Dark Matter During Star Formation

Glossary and prerequisites

Assumes classical mechanics (action-angle variables) and introductory statistical mechanics. The astrophysics terms used below:

• Distribution function - the density of particles in phase space (position-velocity space). For an isotropic system it depends only on energy . The real-space density is its velocity integral, .
• Phase space - the 6D space of positions and velocities . Liouville’s theorem: is conserved along orbits, so it is invariant under slow (adiabatic) changes of the potential.
• Isotropic velocity distribution - velocities have no preferred direction; equivalently a full, unbiased spread of orbital eccentricities. A Maxwellian is the thermal (Gaussian-in-velocity) isotropic case.
• Virialised halo - a self-gravitating system in equilibrium, obeying the virial theorem ; its particles have random (not ordered) velocities.
• Adiabatic invariant - a phase-space integral (, ) conserved when the potential changes slowly compared to the orbital period.
• Cusp / spike - a power-law central density rise, . “Spike” usually denotes the steep enhancement around a compact object.
• Loss cone - the region of the orbit-space whose pericentres reach the central object; such orbits are captured and removed.
• Collisional relaxation - slow evolution of orbits from cumulative two-body gravitational encounters, characterized by a relaxation time .
• Fokker-Planck / detailed balance - the diffusion equation governing under relaxation; its stationary state is fixed by the balance of drift and diffusion (fluctuation-dissipation).
• Dynamical friction - the drag a massive body feels moving through a background of lighter particles (here, DM), which it gravitationally focuses into a trailing wake.
• EMRI - extreme-mass-ratio inspiral: a compact object spiralling into a much larger black hole, a key gravitational-wave source for LISA.

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0. The question

A star forms inside a dark-matter (DM) halo. Initially the DM is spread out in an approximately uniform density profile, . As the baryonic gas collapses to form the star, it deepens the gravitational potential at the centre. Dark matter feels only gravity - no cooling, no radiation, no collisions - so it responds by contracting too.

The questions we answer here, from first principles:

1. What is the final DM density profile ? (It is a power law, .)
2. What is the exponent ?
3. Why do two standard methods give two different answers - and ?
4. What survives a supernova, and what survives billions of years of stellar scattering?
5. What does all this say about whether observed DM spikes imply primordial black holes?

The punchline for (2)-(3): the answer depends entirely on what orbits the DM particles are on. Circular orbits give ; a realistic isotropic (Maxwellian) velocity distribution gives . The difference is a lesson in why “adiabatic contraction” is not a single number.

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1. The tool: adiabatic invariants

When a system’s potential changes slowly compared to the orbital period of a particle (the adiabatic regime), certain phase-space integrals are conserved. For a particle in a spherical potential the two relevant adiabatic invariants are:

where are the peri/apocentre. The physical content of “adiabatic” is: as the potential morphs from initial to final, each particle keeps its fixed, and this determines its new orbit.

Two limits use these differently, and that is the entire origin of the -vs- discrepancy.

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2. The circular-orbit model (Blumenthal et al. 1986)

The simplest and oldest prescription assumes every DM particle is on a circular orbit. A circular orbit has ; it is fully specified by alone. Conservation of for a circular orbit is equivalent to the statement that the quantity

is conserved for each shell, where is the total enclosed mass. This is the Blumenthal invariant. Concretely: label a DM shell by its initial radius ; after the baryons collapse to a central mass, its final radius satisfies

Derivation of the slope

Start from a uniform sphere, , so . For inner shells the collapsed baryons dominate and act like a point mass , so . Then,

Mass conservation shell-by-shell, with and , gives

More generally, for an initial cusp the circular model gives , which is for and rises slowly to for .

The figure below shows a numerical implementation: a uniform profile is contracted through successive stages and the inner slope locks onto -.

** But circular orbits are a fiction.** Real DM in a virialised halo has an isotropic velocity distribution - a full spread of eccentricities. That changes the answer.

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3. The Liouville / phase-space model (Young 1980)

Now we do it honestly. The DM starts with an isotropic Maxwellian velocity distribution inside the uniform sphere. We conserve the full adiabatic invariants , not just .

The initial potential is harmonic

A uniform sphere of total density produces, inside itself, a harmonic potential:

For the harmonic oscillator the radial action has a closed form:

The final potential is Keplerian

After the baryons collapse to a point mass , an inner DM particle sees a Keplerian potential , whose radial action is also closed-form

Equate the invariants - and cancels

Setting (and using that is separately conserved, so the on both sides cancels):

This is an exact, analytic map from a particle’s initial energy to its final Keplerian orbit - no time integration required. The final semi-major axis is and the eccentricity follows from : .

Why the density goes as

Here is the crucial kinematic fact. In a Keplerian potential a particle at small radius moves fast: . The time it spends near radius is . The time-averaged density contribution of any orbit passing through is therefore

This is universal: it does not depend on the orbit’s or , only on the Keplerian potential. Summed over the population of orbits, the inner density is

This is the result of Young (1980) and, in the DM-during-star-formation context, Capela, Pshirkov & Tinyakov (2014) (arXiv:1403.7098).

Numerical confirmation

In order to get a numerical confirmation, we have created a script that samples particles from a uniform Maxwellian sphere, applies the exact invariant map, and reconstructs the density by solving Kepler’s equation (sampling mean anomaly uniformly correctly weights positions by time spent). The measured slope is to three digits. Note the median eccentricity of the final orbits is - the orbits are highly eccentric, which is the whole point.

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4. Why and differ - the physical heart of it

The two models conserve different things because they assume different orbits:

	Circular (Blumenthal)	Eccentric / isotropic (Liouville)
Conserved	only ()	and
Each particle occupies	a fixed radius	a range
Density at built from	the one shell mapped to	time-averaging of many orbits
Result

The one-sentence version: the circular model pins each particle to a radius; the Liouville model puts each particle on an orbit, and an eccentric particle spends most of its time near apocenter - far from the center - which dilutes the inner density and makes the cusp shallower.

Interpolation. If you allowed only orbits up to some maximum eccentricity , you would interpolate: recovers , (full isotropic) gives . The value is essentially a kinematic identity of the Keplerian potential (the velocity law), not a coincidence - which is why, as we will see, it is remarkably robust.

A cleaner bookkeeping: slope ↔ distribution function

For an isotropic population in a Keplerian potential with distribution function , a short calculation (, with ) gives

So (a flat DF in energy - the Young spike); . We will reuse this identity constantly below.

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5. General-relativistic corrections

The Newtonian spike would run to . GR modifies the innermost region:

• Loss-cone cutoff (dominant). Any DM particle whose pericentre falls inside the innermost stable circular orbit is captured. For Schwarzschild, . The spike is simply truncated below : there, outside.
• Black-hole spin. For a maximally spinning Kerr hole, , six times smaller. Since , the density at the inner edge is enhanced by - a large factor for annihilation or gravitational-wave signals.
• The asymptotic slope is unchanged. The GR correction to the effective potential enters at , so ; in the observable region the slope .

The GR-consistent Schwarzschild treatment (Sadeghian, Ferrer & Will 2013) confirms the slope is preserved outside the loss cone with a modest inner-normalisation change.

6. What a supernova does: the slope is invariant

Most stars that leave a compact remnant explode first, ejecting a mass fraction impulsively - fast compared to any DM orbital period. In the impulsive limit the DM positions are frozen while the potential suddenly shallows. A particle at position at the instant of explosion get an energy kick

It escapes if . Writing , the survival condition is

where is the orbital phase. This condition is scale-free - it depends only on the orbital phase and eccentricity, not on the absolute orbit size . Because the eccentricity distribution is the same at every radial scale, the SN removes the same fraction of particles at every radius. Therefore:

Only the normalisation drops (down to 8% survival for a neutron-star-forming ). The classic Blaauw (1961) result - a circular orbit unbinds at - is recovered as the edge case, but eccentric orbits caught near apocenter survive even .

I have created a script to confirm this: the remnant slope stays pinned at across the entire range , while the amplitude falls.

The same invariance holds for adiabatic re-accretion of fallback material: Keplerian-to-Keplerian mass growth rescales every orbit by a common factor and leaves the slope untouched. Neither impulsive mass loss nor subsequent accretion can move away from its formation value in the Keplerian regime.

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7. Billions of years later: stellar scattering and the Bahcall-Wolf attractor

Over cosmic time the DM cusp is stirred by two-body gravitational encounters with stars - collisional relaxation. Bahcall & Wolf (1976, ApJ 209, 214) solved the energy-space Fokker-Planck equation for stars around a massive black hole and found a steady-state attractor: a constant-flux cusp

independent of initial conditions. In our language this is . The exponent is the unique power law for which the inward energy flux is radius-independent - a genuine dynamical fixed point.

Their 1977 sequel (ApJ 216, 883) and Alexander & Hopman (2009) treat multiple masses: heavy species keep , while a light species (like DM, much lighter than stars) relaxes to a shallower . So:

Regime	What scatters the DM	Attractor
DM self-relaxation (single mass)	DM-DM
DM as light species	heavier stars

Emulating relaxation

Rather than solve the full Fokker-Planck PDE, we have emulated relaxation with a stochastic (Langevin) process engineered to have the correct stationary distribution. Working in binding energy , we use the fluctuation-dissipation relation to pick drift and diffusion so the equilibrium is exactly the target attractor . Choosing makes the process a geometric Brownian motion in - every radius relaxes on one timescale - integrated exactly in log-space:

Reflecting boundaries represent resupply by the outer halo and the inner loss cone. The result: both formation slopes converge to the attractor, erasing the memory of the initial condition.

(Caveat: this is an illustrative model whose equilibrium is set by construction - it correctly captures the direction, rate, and endpoint of relaxation, but it is not a first-principles Fokker-Planck solve. For a paper one would cite/use the Alexander-Hopman coefficients.)

The physically important corollary: because DM is much lighter than stars, the relevant attractor is the mass-segregation one, - the same value as the formation slope. Relaxation of DM by stars reinforces ; it does not lift it.

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8. The payoff: do observed spikes imply primordial black holes ?

Several black-hole low-mass X-ray binaries (A0620-00, XTE J1118+480, Nova Muscae 1991) show anomalous orbital-period decay that has been attributed to dynamical friction against a DM spike. Because a light black hole formed by stellar collapse is “not expected” to carry a spike, some authors (e.g. arXiv:2406.07624) argue the primaries could be primordial black holes, which form dense secondary-infall spikes with

Two things sharpen this:

1. How steep must the observed spike be? An N-body analysis with feedback (arXiv:2510.11635) finds the friction requires

2. How steep can a stellar-origin spike be? Everything above gives a hard ceiling:

   • Formation (Young/Liouville): .
   • Supernova: preserves it exactly.
   • Fallback accretion: preserves it (Keplerian rescaling).
   • Relaxation: mass-segregation drives DM toward ; self-relaxation caps at .

The verdict

There is a clean gap: a stellar-origin black hole cannot exceed , while the observation require , which only the primordial secondary-infall channel () naturally reaches. So the quantitative calculation supports the primordial interpretation - but it replaces the hand-wave (“hard to see how a spike survives”) in the literature with an actual number and an actual mechanism-by-mechanism argument.

Honest caveats. (i) This confirms, rather than overturns, the prevailing assumption. (ii) The whole edifice is contingent on the period decays being due to DM at all, which is contested. (iii) The relaxation ceiling should be nailed with real Alexander-Hopman coefficients, not the toy. (iv) One genuinely open effect could change the picture: the natal kick.

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9. The open problem: natal kicks

A supernova imparts a kick velocity to the newborn compact object (from asymmetric ejecta and neutrino emission), typically tens to hundreds of km/s. This moves the black hole out of the center of its own DM spike. Unlike the symmetric mass loss of §6 - which we showed preserves the slope - a kick displaces the potential minimum relative to the frozen DM, and could:

• strip the spike almost entirely if the kick exceeds the DM orbital velocities, or
• leave a lopsided remnant cusp that the black hole slowly re-captures.

Neither the primordial-origin papers nor the pipeline here has modelled it, and it is the one ingredient that could move or destroy the spike outright. It is the natural next calculation - and the impulsive machinery of §6 extends to it directly: add a common velocity boost to every particle’s frame at the SN instant and re-evaluate which orbits stay bound.

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10. Summary

Stage	Mechanism	Effect on slope
Formation (circular approx.)	Blumenthal const
Formation (isotropic, correct)	Liouville / Young 1980
Supernova (mass loss)	impulsive, scale-free	unchanged
Fallback accretion	Keplerian rescaling	unchanged
Relaxation (DM by stars)	mass-segregation
Relaxation (DM self)	Bahcall-Wolf
Stellar-origin ceiling	-
Observations require	dynamical friction (N-body)
Primordial (secondary infall)	Bertschinger 1985

The DM density profile built during star formation is a power law because the final potential is Keplerian; its exponent is , not , because real DM is on eccentric orbits and spends its time near apocentre; that exponent is robust against supernovae, accretion, and stellar relaxation; and the gap between the stellar ceiling () and the observational requirement () is what makes DM spikes around X-ray binaries a genuine probe of primordial black holes.

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how to cite

if this post was useful in your work, please cite it. Two guidelines:

Plain text:

Capela F. (2026). Dark Matter During Star Formation: How a Uniform Halo Becomes a Spike. Available at: https://labfab.io/blog/dark-matter-halo

BibTex:

@misc{capela2026dmspikes,
author		= {Capela, Fabio}
title		= {Dark Matter During Star Formation:How a Uniform Halo Becomes a Spike},
year 		= {2026},
howpublished    = {\url{https://labfab.io/blog/dark-matter-halo}}
}

References

• Blumenthal, Faber, Flores & Primack (1986), ApJ 301, 27 - adiabatic contraction, circular-orbit model.
• Young (1980), ApJ 242, 1232 - phase-space adiabatic growth, .
• Bahcall & Wolf (1976), ApJ 209, 214 - the relaxation cusp.
• Bahcall & Wolf (1977), ApJ 216, 883 - multi-mass / mass segregation.
• Blaauw (1961), BAN 15, 265 - impulsive mass-loss disruption threshold.
• Bertschinger (1985), ApJS 58, 39 - secondary infall, self-similar .
• Gondolo & Silk (1999), PRL 83, 1719 - spikes around massive black holes.
• Sadeghian, Ferrer & Will (2013), PRD 88, 063522 - GR-consistent spike.
• Alexander & Hopman (2009), ApJ 697, 1861 - strong mass segregation.
• Capela, Pshirkov & Tinyakov (2014), arXiv:1403.7098 - adiabatic contraction & primordial black holes.
• arXiv:2406.07624 - primordial-origin interpretation of X-ray-binary spikes.
• arXiv:2510.11635 - N-body critical assessment; requirement.