Tier-4 forecast: the morphology observables

Wave 4 gave the model a morphology sector (the conditional early-type fraction, the sample split, the BH–bulge coupling — see The extended model: implementation of the missing physics, “Wave 4”). The tier-4 study adds the measurements the literature actually delivers for it, and asks:

When galaxy morphology is measured at Euclid scale — fractions, joint morphology–quenching censuses, sizes, AGN-host demographics, morphology-split lensing and intrinsic alignments — what does it teach the halo model, and does the IA self-calibration pay off for cosmology?

Five parameters join the vector (111 in total, TIER4_MORPHOLOGY), all fiducial-preserving; together with the four wave-4 entries they form the nine-parameter morphology sector.

Reproduce with:

JAX_PLATFORMS=cpu python -m hod_mod.scripts.forecasts.run_tier4_forecast \
    --jobs 6
JAX_PLATFORMS=cpu python -m hod_mod.scripts.forecasts.run_tier4_forecast --smoke

The literature basis 

Every observable is anchored to a measurement programme (all arXiv links API-verified before inclusion — the house rule):

Measurement	Key references
f_early(M*, z) at scale	Euclid Q1 morphology catalogue (378k galaxies, →100M) [EuclidQ1Morph2025]; Zoobot ML morphologies [EuclidZoobot2024]; Galaxy Zoo heritage [Skibba2009]
f_early evolution to z ≳ 3	JWST: CEERS [Kartaltepe2023], [Ferreira2023]; COSMOS-Web Hubble sequence [COSMOSWeb2025Hubble]
Morphology-split lensing	SDSS halo masses vs morphology [Mandelbaum2006]
Morphology–quenching joint census	passive red spirals [Masters2010]; two quenching pathways [Schawinski2014]; causality [Bluck2022]
Sizes ↔ haloes	R_e ≈ 0.015 R_200c [Kravtsov2013]; size–mass evolution [vanderWel2014]
Intrinsic alignments ↔ morphology	KiDS-1000: IA driven by morphology more fundamentally than colour [Georgiou2025]
AGN-host morphology	BH–bulge coevolution [Yang2019BHbulge], [KormendyHo2013]; AGN vs inactive hosts at fixed M* [Banerjee2025]

The morphology sector: all nine parameters 

Parameter	Fiducial	Prior	Wave	Meaning
`log10_M_morph`	12.5	1.0	4	early-type Weibull transition mass (centrals)
`beta_morph`	0.8	0.5	4	Weibull shape (transition sharpness)
`f_morph_sat`	0.2	0.3	4	satellite boost toward early types, $f_{E,s} = f_{E,c} + f_{\rm morph,sat}(1 - f_{E,c})$
`mbh_bt_slope`	0.0	0.5	4	BH–bulge coupling: $\langle\log M_{\rm BH}\rangle \mathrel{+}= {\rm slope}\cdot\log_{10}[f_{E,c}(M_h) + 10^{-4}]$ (0 = the pure Powell chain, exactly)
`rho_morph_q`	0.0	0.3	t4	morphology–quenching correlation at fixed M_h: $f_{E\cap Q} = f_E f_Q + \rho\sqrt{f_E(1-f_E)f_Q(1-f_Q)}$ (ρ = 0 is the wave-4 independence, exactly; the square root carries a 1e-24 floor so the Jacobian stays finite where the Weibulls saturate to exactly 0/1)
`log10_f_size`	−1.824	0.3	t4	log10(R_e/R_200c) — the Kravtsov 0.015 relation
`dsize_early`	−0.20	0.3	t4	early-type size offset at fixed M* [dex]
`f_size_zs`	0.0	2.0	t4	size-ratio evolution (a `_Z_EVOL` pair with `log10_f_size`: $\partial d/\partial{\rm zs} = x_{\rm evol}\, \partial d/\partial{\rm base}$ by the chain rule)
`a_ia`	2.0	1.0	t4	NLA IA amplitude carried by the early types ($A_{\rm eff} = a_{\rm IA}\, f_{\rm early}$)

Fixed constants: _SIG_SIZE = 0.2 dex (the Kravtsov scatter, in forecast/tier4.py); the NLA $C_1\rho_{\rm crit}$ = 0.0134 convention is absorbed into the a_ia normalisation.

Architecture: where everything lives 

Location	Contents
`connection/morphology.py`	`f_early_cen(log10m_h, lg_m_morph, beta_morph)` and `f_early_sat(..., f_morph_sat)` — jitted Weibull fractions (the ZM16 `f_red_*_zu16` pattern), ∈ [0, 1] by construction.
`forecast/forward_jax.py`	`_morph_fractions` (marginal f_E per cen/sat); `_morph_q_joint` (joint f_{E∩Q}, tier 4); `_occ_sample` — the 4-way (early/late × SF/Q) partition uses the joint fractions when BOTH `morph` and `sfq` are set, the marginal weights otherwise; `_occ_sfq` = the pre-morphology sample; `_f_early` (conditional on SF/Q-selected samples), `_f_early_q`, `_size_mean`, `_f_early_agn`, `_wgp`; the `mbh_bt_slope` term inside `_agn_kernel_parts`; slices `WAVE4_MORPHOLOGY = PARAM_NAMES[102:106]`, `TIER4_MORPHOLOGY = PARAM_NAMES[106:]`.
`forecast/tier2.py` (hooks, unchanged behaviour)	`include_morph` flag adds `f_early` to every cell; `_cell_extra_obs` / `_shell_extra_obs` / `_extra_blocks` / `_cell_spectro` / `_block_extras` are the extension points the tier-3/4 subclasses consume.
`forecast/tier4.py`	`Tier4Forecast` — flags `include_morphq` / `include_sizes` / `include_ia` / `include_agn_morph` / `include_morph_split` (all default ON), `z_morph_max = 1.2`; builds the `morph_cell` blocks and the tier-4 noise override.
`forecast/noise.py`	`wgp_noise()` (shape noise per annulus, the ΔΣ geometry without the Σ_crit weight); `SpectroSurvey.fmorph_err` = 0.02, `size_err` = 0.02 dex, `fmorph_agn_err` = 0.05.
`scripts/forecasts/run_tier4_forecast.py`	The driver: tier-3 probe groups + `+f_early(z,M*)`, `+E∩Q joint`, `+sizes`, `+AGN hosts`, `+morph-split wp/dS` (selected by block KIND via the `__kind__morph_cell` sentinel — its rows are named wp/ds/n_gal), `+IA w_g+`; `--jobs`, `--smoke`, `--no-morph-split`, `--no-ia`.
`tests/test_forecast_tier4.py`	The 13 gates (below); wave-4 gates live in `tests/test_missing_physics.py`.

The observables 

Joint fractions and the 4-way split. With both morph and sfq set, the occupation weights are the joint-partition set $\{f_{E\cap Q},\ f_E - f_{E\cap Q},\ f_Q - f_{E\cap Q},\ 1 - f_E - f_Q + f_{E\cap Q}\}$ (× the sSFR-cut survivals), which sums to unity by construction — EARLY+LATE+SF/Q additivity is exact at any ρ. The SF/Q-split cells’ f_early data are conditional fractions: $f_{E|Q} = f_{E\cap Q}/f_Q$ and $f_{E|SF} = (f_E - f_{E\cap Q})/(1 - f_Q)$ — at ρ = 0 both reduce to the marginal, so the existing split grid itself measures ρ (at ρ = 0.2 a z = 0.4 cell separates to f_E|Q ≈ 0.30 vs f_E|SF ≈ 0.06). The per-cell f_early_q datum is the base-sample joint fraction — the red-spiral / blue-elliptical census — attached to ONE variant per (z, M*) cell (the SF one) to avoid duplicating a sample-independent number.

Sizes. Per cell (centrals only — satellite sizes do not follow the host halo radius):

\[\langle\log_{10} R_e\rangle = \log_{10} f_{\rm size} + \langle\log_{10} R_{200c}\rangle_{N_c} + \Delta_{\rm size}^{E}\, f_{\rm early}^{\rm cell},\]

with $R_{200c}$ the comoving r_delta of the halo grid [Mpc/h] (the constant unit offset to physical kpc is absorbed by log10_f_size). Cosmology enters through $R_{200c} \propto (M/\rho_{\rm crit})^{1/3}$.

Intrinsic alignments. Per cell, on the rp_ds grid:

\[w_{g+}(r_p) = a_{\rm IA}\, f_{\rm early}^{\rm cell}\; \frac{0.0134\,\Omega_m}{D(z)}\; \Delta\Sigma\text{-transform}\big[b_g P_{\rm lin}\big],\]

reusing _delta_sigma (the J₂-type transform) verbatim and the CPL-aware growth dispatcher for D(z). Because the amplitude is carried by $f_{\rm early}$, the IA contamination of the tomographic shear block shares parameters with the morphology sector — the forecast quantifies IA self-calibration.

AGN hosts. Per shell: the early fraction among soft L_X = 10⁴²–10⁴⁴ AGN, $f_E^{\rm AGN} = \int dM\, \frac{dn}{dM}\, N_{\rm AGN} f_{E,c} \big/ \int dM\, \frac{dn}{dM}\, N_{\rm AGN}$ with the observed-band occupation _agn_occupation_obs. With mbh_bt_slope > 0 the Powell chain shifts AGN into early-type-rich haloes — the Kocevski-style bulge-dominance trend, measured directly.

Morphology-split clustering + lensing. Early/late (w_p, ΔΣ, n̄_g) blocks per (z, M*) cell — [Mandelbaum2006] at Euclid scale. Kind morph_cell; built only where the WIDE spectroscopic tier is complete (imaging morphology needs depth) and z_hi ≤ z_morph_max = 1.2 (where the shear sources also run out): the production grid yields 104 of the naive 156 blocks.

The exact gates 

Every mechanism carries an invariant enforced by tests/test_forecast_tier4.py (13 tests) or the wave-4 section of tests/test_missing_physics.py:

Invariant	Meaning
`f_EQ(ρ=0) ≡ f_E·f_Q` (rtol 1e-14)	tier 4 collapses to wave-4 independence exactly at the fiducial
∂f_EQ/∂ρ = √(f_E(1−f_E)f_Q(1−f_Q)) analytically	the correlation lever arm is exact
4-way partition sums to the unsplit n_gal at ρ ∈ {0, ±0.x}	unity partition by construction, not by tuning
f_E\|Q > f_E\|SF at ρ > 0	the conditional split carries the ρ signal
0 < f_early_q < f_early < 1, ∂/∂ρ > 0	joint bounded by the marginals
∂⟨log R⟩/∂``log10_f_size`` = 1 exactly; `f_size_zs` chain rule = x_evol; nonzero (Ωm, h) response; sizes grow with the M* bin	the size observable is a calibrated ruler
∂ln w_g+/∂``a_ia`` = 1/a_ia exactly; w_g+(θ₂)/w_g+(θ₁) ≡ f_early(θ₂)/f_early(θ₁) under morphology-parameter changes	strict NLA amplitude and strict morphology-carried scaling
∂f_early_agn/∂``mbh_bt_slope`` > 0 at the fiducial; responds to `log10_M_morph`	the BH–bulge coupling is observable
w_g+ obeys the projected r_p scale cut	bookkeeping
tiny `Tier4Forecast` end-to-end: finite d0/J, positive noise, every family present, `morph_cell` rows ∈ {wp, ds, n_gal}, cache round-trip identical	assembly
EARLY+LATE `morph_cell` n_gal ≡ the unsplit cell total	the split blocks add information, not amplitude
parallel == serial BYTE-identical (fresh-interpreter subprocess; batched pools)	the –jobs machinery is exact

Usage 

Model level:

from hod_mod.forecast.forward_jax import ForwardModel
m = ForwardModel(z_eff=0.4, log10m_star_bin=(10.0, 10.4),
                 morph="early", sfq="q")     # the E∩Q sample
out = m.predict(theta, ["wp", "ds", "n_gal"])
m2 = ForwardModel(z_eff=0.4, log10m_star_bin=(10.0, 10.4))
out2 = m2.predict(theta, ["f_early", "f_early_q", "size",
                          "wgp", "f_early_agn"])

Forecast level:

from hod_mod.forecast.tier4 import Tier4Forecast
t4 = Tier4Forecast()                  # all tier-2/3/4 flags default ON
d0, J, meta = t4.data_and_jacobian(t4.fiducial(), cache_dir=cache)
sigma = t4.noise_sigma(t4.fiducial(), d0, meta)
# morph-split rows: meta["kind"] == "morph_cell"

Noise 

Row	σ model	Completeness
`f_early`, `f_early_q`	binomial √(f(1−f)/N) ⊕ `fmorph_err` = 0.02 floor	wide/deep spectro tier of the cell
`size`	`_SIG_SIZE`/√N ⊕ `size_err` = 0.02 dex floor	idem
`wgp`	`wgp_noise()` — shape noise per annulus (ΔΣ geometry, no Σ_crit weight) ⊕ CV floor	idem
`f_early_agn`	binomial over N_AGN(shell) ⊕ `fmorph_agn_err` = 0.05 floor	flagged where the shell lacks the AGN extras
`morph_cell` wp/ds/n_gal	the standard pair-count / shape-noise / count models	wide tier only, z ≤ 1.2 (blocks not built otherwise)

The w_g+ recipe is effective (no IA–clustering cross-covariance) — the tSZ/IM documentation precedent; the AGN-host floor is larger than the galaxy one because point-source light complicates host classification.

Operational record 

Developed in a git worktree on branches feature/wave4-morphology → feature/tier4-morphology while the tier-3 production ran: the --jobs workers re-import the package source at every batch respawn, so in-place edits during a run are forbidden.
The vector growth invalidates all previous per-block caches (PARAM_NAMES enters _spec_repr) — a tier-4 production is a fresh full run.
Production: 396 blocks (260 SF/Q cells, 104 morph-split, 10 shells, global lensing, 10 AGN, 10 SFRD, hi_local), --jobs 6 batched pools (a fresh executor per chunk of jobs × 4 blocks — CPython 3.11’s max_tasks_per_child respawn deadlocks; batching gives the same per-worker memory bound), ~4.5 h wall on 6 workers / 64 GB.

Results (2026-07-04 run)

396 blocks, 65 431 rows (260 SF/Q cells + 104 morph-split blocks — the wide-tier completeness gate trims the naive 156: at z_hi = 1.2 imaging morphology only survives above M* ≈ 10^{10.2}). At $r_\mathrm{min} = 0.1\,h^{-1}$ Mpc, all 111 free:

Parameter	All 111 free	Astrophysics pinned	Degradation
$\Omega_\mathrm{m}$	$4.89\times10^{-5}$	$3.33\times10^{-5}$	×1.5
$\sigma_8$	$7.23\times10^{-5}$	$5.25\times10^{-5}$	×1.4
$w_0$	$7.23\times10^{-4}$	$4.58\times10^{-4}$	×1.6
$w_a$	$3.35\times10^{-3}$	$2.10\times10^{-3}$	×1.6

Answers to the three headline questions:

The morphology sector is fully measured (84 bits of information; no prior-bound directions): the Weibull transition at the sub-permille level (σ(log10_M_morph) = 5.4×10⁻⁴, σ(beta_morph) = 7.2×10⁻⁴), the satellite boost and joint E∩Q correlation at a few ×10⁻³ (σ(rho_morph_q) = 3.0×10⁻³), and the size relation to 1.9×10⁻³ dex with its early-type offset at 5.0×10⁻⁴ (the log10_f_size– dsize_early pair is the sector’s strongest internal degeneracy, ρ = +0.996).
σ(mbh_bt_slope) = 7.7×10⁻⁴ — the BH–bulge coupling is constrained ~650× beyond its prior by the AGN-host early fractions and the morphology-coupled AGN LFs: the coevolution question becomes decisively testable in this data scenario.
The IA amplitude self-calibrates to σ(a_ia) ≈ 0.010 (0.5% of the fiducial A_IA = 2) — the dominant weak-lensing systematic is pinned by the morphology sector for free, with no cost to the shear cosmology.

On the cosmology side, the new information comes almost entirely from the morph-split w_p/ΔΣ blocks, which tighten $\sigma(\Omega_\mathrm{m})$ by 17% (5.89 → 4.89×10⁻⁵) and $\sigma(\sigma_8)$ by 9% over the tier-3 baseline — the scalar morphology fractions and sizes constrain the sector itself rather than cosmology, exactly as designed. The tier progression now reads $\sigma(\Omega_\mathrm{m}) = 2.85 \to 1.82 \to 0.59 \to 0.49 \times 10^{-4}$ across tier-2(61) → tier-2(90) → tier-3(102) → tier-4(111). The tier-3 conditioning caveat stands ($\Sigma m_\nu$ and the agn-sector determinant hit the prior-scaled eigenvalue floor at this dynamic range). SUMMARY, npz and the tier4_forecast__* figure suite live in $HOD_MOD_RESULTS/tier4_forecast/.

Continuation points 

Open items, in rough priority order, with their entry points:

Eigenvalue-floor fix — fisher.constraints drops prior-scaled eigenvalues below rcond·λ_max; at tier-3/4 dynamic range (λ_max > 10¹⁰ in prior units) this sweeps up prior-bounded directions (Σm_ν → ∞, agn-sector slogdet → NaN at r_min = 0.1/0.5). Principled fix: in prior-scaled space every eigenvalue is ≥ ~1 by construction, so FLOOR at 1 instead of dropping. One function (forecast/fisher.py::constraints); changes all tiers’ reported σ slightly — rerun the SUMMARYs from the cached npz (no model recomputation needed: the npz stores d0/J).
Morphology evolution log10_M_morph_zs — one _Z_EVOL pair + a prior; the JWST f_early(z) anchors ([Kartaltepe2023], [COSMOSWeb2025Hubble]) are the data. Trivial with the existing mechanism; adds the 112th parameter.
Satellites in the morphology-coupled kernels — _size_mean, _f_early_agn and the AGN chain are centrals-only (the Powell convention). A satellite term needs a subhalo size/BH prescription.
Assembly-bias hook — order B/T at fixed M_h by formation time via the cosmology-dependent c(M) (cm_relation="diemer15" provides the proxy); the falsifiable Wechsler–Tinker link, roadmap item 3.
IA refinements — luminosity scaling ($A_{\rm IA} \propto (L/L_0)^{\beta}$), a blue-galaxy tidal-torque term, and the IA–shear cross-covariance in the noise model (currently the effective shape-noise recipe).
Morphology → SED coupling — early types carry higher M/L: a dml_early offset in the tier-3 NIR/optical calibrations would tie the two sectors (one parameter, _lf_lognormal weights).
Data-side — the Euclid Q1 catalogue [EuclidQ1Morph2025] is public: a real f_early(M*, z) measurement against a Q1-matched selection would ground the fiducials before the next production.

API reference 

Conditional galaxy morphology — the early-type fraction of the halo.

Missing-physics wave 4 (docs/missing_physics.rst, “Galaxy morphology”): a Weibull conditional early-type fraction mirroring the Zu & Mandelbaum (2016/2017) halo-quenching pattern of hod_mod.connection.hod.zumandelbaum15.f_red_cen_zu16(),

\[f_{\rm early,c}(M_h) = 1 - \exp\!\left[-\left(\frac{M_h}{M_{\rm morph}} \right)^{\beta_{\rm morph}}\right],\]

with a satellite boost toward early types (environmental transformation),

\[f_{\rm early,s}(M_h) = f_{\rm early,c}(M_h) + f_{\rm morph,sat}\,[1 - f_{\rm early,c}(M_h)] .\]

Both are ∈ [0, 1] by construction for $f_{\rm morph,sat} \in [0, 1]$, and EARLY + LATE sums exactly to the unsplit occupation (the SF/Q-split invariant). The mean $f_{\rm early,c}$ also serves as the bulge-to- total proxy that couples morphology to the black-hole sector ($M_{\rm BH} \propto (B/T\,M_*)$-like; Yang et al. 2019) inside the forecast’s Powell chain.

hod_mod.connection.morphology.f_early_cen(log10m_h: Array, lg_m_morph: float, beta_morph: float) → Array[source]

Early-type fraction of central galaxies (Weibull in halo mass).

Parameters:

log10m_h (log10(M_h / [M_sun/h]))
lg_m_morph (log10(M_morph / [M_sun/h]), morphological transition mass)
beta_morph (Weibull shape (transition sharpness))
Output ∈ [0, 1] by construction; → 1 for M_h ≫ M_morph.

hod_mod.connection.morphology.f_early_sat(log10m_h: Array, lg_m_morph: float, beta_morph: float, f_morph_sat: float) → Array[source]: Early-type fraction of satellite galaxies: the central Weibull with an environmental boost toward early types, f_s = f_c + f_morph_sat (1 − f_c) — ∈ [0, 1] for f_morph_sat ∈ [0, 1], and identical to the central fraction at f_morph_sat = 0.

Tier-4 forecast assembly: the morphology observables of the literature.

Extends Tier3Forecast with the measurements that pin the wave-4 morphology sector (see docs/tier4_forecast.rst for the verified literature basis):

f_early_q per cell — the joint early∩quenched fraction (Galaxy Zoo red spirals / blue ellipticals census) measuring the rho_morph_q morphology–quenching correlation;
size per cell — the mean ⟨log10 R_e⟩ of the centrals through the Kravtsov R_e ≈ 0.015 R_200c relation (+ the early-type offset), weighing cosmology through R_200c ∝ (M/ρ_crit)^{1/3};
wgp per cell — the NLA galaxy–intrinsic-alignment cross with the amplitude carried by the early-type fraction (KiDS/DESI: IA is driven by morphology) — the shear IA systematic becomes self-calibrated;
f_early_agn per shell — the bulge-dominance of X-ray AGN hosts, the direct probe of the mbh_bt_slope BH–bulge coupling;
morph_cell blocks — early/late-split w_p, ΔΣ and n̄_g per (z, M*) cell to z ≤ 1.2 (the Mandelbaum-2006 morphology–halo-mass measurement at Euclid scale; wide spectroscopic tier only — morphology needs imaging depth).

class hod_mod.forecast.tier4.Tier4Forecast(include_morphq=True, include_sizes=True, include_ia=True, include_agn_morph=True, include_morph_split=True, z_morph_max=1.2, **kw)[source]

Bases: Tier3Forecast

Tier-4 morphology forecast (see module docstring).

Beyond Tier3Forecast (all of whose flags default ON here):

Parameters:

include_morphq, include_sizes, include_ia, include_agn_morph (bool) – The per-cell f_early_q / size / w_g+ data and the per-shell AGN-host early fraction.
include_morph_split (bool) – Early/late-split (wp, ds, n_gal) blocks per cell up to z_morph_max (wide-tier cells only).
z_morph_max (float) – Morphological-classification depth of the imaging survey (Euclid VIS-like) — also where the shear sources run out for the split ΔΣ.

noise_sigma(fid, d0, meta, verbose=True)[source]: Tier-2 noise for the inherited rows, then the tier-3 families.

hod_mod.forecast.noise.wgp_noise(rp, wgp, z_l, ngal, volume, h, Om, shear: ShearSurvey, spectro: SpectroSurvey)[source]

Absolute σ on w_g+(r_p) [Mpc/h]: shape noise per annulus + CV floor.

The delta_sigma_noise geometry WITHOUT the Σ_crit lensing weight — the intrinsic-alignment estimator correlates density tracers with the SHAPES of the same (or an overlapping) sample, so the noise per r_p bin is σ_e/√(n_shape·A_ann·N_dens) projected over 2π_max. An effective recipe (no IA–clustering cross-covariance), the tSZ/IM documentation precedent.

Parameter	All 111 free	Astrophysics pinned	Degradation
\(\Omega_\mathrm{m}\)	\(4.89\times10^{-5}\)	\(3.33\times10^{-5}\)	×1.5
\(\sigma_8\)	\(7.23\times10^{-5}\)	\(5.25\times10^{-5}\)	×1.4
\(w_0\)	\(7.23\times10^{-4}\)	\(4.58\times10^{-4}\)	×1.6
\(w_a\)	\(3.35\times10^{-3}\)	\(2.10\times10^{-3}\)	×1.6