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Use a monotone linear growth factor $b(t)$ as time, $\mathbf{u}=\dot{\mathbf{x}}/\dot{b}$, and $\psi =a\phi /(Cb)$ with $C=3H_0^2{\Omega }_{m0}/2$ and $a_0=1$. The cosmological Poisson equation becomes ${\nabla }^2\psi =\delta /b$; the continuity equation becomes ${\partial }_b\delta +\nabla \cdot [(1+\delta )\mathbf{u}]=0$. The material derivative is $D_b={\partial }_b+\mathbf{u}\cdot \nabla$. Substitution into the pressureless cosmological Euler equation, followed by the growth equation and $\dot{b}=Hbf$, proves the displayed motion equation, with $f=d\ln b/d\ln a$. The density derivative in the conservative continuity equation is Eulerian, not material.

For pressureless sub-Hubble perturbations, combine the linearized cosmological continuity equation, cosmological Euler equation and gravitational Poisson equation with matter as its source. The expansion contributes a $2H$ friction term and matter self-gravity contributes the negative coefficient of $\delta$. The background expansion can include radiation or dark energy; treating their perturbations as additional sources requires a more complete system.

Differentiate the linearized cosmological continuity equation, remembering the derivative of $1/a$:Insert the linear cosmological Euler equation for the peculiar velocity, givingThus the linear matter perturbation growth equation isThe factor $2H$ includes one expansion term from continuity and one from momentum dilution. This equation uses sub-Hubble, pressureless linear perturbations and the stated matter-only source for the gravitational potential.

Here primes mean conformal time derivatives, $d\tau =dt/a$, and spatial derivatives use comoving coordinates in Newtonian gauge in cosmology. Set ${\theta }_m=\nabla \cdot {\mathbf{v}}_m$. The linear cosmological Euler equation gives ${\theta }_m^{\prime }=-\mathcal{H}{\theta }_m-{\nabla }^2\Phi$, while the continuity equation gives ${\theta }_m=-{\delta }_m^{\prime }+3{\Phi }^{\prime }$.

Differentiate the continuity equation and substitute:Thus the linear matter density equation in Newtonian gauge isThe common potential in these supplied equations assumes that the relevant scalar anisotropic stress is neglected. No Fourier-sign convention is needed for the derivation in position space.

The background Einstein-de Sitter universe satisfies $a^{\prime }=\mathcal{H}a$ and ${\mathcal{H}}^{\prime }=-{\mathcal{H}}^2/2$. Introduce ${\mathcal{F}}_n$ and ${\mathcal{G}}_n$ for the spatial convolutions in the Fourier transform representation with kernels $F_n$ and $G_n$, respectively, so ${\delta }^{(n)}=a^n{\mathcal{F}}_n$ and ${\theta }^{(n)}=-\mathcal{H}{a}^n{\mathcal{G}}_n$. Differentiation givesAt first order the cosmological continuity equation yields $F_1-G_1=0$, while the cosmological Euler equation gives $3F_1/2-3G_1/2=0$. The growing-mode normalization therefore gives $\boxed{G_1=F_1=1}$.

At second order, take the unsymmetrized order of the inputs printed in the paper. Matching the alpha mode-coupling kernel and beta mode-coupling kernel terms givesHenceWith the paper's unsymmetrized convention $\alpha =\beta =(q_1+q_2)/q_1$, one possible unsymmetrized representation isOnly the symmetric part enters a convolution of two identical linear fields. Symmetrizing givesThis is the one-dimensional cosmological density kernel. The physical symmetric one-dimensional Euler kernel is also $(q_1+q_2{)}^2/(2q_1{q}_2)$: symmetrizing the printed $\beta$ reproduces it, so its apparently different unsymmetrized form is not a contradiction. Unsymmetrized kernels themselves are not unique, since antisymmetric parts integrate to zero. Zero input momenta require the usual separate treatment of the homogeneous background.

The cosmological continuity equation expresses conservation of dark-matter mass: $\delta$ is the density contrast, $\mathbf{v}$ is the peculiar velocity, and a prime denotes a conformal time derivative. The cosmological Euler equation expresses momentum conservation: $\mathcal{H}=a^{\prime }/a$ is the conformal Hubble rate, $\varphi$ is the peculiar gravitational potential, $\rho$ is the density, and ${\sigma }_{ij}$ is the velocity-dispersion tensor of collisionless matter. The terms are respectively Hubble drag, convective acceleration, gravity, and velocity-dispersion stress.

For curl-free flow, introduce the peculiar-velocity divergence $\theta =\nabla \cdot \mathbf{v}$ and set ${\sigma }_{ij}=0$. A Fourier transform of the nonlinear continuity term giveswith the alpha mode-coupling kernelTaking the divergence of the Euler equation gives the quadratic velocity kernelThis is the beta mode-coupling kernel; its symmetry follows from the two velocity factors.

The linearized cosmological continuity equation, the divergence of the cosmological Euler equation, and the cosmological Poisson equation giveEliminating $\theta =-{\delta }^{\prime }$ yields the equation for a linear cosmological density perturbation:In an Einstein-de Sitter universe, ${\Omega }_m=1$ and $\mathcal{H}=2/\tau$, soSubstitution of the power-law ansatz $\delta \propto {\tau }^p$ gives $(p-2)(p+3)=0$. HenceThe leading one of the matter-era growing and decaying density modes is $\delta \propto a\propto {\tau }^2$, as direct substitution confirms, and its velocity divergence is

Linearizing the cosmological Euler equation givesTaking the curl removes the gradient force, so the vorticity $\mathbf{w}=\nabla \times \mathbf{v}$ obeysSince $\mathcal{H}=a^{\prime }/a$, its solution isThus expansion dilutes linear vorticity; in an Einstein-de Sitter universe, $a\propto {\tau }^2$ and $\mathbf{w}\propto {\tau }^{-2}$.