Protohalos and their connection to halo assembly, shape and structure

TL;DR

The paper investigates how the initial protohalo structure relates to the assembly history and final morphology of dark matter halos, proposing an Optimal Transport-informed framework to reconstruct protohalo configurations from galaxies. It introduces four tensors (mass, inertia, deformation, energy) and their invariants, along with a new integrated mass accretion history measure $A$ that is compared to the conventional $z_{50}$. Key findings show that the third invariant $U^3/q^3$ of the deformation/energy tensors correlates with assembly timing and final shape, with distinct trends across mass scales, and that energy-based descriptors can link early collapse to higher concentrations and more anisotropic evolution. The work suggests practical avenues for bias estimation and OT-based reconstruction validation, with implications for reducing shape-related systematics in cosmology and for interpreting protohalo-to-halo evolution through tensor invariants.

Abstract

Protohalos, primordial regions in the initial cosmic density field that evolve into dark matter halos, are crucial for understanding cosmic structure formation. Motivated by the potential to reconstruct protohalo positions and shapes from observed galaxies using a novel approach grounded in optimal transport theory, we revisit the relationship between the structural properties of protohalos and the assembly histories, concentrations, and final morphologies of their associated dark matter halos. To better understand halo assembly, we introduce a new estimator defined by an integral over redshifts and compare its performance to $z_{50}$, the commonly used redshift at which half of the final halo mass is formed. We quantify protohalo structure using the three invariants of the inertia, deformation, and energy shear tensors. Although past research has correlated the first two invariants of the deformation and energy tensors with halo formation, our findings reveal that the third invariant also significantly correlates with halo assembly and final shape.

Figures (10)

• Figure 1: Mass accretion histories color coded by $\Delta M_{\rm acc}\equiv A$ of equation (\ref{['eq:MAH']}) (left) and $z_{50}$ (right), for three narrow bins in halo mass (top to bottom). Dashed curve, same in each pair of panels, shows the median mass at each $z$. Solid black curve shows the accretion history of the object with the median $A$ and $z_{50}$; solid red and blue curves show the histories of the objects with the most extreme values.
• Figure 2: Difference from median history as quantified by $A$ is well correlated with that for $z_{50}$.
• Figure 3: Mean halo property (top to bottom shows $z_{50}$, our new integrated measure of assembly history $A$, final concentration and axis ratio $[c/a]$) as a function of halo mass and invariants of the protohalo mass tensor (left to right shows trace, traceless shear, dimensionless ratio of traceless determinant and shear, trace times ellipticity and trace times prolateness). In all cases, the median trend with mass has been removed, so the white band (zero residual) shows the median trend, and the other colors show how the residual from the median relation depends on invariant.
• Figure 4: Same as previous figure, but now for the invariants of the protohalo deformation tensor.
• Figure 5: Same as previous figure, but now for the invariants of the protohalo energy shear tensor.