Which filaments matter: the relative scalings of anisotropic infall

Abstract: Dark-matter haloes do not form in isolation but within the surrounding cosmic web. By the time a halo begins to collapse, its larger-scale environment has typically collapsed along two axes, forming filaments that channel anisotropic infall toward the halo. In this work, we derive from first principles the characteristic Lagrangian scale ratio at which such an anisotropic tidal field most strongly influences halo formation. Specifically, we identify the inflection point of the conditional probability that the tidal field, smoothed on a scale Rsd, undergoes two-dimensional compression, given the presence of a density peak of rarity nu on a smaller scale Rpk. For a standard LambdaCDM cosmology, we find (Rsd/Rpk)infl = 2.2 + (nu-2.5) for Rpk corresponding to a tophat filter of 8Mpc/h. This result implies that the anisotropic tidal influence on a collapsing halo typically extends to 2-3 times the size of its Lagrangian patch. Recast as a function of formation redshift z, the characteristic filament scale around 2.5 sigma peaks can be approximated by Rsd(z) = 31 /(2+(1+z)**2)Mpc/h. We provide practical scaling laws for selecting dynamically relevant smoothing scales in large-scale surveys and for setting initial patch sizes in high-resolution zoom simulations.

• The paper derives a quantitative relation linking halo rarity to the characteristic filament scale that influences anisotropic infall.
• It employs a rigorous statistical framework combining Gaussian Random Field excursion set theory with numerical evaluation on a ΛCDM power spectrum.
• The derived scaling laws provide practical criteria for smoothing scale selection in surveys and zoom-in simulations to enhance cosmic web analysis.

Characteristic Scales of Filamentary Influence on Halo Formation

Introduction

The formation and evolution of dark matter halos is fundamentally determined by their embedding within the multiscale, anisotropic large-scale structure—the cosmic web. The anisotropic tidal fields generated by collapsing filaments funnel matter toward haloes, controlling anisotropic infall, merger rates, and, ultimately, galaxy assembly histories. Despite increasing evidence for the impact of filaments on various halo and galaxy properties, quantitative criteria to relate halo properties to the relative scale of filamentary environments remain poorly established. Most analyses adopt fixed, somewhat arbitrary smoothing scales, lacking first-principle justification for the dynamically relevant range.

This work presents a rigorous statistical framework, rooted in Gaussian Random Field (GRF) and excursion set formalism, to delineate the characteristic scale at which the filamentary tidal field maximally influences halo formation. The authors derive analytic and numerical scaling laws, connecting the Lagrangian size and rarity of haloes to the characteristic surrounding filament scale, calibrated on the $\Lambda$CDM power spectrum. These relations yield practical guidance for the selection of smoothing scales in large-scale structure surveys and initial patch sizes for zoom simulations.

Theoretical Formalism: Conditional Probability of Anisotropic Collapse

The analysis proceeds by evaluating the conditional probability of encountering a two-axis-compressive (filamentary) tidal environment, smoothed on scale $R_\mathrm{sd}$, at the location of a high-density peak (halo) of rarity $\nu$ and smoothing scale $R_\mathrm{pk}$. Formally, this is quantified via

$$P(\rm sd|\rm pk ) = \frac{P(\rm sd, pk)}{P(\rm pk)},$$

where 'sd' denotes the signature of the sorted eigenvalues of the deformation tensor indicating filamentary regions, and 'pk' refers to overdensities above collapse threshold at the peak scale.

The approach leverages rotational invariants and the joint probability distribution of the field and its derivatives, following the extension of the classic peaks formalism to multiscale critical point statistics. The correlation between the density at $R_\mathrm{pk}$ and the tidal field at $R_\mathrm{sd}$, parameterized by

$$\gamma(R_\mathrm{pk},R_\mathrm{sd}) = \frac{4\pi \int dk\, k^2 P_m(k) W(kR_\mathrm{pk})W(kR_\mathrm{sd})}{\sigma_0(R_\mathrm{pk})\sigma_0(R_\mathrm{sd})}$$

fully encodes the power spectrum dependence. The inflection point of $P(\rm sd|\rm pk)$, with respect to the scale ratio $R_\mathrm{sd}/R_\mathrm{pk}$, marks the dynamically relevant filamentary scale.

Numerical Results: Scale Ratio and Rarity Dependence

Numerical evaluation on a three-dimensional $R_\mathrm{sd}$0CDM power spectrum reveals that $R_\mathrm{sd}$1 is suppressed at $R_\mathrm{sd}$2 (full three-axis collapse) and asymptotically approaches the unconstrained filament volume fraction at high scale ratios. The sensitivity to the filamentary scale peaks at a characteristic value of $R_\mathrm{sd}$3, the position of the inflection point.

Figure 1: Conditional probability $R_\mathrm{sd}$4 as a function of the scale ratio for various halo sizes; inflection points (filled circles) trace the characteristic filament scale ratio.

Crucially, the inflection point is neither universal nor constant but depends monotonically on the peak rarity and mass. For typical collapsed haloes, the scale ratio at maximum sensitivity is well-fit by a linear relation:

$R_\mathrm{sd}$5

with $R_\mathrm{sd}$6 over the relevant range.

Figure 2: The characteristic filament-to-halo scale ratio at the inflection point increases monotonically with peak rarity; dashed lines provide linear fits for different masses.

The result implies that the halo response to filamentary tidal fields is realized at scales $R_\mathrm{sd}$7–$R_\mathrm{sd}$8 times the Lagrangian halo size, with more massive (rarer) haloes coupled to filaments over larger physical distances.

Astrophysical Scaling Relations: Mass and Redshift Evolution

By connecting peak rarity and mass to cosmological collapse thresholds and the $R_\mathrm{sd}$9CDM variance, the authors formulate explicit scaling relations predicting the filamentary scale relevant for haloes of given size and redshift:

$\nu$0

with normalization $\nu$1 and $\nu$2 depending on redshift, as detailed in the main work. The mass and redshift dependence is illustrated in the next figure.

Figure 3: The filament-to-halo scale ratio as a function of Lagrangian halo radius (bottom axis) and halo mass (top axis) evolves as a power-law, systematically increasing with mass and redshift.

The filament scale, for instance, reaches $\nu$3 for $\nu$4 haloes at $\nu$5, and contracts with increasing $\nu$6 and decreasing mass. For the most prominent haloes at each epoch ($\nu$7–$\nu$8), the redshift evolution is captured by:

$\nu$9

demonstrating hierarchical growth of the cosmic web.

Figure 4: Characteristic filamentary environment scale for the most massive collapsing peaks as a function of redshift, along with simple analytic fits.

Practical Implications and Future Directions

These scaling laws furnish principled prescriptions for filtering and smoothing scales in both data analysis and simulation initial condition generation. In the context of zoom-in simulations, the findings provide quantitative criteria for buffer region selection, ensuring anisotropic tidal fields are adequately resolved and halo-environment coupling is preserved. For structure and filament finders in galaxy and weak lensing surveys, the results motivate rarity- and mass-dependent smoothing to isolate dynamically influential filaments, improving the interpretability of environmental trends.

On the theoretical front, the explicit dependence of the characteristic scale on the underlying power spectrum slope and the peak rarity opens avenues for constraining cosmology from the statistical properties of the cosmic web and its critical points. The general approach extends to both 2D projected fields and alternative cosmological models, as validated by additional analysis on scale-invariant spectra.

Conclusion

This paper establishes a robust statistical correspondence between halo properties and surrounding filamentary scales, rooted in GRF excursion set theory. The characteristic scale at which filamentary tidal fields dynamically influence halo formation is neither fixed nor arbitrary, but instead scales with halo rarity and mass in a quantifiable manner, typically at $R_\mathrm{pk}$0–$R_\mathrm{pk}$1 times the Lagrangian size. The analytic and numerical scaling relations provide essential tools for the design and interpretation of surveys and simulations targeting anisotropic infall and the cosmic web's role in structure formation. Their adoption will enhance the physical fidelity and consistency of multiscale analyses of the cosmic web and its evolutionary consequences.