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Assume nonrelativistic <pressureless matter>, an initial <irrotational vector field> of growing-mode <linear cosmological density perturbations>, a homogeneous expanding background, and no <shell crossing>. All gradients below refer to initial <comoving coordinates>. Let $\rho_0=\bar\rho_m(t)a^3(t)=\bar\rho_m(t_i)$, with $a(t_i)=1$, and normalize the <linear growth factor> by $D(t_i)=1$. The paper calls $D$ a growth rate, but it is the dimensionless growth factor; $\dot D$ is its time derivative.
Writing $\mathbf x_i$ for the actual position at $t_i$, the <Zeldovich approximation> is, to first order,
$$
\mathbf x(t)=\mathbf x_i+[D(t)-1]\mathbf s(\mathbf x_i),
\qquad
\dot{\mathbf x}=\dot D\,\mathbf s(\mathbf x_i).
$$
This explicitly gives $\mathbf x(t_i)=\mathbf x_i$. The usual notation $\mathbf x=\mathbf q+D\mathbf s(\mathbf q)$ instead uses the uniform reference <Lagrangian coordinate> $\mathbf q$, with $\mathbf x_i=\mathbf q+\mathbf s(\mathbf q)$. The two labels can be interchanged inside a first-order displacement, but not in the unperturbed position without the initial displacement correction.
To determine $\mathbf s$, <mass conservation> between the initial and current positions gives
$$
1+\delta(t)=\frac{1+\delta_i}{\det[\mathbf 1+(D-1)\nabla\mathbf s]}
=1+\delta_i-(D-1)\nabla\cdot\mathbf s+O(\delta_i^2).
$$
Demanding $\delta(t)=D(t)\delta_i$ gives $\nabla\cdot\mathbf s=-\delta_i$. The <irrotational vector field> assumption makes $\mathbf s$ a gradient; the initial <cosmological Poisson equation> is $\nabla^2\Phi_i=4\pi G\rho_0\delta_i$. With the same boundary conditions for these potential equations, and after removing an irrelevant uniform translation,
$$
\mathbf s=-\frac{\nabla\Phi_i}{4\pi G\rho_0},
\qquad
\boxed{\dot{\mathbf x}
=-\frac{\dot D}{4\pi G\bar\rho_m a^3}\nabla\Phi_i.}
$$
The evolution of $D$ follows from the linearized peculiar-motion equation $\ddot{\mathbf x}+2H\dot{\mathbf x}=-\nabla\Phi/a^2$. Since <cosmological Poisson equation> gives $\Phi(t)=(D/a)\Phi_i$ to this order, substitution yields the <linear growth equation>
$$
\ddot D+2H\dot D-4\pi G\bar\rho_mD=0.
$$
Thus the approximation extrapolates the linear growing displacement along a fixed initial direction, even as its density mapping becomes nonlinear.
The early spin of a <dark matter halo> comes from an external gravitational <torque> on its nonspherical initial mass region. A uniform external acceleration moves its <centre of mass> without spinning it; the spatially varying <tidal tensor> exerts different forces on different parts. The <tidal torque theory> requires misalignment of the region's shape and the surrounding tidal field. A local <irrotational vector field> of velocity does not imply zero integrated <angular momentum> for a nonspherical region.
Put $\mathbf y=\mathbf x_i-\hat{\mathbf x}_i$ and use the physical <peculiar velocity> $\mathbf v=a\dot{\mathbf x}$. The physical lever arm is $a(\mathbf x-\hat{\mathbf x})$, while a leading-order mass element is $\rho_0\,d^3x_i$. Substituting the velocity formula into the supplied definition of <angular momentum about the centre of mass> gives
$$
\mathbf J=-\frac{a^2\dot D}{4\pi G}
\int_{V_L}\mathbf y\times\nabla\Phi_i\,d^3x_i.
$$
Terms from the perturbed mass measure are higher order. Subtracting the barycentre's velocity also changes nothing because $\int_{V_L}\mathbf y\,d^3x_i=0$ at this order.
Apply the <divergence theorem> componentwise:
$$
\epsilon_{jkl}\int_{V_L}y_k\partial_l\Phi_i\,d^3x_i
=\epsilon_{jkl}\int_{\Sigma_L}\Phi_i y_k\,dS_l
-\epsilon_{jkl}\int_{V_L}\Phi_i\delta_{kl}\,d^3x_i
=\epsilon_{jkl}\int_{\Sigma_L}\Phi_i y_k\,dS_l.
$$
The antisymmetry of the <Levi-Civita symbol> kills the last term. Consequently, for the ordinary oriented comoving surface element $d\mathbf S$,
$$
\boxed{\mathbf J=-\frac{a^2\dot D}{4\pi G}
\int_{\Sigma_L}\Phi_i\,\mathbf y\times d\mathbf S.}
$$
The original PDF prints a different coefficient, $-\dot D/(4\pi G\bar\rho_m a^2)$. That coefficient does not follow from its own mass measure and velocity equation and has the wrong dimensions for total <angular momentum>. The corrected coefficient above also gives the time dependence consistent with the paper's later tensor expression; this is a source typo, not a TeX transcription issue.
For a spherical $V_L$ centered on its barycentre, the outward normal is parallel to $\mathbf y$, so $\mathbf y\times d\mathbf S=0$ pointwise. Hence
$$
\boxed{\mathbf J=0\quad\text{for a spherical initial region}.}
$$
For an <equipotential surface>, $\Phi_i=\Phi_*$ on $\Sigma_L$, and
$$
\epsilon_{jkl}\int_{\Sigma_L}y_k\,dS_l
=\epsilon_{jkl}\int_{V_L}\partial_l y_k\,d^3x_i
=\epsilon_{jkl}\delta_{kl}\operatorname{Vol}(V_L)=0.
$$
Therefore
$$
\boxed{\mathbf J=0\quad\text{if the initial boundary is an equipotential}.}
$$
The second conclusion holds for any boundary shape at this order, not just a sphere.
To extract the leading tidal torque, expand the initial <peculiar gravitational potential> about the barycentre using its <Taylor series>:
$$
\Phi_i(\hat{\mathbf x}_i+\mathbf y)
=\Phi_0+g_k y_k+\frac12H_{km}y_ky_m+\cdots,
\qquad
H_{km}=\partial_k\partial_m\Phi_i(\hat{\mathbf x}_i).
$$
The constant has no force; $\mathbf g$ has zero integrated torque because the first mass moment vanishes. With the <mass second-moment tensor>
$$
I_{km}=\rho_0\int_{V_L}y_ky_m\,d^3x_i,
$$
we obtain
$$
J_j=-\frac{a^2\dot D}{4\pi G\rho_0}
\epsilon_{jkl}H_{lm}I_{km}
=\frac{\dot D}{4\pi G\bar\rho_m a}
\epsilon_{jkl}H_{km}I_{ml}.
$$
The second equality swaps the indices $k,l$ and uses the symmetry of both tensors. Define the initial acceleration <tidal tensor> by
$$
T_{km}=-\partial_k\partial_m\Phi_i(\hat{\mathbf x}_i)=-H_{km}.
$$
Then the formula in the question has exactly the stated sign:
$$
\boxed{J_j=-\frac{\dot D}{4\pi G\bar\rho_m a}
\epsilon_{jkl}T_{km}I_{ml}.}
$$
If one defines $T$ as the positive <Hessian matrix> of $\Phi_i$ instead, the displayed contraction has a plus sign. Defining the convention is essential. Also, the paper's $I$ is a <mass second-moment tensor>; the mechanical <inertia tensor> is $\operatorname{tr}(I)\mathbf1-I$.
Only the anisotropic parts contribute: a multiple of the identity produces zero contraction with the <Levi-Civita symbol>. In axes where $I=\operatorname{diag}(I_1,I_2,I_3)$, for example,
$$
J_1=\frac{\dot D}{4\pi G\bar\rho_m a}T_{23}(I_2-I_3).
$$
Thus a spherical region or tensors sharing principal axes have zero leading torque; unequal principal moments together with off-diagonal tidal components create spin. The <Taylor series> truncation assumes that higher spatial derivatives of the tidal field are sufficiently small across the region.
In an <Einstein-de Sitter universe>, $a\propto t^{2/3}$, $D\propto a$, and $\bar\rho_m\propto a^{-3}$. The initial $I$ and $T$ are time independent, so
$$
\boxed{\mathbf J\propto a^2\dot D\propto t^{4/3}t^{-1/3}
\propto t.}
$$
This is early-time growth for fixed initial matter, before collapse invalidates the extrapolation. It does not predict indefinite linear spin growth for a virialized halo.
Two reasons for only approximate agreement with simulations are that nonlinear collapse and <shell crossing> change the trajectories and the tidal field, invalidating the first-order displacement and fixed leading <tidal tensor>; and that real halos undergo <dark-matter halo mergers>, anisotropic accretion, and exchange of material and <angular momentum>, so the halo identified at a later time need not be the same isolated collection of initial particles. These processes can change both the magnitude and the direction of the spin.
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