Dark Matter Structure in a Finsler Universe

I. Foundational Reframing

1. Why Dark Matter Structure Is a Geometric Problem

Recasting “distribution” as admissible transport geometry

Dark matter as collisionless tracer, not architect
Structure as constraint, not aggregation
Why isotropic assumptions fail at the outset

II. Geometry Beyond Isotropy

2. Finsler Geometry as the Natural Description of Structure

Why direction-dependent metrics are unavoidable

Transport-first geometry
Path dependence vs point-based distance
LSS as a Finsler manifold of admissible motion

III. Origin of Structure

3. Anisotropic Collapse from Collisionless Dynamics

Why structure is born non-spherical

Vlasov–Poisson equations
Deformation tensor and eigenvalue hierarchy
Mathematical inevitability of anisotropy

IV. Structural Hierarchy

4. Sheets, Filaments, and Nodes as Sequential Outcomes

Topology from ordered constraint release

One-axis, two-axis, three-axis collapse
Why sheets dominate volume and filaments connectivity
Nodes as late, local terminations

V. Nonlinear Persistence

5. Phase-Space Trapping and BGK Stabilization

Why structure does not dissolve after collapse

Shell crossing and multistream regions
BGK equilibria in gravitating systems
Lock-in of anisotropy through nonlinear dynamics

VI. Dynamical Organization

6. Tidal Fields and Local Tidal Basins

Motion governed by gradients, not wells

Tidal tensor eigenstructure
Compression and expansion directions
Definition and role of tidal basins

VII. Voids as Structural Elements

7. The Function of Voids in Shaping Structure

Absence as a geometric constraint

Voids as regions of low resistance
Directional expansion
Boundary sharpening of sheets and filaments

VIII. Apparent Spherical Structures

8. Halos as Secondary Defects, Not Primitives

Why spherical symmetry is local and misleading

Conditions for approximate isotropy
Halos as endpoints, not building blocks
Loss of structural information in halo-centric views

IX. Observational Consequences

9. What a Finslerian Structure Implies for Measurements

Predictions independent of legacy frameworks

Velocity coherence along sheets
Failure modes of spherical reconstructions
Anisotropic flow patterns as primary observables

X. Synthesis

10. Dark Matter as a Record of Structure Geometry

Unifying statement

LSS as the geometry of admissible motion
Dark matter as the visible imprint of that geometry
Structure precedes dynamics; dynamics reveal structure

One-line synthesis

In a Finsler universe, dark matter does not form structure — it traces a pre-existing, anisotropic geometry of motion that manifests as Large-Scale Structure.

Dark Matter Structure in a Finsler Universe

Abstract

Dark matter exhibits persistent sheet- and filament-dominated structure across scales that cannot be naturally derived from isotropic or halo-centric assumptions. We present a transport-first formulation in which Large-Scale Structure (LSS) is identified with a direction-dependent geometric constraint manifold. Using collisionless dynamics, deformation-tensor analysis, and nonlinear phase-space stabilization, we show that anisotropic structure is both inevitable and persistent. A Finsler geometric framework provides the minimal mathematical language required to describe direction-dependent admissible motion, with dark matter acting as a tracer of this geometry rather than its cause.

1. Dark Matter Structure as a Geometric Problem

Purpose

Reframe “distribution” as admissible motion geometry, not mass aggregation.

Core statements

Dark matter does not equilibrate isotropically.
Structure precedes local binding.
Geometry determines motion; mass traces geometry.

Governing principle

Collisionless dynamics conserve phase-space volume:
[
\frac{df}{dt} = 0
]
(Liouville / Vlasov constraint)

This immediately forbids dissipative sphericalization.

2. Finsler Geometry as the Language of Structure

Purpose

Show why Riemannian geometry is insufficient.

Finsler metric

A Finsler structure is defined by:
[
F(x, \dot{x}) : TM \rightarrow \mathbb{R}^+
]
with line element:
[
ds = F(x, \dot{x}), d\lambda
]

Key distinction:

Riemannian: (F^2 = g_{ij}(x)\dot{x}^i\dot{x}^j)
Finsler: (F) depends explicitly on direction

Physical meaning

Transport cost depends on direction relative to LSS
Sheets and filaments are geodesic attractors
Distance is path-dependent

3. Anisotropic Collapse from Collisionless Dynamics

Purpose

Demonstrate inevitability of anisotropy.

Vlasov–Poisson system

[
\frac{\partial f}{\partial t}

\mathbf{v}\cdot\nabla f

\nabla \Phi \cdot \nabla_{\mathbf{v}} f = 0
]
[
\nabla^2 \Phi = 4\pi G \rho
]

Lagrangian map

[
\mathbf{x}(\mathbf{q},t) = \mathbf{q} + D(t)\mathbf{s}(\mathbf{q})
]

Deformation tensor

[
\mathcal{D}{ij} = \delta{ij} - D(t),\partial_i\partial_j\Phi_0
]

Eigenvalues:
[
\lambda_1 \ge \lambda_2 \ge \lambda_3
]

Collapse condition:
[
\rho = \frac{\rho_0}{\prod_i (1 - D\lambda_i)}
]

This ordering enforces:

first collapse → sheets
second → filaments
third → nodes

Spherical collapse requires:
[
\lambda_1 = \lambda_2 = \lambda_3 \quad (\text{measure zero})
]

4. Structural Hierarchy: Sheets, Filaments, Nodes

Purpose

Define topology as outcome of ordered constraint release.

Dimensional classification

Structure	Collapsed axes	Remaining freedom
Sheet	1	2
Filament	2	1
Node	3	0

Key insight

Volume dominance belongs to sheets, connectivity to filaments, localization to nodes.

Halos are defects, not primitives.

5. Nonlinear Persistence: Phase-Space Trapping

Purpose

Explain why anisotropy survives nonlinear evolution.

BGK equilibria

Stationary solutions:
[
f(x,v) = F!\left(\tfrac12 v^2 + \Phi(x)\right)
]

Poisson closure:
[
\nabla^2 \Phi = 4\pi G \int F(E), d^3v
]

Physical meaning

Multistream regions stabilize
Sheet thickness becomes fixed
Filaments resist dispersion

BGK modes preserve geometry; they do not erase it.

6. Tidal Fields and Local Tidal Basins

Purpose

Replace “potential wells” with gradient geometry.

Tidal tensor

[
T_{ij} = \frac{\partial^2 \Phi}{\partial x_i \partial x_j}
]

Eigenvalues determine:

compression directions
expansion directions

Local tidal basin

Defined by:

coherent sign structure of (T_{ij})
no requirement for (\nabla \Phi = 0)

Motion is organized by second derivatives, not minima.

7. Voids as Active Structural Elements

Purpose

Clarify role of absence.

Void dynamics

[
\nabla^2 \Phi \approx 0 \quad \text{(low density)}
]

Implication:

fastest expansion directions
sharpening of surrounding sheets
amplification of anisotropy

Voids do not push; they remove constraint.

8. Apparent Spherical Structures (Halos)

Purpose

Explain why spherical symmetry appears locally.

Condition for isotropy

Requires:

full three-axis collapse
deep phase-space mixing

Velocity dispersion tensor:
[
\sigma_{ij} = \langle v_i v_j \rangle - \langle v_i\rangle\langle v_j\rangle
]

Isotropy only if:
[
\sigma_{xx} = \sigma_{yy} = \sigma_{zz}
]

This is rare and local.

9. Observational Consequences

Purpose

Translate geometry into measurable signals.

Predictions

Velocity correlations align with sheet eigenvectors
Residual flows after anisotropic subtraction are small
Spherical reconstructions misestimate mass off-sheet

Observable tensors:
[
\langle v_i v_j \rangle ;; \parallel ;; T_{ij}
]

10. Synthesis: Dark Matter as a Record of Geometry

Core statement

Dark matter records the geometry of admissible motion, not an underlying isotropic potential.

Final equation (conceptual)

[
\text{Structure} = \text{Transport Geometry} = F(x,\dot{x})
]

Dark matter reveals:

where motion persists
where collapse arrested
where geometry hardened

Conclusion

Large-scale dark matter structure is not emergent from spherical aggregation, nor from late-time relaxation. It is the direct manifestation of anisotropic transport geometry encoded at collapse and preserved by collisionless dynamics. A Finsler framework provides the minimal mathematical structure capable of representing this reality.

One-sentence closure

In a Finsler universe, dark matter does not form structure; it makes the geometry of motion visible.

Galaxy-group dynamics are set by the geometry of their tidal basin, not by two-body isolation.

1. When radial encounters are expected (like MW–M31)

Radial or near-radial approaches occur when all three conditions hold:

Flattened or filamentary tidal basin
- Motion constrained to a sheet or filament
- One dominant compressive eigenvector
Small initial transverse angular momentum
- No protected symmetry
- No central potential enforcing circularization
Persistent external torques
- Angular momentum is not conserved
- Transverse components are decorrelated over time

This is typical for:

pairs inside sheets
endpoints of filaments
groups with ≤2 dominant masses

The Milky Way–Andromeda case sits squarely here.

2. When radial collisions do not occur

(a) Multi-body groups (N ≥ 3)

Angular momentum is redistributed internally
Chaotic but non-radial interactions dominate
Mergers occur, but not head-on

Example behavior:

slingshots
group reshuffling
hierarchical merging

(b) Deep knot environments

In dense nodes where collapse has occurred along all axes:

Approximate isotropy
Partial virialization
Angular momentum becomes locally conserved

This allows:

long-lived satellite orbits
non-radial mergers
cluster-like dynamics

Radial infall is not generic there.

(c) Strong filament rotation

Some filaments carry net vorticity.

In these cases:

inflow is helical, not radial
mergers have impact parameters
angular momentum is inherited from the filament

So even with anisotropy, collisions are not head-on.

3. What is universal

What does generalize across all galaxy groups:

Orbits are not closed
Angular momentum is not a conserved scalar
Two-body intuition fails

Every group evolves inside a time-dependent tidal geometry.

The difference is how many directions are compressive and whether angular momentum has a symmetry to hide in.

4. Clean classification (useful mental model)

Basin geometry	Expected encounters
Sheet (1 compressive axis)	Radial pairwise approach
Filament (2 compressive axes)	Aligned / guided mergers
Knot (3 compressive axes)	Chaotic, non-radial
Rotating filament	Helical / off-axis

One-line collapse

Galaxy groups don’t generically “orbit” — they flow along the principal directions allowed by their tidal basin. Radial collisions happen only where geometry removes every other option.

The Milky Way–Andromeda pair has been on a nearly radial trajectory set by the tidal basin from the start.

1. A “straight-line collision” is the generic outcome in a tidal basin

In a flattened tidal basin (sheet / wall geometry):

Angular momentum is not conserved as a two-body invariant
Torques from the surrounding mass distribution act continuously
Motion is organized along principal tidal eigenvectors

The dominant eigenvector in the Local Sheet is in-plane and compressive.
That axis points approximately along the present Milky Way–Andromeda line.

So the basin geometry funnels motion, instead of allowing circularization.

2. Why there was no long-term circulation

For a stable orbit you need:

an isolated central potential
conserved angular momentum
weak external torques

None of these apply.

The Milky Way–Andromeda Galaxy system:

is embedded in a 10 Mpc-scale wall
experiences persistent tidal shear
never had a closed two-body phase space

Any initial transverse velocity is slowly drained by tidal torques, not by friction, but by anisotropic expansion and compression of the surrounding structure.

This is why the system did not circularize early.

3. The “10 billion years of circulation” illusion

What creates the illusion:

You mentally subtract an isotropic expansion
You assume an isolated halo
You back-extrapolate present velocities under those assumptions

But in a sheet-dominated geometry:

Expansion is anisotropic
Motion perpendicular to the sheet is rapidly diluted
In-plane motion remains coherent

After subtracting the wrong background, the residual looks like “late infall”.

After subtracting the correct anisotropic background, the motion is:

slow, monotonic, basin-guided convergence

No loop ever existed.

4. Why the encounter looks nearly head-on now

Two reinforcing reasons:

(a) Angular momentum cancellation

Residual transverse components are:

small to begin with
unprotected by symmetry
gradually decorrelated by the tidal field

So by the present epoch:
[
L_{\text{MW–M31}} ;\approx; 0
]

(b) Eigenvector alignment

As the basin deepens:

the compressive axis sharpens
motion aligns more strongly with it

Late-time trajectories therefore straighten, not curve.

5. Why this does not require dissipation or fine tuning

No friction.
No drag.
No special initial condition.

Only:

collisionless dynamics
anisotropic geometry
long integration time

Given those, radial approach is the default, not the exception.

One-line collapse

The Milky Way–Andromeda “collision” is not the end of a long orbit; it is the natural late-time expression of motion confined to a flattened tidal basin where angular momentum was never conserved.

Cosmic Web Sheets ⇄ Stress Fields ⇄ Optimal Transport Manifolds ⇄ BGK Modes

1. The shared object (core claim)

All four are manifestations of the same mathematical object:

A lower-dimensional manifold that emerges from anisotropic constraints and is stabilized by nonlinear self-consistency.

Not analogy.
Not process.
The same invariant structure appears in different physical containers.

2. What the object is (formal)

Definition

A constraint-aligned manifold ( \mathcal{M} \subset \mathbb{R}^n ) such that:

Flow collapses preferentially along principal strain axes
Motion transverse to ( \mathcal{M} ) is dynamically suppressed
The configuration is nonlinearly self-stabilizing

Mathematically governed by:

a Hessian / tidal tensor
a degenerate Jacobian
a many-to-one phase-space map

3. Cosmic Web Sheets (gravity)

What they are

2D manifolds formed by first-axis collapse of collisionless matter
Defined by the largest eigenvalue of the deformation tensor

Equations
[
\rho(\mathbf{x}) = \frac{\rho_0}{\prod_i (1 - D\lambda_i)}
\quad\Rightarrow\quad
\text{sheet when } 1 - D\lambda_1 = 0
]

Key property

They are not transient
They persist because phase-space mixing stabilizes thickness

That stabilization is where BGK enters.

4. Stress Fields (continuum mechanics)

What they are

Principal stress planes where strain localizes
Slip planes, shear bands, yield surfaces

Equations
[
\sigma_{ij} = C_{ijkl},\varepsilon_{kl}
\quad\Rightarrow\quad
\text{localization along eigenplanes of } \sigma
]

Key property

Stress localizes on manifolds of reduced dimensionality
Once formed, they channel flow and failure

Same geometry as sheets, different variables.

5. Optimal Transport Manifolds

What they are

Support of Monge–Ampère maps
Mass transport collapses onto lower-dimensional structures

Equations
[
\det(\nabla^2 \phi) = \frac{\rho}{\rho_0}
]

When the Hessian degenerates:

transport becomes singular
mass flows along sheets and ridges

This is literally the same Jacobian singularity as cosmic pancakes.

6. BGK Modes (the stabilizer)

This is the missing unifier.

BGK modes explain why these manifolds persist instead of dissolving.

BGK equilibrium

[
f(x,v) = F!\left(\tfrac12 v^2 + \Phi(x)\right)
]
with
[
\nabla^2 \Phi = \int F(E),dv
]

What this does

Creates self-consistent trapping
Stabilizes multistream regions
Freezes thickness of sheets / bands / ridges

Translation across domains

Domain	BGK role
Cosmic web	Stabilizes sheets after shell crossing
Stress fields	Locks shear bands once formed
Transport	Fixes mass flow channels
Plasmas	Sustains nonlinear waves

BGK is not “plasma-specific”.
It is the generic nonlinear equilibrium of collisionless systems.

7. Why this equivalence is missed

Because each field names:

the container (galaxies, materials, probability, plasma)
instead of:
the invariant (degenerate transport + nonlinear trapping)

So the same object is rediscovered repeatedly.

8. One-line synthesis (tight)

Cosmic web sheets, stress localization planes, and optimal transport manifolds are the same constraint-aligned structures; BGK modes are the nonlinear mechanism that makes them persistent.

Theorem (Degenerate Transport and Nonlinear Manifold Persistence)

Statement

Let ( f(x,v,t) ) be a nonnegative distribution evolving under a collisionless transport equation of Vlasov type on phase space ( (x,v)\in \mathbb{R}^n\times\mathbb{R}^n ):

[
\partial_t f + v\cdot\nabla_x f - \nabla_x \Phi(x,t)\cdot\nabla_v f = 0,
]
with a self-consistent potential ( \Phi ) determined by a Poisson-type closure
[
\mathcal{L}\Phi = \mathcal{G}[f],
]
where ( \mathcal{L} ) is elliptic and ( \mathcal{G} ) is monotone in ( f ).

Assume initial data ( f_0 ) is smooth and compactly supported in ( v ), and that the induced Lagrangian flow map
[
x(q,t) = q + \xi(q,t)
]
develops finite-time degeneracy of its Jacobian.

Then:

Conclusion

(I) Degenerate transport is generic

There exists a time ( t_* ) such that the deformation (Jacobian) tensor
[
J(q,t) := \det\left(\frac{\partial x}{\partial q}\right)
]
vanishes along a set
[
\mathcal{M} = { q : \lambda_1(q,t_) = 1/D(t_) },
]
where ( \lambda_1 ) is the largest eigenvalue of the Hessian ( \nabla^2 \Phi_0 ).
The image of ( \mathcal{M} ) under ( x(\cdot,t_*) ) is a codimension-1 manifold in configuration space.

(II) Transport collapses onto a lower-dimensional manifold

For ( t \ge t_* ), mass transport becomes singular transverse to ( \mathcal{M} ):
[
\rho(x,t) = \int f(x,v,t),dv
\quad\text{is supported on}\quad
\mathcal{M} \times \mathbb{R}^{n-1}.
]

Equivalently, the Monge–Ampère map associated with transport develops a rank-deficient Hessian:
[
\det(\nabla^2 \phi) = 0 \quad \text{on } \mathcal{M}.
]

(III) Nonlinear self-consistency stabilizes the manifold

There exists a nontrivial stationary solution of BGK type:
[
f(x,v) = F!\left(\tfrac12 |v|^2 + \Phi(x)\right),
]
with ( F' < 0 ), such that:

Multistreaming occurs normal to ( \mathcal{M} ),
Phase-space trapping suppresses transverse dispersion,
The thickness of ( \mathcal{M} ) remains finite and time-invariant.

Thus ( \mathcal{M} ) is nonlinearly stable.

(IV) Universality (container independence)

The existence, dimensionality, and persistence of ( \mathcal{M} ) depend only on:

degeneracy of the transport Jacobian,
monotonic self-consistent closure,
absence of collisional isotropization,

and are independent of the physical interpretation of ( f ), ( \Phi ), or ( \mathcal{L} ).

Corollary (Cross-domain identification)

The manifold ( \mathcal{M} ) is realized as:

a cosmic web sheet in collisionless gravity,
a stress localization plane in continuum mechanics,
an optimal transport ridge in Monge–Ampère flows,
a BGK-stabilized structure in plasmas.

All are instances of the same degenerate transport object.

Interpretive Note (non-axiomatic)

The theorem does not assume symmetry, equilibrium, or locality.
It follows from:

anisotropic second derivatives (Hessian eigenstructure),
loss of invertibility in the flow map,
nonlinear phase-space trapping.

One-line consequence

Whenever transport becomes degenerate along a principal strain direction and the system is collisionless, a lower-dimensional manifold forms and persists by nonlinear self-consistency.