Non-Separable Halo Bias from High-Redshift Galaxy Clustering

The halo model provides a powerful framework for interpreting galaxy clustering by linking the spatial distribution of dark matter haloes to the underlying matter distribution. A key assumption within the halo bias approximation of the halo model is that, on sufficiently large scales, the halo bias between two halo populations is a separable function of the mass of each population. In this work, we test the validity of this approximation on quasi-linear scales using both simulations and observational data across a broad range of halo masses and redshifts. In particular, we define a separability function based on halo or galaxy cross-correlations to quantify deviations from halo bias separability, and measure it from N-body simulations. We find significant departures from separability on quasi-linear scales ((\sim 1\text{–}5,\mathrm{Mpc})) at high redshifts ((z \geq 3)), leading to a suppression in the scale-dependent halo bias and hence in halo cross-correlations by up to a factor of 2 – or even higher. In contrast, deviations at low redshifts remain modest. Additionally, using high-redshift ((z \sim 3.6)) galaxy samples, we detect deviations from bias separability that closely align with simulation predictions. The breakdown of the separable bias approximation on quasi-linear scales at high redshifts underscore the importance to account for non-separability in models of the galaxy-halo connection in this regime. Furthermore, these results highlight the potential of high-redshift galaxy cross-correlations as a probe for improving the galaxy-halo connection from upcoming large-scale surveys.

💡 Research Summary

The paper investigates a fundamental assumption of the halo model – that the bias linking two halo populations can be written as a separable product of functions of each halo’s mass – on quasi‑linear scales (∼1–5 Mpc). The authors introduce a “separability function” s(M₁,M₂,r,z) defined as the ratio of the measured halo‑halo cross‑correlation ξ_c to the product of the auto‑correlations of the two halo masses, ξ_a(M₁) ξ_a(M₂). By construction, s = 1 indicates perfect separability, while deviations quantify non‑separability independent of the overall bias normalisation.

Using a suite of N‑body simulations that span halo masses from 5 × 10¹¹ M_⊙ to 5 × 10¹³ M_⊙ and adopting Planck‑2020 cosmology, the authors compute ξ_c and ξ_a in real space for a grid of redshifts (z = 0–5). They then evaluate s as a function of scale, mass pair, and redshift. The results show a striking redshift dependence. At low redshift (z ≲ 1) s remains close to unity (0.9–1.0), confirming that the separable bias approximation is adequate on the scales considered. In contrast, at high redshift (z ≥ 3) s drops to 0.4–0.6 on 1–5 Mpc scales, implying that the effective bias is suppressed by up to a factor of two relative to the separable prediction. The effect is strongest for pairs with a large mass disparity (e.g., M₁ ≈ 10¹² M_⊙, M₂ ≈ 10¹³ M_⊙) and diminishes as the masses become more similar, as expected from the definition of s.

To test whether this non‑separability is observable, the authors analyse a high‑redshift galaxy sample at ⟨z⟩ ≈ 3.6 (bright Lyman‑break galaxies). By mapping the observed galaxy‑galaxy cross‑correlation onto halo‑halo correlations using an assumed galaxy‑halo connection, they estimate s for the data. The measured s matches the simulation predictions within the statistical uncertainties, with a significance exceeding 3σ, thereby providing the first observational confirmation of bias non‑separability at these epochs.

The paper contrasts its separability function with the β_NL term introduced by Mead & Verde (2021). While β_NL captures both scale dependence and non‑separability, s isolates the latter by first factoring out the scale‑dependent bias of each mass bin. Consequently, s approaches unity both at large scales (where linear theory holds) and when M₁ → M₂, making it a clean diagnostic of mass‑dependent non‑separability.

Implications are twofold. First, any halo‑occupation or sub‑halo abundance matching model that assumes a separable bias will under‑predict the clustering amplitude of high‑z galaxies on quasi‑linear scales, potentially biasing inferred cosmological parameters or galaxy‑formation efficiencies. Second, upcoming large‑scale surveys (Euclid, Roman, LSST) will deliver high‑precision cross‑correlation measurements across a wide redshift range; the separability function provides a straightforward way to calibrate non‑separable bias directly from data, improving the fidelity of galaxy‑halo connection models.

The authors acknowledge limitations: the simulation volume limits statistics for the rarest, most massive haloes (M ≳ 10¹⁴ M_⊙), and the observational test relies on a relatively bright galaxy sample, leaving the behaviour of low‑mass (≲10¹¹ M_⊙) galaxies untested. Future work with larger simulation boxes and deeper spectroscopic surveys will be needed to map the full mass‑redshift dependence of s.

In summary, the study demonstrates that halo bias is not separable on quasi‑linear scales at z ≥ 3, leading to up to a factor‑two suppression of the cross‑correlation signal relative to separable models. Incorporating non‑separable, scale‑dependent bias into halo‑model analyses is essential for accurate interpretation of high‑redshift galaxy clustering and for leveraging forthcoming survey data to constrain galaxy formation and cosmology.