M21 CIB HOD Model

• The M21 CIB HOD model is a halo-model framework that connects dark matter halos with dusty star-forming galaxies using a baryonic-accretion-driven star-formation prescription and fixed effective SEDs.
• It employs a one-halo/two-halo decomposition with a lognormal star-formation efficiency, distinguishing centrals from satellites while incorporating luminosity-weighted emissivity in clustering calculations.
• The model robustly fits CIB power spectra yet shows biases in recovering intrinsic parameters, underscoring limitations in using linear approximations for nonlinear clustering effects.

The M21 CIB HOD model is a halo-model framework for cosmic infrared background anisotropies in which dusty star-forming galaxies are connected to dark-matter halos through a baryonic-accretion-driven star-formation prescription and an effective infrared spectral energy distribution. The designation “M21” refers to the model introduced by Maniyar, Béthermin, and Lagache in 2021 and subsequently used as a compact, physically motivated description of the relation between halo growth, star formation, and clustered CIB emission. In the form examined in recent analyses, it combines a standard one-halo/two-halo decomposition with a lognormal star-formation efficiency $\eta(M_h,z)$, explicit treatment of centrals and satellites, and effective SED templates fixed from external calibrations rather than fit as free graybody parameters (Gkogkou et al., 22 Sep 2025).

1. Model definition and conceptual role

The model assumes that all galaxies live in dark matter halos, with the halo clustering signal decomposed into one-halo terms from galaxy pairs within the same halo and two-halo terms from pairs in different halos. Centrals reside at halo centers, satellites inhabit subhalos, and the total clustered CIB anisotropy is obtained by projecting the three-dimensional emissivity power spectrum along the line of sight with the Limber approximation (Gkogkou et al., 22 Sep 2025).

Within the CIB literature, M21 is commonly contrasted with two other approaches. The “S12” family parameterizes a graybody SED and an empirical halo-mass/redshift dependence of infrared luminosity or star formation rate, whereas M21 links the star-formation rate directly to the baryon accretion rate into halos and fixes the SED from externally calibrated templates. The “Y23” hybrid combines M21’s BAR-driven star-formation prescription with S12’s graybody SED parameterization, with the explicit aim of jointly constraining dust SED and star formation (Yan et al., 2023).

A central structural feature of M21 is that emissivity weighting enters the clustering calculation itself, not only the mean CIB window function. In later methodological comparisons, this luminosity-weighted treatment was identified as a key distinction between M21 and simplified number-weighted halo prescriptions in which emissivity fluctuations are taken to trace galaxy-number fluctuations directly (Reischke et al., 2019).

2. Halo growth, star-formation efficiency, and satellite treatment

The core physical assumption is that the star-formation rate is proportional to the baryonic accretion rate onto halos,

$$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$

with

$$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$

The mean halo mass-growth rate is taken from Fakhouri et al. (2010),

$$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$

This construction ties galaxy growth to halo growth through cosmological accretion rather than through a purely empirical luminosity–mass mapping (Gkogkou et al., 22 Sep 2025).

The efficiency function is lognormal in halo mass,

$$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$

and, in the four-parameter form tested against simulated CIB maps, the width evolves as

$$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$

with $z_c$ treated as a pivot redshift fixed as in M21 in that analysis. This parameterization regulates efficiency at both low and high halo masses and yields a peak near $M_h \simeq 10^{12}$–$10^{13}\,M_\odot$, which was described as consistent with the literature (Gkogkou et al., 22 Sep 2025).

Satellite star formation is quenched by taking the minimum of two prescriptions: direct evaluation of the same efficiency law at the subhalo mass, and host-scaled suppression of the form

$$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$

This introduces an explicit environmental quenching rule for satellites while preserving the central role of the BAR$$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$0 mapping (Yan et al., 2023).

In the unWISE–Planck cross-correlation implementation, the model was extended to allow redshift evolution of the peak mass and separate HODs for infrared galaxies and the unWISE galaxy sample. There the peak mass was written as

$$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$1

and $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$2 was left free rather than fixed, with a reparameterization introduced to ensure $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$3 (Yan et al., 2023).

3. Emissivity, effective SEDs, and angular power spectra

M21 converts star-formation rate to infrared luminosity through a Kennicutt-like relation. For a Chabrier IMF, the conversion is

$$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$4

or equivalently $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$5 (Yan et al., 2023).

The frequency dependence is not fitted with free dust parameters inside the basic M21 model. Instead, the model adopts an effective SED $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$6 at each redshift, computed by averaging templates from the Magdis/Bethermin library weighted by their contribution to the infrared luminosity density. In later implementations, the same effective SEDs were combined with the appropriate Planck or Herschel bandpasses to model observed-frame selection and K-corrections (Gkogkou et al., 22 Sep 2025).

The comoving emissivity is built by integrating central and satellite contributions over the halo mass function. In the notation used for the SIDES-Uchuu comparison,

$$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$7

The one-halo emissivity power uses central–satellite and satellite–satellite pair terms with the NFW Fourier profile $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$8, while the two-halo term assumes a scale-independent halo bias $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$9 and a linear matter power spectrum $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$0 (Gkogkou et al., 22 Sep 2025).

The clustered angular CIB power spectrum is then obtained through Limber projection,

$$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$1

A flat shot-noise contribution is added as a nuisance parameter rather than predicted directly from the HOD,

$$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$2

In the SIDES-Uchuu analysis, the auto-power shot-noise levels were varied independently, and the cross shot-noise was specified through a correlation matrix measured from the simulation (Gkogkou et al., 22 Sep 2025).

4. Parameterization and observational implementations

In the four-parameter efficiency sector used to test parameter recovery, the principal physical parameters are $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$3, $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$4, $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$5, and $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$6, with $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$7 fixed as in M21. The example priors reported for that study were uniform in $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$8 over $$\mathrm{BAR}(M_h,z)=\left\langle \dot{M}(M_h,z)\right\rangle \frac{\Omega_b}{\Omega_m}.$$9, uniform in $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$0 over $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$1, uniform in $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$2 over $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$3, and uniform in $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$4 over $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$5. Shot-noise amplitudes were also varied for the four Planck/HFI bands $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$6, $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$7, $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$8, and $$\left\langle \dot{M}(M_h,z)\right\rangle = 46.1\,\mathrm{M_\odot\,yr^{-1}} \left(\frac{M_h}{10^{12}\,\mathrm{M_\odot}}\right)^{1.1} \left(1+1.11\,z\right) \sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}.$$9 GHz (Gkogkou et al., 22 Sep 2025).

Parameter	Meaning
$$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$0	halo mass at peak efficiency
$$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$1	peak efficiency normalization
$$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$2	lognormal width at the pivot
$$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$3	strength of width evolution

For mock-data fitting, the model was applied to CIB auto- and cross-power spectra at the four Planck/HFI bands, using eight $$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$4-bins and the full Gaussian covariance, together with external emission priors from Herschel/SPIRE mean CIB intensities at $$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$5, $$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$6, and $$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$7 and star-formation-rate density as a function of redshift. Inference was performed with emcee, and the likelihood combined the power-spectrum covariance with priors from the CIB monopole and SFRD (Gkogkou et al., 22 Sep 2025).

In the unWISE–Planck cross-correlation study, the same framework was embedded in a broader tomographic analysis. Three unWISE galaxy bins centered at $$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$8, $$\eta(M_h,z)=\eta_{\max}\, \exp\!\left[ -\frac{\big(\ln M_h-\ln M_{\max}\big)^2}{2\,\sigma_{M_h}^2(z)} \right],$$9, and $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$0 were cross-correlated with Planck CIB maps at $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$1, $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$2, and $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$3 GHz over $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$4. The measured pseudo-$$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$5 were corrected for mode mixing with NaMaster, Planck beam and pixel windows were included, photometric color corrections were applied with $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$6, $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$7, and $$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$8, and cosmic magnification of the galaxy maps was modeled through

$$\sigma_{M_h}(z)=\sigma_{M_h,0}-\tau\,\max\!\big(0,\,z-z_c\big),$$9

In that implementation, cross shot noise $z_c$0 was taken to be scale-independent and fitted per frequency–galaxy pair with $z_c$1 free in the prior range $z_c$2 in units of $z_c$3 (Yan et al., 2023).

5. Empirical constraints from cross-correlations and mock-data fits

The tomographic unWISE–Planck analysis reported a $z_c$4 detection from the stacked nine cross-spectra and found that M21, S12, and Y23 gave comparable fit quality. For M21 specifically, the reported fit was $z_c$5 with $z_c$6. The inferred M21 parameters were

$z_c$7

$z_c$8

$z_c$9

These values imply $M_h \simeq 10^{12}$0 with no significant evolution, and a broad efficient halo-mass range at high redshift that narrows somewhat at $M_h \simeq 10^{12}$1 (Yan et al., 2023).

The same analysis also fit separate HODs for infrared galaxies and the unWISE tracer population. For the IR HOD, the quoted constraints were

$M_h \simeq 10^{12}$2

while for unWISE galaxies they were

$M_h \simeq 10^{12}$3

$M_h \simeq 10^{12}$4

The reported interpretation was that the minimum mass for centrals increases with redshift, and that the satellite mass scale $M_h \simeq 10^{12}$5 is weakly constrained because of the Planck beams (Yan et al., 2023).

The SIDES-Uchuu mock-data study reached a different conclusion about parameter recovery. When M21 was fit to the realistic mock, the best-fit parameters were

$M_h \simeq 10^{12}$6

$M_h \simeq 10^{12}$7

with shot-noise amplitudes

$M_h \simeq 10^{12}$8

$M_h \simeq 10^{12}$9

and a global $10^{13}\,M_\odot$0 for $10^{13}\,M_\odot$1 $10^{13}\,M_\odot$2 points (Gkogkou et al., 22 Sep 2025).

6. Recovery tests, degeneracies, and present limitations

The most important recent result is that good spectral fits do not guarantee recovery of intrinsic physical parameters. In the SIDES-Uchuu validation study, M21 matched the mock SFRD in shape and amplitude within $10^{13}\,M_\odot$3–$10^{13}\,M_\odot$4 across redshift and reproduced the differential emissivity within $10^{13}\,M_\odot$5, yet still failed to recover intrinsic parameters, most notably the peak halo mass $10^{13}\,M_\odot$6. The model captured the broad shape of $10^{13}\,M_\odot$7 but systematically overestimated $10^{13}\,M_\odot$8 at $10^{13}\,M_\odot$9, the redshift range dominating the CIB flux (Gkogkou et al., 22 Sep 2025).

The mismatch was traced primarily to the cosmology-side ingredients of the two-halo term rather than to the emission prescription alone. A direct analytic comparison with a simplified simulation built to match the HOD assumptions showed $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$0 agreement in SFRD and differential emissivity, but $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$1 scale-dependent offsets in the two-halo term, increasingly large at higher $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$2 and at $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$3. The reported interpretation was that the use of a linear matter power spectrum and a scale-independent halo bias cannot reproduce the fully nonlinear matter clustering and effective scale-dependent halo bias present in SIDES-Uchuu, especially across the one- to two-halo transition (Gkogkou et al., 22 Sep 2025).

A second limitation is one-halo–shot-noise degeneracy when satellite contributions are important. In the simplified SSU tests, the case with subhalos and $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$4 yielded recovered parameters that deviated by $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$5 from the input values. Removing subhalos improved recovery, with $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$6, $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$7, and $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$8 within $$\mathrm{SFR}_{\mathrm{s}}(m|M)= \min\left\{\mathrm{SFR}(m),\,\frac{m}{M}\times\mathrm{SFR}(M)\right\}.$$9 of the truth, but $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$00 remained offset at $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$01 versus $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$02. Setting $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$03 did not remove the offsets (Gkogkou et al., 22 Sep 2025).

Scatter affects shot noise much more strongly than clustered power. In the SSU analysis at $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$04 GHz, a $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$05 dex lognormal scatter in SFR at fixed halo mass shifted shot noise by $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$06, a $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$07 dex scatter in the mean radiation field $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$08 shifted shot noise by $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$09, and combining both exceeded $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$10 relative to the no-scatter case. By contrast, the clustered component changed by $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$11 for SFR scatter and by $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$12 for SED scatter. The stated explanation was that shot noise scales with flux-squared moments, whereas clustered power depends on mass-weighted emissivity integrals and therefore averages over scatter more efficiently (Gkogkou et al., 22 Sep 2025).

The recommended upgrades follow directly from these diagnostics. The two-halo term was proposed to be generalized from $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$13 to $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$14, and from scale-independent $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$15 to scale-dependent $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$16, with halo exclusion added to suppress small-scale two-halo power. Additional recommended refinements were corrected subhalo-mass-function parameters, more realistic satellite profiles and quenching, self-consistent treatment of SFR scatter, and richer redshift-dependent or distributed effective SED modeling (Gkogkou et al., 22 Sep 2025).

These results support a narrow interpretation of what M21 presently constrains. The model can fit CIB power spectra and external emission priors and can reproduce emission-related quantities robustly, but clustered CIB data alone do not ensure unbiased recovery of $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$17 and especially $$\mathrm{SFR}(M_h,z)=\eta(M_h,z)\,\mathrm{BAR}(M_h,z),$$18 when the two-halo sector retains linear matter clustering and scale-independent bias. A plausible implication is that the principal utility of the current M21 framework is as an efficient emissivity-based phenomenology whose physical inferences must be stabilized by improved cosmological ingredients and complementary external priors (Gkogkou et al., 22 Sep 2025).