Luminosity Function of Galaxies in Voids: A Modification Inspired by Excursion Set Theory

Shant Baghram; Zeynab Ashurikisomi

REVIEW 3 major objections 5 minor 50 references

Luminosity Function of Galaxies in Voids: A Modification Inspired by Excursion Set Theory

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Schechter parameters of void galaxy luminosity functions follow power laws in the excursion-set two-barrier ratio, and this parameterization fits 2dFGRS data better than the previous density-based model.

desk verdict A modest but real empirical improvement to void luminosity function modeling, undermined by an unresolved linear/nonlinear density-coordinate ambiguity in the D parameter. read the letter →

arxiv 2507.01626 v2 pith:RC6TMRXO submitted 2025-07-02 astro-ph.CO

Zeynab Ashurikisomi , Shant Baghram This is my paper · ORCID

classification astro-ph.CO

keywords galaxyluminosityfunctioncosmicvoidsexcursionsettheorySchechterenvironmentdependenceredshiftevolutiongrowth2dFGRS

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the galaxy luminosity function in cosmic voids is controlled by the excursion-set two-barrier ratio $D = |\delta_v|/(\delta_c+|\delta_v|)$, not merely by the local density contrast. Writing the Schechter parameters as power laws in $(1+D)$ and fitting to 2dFGRS data in three underdense environments gives $\chi^2 = 20.8$, compared with 22.2 for the previous model with $D$ substituted for the density and 34.4 for the original density-based model, and the AIC/BIC ranking follows the same order. The paper further claims that the redshift evolution of the void luminosity function enters through the linear growth function $D(z)$ as $\Phi(M,z)\propto \exp(-\eta/D(z)^2)/D(z)$, with $\eta$ fitted to GAMA void data. If these claims hold, the luminosity function becomes a direct observable link between galaxy properties in voids and the physics of halo and void formation, and a possible cosmological probe.

What carries the argument

The load-bearing object is the ratio $D\equiv|\delta_v|/(\delta_c+|\delta_v|)$, which measures how close the void-formation barrier $\delta_v\approx -2.81$ is to the halo-collapse barrier $\delta_c\approx 1.686$; in excursion set theory this ratio controls which voids survive being swallowed by collapsing halos. The paper makes $D$ the environmental variable in the Schechter function, so the three Schechter parameters become power laws in $(1+D)$. The second mechanism is the linear growth function $D(z)$, imported through the halo mass function, which turns the redshift dependence of the luminosity function into $\Phi(M,z)\propto\exp(-\eta/D(z)^2)/D(z)$ and pushes all of that dependence into $\phi_*$.

What would settle it

Use Eq. (2) to convert the nonlinear contrasts $\delta_v=-0.90,-0.75,-0.43$ to linear values, recompute $D$, refit Eq. (14) to the same 2dFGRS data, and compare $\chi^2$, AIC, and BIC; if the advantage over the density-based model disappears, the central environmental claim is not supported.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the environment dependence of the void galaxy luminosity function is not an arbitrary empirical trend but follows from the distance between the two formation barriers in excursion set theory. The claimed result is $\phi_* = \zeta_1(1+D)^{\zeta_2}$, $M_* = \zeta_3(1+D)^{\zeta_4}$, $\alpha = \zeta_5(1+D)$, where $D\equiv|\delta_v|/(\delta_c+|\delta_v|)$ and the $\zeta_i$ are fitted to 2dFGRS; this form outperforms the earlier density-based Schechter model by $\chi^2$, AIC, and BIC. For redshift, the claim is that the halo mass function conveys the evolution through the growth factor, giving $\Phi(M,z)\propto \exp(-\eta/D(z)^2)/D(z)$ and concentrating the redshift dependence in $\phi_*$, with $\eta = 0.068^{+0.017}_{-0.018}$ from GAMA void luminosity functions.

Load-bearing premise

The load-bearing premise is that the observed nonlinear void density contrasts in Table 2 can be identified directly with the linear barrier $\delta_v$ when forming $D=|\delta_v|/(\delta_c+|\delta_v|)$; if the linear-to-nonlinear mapping must be applied first, the fitted D values and the claimed model superiority lose their stated physical meaning.

Editorial extensions

If this is right

Void luminosity functions in sparse, populous, and underdense regions can be described by one shared parameterization in the two-barrier ratio rather than fitted separately.
The redshift dependence of the void luminosity function is concentrated in $\phi_*$, so measuring $\phi_*$ in redshift bins is a direct route to the growth function and hence to the cosmological model.
The reported AIC/BIC improvement implies that in underdense regions the two-barrier ratio is a more informative environmental coordinate than the Eulerian density contrast.
Neglecting satellite galaxies is a safe simplification for voids, so the model connects cleanly to the halo mass function there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same $D$-parameterization could be tested in larger void samples from current spectroscopic surveys to see whether the fitted exponents $\zeta_2$ and $\zeta_4$ are universal or survey-dependent; this is an extension, not a paper claim.
Because the paper never applies its own linear-to-nonlinear mapping to the Table 2 density contrasts, the numerical value of $D$ feeding the fit is ambiguous; a recalibrated fit might shift the exponents even if the qualitative ranking of models survives.
If the redshift formula holds, the luminosity function of voids could be used as a growth-rate probe, complementing clustering-based growth measurements; this direction is implicit in the paper rather than demonstrated there.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper gives a new, compact parameterization of the void galaxy luminosity function in terms of the EST two-barrier ratio D, and it genuinely outperforms the McNaught-Roberts model on the same 2dFGRS data with the same number of free parameters. That part holds up. The weak spot is that the D values used in the fit are built from nonlinear density contrasts while the EST barriers are linear, so the theoretical interpretation is shaky.

What is actually new: the D-dependence of the Schechter parameters (Eq. 14) and the exp(-eta/D(z)^2)/D(z) redshift form (Eq. 15) are not in the cited McNaught-Roberts model. The authors also show that simply swapping delta_m for D in the old model cuts chi^2 by 35%, which is a nice consistency check. The AIC/BIC comparison is fair: same data, same number of free parameters. They use real survey data (2dFGRS, GAMA) and report error bars on the fits.

The soft spots, in proportion: first, the D ambiguity. Table 2 lists the environment density contrasts as nonlinear (Eulerian) values, delta_v = -0.90, -0.75, -0.43, while the EST two-barrier ratio is derived from linear thresholds, delta_v ~ -2.81 and delta_c ~ 1.686. The paper gives the Lagrangian-to-Eulerian mapping in Eq. 2 but never applies it. If you plug the nonlinear values directly into D, you are not computing the quantity EST motivates. If you instead use the constant linear threshold, D stops being environment-dependent and the model has no environmental lever at all. This is not just bookkeeping; it determines what the fitted exponents zeta2, zeta4, zeta5 mean physically. The empirical fit may still be better than the alternatives, but the theoretical story needs to be repaired, either by converting to linear contrasts or by reframing the model as a purely empirical fitting form.

Second, the redshift model is a fit, not a prediction. eta is a free parameter fitted to the same GAMA data that Fig. 4 compares against, so the agreement is a fitting result. The authors are honest about the faint-end deviation, but the word 'theory' is doing too much work here. This is a minor issue if the model is presented as an empirical fitting formula; it is a larger issue if the goal is to use this as a cosmological probe.

Third, no code or data release, which makes it harder to check the fitting procedure. That is a smaller point.

Bottom line: this is a subfield paper with limited scope, but the empirical claim is real and the data comparison is fair. The density-coordinate issue is the main thing standing between a conditional acceptance and a clean one. I'd send it to a competent referee focused on large-scale structure and voids.

Recommendation: accept with major revision, or conditional accept, pending the authors either applying the linear-to-nonlinear mapping or explicitly recasting Eq. 14 as an empirical fit with no EST-based physical interpretation.

Referee Report

3 major / 5 minor

Summary. The paper proposes an excursion-set-inspired parameterization of the Schechter luminosity function for galaxies in voids, with the Schechter parameters written as functions of the two-barrier ratio D = |δv|/(δc+|δv|) and, separately, a redshift dependence of the form Φ(M,z) ∝ exp(−η/D(z)^2)/D(z), where D(z) is the linear growth function. The environmental model is fitted to 2dFGRS luminosity functions in three underdense environments and compared against the McNaught-Roberts et al. (2014a) model and a modified version of it, reporting lower χ², AIC, and BIC. The redshift model is fitted to GAMA void luminosity functions in three redshift bins. The central claim is that EST-motivated environmental and redshift dependencies provide a better description of the void galaxy luminosity function than previous empirical parameterizations.

Significance. If the environmental model holds, it offers a compact, EST-inspired description of void galaxy luminosity functions and a modest improvement over an existing empirical model using the same data and the same number of free parameters; the AIC/BIC comparison in Fig. 2 is therefore a fair model-selection exercise. The redshift part is more exploratory: the GAMA data provide only one void density interval, as the paper acknowledges, and the single fitted parameter η makes Fig. 4 a demonstration of a functional form rather than an out-of-sample prediction. The paper ships no code or machine-checked derivations; its value rests on the clarity and correctness of the physical mapping between the fitted density contrasts and the EST barrier ratio, which is currently unresolved.

major comments (3)

[§3.1, Eq. (14), Table 2, Eq. (2)] The parameter D = |δv|/(δc+|δv|) is introduced in §2.1 as an EST two-barrier ratio built from the linear thresholds δv ≈ −2.81 and δc ≈ 1.686, but Table 2 and Fig. 1 label the fitted environments by nonlinear (Eulerian) density contrasts δv = −0.90, −0.75, −0.43. The manuscript never applies the Lagrangian-to-Eulerian mapping of Eq. (2) to these values before forming D. If the nonlinear values are inserted directly, D is not the EST barrier ratio and the fitted exponents ζ2, ζ4, ζ5 in Table 3 lose their stated theoretical meaning; if one instead uses the constant linear threshold δv = −2.81, D becomes environment-independent and Eq. (14) has no environmental dependence at all. Please report the D values actually used, state explicitly whether δv is linear or nonlinear, and if the values in Table 2 are nonlinear, convert each one with Eq. (2) and refit; the χ², AIC, and BIC results in Fig. 2 must be rechecked under the corrected D.
[§3.2, Eq. (15), Fig. 4] The redshift model is presented as a test of Eq. (15), but η is a free parameter fitted to the same GAMA void luminosity functions shown in Fig. 4; the agreement is therefore a fitting result, not an independent validation. The derivation of Eq. (15) from Eq. (5) also relies on unstated assumptions, namely that δc(z) − δv(z) scales as 1/D(z) and that the relevant variance combination is redshift-independent. The paper itself acknowledges a deviation at the faint end. Please provide a quantitative comparison of Eq. (15) against, for example, a no-evolution model or the redshift-dependent parameterization used by Loveday et al. (2012), and state explicitly whether the model is intended as a description or as a predictive test until independent data are used.
[§2.2–§2.3, Eqs. (5), (7), (9), (14)] The halo-in-void mass function in Eq. (5) is introduced as the theoretical basis, but it is not used in the fits: Eq. (9) employs the global Sheth-Tormen mass function n(M), and the environmental dependence in Eq. (14) enters only through the ad hoc replacement of the density contrast by D. The mapping from Eq. (5) to the specific functional forms in Eq. (14) is not derived. Either derive Eq. (14) from the halo-in-void mass function under stated approximations, or clearly present Eq. (14) as an empirical fitting function whose parameters are merely inspired by EST; the current text claims an EST grounding that the equations do not actually provide.

minor comments (5)

[§3.1–§3.2, Eqs. (14) and (15)] The symbol D denotes two different quantities: the barrier ratio D ≡ |δv|/(δc+|δv|) in Eq. (14) and the linear growth function D(z) in Eq. (15). This is confusing in §3.2, where both appear nearby; please rename one of them, for example using B for the barrier ratio.
[§2.1, Eq. (1)] Eq. (1) defines n(M,z) as the cumulative number density of halos with mass greater than M, but Eqs. (7) and (9) treat n(M) as a differential mass function. Please clarify the notation and use dn/dM where appropriate.
[§3.1, Fig. 2] The table and figure block around Fig. 2 repeats the AIC/BIC values and contains a stray “Model 2” label; please fix the typesetting so the reader can unambiguously identify which row corresponds to the new model, the McNaught-modified model, and the original McNaught-Roberts model.
[§3.2, Fig. 4] The caption of Fig. 4 states the void selection as −1 ≤ δv ≤ −0.75, but Table 2 lists discrete values −0.90, −0.75, and −0.43; please clarify how the redshift-binned GAMA sample maps onto the environment definitions used in the 2dFGRS analysis.
[References] The references to McNaught-Roberts et al. (2014a) and (2014b) appear to cite the same MNRAS volume and page; please verify that the two citations are indeed distinct works and cite them correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the models are explicit fits to the data, and the model-comparison statistics are genuine same-data comparisons.

full rationale

The paper does not claim an independent prediction that reduces by construction to a fitted input. The environmental model of Eq. 14 uses five free parameters zeta1 through zeta5, all explicitly fitted to the 2dFGRS data, and Figure 1 is described as 'the result of fitting our model (equation 14) to this data.' The AIC, BIC, and chi-squared comparisons against the McNaught-Roberts and McNaught-modified models are legitimate same-data model-selection exercises with equal numbers of free parameters; they test functional forms, not circularly enforced outcomes. The redshift model of Eq. 15 contains a free parameter eta that is fitted to the GAMA data, and the paper honestly describes Figure 4 as 'our model of redshift-dependence (equation 15) to GAMA data points,' so the agreement is a fit, not a claimed prediction. The EST background in Eqs. 1-5 provides motivation for the functional form D = |delta_v|/(delta_c + |delta_v|), but the coefficients are not derived from EST, so the theory is not used to force the final parameter values. Self-citations such as Kameli and Baghram (2022, 2025) and Parkavousi et al. (2023) are background references and are not load-bearing for the central fitting claims. The main caveat is a physical-interpretation inconsistency: Table 2 labels delta_v as the nonlinear density contrast while D is motivated from linear EST barriers, and Eq. 2 is not applied before forming D. That ambiguity affects the physical meaning of the fitted exponents but is not a circularity, because the model would still be the same empirical fit under either interpretation. No step reduces by definition or by self-citation to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard halo mass function, the two-barrier excursion set picture, and a series of fits: beta, zeta1 through zeta5, and eta are all fitted to the same observational datasets they are used to describe. The paper contributes a functional form, not a parameter-free prediction.

free parameters (3)

zeta1 through zeta5 = -0.661 +/- 0.032, -19.547 +/- 0.074, -0.081 +/- 0.010, 0.0149 +/- 0.0012, -3.61 +/- 0.24
Five coefficients in Eq. 14 fitted to 2dFGRS void luminosity function data; they set the normalization, break magnitude, and faint-end slope of the Schechter function.
beta = 0.000406, 0.000561, 0.001361, 0.004450 for the four environments in Table 2
Normalization fitted per environment to absorb the environmental dependence of the luminosity-mass relation in Eq. 9.
eta = 0.068 +0.017 -0.018
Coefficient in the redshift dependence model Eq. 15, fitted to GAMA void luminosity function data across three redshift bins.

assumptions (5)

standard math The Sheth-Tormen mass function gives the number density of dark matter halos.
Used in Eq. 7 and Eq. 9 to convert halo abundance into the galaxy luminosity function. Cited from Sheth and Tormen 2002.
domain assumption Spherical collapse threshold delta_c = 1.686 and void shell-crossing threshold delta_v = -2.81 are the correct linear density contrasts.
Used in the definition of D in Eq. 14 and in the void distribution. Standard excursion set values but assumed throughout.
domain assumption All galaxies in voids can be treated as central galaxies with a delta-function conditional luminosity function.
Basis for Eq. 9, justified by the small satellite fraction in voids according to Habouzit et al. 2020 and Rosas-Guevara et al. 2022.
domain assumption Halo abundance matching gives a monotonic, scatter-free luminosity-mass relation (Eq. 11 with Vale and Ostriker parameters).
Used to map halo mass to galaxy luminosity; ignores the roughly 0.15 dex scatter mentioned in the text.
ad hoc to paper Redshift dependence of the halo-in-void mass function enters only through the linear growth function D(z), giving Phi(M,z) proportional to exp(-eta/D(z)^2)/D(z).
Eq. 15 is presented as a consequence of excursion set theory, but eta is fitted to the same data rather than computed from theory.

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Pith. "Pith review of Luminosity Function of Galaxies in Voids: A Modification Inspired by Excursion Set Theory." pith.science (2026). https://pith.science/paper/RC6TMRXO

@misc{pith2026250701626,
  author       = {Pith},
  title        = {Pith review of: Luminosity Function of Galaxies in Voids: A Modification Inspired by Excursion Set Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RC6TMRXO}},
  note         = {Machine review of arXiv:2507.01626}
}

abstract

In the standard picture of cosmology, the galaxies reside in dark matter (DM) halos. DM halos are distributed in the cosmic web in different environments. The luminosity of the galaxies in different environments can be used as a probe to assess a cosmological model. This study focuses on the properties of galaxies in void regions, where halos typically do not experience extreme conditions. By examining the galaxy luminosity function, we aim to understand the dependence of galaxy properties on their environment and redshift so that later, we can use this as a tool to evaluate cosmological models. We employ the excursion set theory to incorporate parameters related to the number density of DM halos into the luminosity function. Using the Galaxy and Mass Assembly (GAMA) survey and 2dFGRS datasets, we fit our theoretical models to observational data, examining the environmental and redshift dependence of the galaxy luminosity function. Our results indicate that we model the galaxy luminosity function in voids effectively by considering the linear density contrast of the environment and the growth function $D(z)$ for redshift dependence. This study provides a model for the environmental dependence of galaxy luminosity function that offers an improvement in the $\chi ^2$ parameter compared to the previously proposed model in \cite{mcnaught2014galaxy}. Both Bayesian information criterion (BIC) and Akaike information criterion (AIC) tests support the superiority of this model for the void region.

Figures

Figures reproduced from arXiv: 2507.01626 by the authors.

**Figure 2.** AIC and BIC test results comparing our environmental dependence model of the galaxy luminosity function, McNaught-Roberts-modified, and McNaught-Roberts models using 2dFGRS data and ADAM optimizer Kingma (2014). The results show an apparent difference in the compatibility of the models to the data [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 1.** Galaxy luminosity function as a function of magnitude for three different environments labeled by their density contrast. All of the environments are in the under dense region so we can relate the galaxy luminosity function to the critical density of the host void. The nonlinear (Eulerian) value of the density contrast is mentioned in the figure. The data points are from 2dfGRS (Croton et al. 2005). ple, McNaught-… view at source ↗

**Figure 3.** The confidence level of free parameters of Schechter function defined in equation 14. Dark and light blue are 68% and 95% confidence regions. in which D ≡ |δv| δc+|δv| , where δc is the critical density for halo formation and δv is the threshold of the formation of void (as mentioned in Section 2.1). Additionally, ζ1 to ζ5 are free parameters and their best-fit values (calculated using 2dFGRS data) are mentioned i… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Galaxy luminosity function as a function of rband magnitude for three different redshift intervals inside the void region (−1 ≤ δv ≤ −0.75). The lines are produced by fitting our model of redshift-dependence (equation 15) to GAMA data points. The data points are take…

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