Cosmology · Dark-matter candidates and discriminating tests

Cold, warm and self-interacting scenarios

Separate momentum-driven smoothing from collisions using two explicitly different finite teaching calculations.

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

Back to the chapter

After reading, you can

Explain why a momentum distribution and epoch enter cold or warm descriptions.
Follow a free-flight density amplitude and a separate first-collision probability.
Keep total scattering and momentum-transfer cross sections distinct.

Separate motion from collisions

A collisionless constituent can still have a nonzero momentum distribution. Particles moving between regions can reduce a density pattern without exchanging energy through collisions.

Cold and warm descriptions refer to the importance of motion in a specified cosmological history and on specified scales. A particle mass alone does not fix that motion: the production process and momentum distribution also enter.

Self-interactions add a different process, collisions among the constituents. Their effects depend on relative speeds, angular scattering and the distribution being evolved. A self-interacting model can also possess an initial momentum distribution.

Two separate reference calculations below isolate these mechanisms. They are not combined into a cosmological simulation, and neither supplies an observed smoothing scale or preferred interaction strength.

Let a small pattern travel without forces

Consider a one-dimensional, static, nonrelativistic system with no force or collisions. Supply velocities with normalized density f(v) = (1−|v|/v_max)/v_max for |v| ≤ v_max, zero outside, where 0 < v_max ≪ c.

At the initial time, every velocity class shares a small density contrast δ₀ cos(kx), with δ₀ = 0.01 and k > 0. During elapsed time Δt, a particle moves by vΔt; the pattern from that class shifts accordingly.

Integrating all shifted patterns gives a contrast amplitude δ = δ₀W(q), where q = kv_maxΔt and W(q) = [sin(q/2)/(q/2)]², with W(0) = 1 by continuity.

The expression follows because this triangular velocity distribution is the sum of two independent uniform contributions, each on [−v_max/2, v_max/2]. Each contributes a sinc factor to the spatial mode. W is a density-amplitude ratio, not a power ratio.

One supplied Fourier mode under unforced free flight
q = kv_maxΔt	Amplitude W = δ/δ₀	Contrast amplitude δ
0	1	0.010000
π/2	8/π² ≈ 0.810569	0.008106
π	4/π² ≈ 0.405285	0.004053
2π	0	0

Smoothing does not remove the particles

At q = π, particles remain present but their shifted patterns partly cancel. The chosen mode’s amplitude falls to about 0.405285 of its initial value.

At q = 2π this particular Fourier amplitude is zero. That does not mean the total mass vanishes, every possible pattern disappears or particles stop moving.

The free-flight model omits expansion and gravity. A cosmological transfer calculation must evolve the distribution with the appropriate expansion, perturbations and couplings; this static expression is not that transfer function.

Different supplied velocity distributions give different mode responses. This is why a mass label without a production and momentum model is insufficient to predict small-scale structure.

A column tests collision probability

Now use an independent beam reference: a dilute set of incident particles traverses a static target mass column Σ. Each target has mass m, and the total elastic cross section is a supplied constant σ_tot.

Assume uncorrelated targets, negligible depletion or backreaction, and a straight incident trajectory until the first collision. The expected collision optical depth along that initial path is τ = Σσ_tot/m.

The probability of no collision is e⁻ᵗᵃᵘ, so the probability of at least a first collision is 1−e⁻ᵗᵃᵘ. These assumptions determine a first-collision count, not a particle’s final velocity or a halo density profile.

The cross section here is explicitly total. Momentum-transfer and viscosity cross sections weight scattering angles differently. They cannot be substituted silently, especially for forward-peaked scattering.

Separate static beam reference; unrelated to the q samples above
τ	No-collision probability	First-collision probability
0	1	0
0.25	0.778801	0.221199
0.5	0.606531	0.393469
1	0.367879	0.632121

Two distinct original deterministic teaching systems occupy separate panels. They are not a combined warm or self-interacting cosmological calculation. Panel A is a one-dimensional static nonrelativistic unforced system with no collisions, normalized triangular velocity density f(v) = (1−|v|/v_max)/v_max for |v| ≤ v_max, zero outside, and supplied positive v_max much smaller than c. Every velocity class initially shares a density contrast δ₀ cos(kx), with δ₀ = 0.01. The dimensionless phase is q = kv_maxΔt. Direct averaging of shifted density patterns gives amplitude W = δ/δ₀ = [sin(q/2)/(q/2)]², with W(0) = 1. Panel A has separate linear axes q from 0 to 2π and dimensionless ratio from 0 to 1. Solid blue circles show amplitude W; dashed violet squares show the corresponding single-mode power ratio W². The latter is not a second density amplitude or a probability of survival. At q = 0, π/2, π, 2π, amplitudes are exactly 1, 8/π², 4/π², 0, approximately 1, 0.810569, 0.405285, 0. Their contrast amplitudes δ₀W are 0.01, approximately 0.008106, 0.004053, 0. The plotted power values are their squared amplitude ratios, approximately 1, 0.657023, 0.164256, 0. Coincident endpoint markers are intentional. Cancelling this Fourier mode removes neither particles nor mean mass; no monotonic extrapolation beyond the displayed first zero is asserted. Panel B instead describes an independent dilute incident beam through a static target mass column Σ. Each target has mass m, with chosen total elastic cross section σ_tot, uncorrelated positions, negligible depletion and a straight incident path until its first collision. Optical depth τ = Σσ_tot/m is dimensionless; it has a linear horizontal axis from 0 to 1 unrelated to q, and a separate probability ordinate from 0 to 1. Dashed green circles show exp(−τ), the no-collision probability. Solid orange squares show 1−exp(−τ), the probability of at least the first collision. At τ = 0, 0.25, 0.5, 1, the two probabilities are (1, 0), approximately (0.778801, 0.221199), (0.606531, 0.393469), (0.367879, 0.632121). Total cross section is not silently replaced by momentum-transfer or viscosity cross sections. Mass column is kg m⁻², cross section m² and target mass kg. Every series has a distinct dash or marker shape and direct legend without relying on color. No expansion, gravity, thermal relic distribution, coupled collision evolution, halo core, current preferred interaction, observed cutoff or empirical bound is supplied. Original Ark teaching model.

Connect the mechanism to a specified observable

A real structure prediction must state which momentum distribution, interaction law and gravitational initial conditions are evolved. The two reference panels deliberately omit that joint dynamics.

Visible matter can also alter structure through gravity, cooling and feedback. Distinguishing a dark-sector mechanism requires forward predictions that include those competing responses and how the observed tracers are selected.

The free-flight calculation demonstrates phase cancellation without collisions. The beam calculation demonstrates a probability that depends on column density and total cross section per target mass.

Neither reference declares a preferred candidate, predicts a measured halo core or excludes a physical parameter. It shows which additional quantities a discriminating test must specify.

Worked reasoning

Interpret each reference within its own assumptions

What do q = π and τ = 0.5 imply in their separate models?

Free flight gives W = 4/π² ≈ 0.405285, so δ ≈ 0.004053.
The beam has no-collision probability e⁻⁰·⁵ ≈ 0.606531.
Its first-collision probability is about 0.393469; this is not the amount of density-mode damping.

One result describes a mode amplitude; the other describes a first-collision probability in a different experiment.

Try the reasoning

Is W = 4/π² the mode’s power ratio?
Show the answer and reasoning
No. Its power ratio would be W² for this single amplitude comparison.
The density perturbation is linear in its amplitude; power is quadratic.
Why is τ = Σσ_tot/m dimensionless?
Show the answer and reasoning
Mass per area times area divided by mass has no units.
Σ/m is a number column and multiplication by a total cross section gives the expected initial-path collision count.
Can the two tables be read as a single predicted warm, self-interacting halo?
Show the answer and reasoning
No.
They use separate initial systems and omit the coupled gravitational, expanding and collisional evolution.

What to carry forward

Momentum distributions and interactions describe distinct physical ingredients.
Unforced motion can suppress a mode through phase cancellation without particle loss.
A collision probability needs a stated column and cross-section convention.

References

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