ρ (x) = ρ (x)

. Write both sides as background plus perturbation and expand the background time shift:

\overset{ρ}{ˉ} (τ) + δ ρ (τ, x) = \overset{ρ}{ˉ} (τ + T) + δ ρ (τ + T, x + \nabla L) = \overset{ρ}{ˉ} (τ) + \overset{ρ}{ˉ}^{'} T + δ ρ (τ, x) .

The coordinate shift acting on an already first-order perturbation contributes only at second order. Thus the density perturbation gauge transformation is

δ ρ = δ ρ - \overset{ρ}{ˉ}^{'} T .

ζ = - C + \frac{1}{3} \nabla^{2} E + H \frac{δ ρ}{ρ ˉ ^{'}} .

\overset{ρ}{ˉ}^{'}

is essential and is present in the PDF; the TeX transcription drops it. The scalar-curvature combination transforms as

- C + \frac{1}{3} \nabla^{2} E = - C + \frac{1}{3} \nabla^{2} E + H T,

while the density term changes by

- H T

. Hence the two time-slicing changes cancel:

ζ = ζ .

On a uniform-density slice,

δ ρ = 0

and this variable is the signed spatial-curvature perturbation. The construction assumes

\overset{ρ}{ˉ}^{'} \neq = 0

; a pure constant-density cosmological constant does not define such a time slicing.

E = B = 0

and

C = - Φ

\overset{ρ}{ˉ}^{'} = - 3 H (\overset{ρ}{ˉ} + \overset{ˉ}{P})

D = \overset{ρ}{ˉ} + \overset{ˉ}{P}

ζ = Φ - \frac{δ ρ}{3 D} .

c_{a}^{2} = \overset{ˉ}{P}^{'} / \overset{ρ}{ˉ}^{'}

. Then

D^{'} = - 3 H D (1 + c_{a}^{2})

. Differentiate the previous expression and insert the perturbed energy-conservation equation:

ζ^{'} = Φ^{'} - \frac{δ ρ ^{'}}{3 D} + \frac{δ ρ D ^{'}}{3 D ^{2}} = \frac{H}{D} (δ ρ + δ P) + \frac{\nabla \cdot q}{3 D} - \frac{H}{D} (1 + c_{a}^{2}) δ ρ .

Consequently the curvature evolution equation in this sign convention is

ζ^{'} = \frac{H}{ρ ˉ + P ˉ} δ P_{nad} + \frac{\nabla \cdot q}{3 ( ρ ˉ + P ˉ )} = \frac{H}{ρ ˉ + P ˉ} δ P_{nad} + \frac{1}{3} \nabla \cdot v .

The last equality defines the total energy-frame velocity by

q = D v

. For adiabatic cosmological perturbations, the non-adiabatic pressure perturbation vanishes. On superhorizon scales, the stated suppression of the velocity divergence makes the remaining gradient term negligible. Thus $ζ$ is conserved to leading order in $k / H$ . The positive sign of the pressure source follows from the paper's definition of