Dependency of the Bar Formation Timescale On The Halo Spin

TL;DR

Bars are common in disk galaxies and their formation timescales encode a key aspect of secular evolution. The authors use a suite of 60 N-body simulations with varying disk fractions, Toomre Q, and halo spin to test how halo spin affects established bar formation relations. They find that halo spin can modestly modulate the bar formation timescale: prograde spins tend to accelerate bar onset for intermediate disks, while retrograde spins tend to slow it, with negligible effects for the fastest cases and more stochastic variations for the slowest ones. The results support a scenario in which angular momentum transfer between the disk and halo mediates bar formation and help explain the presence of early bars at high redshift observed by JWST.

Abstract

Bars are among the most prominent structures in disk galaxies. While the widely accepted swing-amplification theory provides a qualitative framework for their formation, the detailed physical processes remain incompletely understood. Previous studies have shown that the bar formation timescale in isolated galaxies depends exponentially on the disk mass fraction (the so-called "Fujii relation") and linearly on disk hotness and thickness. However, the influence of dark matter halo spin on bar formation has not been systematically investigated. In this work, we construct a suite of $N$-body models of disk and halo with varying disk mass fractions and amounts of random motions. By introducing prograde and retrograde spins in the dark matter halo, we explore how halo spin modifies the established empirical relations governing bar formation timescales. We find that these relations remain valid in both prograde and retrograde halo spin models. For rapid bar formation (short timescale), the effect of halo spin is nearly negligible. In contrast, for moderately slow bar formation, prograde (retrograde) halo spin tends to accelerate (suppress) bar onset. In cases of extremely slow bar formation, halo spin introduces a stronger but more stochastic influence. These trends might arise from the exchange of angular momentum between the stellar disk and the dark matter halo.

Figures (6)

• Figure 1: Face-on views of the forming bars near the saturation of their exponential growth for several models with $Q = 1.4$. From bottom to top, $f_\mathrm{disk}$ increases. From left to right: models with retrograde, zero, and prograde halo spin.
• Figure 2: Time evolution of bar strength $A_2(t)$ for models with different halo spins: $\lambda = 0$ (black), $\lambda = 0.10$ (green), and $\lambda = -0.10$ (red). From top to bottom, panels are ordered by decreasing $f_\mathrm{disk}$; from left to right, by increasing Toomre $Q$. For models exhibiting effective bar formation ($\operatorname{max}\{A_2(t)\} \geq 0.15$), dashed lines indicate exponential fits to the initial growth phase of $A_2(t)$ in the corresponding color.
• Figure 3: The "Fujii diagram": distribution of $\tau_\mathrm{bar}$ versus $f_\mathrm{disk}$ for models with effective bar formation. Data point colors indicate Toomre $Q$ values. Marker shapes denote halo spin: circles for $\lambda = 0$, upward-filled triangles for $\lambda = 0.10$, and downward-unfilled triangles for $\lambda = -0.10$. For comparison, we show the Fujii relation from the high halo mass models of bland_etal_2023 in a black dotted-dashed line (Equation \ref{['eq: BH formula']}) and the empirical relation from che_she_2025 in colored solid lines (Equation \ref{['eq: fit in CS2025']}). Overall, models with different spin values follow the general empirical trends, with halo spin introducing additional variations in $\tau_\mathrm{bar}$.
• Figure 4: Left panel: variation in bar formation timescale $\Delta\tau_\mathrm{bar}$ as a function of $f_\mathrm{disk}$, where $\Delta\tau_\mathrm{bar}$ is defined as the difference between the prograde (or retrograde) spin models and their zero-spin counterparts. Upward-filled triangles denote prograde models; downward unfilled triangles denote retrograde models. Only models with well-defined $\Delta\tau_\mathrm{bar}$ values are included. Point colors indicate the Toomre $Q$ of each model. Small horizontal offsets are applied to reduce overlap among points with the same $f_\mathrm{disk}$. Right panel: zoom-in view of the left panel, limited to $|\Delta\tau_\mathrm{bar}| \leq 1\ \mathrm{Gyr}$.
• Figure 5: Left panel: time evolution of $A_2$ (solid lines) for model $\mathrm{fd0.60Q1.6}$ and its prograde and retrograde halo spin counterparts, with colors indicating halo spin. The corresponding variations of the vertical angular momentum $\Delta L_z$ of the disk (dotted-dashed lines) and DM halo (dashed lines) are also shown. Right panel: zoom-in on the early evolution; for better visualization, we restrict the range of $A_2$ to $[0,\ 0.25]$ and $\Delta{L}_z$ to $[-100,\ 0]$. Dotted lines show $\Delta{L}_z$ for the inner disk ($R<10\ \mathrm{kpc}$, $|z|<2.5\ \mathrm{kpc}$).