EFT-inspired IA models
• Tidal-field alignment models
• Different tracer populations (halos, centrals, satellites)