Environment-dependent IMF validation - progenax — differentiable initial conditions for N-body simulations

The cluster birth environment ( $\rho_{\rm cl}$ , $[\mathrm{Fe/H}]$ , $M_{\rm ecl}$ , SFE) sets the IMF slopes via the Marks et al. (2012) Fundamental Plane and the Jeřábková et al. (2018) IGIMF, exposed through BirthEnvironment, env_to_imf_params, alpha3_marks_plane, and lowmass_slopes_metallicity. Test files: tests/validation/test_environment_physics.py (vs published tables) and tests/unit/imf/test_environment.py (see the test dashboard for the live per-suite counts); figures: scripts/validate_environment.py. Theory at Environment-dependent IMFs.

Key result: the predicted top-heaviness matches named globular clusters

The Fundamental Plane reproduces the published high-mass slope $\alpha_3$ for the four Marks et al. (2012) Table 1 globular clusters within the table’s own $\sim0.15$ scatter: NGC 7078 (M15) is correctly the most top-heavy ( $\alpha_3=0.84$ vs published 0.76), NGC 104 the least (1.50 vs 1.34). The top-heaviness is driven by density, not metallicity — a 1-dex change in $\rho_{\rm cl}$ moves $\alpha_3$ by 0.40 versus only 0.06 for a 1-dex change in $[\mathrm{Fe/H}]$ (a $7{:}1$ ratio, matching the Fundamental-Plane $\sin\theta/\cos\theta$ ). The metallicity-dependent low-mass slopes reproduce Table 4 exactly, and every relation is differentiable for gradient-based inference.

What is verified¶

Rows map to test_environment_physics.py; Measured values are regenerated by scripts/validate_environment.py.

Property	Tolerance (as tested)	Measured	Anchor
$\alpha_3$ (NGC 104) Fundamental Plane	$\|\Delta\| < 0.20$	1.500 (pub 1.34)	Marks et al. (2012) Table 1
$\alpha_3$ (NGC 6341)	$\|\Delta\| < 0.20$	1.076 (pub 1.11)	Marks et al. (2012) Table 1
$\alpha_3$ (NGC 6752)	$\|\Delta\| < 0.20$	1.244 (pub 1.27)	Marks et al. (2012) Table 1
$\alpha_3$ (NGC 7078 / M15)	$\|\Delta\| < 0.20$	0.844 (pub 0.76)	Marks et al. (2012) Table 1
NGC 7078 is the most top-heavy	min $\alpha_3$ of the four	True (0.844 is min)	M15 starburst birth
Density dominates metallicity	$\|\partial_{\log\rho}\| > 3\,\|\partial_{\rm[Fe/H]}\|$	0.40 vs 0.06 (ratio 7.1)	Fundamental-Plane $\sin/\cos$
Low-mass slopes $\alpha_1,\alpha_2$ ([Fe/H])	$\|\Delta\| < 0.02$ (5 anchors)	0.0000 (exact)	Marks et al. (2012) Table 4, Eq. 12
Erratum-corrected Marks $\equiv$ Jeřábková	max $\|\Delta\alpha_3\| < 0.05$	0.008 (rounding only)	same -0.87 relation (2014 erratum)
$\alpha_3$ differentiability (AD vs FD)	rel err $< 10^{-3}$	$9\times10^{-8}$	$\partial\alpha_3/\partial$ ([Fe/H], $\log\rho$ )

Figures¶

Generated by scripts/validate_environment.py (PASS/FAIL per panel; PNG + PDF vector).

Globular-cluster anchors (headline). Predicted \alpha_3 from the Fundamental
Plane vs the published Table 1 values for four GCs, all within the
table’s \pm0.20 intrinsic scatter (shaded). NGC 7078 / M15 (vermilion) is correctly
the most top-heavy; NGC 104 sits at the band edge (|\Delta|=0.16, the only one
near the scatter limit). — Figure 1:**Globular-cluster anchors (headline).** Predicted $\alpha_3$ from the Fundamental Plane vs the published Marks *et al.* (2012) Table 1 values for four GCs, all within the table’s $\pm0.20$ intrinsic scatter (shaded). NGC 7078 / M15 (vermilion) is correctly the most top-heavy; NGC 104 sits at the band edge ( $|\Delta|=0.16$ , the only one near the scatter limit).

The Fundamental Plane (erratum-corrected). \alpha_3 over
(\log\rho_{\rm cl},\,[\mathrm{Fe/H}]) with the four GCs overplotted, using the
2014-erratum threshold \hat x \ge -0.87 (continuous; the originally printed +0.87
was a typo — see ). The near-vertical contours show \alpha_3 responds
chiefly to density: a 1-dex change in \rho_{\rm cl} shifts \alpha_3 by 0.40
versus 0.06 for [\mathrm{Fe/H}] — the \sim7{:}1 ratio that places dense,
metal-poor M15 deep in the top-heavy regime, while the diffuse low-density field
(\log\rho_{\rm cl}/10^6 \lesssim -0.9, left edge) smoothly recovers the canonical
\alpha_3 = 2.3. — Figure 2:**The Fundamental Plane (erratum-corrected).** $\alpha_3$ over $(\log\rho_{\rm cl},\,[\mathrm{Fe/H}])$ with the four GCs overplotted, using the 2014-erratum threshold $\hat x \ge -0.87$ (continuous; the originally printed +0.87 was a typo — see Figure 4). The near-vertical contours show $\alpha_3$ responds chiefly to **density**: a 1-dex change in $\rho_{\rm cl}$ shifts $\alpha_3$ by 0.40 versus 0.06 for $[\mathrm{Fe/H}]$ — the $\sim7{:}1$ ratio that places dense, metal-poor M15 deep in the top-heavy regime, while the diffuse low-density field ( $\log\rho_{\rm cl}/10^6 \lesssim -0.9$ , left edge) smoothly recovers the canonical $\alpha_3 = 2.3$ .

Metallicity-dependent low-mass slopes. \alpha_1 (0.08–0.5\,M_\odot) and
\alpha_2 (0.5–1\,M_\odot) vs [\mathrm{Fe/H}] ( Eq. 12),
passing exactly through the five Table 4 anchor points (markers). Metal-poor clusters
are bottom-light (shallower low-mass slopes); both slopes reach the canonical 2.3
near and above solar. — Figure 3:**Metallicity-dependent low-mass slopes.** $\alpha_1$ (0.08– $0.5\,M_\odot$ ) and $\alpha_2$ (0.5– $1\,M_\odot$ ) vs $[\mathrm{Fe/H}]$ (Marks *et al.* (2012) Eq. 12), passing exactly through the five Table 4 anchor points (markers). Metal-poor clusters are bottom-light (shallower low-mass slopes); both slopes reach the canonical 2.3 near and above solar.

The corrected Marks plane is the Jeřábková IGIMF relation. With the 2014
erratum applied (threshold \hat x \ge -0.87), the Fundamental
Plane (solid lines) and the density-based IGIMF (circles)
coincide for every [\mathrm{Fe/H}] — max |\Delta\alpha_3| = 0.008, purely the
-0.4072-vs--0.41 rounding (Jeřábková Eq. 6 simply adopts the erratum-corrected
relation). The dotted grey curve shows the literally printed Marks+2012 Eq. 14
with its missing-minus-sign typo (\hat x \ge +0.87): it spuriously pins
\alpha_3 = 2.3 out to \hat x = +0.87 and then drops discontinuously to 1.58 —
the artifact the erratum (and Marks+2012 Fig. 6) corrects. — Figure 4:**The corrected Marks plane is the Jeřábková IGIMF relation.** With the 2014 erratum applied (threshold $\hat x \ge -0.87$ ), the Marks *et al.* (2012) Fundamental Plane (solid lines) and the Jeřábková *et al.* (2018) density-based IGIMF (circles) **coincide** for every $[\mathrm{Fe/H}]$ — max $|\Delta\alpha_3| = 0.008$ , purely the -0.4072-vs--0.41 rounding (Jeřábková Eq. 6 simply adopts the erratum-corrected relation). The dotted grey curve shows the **literally printed Marks+2012 Eq. 14** with its missing-minus-sign typo ( $\hat x \ge +0.87$ ): it spuriously pins $\alpha_3 = 2.3$ out to $\hat x = +0.87$ and then drops discontinuously to 1.58 — the artifact the erratum (and Marks+2012 Fig. 6) corrects.

Gradient validation. Autodiff (line) vs central finite difference (circles) for
(a) \partial\alpha_3/\partial[\mathrm{Fe/H}] and (b) \partial\alpha_3/\partial\log\rho
(smoothed-threshold mapping), agreeing to max rel err 9\times10^{-8} — so the birth
environment (\rho_{\rm cl}, [\mathrm{Fe/H}], M_{\rm ecl}) can be inferred by
gradient descent / HMC through the IMF. — Figure 5:**Gradient validation.** Autodiff (line) vs central finite difference (circles) for (a) $\partial\alpha_3/\partial[\mathrm{Fe/H}]$ and (b) $\partial\alpha_3/\partial\log\rho$ (smoothed-threshold mapping), agreeing to max rel err $9\times10^{-8}$ — so the birth environment ( $\rho_{\rm cl}$ , $[\mathrm{Fe/H}]$ , $M_{\rm ecl}$ ) can be inferred by gradient descent / HMC through the IMF.

How to run¶

# published-table physics tests (< 1 s)
pytest tests/validation/test_environment_physics.py -q

# regenerate the five figures with PASS/FAIL tables
python scripts/validate_environment.py

Honest scope¶

The GC $\alpha_3$ tolerance (0.20) is the intrinsic scatter of Marks et al. (2012) Table 1 about its own best-fit plane; NGC 104 sits near that limit ( $|\Delta|=0.16$ ).
alpha3_marks_plane applies the 2014 erratum (Marks et al. 2014, MNRAS 442, 3315; PDF held): the threshold is $\hat x \ge -0.87$ , not the +0.87 printed in the 2012 Eq. 14/15 (a missing-minus-sign typo the authors did not use). With this correction the Marks plane and the Jeřábková density relation are the same relation (Figure 4, $\Delta \le 0.008$ ); the GC anchors are unaffected because all four GCs lie at high density where +0.87 and -0.87 give identical $\alpha_3$ .
The turbulence diagnostics on BirthEnvironment (turbulent_mach, sigma_ln_rho, spectral_slope) are not inputs to $\alpha_3$ here and are not validated on this page.

References¶

Marks et al. (2012) (Fundamental Plane, Tables 1 & 4), Marks & Kroupa (2012) ( $r_h$ – $M_{\rm ecl}$ relation), Jeřábková et al. (2018) (IGIMF). Theory and the full coefficient derivations at Environment-dependent IMFs; per-paper notes in the bibliography.

References¶

Marks, M., Kroupa, P., Dabringhausen, J., & Pawlowski, M. S. (2012). Evidence for top-heavy stellar initial mass functions with increasing density and decreasing metallicity. Monthly Notices of the Royal Astronomical Society, 422, 2246–2254. 10.1111/j.1365-2966.2012.20767.x
Jeřábková, T., Kroupa, P., Dabringhausen, J., Hilker, M., & Bekki, K. (2018). Impact of metallicity and star formation rate on the time-dependent, galaxy-wide stellar initial mass function. Astronomy and Astrophysics, 620, A39. 10.1051/0004-6361/201833055
Marks, M., & Kroupa, P. (2012). Inverse dynamical population synthesis. Constraining the initial conditions of young stellar clusters by studying their binary populations. Astronomy and Astrophysics, 543, A8. 10.1051/0004-6361/201118231