Frequent Small-Angle Self-Interactions (fSIDM)

• fSIDM is a dark matter regime characterized by highly forward-peaked, small-angle scatterings that individually impart tiny momentum kicks but collectively induce drag, diffusion, and heat conduction.
• Simulation methodologies for fSIDM include SPH-based conduction models and pairwise drag with momentum diffusion, accurately capturing phenomena such as core formation and halo shape rounding.
• Cosmological analyses reveal that fSIDM leads to distinct outcomes in subhalo depletion, merger-induced offsets, and inner density profiles compared to rare self-interacting dark matter.

Frequent small-angle self-interactions (fSIDM) denote the limit of self-interacting dark matter in which the differential cross section is strongly forward-peaked, so that each collision transfers only a tiny momentum kick while the scattering rate is high. In contrast to rare self-interacting dark matter (rSIDM), where scatterings are individually infrequent but can produce large deflections, fSIDM is typically associated with long-range or light-mediator interactions and is modeled through cumulative transport effects such as drag, diffusion, and heat conduction rather than as isolated large-angle events. Across the recent simulation literature, fSIDM reproduces many of the large-scale behaviors of rSIDM when compared at fixed transport cross section, but it differs most clearly in subhalo depletion, halo roundness, and merger-induced offsets (Fischer et al., 2022).

1. Definition and microphysical characterization

In the SIDM framework, dark-matter particles scatter elastically with differential cross section $d\sigma/d\Omega(\theta)$. The standard distinction is between “rare” SIDM, for which $d\sigma/d\Omega$ is almost isotropic and each collision transfers a large fraction of momentum, and “frequent” SIDM, for which scatterings are highly forward-peaked, $\theta \ll 1$, so that each collision produces only a small deflection but the interaction rate is large. The latter is the regime expected for long-range or light-mediator interactions (Fischer et al., 2022).

A convenient particle-physics realization is Yukawa or Rutherford-like scattering. In the small-angle limit, one commonly encounters a differential cross section of the schematic form

$$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$

or, in Born-limit parameterizations, forms proportional to $[1+r\sin^2(\theta/2)]^{-2}$ with $r\equiv m_\chi^2 v^2/m_\phi^2$. These expressions are strongly peaked at $\theta\to 0$ and imply that the total cross section can be dominated by very small deflections (Fischer et al., 2022, Arido et al., 2024).

For fSIDM, the relevant observables are transport-weighted cross sections rather than the total cross section. The most common are the momentum-transfer cross section

$$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$

the modified transfer cross section for identical particles,

$$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$

and the viscosity cross section

$$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$

In the isotropic limit, $d\sigma/d\Omega$0, whereas in the strongly forward-dominated limit $d\sigma/d\Omega$1 and $d\sigma/d\Omega$2 formally (Arido et al., 2024, Fischer et al., 2020).

This distinction is not merely notational. In merger dynamics, deceleration is tied to $d\sigma/d\Omega$3 or $d\sigma/d\Omega$4, while gravothermal evolution and conductive heat transport are controlled by $d\sigma/d\Omega$5 or effective conductivity parameters derived from it. A plausible implication is that matching one transport cross section does not, by itself, guarantee identical halo phenomenology across all observables.

2. Kinetic regime, drag, diffusion, and heat conduction

The frequent-scattering limit is reached when the per-particle scattering rate is high and the mean deflection angle per collision is small. In the cosmological treatment of frequent scattering, this is summarized by the condition

$d\sigma/d\Omega$6

together with a small per-collision angle, so that the cumulative effect is better approximated as a drag or diffusion term in the Boltzmann equation than as discrete kicks (Fischer et al., 2022).

In this limit, several papers recast the collision operator into a Fokker–Planck or fluid description. For isolated halos, Kummer et al. model the dark matter as a pressureless fluid with an effective heat flux

$d\sigma/d\Omega$7

so that energy conservation becomes

$d\sigma/d\Omega$8

They identify temperature through the one-dimensional velocity dispersion $d\sigma/d\Omega$9 via $\theta \ll 1$0, and derive an SPH discretization for the evolution of $\theta \ll 1$1 (Kummer et al., 2019).

The conductivity depends on the heat-transfer mean free path $\theta \ll 1$2 and interpolates between a short-mean-free-path regime and a long-mean-free-path regime. In the isolated-halo application of Kummer et al., all radii satisfy the long-mean-free-path limit, so that $\theta \ll 1$3 and core-formation rates scale proportionally to $\theta \ll 1$4 (Kummer et al., 2019).

A complementary formulation emphasizes drag and momentum diffusion. For a particle moving through a homogeneous background of density $\theta \ll 1$5, Sabarish et al. quote the deceleration law

$\theta \ll 1$6

which is the same deceleration structure derived analytically for frequent small-angle interactions in cluster collisions by Kahlhoefer et al. (Sabarish et al., 2023, Kahlhoefer et al., 2013). The collisionless-tracer interpretation is then straightforward: dark matter decelerates coherently, while galaxies or BCGs do not, which can generate tracer–DM offsets even when most of the mass remains bound.

3. Numerical implementations in $\theta \ll 1$7-body simulations

Two numerical strategies dominate the fSIDM literature. One treats frequent scattering as conductive energy transport in an SPH-like framework. The other implements an effective pairwise drag force supplemented by transverse momentum diffusion to preserve energy and momentum statistically or exactly, depending on the scheme (Kummer et al., 2019, Fischer et al., 2020).

Paper	fSIDM implementation	Scope
(Kummer et al., 2019)	SPH-conduction module in gadget-2	Isolated halos
(Fischer et al., 2020)	Pairwise drag + perpendicular heating in gadget-3	Isolated halos and cluster mergers
(Fischer et al., 2022)	Continuous momentum-transfer force in Gadget-3	First cosmological simulation of frequent self-interactions
(Arido et al., 2024)	Hybrid scheme with critical angle $\theta \ll 1$8	Realistic angular dependence in cluster mergers
(Fischer et al., 10 Mar 2026)	Public OpenGadget3 implementation with drag + diffusion	Cosmological SIDM code infrastructure

In the drag-based formulation of Fischer et al., the pairwise force is

$\theta \ll 1$9

where $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$0 is the kernel-overlap integral. The deterministic momentum loss is accompanied by a random kick perpendicular to $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$1, with magnitude chosen so that the lost kinetic energy is re-injected locally (Fischer et al., 2020). The 2026 OpenGadget3 implementation generalizes this to velocity- and angle-dependent cross sections and updates velocities pairwise so that linear momentum and energy are conserved explicitly, with only machine-precision round-off accumulation (Fischer et al., 10 Mar 2026).

The cosmological implementation of Fischer et al. in Gadget-3 uses a continuous momentum-transfer update,

$$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$2

with

$$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$3

where $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$4 and $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$5 are estimated from a kernel over the 64 nearest neighbors (Fischer et al., 2022). This realizes the same heat-conduction behavior as many unresolved small-angle scatterings.

Hybrid methods aim to bridge realistic angle dependence. Arido, Fischer, and Garny split the differential cross section at a technical cutoff angle $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$6: scatterings with $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$7 are treated by effective drag plus diffusion, while $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$8 is sampled explicitly via Monte Carlo. Within merger simulations, results are unchanged at the few-percent level when $$\frac{d\sigma}{d\Omega}(\theta)\propto \frac{1}{(\theta^2+\theta_0^2)^2},$$9 is varied by factors of 2, for example between $[1+r\sin^2(\theta/2)]^{-2}$0 and $[1+r\sin^2(\theta/2)]^{-2}$1, provided the validity criterion for the small-angle approximation is respected (Arido et al., 2024).

These implementations impose distinct numerical constraints. Kummer et al. require unusually large SPH neighbor numbers to reproduce analytic heat fluxes, finding $[1+r\sin^2(\theta/2)]^{-2}$2 for $[1+r\sin^2(\theta/2)]^{-2}$3 particles and $[1+r\sin^2(\theta/2)]^{-2}$4 for $[1+r\sin^2(\theta/2)]^{-2}$5 (Kummer et al., 2019). By contrast, OpenGadget3 emphasizes pairwise accuracy and time-step control, with $[1+r\sin^2(\theta/2)]^{-2}$6–$[1+r\sin^2(\theta/2)]^{-2}$7 and the condition $[1+r\sin^2(\theta/2)]^{-2}$8 so that the kernel scale remains smaller than the mean free path (Fischer et al., 10 Mar 2026).

4. Halo evolution, core formation, and cosmological structure

The isolated-halo problem established the basic gravothermal phenomenology of fSIDM. Kummer et al. showed that after a few relaxation times the inner cusp is heated, the velocity dispersion flattens, and a constant-density core emerges. For their cored-Hernquist fits, the growth curves obey a universal scaling in the long-mean-free-path regime: for $[1+r\sin^2(\theta/2)]^{-2}$9, $r\equiv m_\chi^2 v^2/m_\phi^2$0, and at $r\equiv m_\chi^2 v^2/m_\phi^2$1 they find $r\equiv m_\chi^2 v^2/m_\phi^2$2 at $r\equiv m_\chi^2 v^2/m_\phi^2$3 (Kummer et al., 2019).

Fischer et al. extended this to an effective drag implementation in isolated Hernquist halos with $r\equiv m_\chi^2 v^2/m_\phi^2$4 particles. For $r\equiv m_\chi^2 v^2/m_\phi^2$5, the core grows to $r\equiv m_\chi^2 v^2/m_\phi^2$6 and then gradually collapses, with the collapse time shortening as the cross section increases. In direct comparison with isotropic rare scattering, fSIDM and rSIDM produce nearly identical maximum core sizes, but fSIDM reaches the maximum slightly earlier, by $r\equiv m_\chi^2 v^2/m_\phi^2$7–$r\equiv m_\chi^2 v^2/m_\phi^2$8 (Fischer et al., 2020).

Velocity dependence alters that picture. In the idealized and cosmological study of Fischer et al., velocity-dependent fSIDM with $r\equiv m_\chi^2 v^2/m_\phi^2$9 produces larger maximum cores and delayed collapse relative to velocity-independent models; in an NFW halo, strongly velocity-dependent scattering with $\theta\to 0$0 yields $\theta\to 0$1 larger $\theta\to 0$2 and $\theta\to 0$3 instead of $\theta\to 0$4 for constant cross section (Fischer et al., 2023).

The first cosmological simulation of frequent DM self-interactions then quantified how much of this behavior survives hierarchical structure formation. At $\theta\to 0$5, both rSIDM and fSIDM with $\theta\to 0$6 suppress small-scale power by $\theta\to 0$7–$\theta\to 0$8 for $\theta\to 0$9–$$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$0 relative to CDM, with differences between the two scattering regimes $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$1. Halo abundances above $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$2 are nearly unchanged, but at smaller masses fSIDM reduces low-mass halo counts by $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$3–$$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$4, and the subhalo mass function is more strongly suppressed: $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$5 at $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$6 for fSIDM versus $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$7 for rSIDM (Fischer et al., 2022).

The strongest cosmological distinctions appear inside halos. In the same simulations, fSIDM with $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$8 reduces $$\sigma_T=\int d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega},$$9 by factors $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$0–$$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$1 relative to CDM, compared to a milder factor $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$2 for rSIDM. Shape profiles are also more strongly affected: CDM halos have $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$3–$$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$4, while fSIDM halos reach $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$5, and at $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$6 the axis ratio $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$7 is larger by $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$8 than in rSIDM (Fischer et al., 2022).

Cluster zoom simulations with baryons complicate the interpretation. Ragagnin et al. find that dark-matter-only fSIDM and rSIDM both develop $$\sigma_{\widetilde T}= \int d\Omega\,(1-|\cos\theta|)\,\frac{d\sigma}{d\Omega},$$9 cores with $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$0–$$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$1 lower central density than CDM below $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$2, but in full-physics runs the baryonic potential can dominate: in fSIDM full-physics simulations, the central dark-matter density is nevertheless $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$3 higher than in collisionless full-physics runs because the clusters remain in their core-formation phase (Ragagnin et al., 2024). This suggests that cluster constraints based on total density profiles alone are not portable between dark-matter-only and full-physics settings.

5. Mergers, offsets, and the angular dependence of self-interactions

The merger problem isolates the dynamical consequence of coherent drag. Kahlhoefer et al. showed analytically that for frequent small-angle scattering the deceleration and cumulative evaporation rates are both governed by

$$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$4

In this picture, the dark-matter halo slows while collisionless galaxies continue nearly ballistically. The resulting apparent offset is dominated by escaping tails rather than by a wholesale separation of the main peaks: the highest-density peaks of the dark-matter and galaxy distributions remain spatially coincident, while centroid-based or tail-sensitive measures can show separations of order $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$5–$$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$6 for $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$7 in Bullet-like systems (Kahlhoefer et al., 2013).

Direct $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$8-body merger simulations sharpened this difference. In equal-mass cluster mergers, Fischer et al. found maximum late-time offsets of $$\sigma_V=\int d\Omega\,(1-\cos^2\theta)\,\frac{d\sigma}{d\Omega}.$$9 for fSIDM at $d\sigma/d\Omega$00, compared to $d\sigma/d\Omega$01 for rSIDM at $d\sigma/d\Omega$02. For $d\sigma/d\Omega$03, fSIDM offsets exceed rSIDM by a factor $d\sigma/d\Omega$04–$d\sigma/d\Omega$05 (Fischer et al., 2020).

Unequal-mass mergers amplify these distinctions but also expose observational systematics. Fischer et al. report that results are sensitive to the peak-finding method, especially for minor mergers. Using the gravitational-potential peak finder, they obtain for a 1:10 cluster merger with $d\sigma/d\Omega$06 a maximum subhalo offset of $d\sigma/d\Omega$07 in fSIDM versus $d\sigma/d\Omega$08 in rSIDM; the 1:5 and 1:1 cases show the same ordering, and the fSIDM subhalo dissolves more quickly (Fischer et al., 2021). This is one reason merger constraints on the angular dependence of the cross section remain method-dependent.

Velocity dependence shifts the relevant scales. In the equal-mass and 5:1 cluster mergers of Sabarish et al., upper-bound models give dark-matter–BCG offsets just after first pericenter of order $d\sigma/d\Omega$09, but later apocenters reach $d\sigma/d\Omega$10–$d\sigma/d\Omega$11 for fSIDM and $d\sigma/d\Omega$12–$d\sigma/d\Omega$13 for rSIDM. They therefore argue that late-time BCG oscillations in relaxed clusters, rather than immediate post-pericenter offsets, may provide the more viable statistical probe (Sabarish et al., 2023).

Hybrid angular treatments show that the isotropic and purely forward limits do not bracket all realistic cases tightly. In idealized cluster mergers with Møller scattering, Arido, Fischer, and Garny find that for $d\sigma/d\Omega$14 the maximal DM–galaxy offset matches the isotropic result, for $d\sigma/d\Omega$15 it matches the purely forward-dominated result, and in the intermediate regime the offset can differ by up to a factor $d\sigma/d\Omega$16–$d\sigma/d\Omega$17. In the strongly forward case $d\sigma/d\Omega$18, the maximal offset between first and second pericenter scales roughly linearly as

$d\sigma/d\Omega$19

This suggests that merger observables are sensitive not only to the normalization of the self-interaction but also to its full angular structure (Arido et al., 2024).

6. Constraints, degeneracy breaking, and recent extensions

The first cosmological study of frequent self-interactions also derived upper limits on the fSIDM cross section by transferring cluster and group constraints previously formulated for rSIDM. Using shape and core-size comparisons, the adopted limits are $d\sigma/d\Omega$20 for groups and $d\sigma/d\Omega$21 for clusters at 95% CL from Sagunski et al., together with $d\sigma/d\Omega$22 from cluster-shape constraints following Peter et al. Because fSIDM produces stronger core formation and stronger shape rounding than rSIDM at fixed $d\sigma/d\Omega$23, these are conservative upper bounds for fSIDM (Fischer et al., 2022).

For velocity-dependent models, Sabarish et al. generalize the matching procedure through an effective cross-section construction and derive central-density-matched upper bounds for fSIDM: $d\sigma/d\Omega$24, $d\sigma/d\Omega$25, $d\sigma/d\Omega$26, $d\sigma/d\Omega$27, and $d\sigma/d\Omega$28 for $d\sigma/d\Omega$29, $d\sigma/d\Omega$30, $d\sigma/d\Omega$31, $d\sigma/d\Omega$32, and $d\sigma/d\Omega$33, respectively. The corresponding rSIDM values are $d\sigma/d\Omega$34, $d\sigma/d\Omega$35, $d\sigma/d\Omega$36, $d\sigma/d\Omega$37, and $d\sigma/d\Omega$38 (Sabarish et al., 2023). These numbers make explicit that matched core evolution does not imply equal microscopic normalization across angular regimes.

A central result of the cosmological literature is that single probes are degenerate. Density cores, halo shapes, or satellite counts can each be matched by rare and frequent scattering at different cross sections. The degeneracy can be broken by combining multiple observables: satellite abundance as a function of radius and $d\sigma/d\Omega$39, inner density slope or core size, and halo shape at $d\sigma/d\Omega$40. In the simulations of Fischer et al., for a given number of satellites within $d\sigma/d\Omega$41, rSIDM predicts lower central densities and rounder shapes than fSIDM (Fischer et al., 2022). This suggests that the angular dependence of the dark-matter self-interaction cross section is, in principle, observable.

Recent work has extended the framework beyond purely elastic single-species models. The public OpenGadget3 SIDM implementation now supports strongly anisotropic cross sections and two-species interactions, while emphasizing remaining numerical challenges such as time-step constraints and the difficulty of GPU acceleration for lock-based pairwise updates (Fischer et al., 10 Mar 2026). Dissipative extensions show that cooling can qualitatively change gravothermal evolution: sufficiently strong central cooling can suppress isothermal-core formation, keep conduction directed inward, and shorten the collapse time according to

$d\sigma/d\Omega$42

with $d\sigma/d\Omega$43 and $d\sigma/d\Omega$44 (Schmidt et al., 17 Jun 2026). In that framework, weakly dissipative self-interactions with $d\sigma/d\Omega$45 reduce the required cross section or evolution time for the strong lens perturber in JVAS B1938+666 by $d\sigma/d\Omega$46 (Schmidt et al., 17 Jun 2026).

The satellite problem has also been reformulated with environmental scattering treated explicitly. Using virtual host particles and Eddington-inverted host velocity distributions, recent simulations find that scattering-induced subhalo–halo interactions strongly reshape the evolution of SIDM satellites; in the forward-dominated limit, subhalos can develop larger cores, later collapse, and on the closest orbits complete tidal disruption (Klemmer et al., 19 Mar 2026). Together with the cosmological and cluster studies, this reinforces a broad conclusion: fSIDM is not a merely technical limit of SIDM, but a transport regime with its own hierarchy of observables, numerical methods, and constraints.