Department of Physics · UDESC

Seminar · 2026

Dark matter in neutron stars

Simulating
the invisible

How a neutron star captures dark matter one collision at a time — and what state the captured particles end up in.

Outline

Two papers, one code

I

OBSCURA: every collision

A relativistic Monte Carlo of dark-matter capture. Each particle follows the interior geodesic of the star, collides, changes direction and energy — and the code reduces exactly to the optical-depth closure when that extra physics is switched off. What does the closure miss?

de Lima & Rueda, A Monte Carlo relativistic framework for dark matter capture by neutron stars (in preparation)

II

Degeneracy before self-gravity

After capture: is the population a classical thermal tracer, a degenerate Fermi gas, or a self-gravitating dark component? Two independent diagnostics — S for gravity, Θ for statistics — and a map of where their thresholds fall.

de Lima & Rueda, Degeneracy Before Self-Gravity: Dark Matter Regimes in Neutron Stars (draft)

The premise

A neutron star is a dark-matter detector

As it crosses the Galaxy, a neutron star sweeps up dark matter: its potential well focuses the particles, its matter stops them.

halo→focusing→collisions→capture→accumulation→regime→heating · collapse · structure

A deep well

Escape speed at the surface √(2GM/R) = 0.59 c for 1.4 M☉ and 11.7 km. Slow halo particles are bent toward the star from a thousand radii away.

A dense target

n_b ≈ 5×10³⁸ cm⁻³ at the centre: with σ = 10⁻⁴⁴ cm² a particle meets τ ≈ 6–9 collisions in one crossing.

An old, cold star

What accumulates can heat it, collapse into a black hole, or change its structure — if we know how much is captured and in what state.

I

Part one

OBSCURA:
every collision

A relativistic Monte Carlo that follows each dark-matter particle through the interior metric of the star, collision by collision — and recovers the optical-depth closure when its extra physics is switched off.

the dark-matter backgroundfocusingthe optical-depth closurethe transportwhat the closure missescount lawscooling

Bullet Cluster, 1E 0657−56 · NASA/CXC/M. Weiss · public domain

1933 – 2006

Matter we only see through gravity

Fritz Zwicky, 1960 — **Fritz Zwicky**1933: the Coma cluster moves too fast for its visible mass — *dunkle Materie*

**Vera Rubin**1970s: rotation curves of spiral galaxies stay flat far beyond the light

2006, the Bullet Cluster: the lensing mass (blue) is offset from the X-ray gas (pink). Today: about 85% of the matter, a local density ρ_χ ≈ 0.4 GeV cm⁻³ — and no idea what it is made of.

The dark-matter background

Two ways to fill the halo

Standard halo · Maxwell–Boltzmann

f(v) ∝ v² e^{−v²/v₀²}, v < v_escv₀ = 220 km/s, v_esc = 544 km/s

The usual WIMP benchmark: particles of GeV to TeV, a few hundred km/s, isotropic. Used for the Maxwell–Boltzmann results of Part II.

RAR fermions · truncated Fermi–Dirac

f ∝ 1 − e^{(E − E_c)/kT}e^{(E − μ)/kT} + 1, E ≤ E_cRuffini–Argüelles–Rueda

Self-gravitating fermions in general-relativistic equilibrium: a degenerate core at the Galactic centre and a Boltzmann halo. Rotation curves and the S-stars point to ~60–380 keV; we take 100 keV.

Forty years of capture

From the Sun to neutron stars

1985

Press & Spergel
capture of WIMPs by the Sun

1987

Gould
the capture-rate formalism

1989

Goldman & Nussinov
WIMPs and neutron stars

2008–10

Kouvaris; Kouvaris & Tinyakov
heating by annihilation; collapse to a black hole

2017

Baryakhtar et al.
dark kinetic heating

2020

Bell et al.
relativistic capture with Pauli blocking, opacity, multiple scattering

2026

OBSCURA
collision histories through the interior metric

A short history. Press and Spergel computed capture by the Sun in 1985; Gould built the formalism of the capture rate two years later; Goldman and Nussinov applied it to neutron stars in 1989. Around 2008 to 2010 Kouvaris and Tinyakov turned it into constraints: annihilation heating, and collapse of the captured dark matter into a black hole. In 2017 Baryakhtar and collaborators proposed kinetic heating — the energy deposited by the capture itself. And around 2020 Bell and collaborators put the relativistic kinematics, Pauli blocking, opacity and multiple scattering into one capture treatment. What OBSCURA adds is not a new ingredient in that list; it is to follow individual collision histories through the metric, and use them to test the deterministic approximations of an optical-depth closure.

Gravitational focusing

The star looks a thousand times wider

Schwarzschild geodesics · livereaches the surfacemisses2GM/c²

compactness GM/Rc²

speed at infinity (drawing)

b_maxR = 1γv √(γ² − 1 + 2C) / (1 − 2C)from ℒ < ℒ_c(ℰ): the orbit must reach R

The first effect is purely gravitational. A particle from infinity with speed v has conserved energy gamma and angular momentum gamma v b; it reaches the surface only if its angular momentum is below a critical value set by the metric at R. That gives the maximum impact parameter on the right. On the left, real Schwarzschild geodesics: violet ones hit, grey ones miss, and the bars on the left compare the relativistic capture radius with the Newtonian one. The drawing runs at a speed you can see; at the halo speed, 220 km/s, b_max is a thousand stellar radii — the readout gives 1.01 times ten to the three — and general relativity makes it 24 per cent larger than Newton, the factor one over the square root of one minus 2C. Drag the compactness: the more compact the star, the larger the gain.

Supply × probability

What reaches the star, and what stays

ds² = −e^ν(r)dt² + e^λ(r)dr² + r²dΩ²the TOV star: SLy, 1.40 M☉, 11.70 km, C = 0.177

d²Ṅ_indℰ dℒ = 8π² m³ f(ℰ) ℒ, ℒ < ℒ_c(ℰ)particles entering the surface, per unit coordinate time

Ṅ_χ = 8π²m³ ∫∫ P_cap(ℰ, ℒ) f(ℰ) ℒ dℒ dℰthe capture rate: the kernel P_cap is what OBSCURA computes

ℰ and ℒ are the energy and angular momentum per unit mass, measured at infinity; between collisions they are conserved, and the particle follows the interior geodesic.

Captured

ℰ_final = γ_final e^ν(r_coll)/2 < 1

after a collision the particle can no longer reach infinity: γ is its Lorentz factor for a static observer at the collision, e^ν/2 the lapse there.

The optical-depth closure

A Poisson count and three frozen choices

P_cap = 1 − e^−τ Σ_{k < N_min} τ^kk!the number of collisions is Poisson with mean τ

τ = 2 ∫_{r_tp}^R e^(ν+λ)/2 Γ_nχ√(ℰ² − V_eff²) drcollisions along the incident geodesic, one crossing

The closure of Sirvente & Rueda (submitted): relativistic and covariant. OBSCURA was built to test it, not to replace it — and reduces to it exactly.

1 · Frozen trajectory

Collisions change the energy, never the path: every particle keeps the incident geodesic.

2 · Deterministic loss

Each collision removes a fixed amount — head-on or the isotropic mean — so capture needs exactly N_min of them.

3 · Evaluated at the surface

N_min is computed with the lapse at R, wherever the collision actually happens.

100 keV on a neutron of 939 MeV

Capture decided in a narrow window

N_min from the event kinematics · livehead-on, surface lapseisotropic mean, surfacehead-on, central lapse

dark-matter mass

energy, 10³(ℰ − 1)

TK = m_nm_χ(γ + 1)(1 − cos θ*)skinetic energy given to the neutron, exact

For m_χ ≪ m_n each collision removes a fraction ~2(m_χ/m_n)(γ + 1) ≈ 5×10⁻⁴ at most, while it must lose up to a per cent of it: tens of collisions.

Deeper in the star the lapse is smaller and the same collision removes more of ℰ: N_min halves at the centre.

Slide the mass to the GeV range: one collision suffices — the WIMP limit.

Why 100 keV is the hard case. In an elastic collision on a neutron at rest, the fraction of kinetic energy the neutron takes is given exactly by the formula on the right, with s the invariant mass squared. For a 100 keV particle that is at most five parts in ten thousand. Meanwhile, to be captured the particle must lose about one per cent of its kinetic energy at the surface. So the closure needs N_min of tens — the red and orange staircases: head-on losses and isotropic-mean losses, both with the surface lapse. The dashed curve is the same head-on loss evaluated with the lapse at the centre: the same collision removes more conserved energy where the well is deeper, and N_min halves. When capture is decided by tens of small steps, the distribution of the number of collisions — and where they happen — is the whole story. Move the mass slider up to a GeV: N_min drops to one, and none of this matters.

The transport

One history, collision by collision

Draw a target optical depth −ln U and fly along the interior geodesic with (ℰ, ℒ) conserved, accumulating dτ = n_bσ dl (step R/4096).
At the collision, an elastic two-body event on a neutron at rest, isotropic in the centre of mass: energy loss and deflection from the same angles.
New γ and direction → new (ℰ, ℒ): the orbit is rebuilt.
Captured if ℰ < 1; escaped if it leaves unbound; otherwise fly again.

T = p*²m_n(1 − cos θ*), p* = m_np_χ√sexact, from the invariant t

Every approximation has a name

GEO-8 interior tetrad transport · COL-3 relativistic event · PAULI-1 blocked rate · STAT-1 Wilson intervals — each documented, cited in the source, and switchable.

Switch off the release, the sampling and the local lapse, and the Monte Carlo is the closure.

What one history is. Draw a target optical depth, minus log of a uniform number, and fly along the interior geodesic until you accumulate it; the production step is R over 4096. At the collision, a relativistic elastic event on a neutron at rest, isotropic in the centre of mass. Energy and direction come from the same pair of angles, so a collision that removes little energy also deflects little — they are correlated as in the physical event. The new Lorentz factor and direction give a new energy and angular momentum, and a new orbit. Then the capture test. Every physical approximation in the code has an identifier, documented in one file and cited in the source where it applies — a discipline we adopted after the code misdescribed its own method once. And there are switches to undo each released ingredient, which is how we recover the closure exactly.

Three modes, one star · live

Releasing the trajectory broadens the transition

capturedescaped

energy drawn, 10³(ℰ − 1)

ℒ/ℒ_c

analytic closureclosure realised (F_iS)loss at the collision (F_sL)transport (R_sL)

This panel runs the transport in the browser, live: same star, 100 keV, ten to the minus forty-four, no Pauli blocking; the step here is R over 300 instead of R over 4096, so it is a re-run, not the production numbers. Three modes, with the names of the paper. Grey: the closure realised by the Monte Carlo — frozen trajectory, isotropic-mean loss at the surface lapse. It sits on the dotted analytic Poisson closure: that is the reduction test, live. Pink: still frozen, but the loss is sampled and evaluated at the collision radius — the transition moves to higher energy, because deep collisions remove more conserved energy. Orange: the full transport — the transition broadens: fewer captures at low energy, more in the tail. On the left, the histories themselves at the energy you choose; switch between closure and transport and look at what the collisions do to the path.

Reduction

Switch it off, and it is the closure

switch off	recovers
trajectory update at collisions	the incident geodesic of τ
sampled energy loss	the recurrence behind N_min
loss at the collision radius	the cascade at the surface lapse
histories beyond one crossing	the single-crossing convention
neutrons at rest in the rate	not switched: up to 0.035 in P_cap

N_min

every capture of the realised closure happens at exactly N_min

0.2%

mean count vs σ∫n_bdl by an independent quadrature

2.8σ

largest deviation of P_cap from the Poisson tail, 123 points × 5000 histories

10⁻¹⁰

a sampled event vs an independent closed form; head-on = one step of the recurrence

The decomposition · 2³ × 2 controls, order-averaged

Two effects of opposite sign

Decomposition of the capture probability — Benchmark star, 100 keV, σ = 10⁻⁴⁴ cm², 5000 histories per point, no blocking.

switch	mean ΔP_cap
● evaluation point	+0.09 … +0.14
● loss dispersion	\|·\| < 0.01
● trajectory	−0.09 … −0.11
total (signed mean)	+0.00 … +0.03
total, RMS over the transition	0.16 … 0.17

They nearly cancel in the mean but not in shape: the first shifts the transition up, the second broadens it. Single energies differ by up to ±0.29.

An attribution with interactions, not three independent corrections.

The production result, on the benchmark star of the closure. We run all eight combinations of the three switches, for both deterministic variants, and attribute to each switch its increment averaged over the six orders in which they can be applied, so the three add up exactly to the total. Evaluating the loss at the collision raises the capture probability by 0.09 to 0.14 on average over the transition. Releasing the trajectory lowers it by 0.09 to 0.11. The dispersion of the loss about its mean does almost nothing on average. The two big effects nearly cancel in the signed mean — but the RMS difference stays at 0.16, 0.17, and single energies differ by almost 0.3. The closure gets the average right for the wrong reason. The bands in the lower panels are the order dependence, not error bars: the switches interact.

Mechanism · what the first collision does

Half the particles turn back

48–53%

leave the first collision moving outward — frozen chord: 2–8%

≈ 3.1

mean column left to the surface, spread over 0.2–6.6 (10–90%) — frozen: 4.4–6.5, narrow

16–28%

never collide again — frozen: at most 3%

After the first collision: direction, remaining column, further collisions — Frozen (F_sL) vs released (R_sL) after the first collision, at 10³(ℰ − 1) ≈ 1.8, for ℒ/ℒ_c = 0.1 and 0.9.

Why does releasing the trajectory lower the capture probability? The histories let us measure it instead of guessing. For a 100 keV particle hitting a neutron, the deflection is large — the light particle bounces off a heavy wall. After the first collision, half of the particles are moving outward; on the frozen chord, almost none. The column left to the surface halves on average, and spreads enormously. A fifth of the released particles never collide again, against three per cent on the frozen chord, while a tail of particles that turn inward collide many more times. So the released trajectory makes the number of collisions more often small and occasionally large: the first lowers capture where few collisions are needed, the second raises it in the tail. The orbit also contracts after each loss, which alone would raise capture; at 100 keV the change of direction wins.

How many collisions · live

A count that is not Poisson

ℒ/ℒ_c

transportPoisson, same meannegative binomial, same mean and variance

energy, 10³(ℰ − 1) — above the transition, few captures

Above the transition almost nothing is captured, so each count is complete: the history ends when the particle leaves. The closure assumes Poisson with the mean τ. The mean is right; the shape is not.

P_NB(k; r, p), r = m²v − m, p = mvPoisson count with a Gamma-distributed τ

In the paper: variance 8–13 times the mean. This is a random walk out of an optically thick sphere, not a straight chord.

Count-law closures · calibrated and tested out of sample

Better laws, honest residuals

Count-law closures tested out of sample — SLy 1.40 M☉, 11.70 km; calibrated above the transition on independent runs, tested inside it.

mean \|prediction − MC\|	our SLy	benchmark
Poisson, measured mean	0.15–0.19	0.16–0.18
negative binomial	0.08–0.11	0.057–0.064
zero-and-one-inflated NB	0.035–0.057	0.044–0.064

Calibration and test share no histories. The zero-and-one-inflated form removes the spurious zero-collision mass of the plain NB.

Residuals stay 6–23 times the combined Monte Carlo and calibration errors: closed forms that help, tested at fixed parameters — not yet a transferable law.

A more compact star · SLy at maximum mass, 2.05 M☉

Broadening is not a detail

Transport against the analytic closure on the maximum-mass SLy star — Points: the transport. Red: closure with head-on N_min; orange dashed: isotropic-mean N_min. 100 keV, σ = 10⁻⁴⁴ cm², no blocking.

Optical depth of the closure: τ ≈ 25–27, three times the canonical star.

The transport already falls below 0.95 at 10³(ℰ − 1) = 0.1–0.2, and keeps 11–14% at 20, where both closures vanish.

Mean |difference| from the isotropic-mean closure: 0.28–0.29; single energies up to 0.6–0.7.

Same five particles · different stars

The equation of state decides the fate

The same five incident particles in an SLy and an SQM3 star — Identical ℰ, ℒ, speed and impact parameter in both panels; σ = 10⁻⁴⁴ cm². Squares: entry; dots: collisions; crosses: escape.

Five incident particles, the same phase-space points, sent into the maximum-compactness stars of a hadronic (SLy) and a strange-quark (SQM3) equation of state.

The same orbit crosses a different column and collides at a different depth: five captures against none.

Selected cases, not an ensemble average — an illustration of where the EOS dependence of P_cap comes from.

Beyond the benchmark

Pauli blocking and cooling

Pauli blocking · implemented, not yet in the campaigns

The recoiling neutron must leave the Fermi sea: the blocked optical depth is 7×10⁻⁵ of the unblocked one.

But the collisions that do happen are violent: on average 57% of the kinetic energy, against 2×10⁻⁴ for a static target.

So a blocked τ can never be paired with a static N_min: both must come from the same blocked kernel. The hierarchy has to be measured again, not rescaled.

Rate tabulated in (ln n_b, ln(γ − 1)) to 10⁻³, the Fermi-surface shell of width ~3×10⁻⁴ resolved by bisection.

Cooling to 3kT/2 · after capture

	10⁷ K	10⁸ K
collisions to capture	5	5
collisions to the threshold	4494	1374
re-entries into the star	396	191
coordinate time	5.1 s	6.8 s
time spent outside the star	97%	99%

Medians. A first-passage time in a cooling-only model — no heating, no detailed balance: not thermal equilibrium.

Two things beyond the controlled benchmark. Pauli blocking is implemented: the recoiling neutron must end outside the Fermi sea, and that suppresses the optical depth by almost five orders of magnitude. But the few collisions that are allowed remove more than half of the kinetic energy each. The lesson is methodological: you cannot multiply the unblocked optical depth by a suppression factor and keep the old N_min — both must come from the same blocked kernel. Those campaigns are next. On the right, what happens after capture: we keep following captured particles until their local kinetic energy drops below three halves kT. Capture costs five collisions; cooling costs thousands, with hundreds of excursions out of the star — and takes only seconds, almost all of it outside. This is a cooling time in a model with no heating, not thermalisation. And that brings us to the question of part two.

Under the hood

Validated stage by stage

check	agreement
interior transport vs independent orbit quadrature (path, azimuth)	1–2×10⁻⁴
P_cap with the step refined from R/4096 to R/16384	≤ 2×10⁻⁴
exterior periapsis vs analytic turning point	< 1.4×10⁻¹³
GR / Newtonian focusing at v → 0	1.244
surface hits: quadrant asymmetry, 10⁶ samples	3×10⁻³
CPU vs CUDA, 60 cases × 10⁵ histories	4.5×10⁻³
population equation vs N = C_χt	10⁻¹⁶
Fermi EOS: Γ_eff at the two ends	1.6667 · 1.3333

3

independent implementations of the N_min recurrence, same integers

425

assertions in 26 test suites

C++17 with OpenMP; the same engine compiled to WebAssembly for an interactive lab.

Part I · in short

What the closure misses

The Monte Carlo reduces exactly to the Poisson closure when its released ingredients are switched off.
Where the loss is evaluated and whether the trajectory is released move P_cap by ±0.1 each, in opposite directions: right on average, 0.16–0.18 RMS off in shape.
Collision counts are strongly overdispersed; negative-binomial laws calibrated out of sample do much better than Poisson.
Pauli blocking changes rate and kernel together: it must be measured, not rescaled.

A correct mean number of collisions is not enough for a Poisson closure.
the one-line summary of Part I

II

Part two

Degeneracy before
self-gravity

A captured population can be far too light to change the star — and still too dense to be a classical gas. Two diagnostics, and a map of where their thresholds fall.

tracer, Fermi gas or dark core?S and Θthe fiducial populationthe regime mapwhat comes next

Neutron star over Manhattan · NASA's Goddard Space Flight Center · public domain

The question

After capture, what is it?

capture supplies N_χ(t)→which regime?→the right next model

A classical tracer

The usual assumption: the particles thermalise into a Boltzmann cloud in the fixed potential of the star. Right for a dilute gas.

n ∝ e^−mΔΦ/kT

A degenerate tracer

Fermions in a deep well can reach a phase-space density where Pauli matters — while still too light to change the gravity.

E_F ≳ kT

A dark component

Dark-matter admixed neutron stars: the dark fluid takes part in hydrostatic balance and shifts the mass–radius relation.

two-fluid TOV

Two independent diagnostics

One for gravity, one for statistics

S = M_χ(r_χ)M_b(r_χ)self-gravity · inside the region the dark matter occupies

S	regime	omitted gravity S/(1 + S)
< 10⁻³	tracer	< 10⁻³
10⁻³ – 10⁻¹	transitional	up to 9%
≥ 10⁻¹	self-gravitating	≥ 9%

Local, not the global M_χ/M_★: r_χ = r₉₀ holds 90% of the particles.

Θ = TT_F = k_BTE_Fclassicality · thermal energy over Fermi energy

Θ	regime	Fermi share 1/(1 + Θ)
> 10	thermal	< 0.09
0.1 – 10	transitional	0.09 – 0.91
< 0.1	degenerate	> 0.91

There is no general reason for the two sets of thresholds to coincide.

The analysis rests on two numbers. S compares the dark mass with the baryonic mass enclosed in the same central region, the radius that holds ninety per cent of the particles: it asks whether the dark matter contributes to the local gravitational field. Since S is the ratio of the two local accelerations, the thresholds have a direct meaning: below ten to the minus three, neglecting the dark matter's gravity is a per-mille error. Theta compares the thermal energy with the Fermi energy of the captured particles: it asks whether Maxwell–Boltzmann statistics are adequate. Large Theta, classical; small Theta, degenerate. Note the convention: Theta is the inverse of the usual degeneracy parameter. The thresholds are operational, not new constants. The point of separating them is that they answer different questions — and there is no reason for them to be crossed together.

The two closures

A Boltzmann cloud, and the Fermi limit

n_χ(r) = n₀ exp[−m_χΔΦ(r)k_BT]thermal tracer in the fixed potential of the star

N_χ = 4π ∫₀^R n_χ r² √g_rr drfixes n₀ · r₉₀ holds 0.9 N_χ

The cloud has no edge: r₉₀ ignores the dilute tail and measures where the particles actually are.

p_F = ħ (3π²n_χ)^1/3, x_F = p_Fm_χcspin ½, g = 2

E_F = m_χc² (√(1 + x_F²) − 1)P ∝ ρ^5/3 for x_F ≪ 1 · P ∝ ρ^4/3 for x_F ≫ 1

Θ is a diagnostic ratio, not a finite-temperature chemical potential: it says when a Fermi–Dirac treatment is needed.

The two diagnostics · live

Move a cloud through the plane

presets

baryonscaptured dark mattersame m, T, any masspaper fiducial

Dashed lines: the thresholds S = 10⁻³, 10⁻¹ and Θ_c = 10, 0.1. The bands between them are transitional.

IBoltzmann tracer S < 10⁻³, Θ_c > 10 IIBoltzmann, self-gravitating S ≥ 10⁻¹, Θ_c > 10 IIIFermi tracer S < 10⁻³, Θ_c < 0.1 IVFermi, self-gravitating S ≥ 10⁻¹, Θ_c < 0.1

particle mass m_χ

core temperature

captured mass M_χ

This panel computes the thermal cloud live, the way OBSCURA does: the Boltzmann profile in the potential of the SLy star, r90, S, and Theta at the centre. It starts at the paper's fiducial, a 100 keV particle at ten to the nine kelvin with the mass captured in a billion years: S comes out 6.4 times ten to the minus sixteen and Theta 3.46, as in the paper. The cloud fills the star. The dashed line is what happens if you change only the captured mass: the profile keeps its shape, S grows in proportion, Theta falls as S to the minus two thirds, then minus one third once the Fermi momentum becomes relativistic. Now pick the 1 GeV WIMP at ten to the five kelvin: the cloud is a few metres wide, and Theta crosses one at S around ten to the minus eleven — eight orders of magnitude before self-gravity. Push the mass up to ten to the six GeV and the order reverses.

Why the thresholds do not coincide · a back-of-the-envelope

Pauli comes first for light particles

A compact cloud sits in the harmonic core, ΔΦ = (2π/3)Gρ_cr². Its size cancels between S and the central density:

n₀ ≈ 4.6 S ρ_cm_χindependent of T

Θ_c ∝ T m_χ^5/3 S^−2/3 ⇒ S(Θ_c = 1) ∝ m_χ^5/2 T^3/2ρ_cnon-relativistic Fermi gas

Light particles are many per gram: they fill phase space long before they weigh anything.

particle, core	S at Θ_c = 1
1 keV, 10⁶ K	10⁻²⁴
100 keV, 10⁹ K	10⁻¹⁴
1 GeV, 10⁵ K	10⁻¹¹
1 GeV, 10⁷ K	10⁻⁸
1 TeV, 10⁵ K	10⁻⁴

Self-gravity starts at S ~ 10⁻³. For anything lighter than about a TeV, the gas turns degenerate first.

Full thermal profile in the SLy star, as in the previous panel.

Why should degeneracy come first? A back-of-the-envelope that I find useful, not in the paper in this form. A cloud much smaller than the star sits in the harmonic core of the potential. Its radius scales as the square root of T over m, but it cancels between S and the central density: the central density is about 4.6 times S times rho_c over m, whatever the temperature. Put that in the non-relativistic Fermi energy and Theta scales as T times m to the five thirds times S to the minus two thirds. So the value of S at which the gas becomes degenerate grows as m to the five halves. The table is from the full calculation of the previous panel: for a GeV particle at ten to the five kelvin, ten to the minus eleven; for a keV fermion, ten to the minus twenty-four. Only around a TeV does the degeneracy threshold approach the self-gravity one.

The pipeline

From the star to the diagnostics

baryonic TOV star→b_max(v)→Γ_inc→f_cap (Monte Carlo)→C_χ→N_χ(t)→ρ_χ(r)→(S, Θ)

Γ_inc = n_χπ ∫ b_max²v f(v) dvfocused flux onto the surface

C_χ = Γ_inc f_capf_cap: captured over sampled

dN_χdt = C_χ − C_AN_χ² − C_EN_χannihilation, evaporation: off in the fiducial

The order matters: capture supplies the particle budget, S tests whether the fixed gravitational background is adequate, Θ tests whether the classical closure is.

The fiducial population · after 10⁹ years

Gravitationally invisible, not quite classical

	Maxwell–Boltzmann	RAR–FD
incidence rate Γ_inc	4.36×10²⁹ s⁻¹	2.33×10²⁹ s⁻¹
capture fraction f_cap	0.962	0.959
captured particles N_χ	1.32×10⁴⁶	7.06×10⁴⁵
captured mass M_χ	2.36×10¹⁸ g	1.26×10¹⁸ g
cloud radius	11.3 km (r₉₀)	11.7 km (support)
self-gravity S	6.4×10⁻¹⁶	3.8×10⁻¹⁶
classicality Θ_c	3.46	5.07

Fiducial: m_χ = 100 keV, σ = 10⁻⁴⁴ cm², ρ_χ = 0.4 GeV cm⁻³, T_core = 10⁹ K, t = 10⁹ yr, SLy 1.40 M☉.

M_χ/M_★ ≈ 10⁻¹⁵: no mass–radius shift to look for. Yet Θ_c ≈ 3–5 is in the transitional band: corrections of order one to Boltzmann.

Changing the halo from MB to RAR–FD changes the supply by 47% and the microscopic capture by 0.3%.

The fiducial model of the paper: a 100 keV particle, ten to the minus forty-four, the local density, a hot young core at ten to the nine kelvin, a billion years. With the Maxwellian halo, about ten to the forty-six particles, two times ten to the eighteen grams — ten to the minus fifteen of the star. S is ten to the minus sixteen: an extreme tracer, no structural effect at all. But Theta at the centre is 3.5: not classical, not degenerate — the transitional band, where Fermi–Dirac corrections are of order one. With the RAR halo the incidence rate drops by almost half, because the distribution is different, while the capture fraction hardly moves; the classification does not change. One honest remark: an earlier draft found this point strongly degenerate; with the updated multi-scatter capture it is transitional, and the degenerate tracers come from the survey.

The regime map · every model of the survey

The two boundaries do not coincide

colour

tracks

I

Boltzmann tracer

the usual assumption

II

Boltzmann, self-gravitating

heavy, classical, massive

III

Fermi tracer

degenerate, gravitationally passive

IV

Fermi, self-gravitating

toward a dark core

The central figure of the paper, point by point: each dot is one model of the survey over ambient density, particle mass, cross section, accumulation time and core temperature; the coordinates are derived diagnostics, not inputs. Vertical lines: the gravitational thresholds; horizontal lines: the statistical ones. The colour is one input at a time — switch it and you see that no single input decides the quadrant. Hover over a point to see its inputs. The star is the fiducial, far to the left. The key observation: the survey has many models in quadrant three, degenerate yet gravitational tracers — degeneracy before self-gravity — and in the Fermi–Dirac map the correlation between the two coordinates is weak, minus 0.15: they are independent diagnostics. The track buttons isolate the straight families, next slide.

Reading the map

Two slopes, one Fermi energy

Hold m_χ, σ, t, T fixed and change ρ_χ: the profile keeps its shape, S ∝ n₀ ∝ N_χ, and p_F ∝ n₀^1/3:

Θ_c ∝ S^−2/3x_F ≪ 1 · E_F ≈ p_F²/2m

Θ_c ∝ S^−1/3x_F ≫ 1 · E_F ≈ p_Fc

RAR–FD survey	tracks	points	median slope
non-relativistic family	352	2112	−0.667
relativistic family	213	1278	−0.334

The mass trend

Heavier particles are less degenerate: d log Θ_c/d log m_χ ≈ 0.49 (MB) and 0.42 (RAR–FD). The degenerate quadrants hold 85–89% of the lightest models and 21–28% of the heaviest.

Not a change of particle

III → IV is the same Fermi gas becoming heavier where it sits: the growth of S, not of the particle mass.

MB vs RAR–FD: corr(log S, log Θ_c) = −0.652 vs −0.154.

What comes next

The right model for each quadrant

I

S ≪ 1, Θ ≫ 1

a Boltzmann cloud in the fixed star — the standard assumption, justified

II

S ≳ 0.1, Θ ≫ 1

classical, but it shapes its own potential

III

S ≪ 1, Θ ≲ 1

a capture-normalised Fermi tracer in the fixed star — finite-T if Θ ~ 1–10

IV

S ≳ 0.1, Θ ≪ 1

only here: a two-fluid TOV, the dark-matter admixed star

Capture calculations and admixed-star models are not interchangeable endpoints. A small M_χ/M_★ or S says the structure is unchanged; it does not justify a Boltzmann profile.

Not yet included

Annihilation heating, collapse and black-hole formation, rotation, magnetic fields, and a self-consistent time evolution of the captured Fermi gas. Θ is a diagnostic, not a finite-temperature solution.

Conclusions

Capture, then character

OBSCURA follows dark-matter collision histories through the interior metric and reduces exactly to the optical-depth closure.
At 100 keV, the frozen trajectory and the surface evaluation each move P_cap by ~0.1, in opposite directions; the count of collisions is overdispersed.
After capture, S and Θ answer different questions: the fiducial population is an extreme tracer (S ~ 10⁻¹⁶) in the transitional band (Θ_c ≈ 3–5).
For light particles degeneracy comes before self-gravity: the thermalised-WIMP assumption has to be tested in (S, Θ), not by M_χ/M_★.

Next steps

Pauli-blocked capture campaigns with the consistent blocked kernel.
Capture rates over the full incident phase space, RAR halo included.
Thermalisation with heating and detailed balance, then a finite-T Fermi tracer.
Two-fluid structure once S leaves the tracer regime.

The panels in this seminar run in the browser, on the same physics as OBSCURA: rafael-lima.pages.dev/talks.

Image credits

Freely licensed photos and figures

Bullet Cluster, 1E 0657−56 — NASA/CXC/M. Weiss · public domain

Fritz Zwicky, 1960 — Comet Photo AG (Zürich), ETH-Bibliothek · CC BY-SA 4.0

Vera Rubin — AIP Emilio Segrè Visual Archives, AIP Gallery of Member Society Presidents · attribution

The centre of the Milky Way — ESO · CC BY 4.0

Neutron star over Manhattan — NASA's Goddard Space Flight Center · public domain

Paper figures — de Lima & Rueda, A Monte Carlo relativistic framework for dark matter capture by neutron stars (in preparation); de Lima & Rueda, Degeneracy Before Self-Gravity: Dark Matter Regimes in Neutron Stars (draft)

Regime-map panel — the survey data of the second paper, from the OBSCURA pipeline

Neutron star — SLy equation of state (Douchin & Haensel 2001) in the piecewise-polytropic fit of Read, Lackey, Owen & Friedman, PRD 79, 124032 (2009)

OBSCURA logo — OBSCURA project

Thank you

Questions?

Slides and live panels, on the same physics as OBSCURA, at rafael-lima.pages.dev/talks