Confidently Wrong: Why Ignoring Binaries Biases IMF Inference at Large Sample Sizes

Abstract

The stellar initial mass function (IMF) high-mass slope $α$ is routinely measured by fitting single-star models to photometric samples that contain 20-90% unresolved binaries. This practice introduces a systematic negative bias on $α$ that is constant with sample size $N$. Because posterior credible intervals shrink as $1/\sqrt{N}$, at sufficiently large $N$ the bias exceeds the reported uncertainty and the true value falls outside the credible interval - a regime we call "confidently wrong." We bracket this bias between two limiting observation operators: mass-addition $(m_\text{obs} = m_1 + m_2)$, a formal upper bound on unresolved-system mass overestimation, and luminosity-addition $(m_\text{obs} = L^{-1}(L_1 + L_2))$, an idealized lower-bias photometric case based on the ZAMS mass-luminosity relation. Across four astrophysical environments spanning $α= 1.60-2.30$, we find: (1) mass-addition bias of $0.054-0.086$ with crossover to confidently wrong at $N_\text{cross} \sim 5{,}000-10{,}000$; (2) luminosity-addition bias of $0.011-0.021$ with $N_\text{cross} \sim 75{,}000-150{,}000$; and (3) a binary-aware mixture likelihood that marginalizes over the Moe & Di Stefano (2017) binary population model recovers the true slope in the synthetic tests presented here. Published single-star IMF slopes can therefore plausibly carry systematic errors of order $0.01-0.09$ if unresolved binaries are not modeled, comparable to or exceeding reported uncertainties in some regimes. Since current and upcoming surveys (Gaia, JWST, Roman, LSST) will deliver $N = 10^4-10^6$ resolved stars per rich cluster, binary-aware inference is likely necessary to avoid binary-driven systematic bias in the large-$N$ single-star-fitting regime.

Figures (5)

• Figure 1: Literature context for the benchmark IMF slopes used in this paper. Published high-mass IMF slopes from the compilation of hennebelle2024 are plotted against the IMF-depth proxy $1/f(m_\mathrm{lo})$, defined here as the number of total IMF stars expected per star above a study's lower-mass fitting limit $m_\mathrm{lo}$. Larger values therefore correspond to shallower surveys that constrain $\alpha$ using only the rare high-mass tail. Points are color-coded by broad environment class, and diamond markers denote measurements that include explicit binary corrections. Horizontal dotted lines mark the four benchmark $\alpha$ values used in this paper.
• Figure 2: Binary population model from moe2017. (a) Mass-dependent binary fraction $f_\mathrm{b}(m_1)$, showing the step-function dependence from 22% for late M dwarfs to 90% for O stars. Colored bands mark spectral type regions (M, K, G/F, A, B, O). (b) Mass-ratio distribution $p(q \mid m_1)$ for four representative primary masses spanning the full range of power-law index $\gamma$. The narrow peak at $q \approx 1$ is the "twin excess." (c) Distortion of the observed system mass function under mass-addition (dashed) relative to the true single-star IMF (solid) for all four environments. The shaded region highlights the mass overestimate from unresolved binaries.
• Figure 3: Parameter recovery across four astrophysical environments at $N = 10{,}000$ using the mass-addition operator. (a) True vs. recovered $\alpha$: naive estimates (diamonds, with 95% CIs) are systematically biased low, while binary-aware estimates (circles) are consistent with the truth (dashed line). (b) Residual posterior distributions: naive posteriors (dashed) are shifted negative, confirming systematic bias; binary-aware posteriors (solid) are centered on zero.
• Figure 4: The "confidently wrong" regime across four astrophysical environments. Each panel shows the 95% credible interval width (solid lines) and absolute bias $|\hat{\alpha}_\mathrm{naive} - \alpha_\mathrm{true}|$ (dotted lines) as functions of sample size $N$ on log-log axes. Four inference configurations are compared: mass-addition naive (orange diamonds), luminosity-addition naive (amber squares), binary-aware mass-addition (blue circles, solid), and binary-aware luminosity-addition (blue triangles, dashed). The gray dashed line shows the $1/\sqrt{N}$ Bernstein--von Mises scaling. The shaded region marks where the mass-addition bias exceeds the CI width --- the "confidently wrong" regime. Both binary-aware models track the $1/\sqrt{N}$ scaling with bias consistent with zero.
• Figure 5: The "confidently wrong" posteriors under mass-addition for all four environments. Naive posteriors (solid lines) at $N = 1{,}000$, $5{,}000$, $30{,}000$, and $100{,}000$ are shown with progressively darker shading; binary-aware posteriors (dashed) at $N = 100{,}000$ recover the truth (dotted vertical line) in every environment. As $N$ increases, naive posteriors narrow onto the biased value rather than the truth. Bias magnitudes decrease from Solar (0.086) to Low-$Z$ starburst (0.054), but all environments enter the confidently wrong regime. Posteriors are Gaussian approximations from HMC summary statistics.