Caustic fringes for wave dark matter

TL;DR

This paper analyzes how wave dark matter near caustics—regions where collisionless matter piles up at apogee—forms interference fringes described by Airy functions. By modeling the gravitational potential as locally linear near caustics, the authors derive a Fringe pattern whose scale depends on the particle mass and halo parameters, and they validate the theory with self-consistent Schrödinger-Poisson simulations in 1D and 3D, alongside N-body comparisons. The leading fringes near caustics are shown to be well captured by the Airy description, with fringe separations that can exceed the naive de Broglie scale, enabling potential observational constraints on ultra-light dark matter through stacking data across systems. The work highlights the observational relevance for splashback in clusters and tidal shells in galaxies and discusses strategies and challenges for detecting caustic fringes with current and upcoming surveys.

Abstract

Wave dark matter is composed of particles sufficiently light that their de Broglie wavelength exceeds the average inter-particle separation. A typical wave dark matter halo exhibits granular substructures due to wave interference. In this paper, we explore the wave interference effects around caustics. These are locations of formally divergent density in cold collisionless systems. Examples include splashback in galaxy clusters, and tidal shells in merging galaxies, where the pile-up of dark matter close to apogee gives rise to caustics. We show that wave interference modifies the density profile in the vicinity of the caustics, giving rise to a fringe pattern well-described by the Airy function. This follows from approximating the gravitational potential as linear close to apogee. This prediction is verified in a series of numerical simulations in which the gravitational potential is computed exactly. We provide a formula expressing the fringe separation in terms of the wave dark matter mass and halo parameters, which is useful for interpreting and stacking data. The fringe separation near caustics can be significantly larger than the naive de Broglie scale (the latter set by the system's velocity dispersion). This opens up the possibility of detecting caustic fringes for a wide range of wave dark matter masses.

Figures (6)

• Figure 1: A plot of the Airy function ${\rm Ai} (z)$ (upper panel) and its square (lower panel). The Airy function solves the Schrödinger equation with a linear potential, with energy chosen such that $z=0$ corresponds to the turning point of a particle (dashed line).
• Figure 2: A simulation of the collapse of an initial overdensity in a single spatial dimension. Each column corresponds to a different snapshot at time $T$. Top row. We plot the density of a wave dark matter simulation (red) and an N-body particle simulation (blue). Broadly speaking, the two evolve similarly, but the wave dark matter simulation has interference fringes. Bottom row. We show the corresponding particle phase space at each snap shot. In the second and third columns, the dashed black line indicates the location of a caustic. Note how the velocity at the caustic nearly vanishes, i.e. it is close to apogee. Here $\hbar/m = 4 \times 10^{-4}$. All quantities, $\rho, v, x, T$, are shown in dimensionless code units.
• Figure 3: Here we zoom in on the caustic depicted in Figure \ref{['fig:evolution1D']}. In both panels the caustic location is labeled with a vertical dashed black line. Left. A plot of the particle phase space (blue) describing the collapse of an over-density due to gravity in a single spatial dimension. The red line indicates an approximation to the phase space in the vicinity of the caustic (equation \ref{['vvc']}). The acceleration $a$ is the local gravitational acceleration at the caustic. Right. A plot of the wave dark matter density (suitably normalized) in the vicinity of the caustic, for a series of wave simulations with the mass $m$ varying by a factor of $128$. Each colored solid line represents a different $m$. The normalization of the y-axis is chosen so the smallest $m$ is at the bottom (lightest blue) and the largest $m$ is at the top (darkest blue). The Airy function, suitably normalized, provides a good match to all of them. To illustrate, we show with a dashed line an example that matches the top solid curve (modulo small wiggles; see text). The Airy function also gives the correct prediction for the fringe separation (red) and fringe width (green), in accordance with equations (\ref{['eqn:r_fringe']}) and (\ref{['eqn:caustic_width']}).
• Figure 4: Two 1D wave simulations of the gravitational collapse of an initial overdensity for two different masses differing by a factor of $16$. The blue/orange line indicates the density profile for the higher/lower mass. The left panel shows the overall density profile in the simulation box. The right panel provides a zoom-in of the red region in the left panel. The horizontal lines indicate the respective expected fringe separation given by equation \ref{['eqn:r_fringe']}. It provides an accurate description of the simulation results.
• Figure 5: A 3D wave simulation of a spherically symmetric Gaussian blob undergoing gravitational collapse. Left. We show the projected log density. The collapse is spherically symmetric so we can see concentric rings from shells of materials and fringe structures. Center. We zoom in on the region with fringes (shown outlined in red in the left panel). We plot the expected separation in red superimposed on the fringes. We can see that the separation predicted in equation \ref{['eqn:r_fringe2']} accurately describes the separation of the fringes. Here, the acceleration is $a=GM_h/r_c^2$, where $r_c$ is the caustic radius and $M_h$ is the mass enclosed. Right. We plot the simulated density in blue and the Airy function prediction (equation \ref{['psiAicaustic3']}) in magenta. Here $\tilde{r} \equiv (2m^2 GM_h / \hbar^2 r_c^2)^{1/3} (r - r_c)$, where $r_c$ is the caustic radius. We can see that the Airy function describes the location and shape of the leading two fringes well but becomes inaccurate after this. In this simulation the total mass is $M_\mathrm{tot} = 4 \times 10^8 \, M_\odot$, the field mass $m = 5 \times 10^{-22} \, \mathrm{eV}$, and the grid is $384^3$.