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 Incoming links: Velocity-dispersion tensor of collisionless matter

δ

v

H = a^{'} / a

is the conformal Hubble rate,

ϕ

ρ

is the density, and

σ_{ij}

θ = \nabla \cdot v

and set

σ_{ij} = 0

. A Fourier transform of the nonlinear continuity term gives

δ^{'} (k) + θ (k) = - \int \frac{d ^{3} q}{( 2 π ) ^{3}} α (q, k - q) θ (q) δ (k - q),

α (k_{1}, k_{2}) = \frac{( k _{1} + k _{2} ) \cdot k _{1}}{k _{1}^{2}} .

Taking the divergence of the Euler equation gives the quadratic velocity kernel

β (k_{1}, k_{2}) = \frac{∣ k _{1} + k _{2} ∣ ^{2} ( k _{1} \cdot k _{2} )}{2 k _{1}^{2} k _{2}^{2}} .

The displayed kernel is the growing-mode result of standard perturbation theory in cosmology. Its assumptions are a Newtonian, weak-field, subhorizon treatment of pressureless cold matter; a single-stream, irrotational velocity field; and negligible velocity-dispersion tensor of collisionless matter, so

σ_{ij} = 0

. The background is the Einstein-de Sitter universe, which makes the growing mode proportional to

a

and permits the separated powers

a^{n} F_{n}

. The density contrast is assumed perturbatively small and the decaying solutions are discarded. Gaussian initial conditions are not needed to derive

F_{2}

, but they are needed for the loop contractions in the later parts.

ρ = m a^{- 3} \int f d^{3} p

, the zeroth moment is

ρ^{'} + 3 H ρ + \partial_{i} (ρ v^{i}) = 0 .

Multiplying by

p^{i} / (am)

before integrating gives the first moment. Decompose the second velocity moment as

⟨ u^{i} u^{j} ⟩ = v^{i} v^{j} + σ^{ij}

v^{i'} + H v^{i} + v^{j} \partial_{j} v^{i} = - \partial^{i} ϕ - \frac{1}{ρ} \partial_{j} (ρ σ^{ij}) .

The single-stream pressureless-fluid equations follow when

σ^{ij} = 0