Cosmology · From density fluctuations to growing structure

Pressure support, particle motion and smoothing

Compare a standing sound mode with collisionless phase mixing in two separate supplied systems, keeping their mechanisms and limitations distinct.

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 pressure-driven sound and collisionless particle motion are different mechanisms.
Calculate two mode coefficients on a common supplied dimensionless time interval.
Distinguish an instantaneous zero or reduced density coefficient from damping, particle loss or a full cosmological growth prediction.

Matter motion matters as well as gravitational attraction

Gravity can promote the growth of overdensities, while pressure gradients and particle motion can redistribute a density pattern. Expansion and the perturbation scale also affect that competition in a cosmological calculation.

Pressure-driven sound in a fluid involves a relation between pressure and density together with collective motion. Collisionless particles instead carry phase-space density along their individual trajectories without requiring local collisional equilibrium.

A declining density contrast can therefore have different physical explanations. It is useful to isolate the mechanisms before combining them in a gravitational and expanding background.

This lesson compares two separate static test systems with no self-gravity or expansion. Their supplied equal speed scales make the arithmetic comparable; they do not turn the fluid and particle population into the same substance or solve their competition with gravity.

A fluid perturbation can oscillate without damping

In the first system, let a linear fluid density contrast satisfy ∂²δ/∂t² = c_s²∂²δ/∂x² with positive constant sound speed c_s. Choose initial δ(x,0) = δ_0cos(kx) and zero initial time derivative, with δ_0 = 0.01.

The resulting solution is δ(x,t) = δ_0cos(kx)cos(u), where u = kc_sΔt. Here k is a physical wavenumber in inverse metres, c_s is in metres per second and Δt is elapsed time in seconds.

The coefficient of the spatial pattern is cos(u). It begins at one, crosses zero at u = π/2 and reaches minus one at u = π. A position that began overdense is then underdense relative to the same static background.

This is an undamped standing solution of the stated linear wave equation. The density mode’s zero at one instant does not erase the fluid motion or prove dissipation; the mode continues oscillating within this ideal account.

Different trajectories can dephase a spatial density mode

In the second system, prescribe passive collisionless particles with one speed v_0 and isotropically distributed directions. Initially every direction has the same small spatial modulation 1 + δ_0cos(kx). No force, collision or expansion is included.

Let μ be the velocity direction cosine along x, uniformly distributed from −1 to 1. A trajectory advances by v_0μΔt along that coordinate. Averaging the translated cosine over μ gives a density coefficient sinc(kv_0Δt), where sinc(w) = sin(w)/w and sinc(0) = 1.

The angular average follows directly from (1/2)∫₋₁¹cos(kv_0μΔt)dμ; odd sine terms cancel. Different directional streams contribute different spatial phases, reducing their combined density-mode coefficient over the displayed interval.

Choose v_0 = c_s solely for the comparison, with both speeds much smaller than c. Then the same u labels the two coefficients. A mono-speed angular average is not a thermal neutrino distribution or a cosmological collisionless transfer function.

An oscillation and an angular average need not vanish together

Over 0 ≤ u ≤ π, compare cos(u) for sound with sinc(u) for the collisionless mode. At u = π/2, the sound density coefficient is zero, while the ballistic coefficient is 2/π ≈ 0.636620.

Multiplying by δ_0 = 0.01 gives a remaining ballistic density-mode amplitude of about 0.00636620 at that instant. At u = π, the sound coefficient is −1 and the ballistic coefficient is zero.

The table lists coefficients relative to the common initial amplitude, rather than full local density ratios. In either system, ρ/ρ̄ = 1 + δ remains at least 0.99 in this finite example; a negative mode coefficient is not a negative mass density.

The selected sinc interval does not justify claiming monotonic decrease for all future times. Outside this domain the function changes sign and has further oscillations. The full particle trajectories still retain information that is absent from one integrated density moment.

Two separate non-expanding, non-self-gravitating test systems with δ_0 = 0.01 and v_0 = c_s; columns are mode coefficients relative to δ_0
u	Sound coefficient cos(u)	Collisionless coefficient sinc(u)	Meaning at the stated time
0	1	1	Common initial density coefficient
π/4	0.707107	0.900316	Different finite responses
π/2	0	0.636620	Only the sound density coefficient vanishes
3π/4	−0.707107	0.300105	The sound pattern has reversed sign
π	−1	0	Only the ballistic density coefficient vanishes

Two distinct static teaching systems exclude self-gravity and expansion. In the linear fluid test, constant sound speed and initial density contrast δ_0 cos(kx) with zero time derivative give δ(x, t) = δ_0 cos(kx)cos(u), where u = kc_s Δt and δ_0 = 0.01. Physical k has inverse-metre units, c_s is metres per second and elapsed time is seconds. In the separate passive collisionless test, particles have one nonrelativistic speed v_0 and isotropic directions. Initially every direction shares the same small spatial modulation. A direction cosine μ uniformly distributed from −1 to 1 translates it by v_0 μ Δt. Averaging translated phases gives the density coefficient sinc(kv_0 Δt) = sin(kv_0 Δt)/(kv_0 Δt), with limit one at zero. The supplied equality v_0 = c_s makes the same dimensionless u compare the coefficients without equating their physical descriptions. The finite linear plot spans u from zero to π and signed coefficients from −1 to one. Solid blue circle samples show cos(u); dashed orange square samples show sinc(u). At u = 0, π/4, π/2, 3π/4, π, sound coefficients are 1, 0.707107, 0, −0.707107, −1 and collisionless coefficients are 1, 0.900316, 0.636620, 0.300105, 0. At π/2 the remaining ballistic density-mode amplitude is 0.01 times 2/π, approximately 0.00636620. The initial symbols coincide because both coefficients are one. Coefficients are not full density ratios: ρ/ρ̄ = 1+δ remains at least 0.99. An instantaneous density zero does not dissipate the undamped sound solution, and a reduced collisionless moment does not remove particles or their phase-space information. sinc changes sign and oscillates outside this chosen interval. This is not an expanding gravitational instability calculation, thermal-neutrino distribution, damping-length fit, measured mass bound or cosmological transfer function. Original Ark teaching model.

A cosmological suppression calculation needs the full system

Real structure growth combines gravity, expansion and the relevant matter properties. A quantitative pressure or free-streaming scale must be derived for that stated background and distribution, rather than read from these gravity-free curves.

A collisionless cosmological calculation includes a momentum distribution and gravitational or metric forcing as well as expansion. Treating it as the fluid sound equation would change the physical model.

Likewise, an instability threshold from a static self-gravitating uniform background needs its consistency assumptions examined. The static fluid used here avoids that issue by excluding self-gravity rather than silently assuming a cosmological equilibrium.

The comparison establishes two mechanisms: an ideal collective sound oscillation and dephasing of a collisionless density moment. The next lesson restores gravitational growth for an explicitly pressureless matter background, where initial density and initial motion both matter.

Worked reasoning

Compare the modes at u = π/2

Do the two density coefficients vanish at the same dimensionless time u = π/2?

The standing sound coefficient is cos(π/2) = 0.
The collisionless coefficient is sin(π/2)/(π/2) = 2/π ≈ 0.636620.
With δ_0 = 0.01, the ballistic mode amplitude is about 0.00636620 while the sound density coefficient is zero at that instant.

No. The different mechanisms produce different finite responses despite the chosen equal speed scales.

Try the reasoning

Does a reduced ballistic density coefficient mean that particles disappeared?
Show the answer and reasoning
No.
The directional trajectories dephase the spatial Fourier mode while preserving the particle population in the stated transport account.
Has the ideal sound wave been damped because its density coefficient reaches zero?
Show the answer and reasoning
No.
The coefficient continues its undamped cosine oscillation; one instantaneous density zero does not erase the wave’s motion.
Is sinc(u) here an inferred cosmic neutrino damping function?
Show the answer and reasoning
No.
It describes a chosen static, force-free mono-speed angular average, excluding the distribution and gravitational evolution of a cosmological neutrino calculation.

What to carry forward

Fluid sound and collisionless phase mixing act through different physical descriptions.
An instantaneous density zero or reduced mode coefficient does not by itself mean damping or particle loss.
Cosmological pressure and particle-motion effects require the relevant expanding gravitational background and populations.

References

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