Binary-aware IMF validation - progenax — differentiable initial conditions for N-body simulations

Binary-aware IMF recovery validation¶

Two validated layers. The orbital mechanics (progenax.binaries) are covered by tests/validation/test_binary_physics.py (Kepler’s laws, periapsis/apoapsis, energy, momentum, differentiability; see the test dashboard for the live per-suite count). The binary-aware IMF recovery — the headline — is regenerated by scripts/validate_binaries.py using a fast differentiable MLE of the marginalised likelihood (no MCMC), so it runs in seconds and prints PASS/FAIL. The full Bayesian (numpyro NUTS) cross-check remains offline at validation/imf/validate_binary_aware_recovery.py. Theory at Binary-aware IMF recovery; likelihood derivation at Binary-aware likelihood.

What is verified¶

Orbital rows map to test_binary_physics.py; recovery rows are regenerated by scripts/validate_binaries.py (Maschberger $\alpha=2.30$ , Moe+17 binaries, $f_{\rm bin}\simeq0.27$ ; binary-aware MLE + observed-Fisher $1\sigma$ ).

Property	Tolerance (as tested)	Measured	Anchor
Kepler III: $T = 2\pi\sqrt{a^3/GM}$	rel $< 10^{-9}$	0 (machine)	Kepler’s third law
$a \leftrightarrow T$ roundtrip	$< 10^{-9}$	0 (machine)	inverse consistency
Orbit geometry $r_{\rm peri}=a(1-e)$ , $r_{\rm apo}=a(1+e)$	$< 10^{-6}$	exact (1.000, 3.000)	Kepler’s equation
Orbital energy $E=-Gm_1m_2/2a$	rel $< 10^{-3}$	$4.2\times10^{-16}$	vis-viva / energy
Moe+17 $q$ sampler vs implemented PDF (KS)	$D < D_{\rm crit}=0.0048$	0.0021 / 0.0019	Moe & Di Stefano (2017) Table 13
Mass-dependent twin fraction ( $q>0.95$ )	decreases with $M_1$	$15.2\%$ ( $1\,M_\odot$ ) $\to 6.3\%$ ( $10\,M_\odot$ )	Moe+17 twin excess
Naive fit “confidently wrong” at $N=10^5$	$\|\hat\alpha-\alpha\|/\sigma \gg 3$	$\hat\alpha=2.21$ , $17.8\sigma$	biased single-star model
Binary-aware recovery at $N=10^5$	within $2.5\sigma$ of truth	$\hat\alpha=2.298$ ( $0.4\sigma$ )	marginalised likelihood
Bias vanishes at $f_b=0$	$\|{\rm bias}\|<0.03$ , monotonic in $f_b$	-0.006 ( $f_b{=}0$ ) $\to -0.106$ ( $f_b{=}0.7$ )	bias is binary contamination
Recovery-likelihood differentiability (AD vs FD)	rel err $< 10^{-3}$	$8\times10^{-6}$	$\partial\,$ NLL $_{\rm aware}/\partial\alpha$

Figures¶

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

The “confidently wrong” result. Recovered slope \hat\alpha vs sample size for a
naive single-star fit (vermilion) and the binary-aware marginalised likelihood (blue),
\pm1\sigma bands. Both CIs shrink as 1/\sqrt{N}, but the naive estimate shrinks
around the wrong value — 17.8\sigma from the truth at N=10^5 — while the
binary-aware estimate tracks the true \alpha=2.30 (dashed) to within 1.3\sigma at
every N. — Figure 1:**The “confidently wrong” result.** Recovered slope $\hat\alpha$ vs sample size for a naive single-star fit (vermilion) and the binary-aware marginalised likelihood (blue), $\pm1\sigma$ bands. Both CIs shrink as $1/\sqrt{N}$ , but the naive estimate shrinks around the *wrong* value — $17.8\sigma$ from the truth at $N=10^5$ — while the binary-aware estimate tracks the true $\alpha=2.30$ (dashed) to within $1.3\sigma$ at every $N$ .

Why, and the cure’s reach. (a) The “wrongness significance” |\hat\alpha-\alpha|/\sigma
grows \propto\sqrt{N} for the naive fit (the bias is fixed while \sigma shrinks),
crossing 3\sigma near N\sim3000; the binary-aware fit stays at \mathcal{O}(1).
(b) The naive bias scales with binary fraction f_b and extrapolates to zero at
f_b=0 — confirming the bias is binary contamination, not a sampling artefact. — Figure 2:**Why, and the cure’s reach.** (a) The “wrongness significance” $|\hat\alpha-\alpha|/\sigma$ grows $\propto\sqrt{N}$ for the naive fit (the bias is fixed while $\sigma$ shrinks), crossing $3\sigma$ near $N\sim3000$ ; the binary-aware fit stays at $\mathcal{O}(1)$ . (b) The naive bias scales with binary fraction $f_b$ and extrapolates to **zero at $f_b=0$** — confirming the bias is binary contamination, not a sampling artefact.

Orbital-mechanics oracles. (a) Kepler’s third law T=2\pi\sqrt{a^3/GM} over five
decades in a (max rel error 0, machine precision). (b) A bound e=0.5 orbit with
its focus at the binary COM; periapsis a(1-e)=1 and apoapsis a(1+e)=3 recovered
exactly. (c) The resolved two-body energy equals -Gm_1m_2/2a to 4\times10^{-16}. — Figure 3:**Orbital-mechanics oracles.** (a) Kepler’s third law $T=2\pi\sqrt{a^3/GM}$ over five decades in $a$ (max rel error 0, machine precision). (b) A bound $e=0.5$ orbit with its focus at the binary COM; periapsis $a(1-e)=1$ and apoapsis $a(1+e)=3$ recovered exactly. (c) The resolved two-body energy equals $-Gm_1m_2/2a$ to $4\times10^{-16}$ .

Moe & Di Stefano (2017) mass-ratio distribution. Sampled q=m_2/m_1 (filled)
overlays the implemented g(q\,|\,M_1) (solid) for (a) a 1\,M_\odot and (b) a
10\,M_\odot primary; KS D<D_{\rm crit} in both. The excess at q\to1 (the “twin”
fraction) drops from 15.2\% to 6.3\% with primary mass — the mass-dependent twin
excess that the period-averaged Moe+17 model encodes. — Figure 4:**Moe & Di Stefano (2017) mass-ratio distribution.** Sampled $q=m_2/m_1$ (filled) overlays the implemented $g(q\,|\,M_1)$ (solid) for (a) a $1\,M_\odot$ and (b) a $10\,M_\odot$ primary; KS $D<D_{\rm crit}$ in both. The excess at $q\to1$ (the “twin” fraction) drops from $15.2\%$ to $6.3\%$ with primary mass — the mass-dependent twin excess that the period-averaged Moe+17 model encodes.

Gradient validation. Autodiff (line) vs central finite difference (circles) for
(a) the Kepler transform \partial|v|^2/\partial a (rel err 8\times10^{-9}) and (b)
the binary-aware recovery likelihood \partial\,NLL/\partial\alpha (rel err
8\times10^{-6}) — so the binary-aware slope can be inferred by gradient descent / HMC. — Figure 5:**Gradient validation.** Autodiff (line) vs central finite difference (circles) for (a) the Kepler transform $\partial|v|^2/\partial a$ (rel err $8\times10^{-9}$ ) and (b) the binary-aware recovery likelihood $\partial\,$ NLL $/\partial\alpha$ (rel err $8\times10^{-6}$ ) — so the binary-aware slope can be inferred by gradient descent / HMC.

How to run¶

# orbital-mechanics physics tests (~3 s on CPU)
pytest tests/validation/test_binary_physics.py -q

# regenerate the five figures with PASS/FAIL tables (seconds; MLE, no MCMC)
python scripts/validate_binaries.py

# optional full Bayesian (numpyro NUTS) cross-check (offline; minutes-hours)
python validation/imf/validate_binary_aware_recovery.py

Honest scope¶

The headline figure uses a maximum-likelihood point estimate with an observed-Fisher $1\sigma$ ; the offline numpyro script provides the full posterior (credible intervals, $\hat R$ , ESS) and reaches the same conclusion.
The recovery fixes the Moe+17 binary statistics ( $f_b$ , $q$ ) and infers only the IMF slope $\alpha$ ; jointly inferring the binary parameters is future work.
KS rows validate the sampler against the implemented Moe PDF, which is itself grounded in Moe+17 Table 13 (held PDF) — not against raw survey histograms.

References¶

Moe & Di Stefano (2017) (binary statistics), Moe et al. (2019) (metallicity caveat), Sana et al. (2012) (O-star binaries), Maschberger (2013) (IMF). The methodology chapter is Binary-aware IMF recovery; the likelihood derivation Binary-aware likelihood; the Moe calibration Multiplicity statistics (Moe & Di Stefano 2017).

References¶

Moe, M., & Di Stefano, R. (2017). Mind your Ps and Qs: The interrelation between period (P) and mass-ratio (Q) distributions of binary stars. The Astrophysical Journal Supplement Series, 230, 15. 10.3847/1538-4365/aa6fb6
Moe, M., Kratter, K. M., & Badenes, C. (2019). The close binary fraction of solar-type stars is strongly anticorrelated with metallicity. The Astrophysical Journal, 875, 61. 10.3847/1538-4357/ab0d88
Sana, H., de Mink, S. E., de Koter, A., Langer, N., Evans, C. J., Gieles, M., Gosset, E., Izzard, R. G., Le Bouquin, J.-B., & Schneider, F. R. N. (2012). Binary interaction dominates the evolution of massive stars. Science, 337, 444–446. 10.1126/science.1223344
Maschberger, T. (2013). On the function describing the stellar initial mass function. Monthly Notices of the Royal Astronomical Society, 429, 1725–1733. 10.1093/mnras/sts479