Cosmology · The evidence motivating dark matter

Gravitational evidence does not identify a particle

Keep the same positive collisionless mass distribution while changing the mass and number of its hypothetical particles, exposing a restricted limit of smooth gravitational inference.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

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After reading, you can

Relate mass density to particle mass and number density.
Explain what remains unchanged when a supplied mass distribution is held fixed.
Identify additional physics that a microscopic candidate must satisfy.

The same mass can have different microscopic inventories

In a nonrelativistic population of identical particles, mass density is ρ = mn, where m is the individual particle mass and n is number density.

At a fixed supplied ρ, doubling m halves n. A smooth gravitational source that depends on the mass density alone does not distinguish those two inventories through that quantity.

This is a statement about a specified gravitational description. It does not say that any chosen particle mass has an acceptable production history, quantum distribution or interaction model.

Construct an explicit stationary collisionless example to make the limitation precise. The particles are numerous enough for a smooth mean-field treatment, and effects of their discreteness are excluded.

Hold the positive spatial mass profile fixed

Choose positive total mass M and length b, and write x = r/b. Supply the Plummer potential Φ = −GM/√(r²+b²) and density ρ = [3M/(4πb³)](1+x²)⁻⁵ᐟ².

Assume Newtonian weak gravity with GM/(bc²) ≪ 1, nonrelativistic velocities and stationary isotropic collisionless motion. The density is positive and integrates to finite total mass M, although it extends to arbitrarily large radius.

Let V = √(GM/b) and Ψ = −Φ, the positive relative potential in speed-squared units. The common radial-component variance is σ_r² = Ψ/6.

Compare hypothetical particle masses m_*, 2m_* and 4m_*, where m_* is an arbitrary microscopic mass unit chosen small enough for the stipulated continuum treatment. The same total M, spatial density and mass-normalized velocity distribution are supplied for all three.

Changing the particle mass changes the number density

Normalize ρ by M/b³ and n by M/(m_*b³). For the three particle masses, the normalized number densities are respectively the common normalized ρ, half of it and one quarter of it.

At x = 1, ρ/(M/b³) ≈ 0.0422023. The corresponding normalized number densities are about 0.0422023, 0.0211012 and 0.0105506.

Multiplying each number density by its own particle mass returns the same physical mass density. The smooth gravitational potential and the supplied mean velocity moments therefore remain unchanged in this comparison.

A measurement of the common smooth mass profile would not select one of these particle masses by itself. The table contains mathematical profiles, not inferred particle densities or observations of a real halo.

One common mass density with three supplied microscopic masses; density and number use the distinct units defined above
x=r/b	ρ/(M/b³)	n for m_*	n for 2m_*	n for 4m_*
0	0.238732	0.238732	0.119366	0.0596831
0.5	0.136658	0.136658	0.0683292	0.0341646
1	0.0422023	0.0422023	0.0211012	0.0105506
2	0.00427058	0.00427058	0.00213529	0.00106764
4	0.000200350	0.000200350	0.000100175	0.0000500875
6	0.0000286686	0.0000286686	0.0000143343	0.00000716716

This is a restricted supplied classical smooth gravitational system, not an observed halo or viable particle-candidate selection. Choose M > 0, b > 0, x = r/b and V² = GM/b, with GM/(bc²) ≪ 1. The common positive Plummer mass density is ρ = 3M/[4πb³](1+x²)^(−5/2), potential Φ = −GM/√(r²+b²) and relative potential Ψ = −Φ. The total spatial mass is finite M even though the profile extends to arbitrarily large radius. Only 0 ≤ x ≤ 6 is displayed. Both panels use a linear radius axis, but their vertical axes are explicitly base-10 logarithms of separately normalized densities, from −6 to zero. A negative logarithm means a positive density smaller than its reference unit; one log unit means a factor of ten, and these are not negative physical densities. Panel A draws ONE shared solid green mass-density curve with circles: log10[ρ/(M/b³)]. It is not three artificially separated coincident lines. Panel B shows log10[n/(M/(m_*b³))] using the common arbitrary microscopic reference mass m_*. Solid blue circles correspond to individual mass m_*, dashed orange squares to 2m_* and dotted violet diamonds to 4m_*. Each normalized number density equals the normalized mass density divided by its own mass factor one, two or four. At x = 1, the mass ordinate before the logarithm is about 0.0422023; number values are about 0.0422023, 0.0211012 and 0.0105506 in their distinct number unit. At x = 6 all three remain positive and are shown without a false zero cutoff. Multiplication by each population’s own particle mass returns exactly the same physical ρ, so the supplied smooth potential and mass-weighted velocity distribution remain common. Particle mass units are arbitrary and chosen small enough for the many-particle continuum; no measured microscopic mass is assigned. The optional stationary realization is a mass phase-space density F = Cℰ^(7/2) for specific binding energy ℰ = Ψ−v²/2 > 0, zero otherwise, with C = 24√2 b²/(7π³G⁵M⁴). F integrates over velocity to mass density and has SI units kg s³ m⁻⁶ under d³x d³v. It is not an energy-bin probability or particle-number DF; number phase-space density is F/m. It gives common radial-component variance Ψ/6 over 0 ≤ v < √(2Ψ). These classical positive stationary distributions do not test quantum occupancy, primordial production, free streaming, interactions, decay, scattering, annihilation or discreteness. DM06’s mechanism extension and DM-R01’s narrow quantum/production caveat support only those excluded-physics limitations; no historical bound, current status or candidate viability window is imported. Equal imposed smooth mass sources therefore do not prove equal full cosmological or laboratory predictions. Original Ark teaching model.

A candidate must pass tests beyond this imposed continuum

A production mechanism determines how a population acquires its abundance and momentum distribution. Thermal history and free streaming can connect a microscopic mass to cosmological structure in ways that this fixed-distribution comparison does not calculate.

Interactions can produce scattering, annihilation, decay or other signals. Quantum statistics and phase-space constraints can also limit which particle masses can realize a given distribution.

A sufficiently discrete population can alter gravitational fluctuations or encounters even at the same average density. Those effects are absent from the smooth mean-field assumptions here.

The example thus establishes a restricted degeneracy of a supplied collisionless gravitational system. It proves neither the viability of all three populations nor equality of their cosmological or laboratory predictions.

Optional: realize the common profile with a positive distribution

For a mathematical consistency check, define specific binding energy ℰ = Ψ−v²/2. A mass phase-space density F integrated over velocity gives ρ = ∫F d³v; F is not a particle-number distribution or an energy-bin probability.

Choose F = Cℰ⁷ᐟ² for ℰ > 0 and zero otherwise, with C = 24√2 b²/(7π³G⁵M⁴). Since F is a nonnegative function of conserved specific energy, it is stationary in the supplied potential.

Integrating over 0 ≤ v < √(2Ψ) recovers the stated Plummer density and radial variance Ψ/6. F has SI units kg s³ m⁻⁶ under the d³x d³v measure; Ψ and ℰ have units m² s⁻².

For particle mass m, use number phase-space density F/m. Its number integral is ρ/m, while the common mass distribution remains F. This supplies a positive realization of the comparison without requiring the optional derivation for the basic mass–number reasoning.

Worked reasoning

Keep the mass source fixed at r = b

What happens to number density when the supplied individual mass changes from m_* to 4m_* at r = b?

The common normalized density there is about 0.0422023 in units M/b³.
Divide by the mass factor four to obtain normalized n ≈ 0.0105506 in units M/(m_*b³).
Multiply by 4m_* to recover the same physical ρ; the stipulated smooth gravitational source is unchanged.

The number density becomes one quarter as large while the mass density remains fixed.

Try the reasoning

At fixed ρ, how does n change when m doubles?
Show the answer and reasoning
It halves.
The product mn must still equal the supplied mass density.
Is F in the optional construction a normalized probability density over energy bins?
Show the answer and reasoning
No.
It is a mass phase-space density under the spatial and velocity measure, and its velocity integral gives ρ.
Do the common gravity and velocity moments establish acceptable quantum physics and production for every chosen mass?
Show the answer and reasoning
No.
Those additional physical constraints were excluded and require separate predictions and tests.

What to carry forward

Smooth mass density does not by itself fix individual mass and particle number.
A positive collisionless distribution makes this restricted gravitational degeneracy explicit.
Physical identity requires additional information about production, interactions, quantum behavior or other observables.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.