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The condition is that the velocity be the pure irrotational growing-mode velocity, with the homogeneous component in part (c) absent. It is enough to impose ${\mathbf{u}}_{\mathrm{in}}=-\nabla {\psi }_{\mathrm{in}}$ and exclude additional harmonic or transverse motions through the boundary conditions. The linear equation for $\mathbf{u}+\nabla \psi$ then keeps that difference zero. A time-independent velocity also forces the same equality in the linear motion equation when its coefficient is nonzero.

At first order the gradient can be evaluated at the initial position: shifting its argument by the first-order displacement changes the velocity only at second order. Integrate $d\mathbf{x}/db=-{\nabla }_{{\mathbf{x}}_{\mathrm{in}}}{\psi }_{\mathrm{in}}$ from $b_{\mathrm{in}}$ to $b$, givingThe trajectory is straight in comoving position as a function of growth time. The physical trajectory $a(t)\mathbf{x}(t)$ need not be straight or have constant physical peculiar velocity. This linear displacement relation is the basis of the Zeldovich approximation.

For the Zeldovich approximation, form the Jacobian matrix of the Lagrangian map:Let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of this potential Hessian matrix, with the convention that positive $\lambda$ means compression. Before shell crossing, conservation of the mass in a Lagrangian volume givesIf ${\lambda }_1>0$ and is distinct from the other eigenvalues, the first singularity occurs at $b=1/{\lambda }_1$. One principal length collapses while the other two remain nonzero. The resulting cosmological caustic is locally a sheet, conventionally a pancake, rather than a simultaneous spherical point collapse. Degenerate equal eigenvalues can instead give line- or pointlike special cases.

The predicted density divergence signals failure of the single-stream description. If the straight Lagrangian displacements are simply continued, particles cross the sheet and keep moving along their original displacement directions. They are not slowed and trapped by the evolved self-gravitating multistream structure. The streams therefore separate again, producing excessively broad pancakes at late times. In an N-body simulation, the changing gravitational field pulls crossing streams back, produces repeated crossings and binds matter into more compact structures. This is the limitation described by Zeldovich pancake thickness after shell crossing. The initial-displacement extrapolation is useful through the onset of collapse, but does not supply the self-consistent post-crossing gravitational dynamics or virialization. After crossing, the physical density would require summing inverse absolute Jacobians over all streams, not using one negative determinant as a density.

Mass conservation in the Zeldovich approximation gives $\rho d^{3}r=\bar{\rho }{a}^3{d}^{3}q$. Expanding the trajectory Jacobian determinant to first order therefore yields $\boxed{\delta =-{\nabla }_q\cdot \Psi }$. At the same order, ${\nabla }_r=a^{-1}{\nabla }_q$ andThe last equality uses ${\nabla }_q\times \Psi =0$ and the vector calculus $\nabla \mathrm{div}={\nabla }^2+\nabla \times \nabla \times$.

Split the gravitational potential into its homogeneous-background and perturbed parts. The background has ${\nabla }_r{\Phi }_0=(4\pi G\bar{\rho }/3)r$. For the perturbation, Poisson equation and irrotationality givewith a harmonic or uniform-acceleration contribution set to zero by the perturbation boundary conditions. Its divergence is $-4\pi G\bar{\rho }{\nabla }_q\cdot \Psi =4\pi G\bar{\rho }\delta$, as required. Insert this and the pressure gradient into $\ddot{r}=\ddot{a}(q+\Psi )+2\dot{a}\dot{\Psi }+a\ddot{\Psi }$. The background acceleration cancels using $\ddot{a}/a=-4\pi G\bar{\rho }/3$, leavingTake minus the comoving divergence, and expand $\delta (q,t)={\sum }_k{\delta }_k(t)e^{ik\cdot q}$. Since ${\nabla }_q^2$ acts on a Fourier mode by $-k^2$, each mode satisfiesHere the expansion coordinate is comoving; confusing it with the physical coordinate would lose the factor $a^{-2}$.

A fixed comoving volume $d^{3}q$ labels a fixed collection of cold dark matter particles. Their background mass is $a^3\bar{\rho }{d}^{3}q$, which is constant by mass conservation. At the perturbed positions it is $\rho d^{3}r$. The Jacobian determinant of the Zeldovich approximation map isTaking its reciprocal and using mass conservation givesFor the perturbed field the Poisson equation must be ${\nabla }_r^2\Phi =4\pi G\rho =4\pi G\bar{\rho }(1+\delta )$. The displayed background-only right-hand side in the PDF would give no clumping force; it is the background equation, not the full perturbed equation.

Write $\Phi ={\Phi }_b+\phi$, where ${\nabla }_r{\Phi }_b=(4\pi G/3)\bar{\rho }\mathbf{r}$. To linear order,For the longitudinal, curl-free displacement used in the growing Zel'dovich mode, this integrates to ${\nabla }_q\phi =-4\pi G\bar{\rho }{a}^2\mathbf{\psi }$, after removing external homogeneous harmonic fields. More generally the force involves only the longitudinal part ${\mathbf{\psi }}_{\parallel }$, which has the same divergence as $\mathbf{\psi }$.

Differentiate $\mathbf{r}=a(\mathbf{q}+\mathbf{\psi })$ twice at fixed $\mathbf{q}$ and substitute $\ddot{\mathbf{r}}=-{\nabla }_r\Phi$. The terms proportional to $\mathbf{q}+\mathbf{\psi }$ cancel using $\ddot{a}=-(4\pi G/3)\bar{\rho }a$, leavingFor a longitudinal displacement the right side is $4\pi G\bar{\rho }\mathbf{\psi }$. Taking minus the comoving divergence gives the general scalar density contrast equation

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,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 givesDemanding $\delta (t)=D(t){\delta }_i$ 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,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 equationThus 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 ${\rho }_0{d}^3{x}_i$. Substituting the velocity formula into the supplied definition of angular momentum about the centre of mass givesTerms from the perturbed mass measure are higher order. Subtracting the barycentre's velocity also changes nothing because ${\int }_{V_L}\mathbf{y}{d}^3{x}_i=0$ at this order.

Apply the divergence theorem componentwise:The antisymmetry of the Levi-Civita symbol kills the last term. Consequently, for the ordinary oriented comoving surface element $d\mathbf{S}$,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. HenceFor an equipotential surface, ${\Phi }_i={\Phi }_{\ast }$ on ${\Sigma }_L$, andThereforeThe 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:The constant has no force; $\mathbf{g}$ has zero integrated torque because the first mass moment vanishes. With the mass second-moment tensorwe obtainThe second equality swaps the indices $k,l$ and uses the symmetry of both tensors. Define the initial acceleration tidal tensor byThen the formula in the question has exactly the stated sign: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 $\mathrm{tr}(I)1-I$.

Only the anisotropic parts contribute: a multiple of the identity produces zero contraction with the Levi-Civita symbol. In axes where $I=\mathrm{diag}(I_1,I_2,I_3)$, for example,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, soThis 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.

Write physical position as $\mathbf{r}=a(t)\mathbf{x}$ and physical velocity asSubtract the homogeneous expanding background from the pressureless Euler equation. To linear order in peculiar velocity and density contrast, the convective term is negligible andorHere $\nabla$ differentiates with respect to comoving position and $\Phi$ is the peculiar gravitational potential.

The linear continuity and Poisson equations areFor the growing mode $\delta (\mathbf{x},t)=D(a){\delta }_i(\mathbf{x})$, matter conservation gives ${\bar{\rho }}_m\propto a^{-3}$, so the potential scales asafter absorbing the fiducial normalization into ${\Phi }_i$. Integrating the linear Euler equation with a negligible decaying mode gives

Taking the divergence of Euler and using continuity and Poisson yields the linear growth equationIn the fiducial normalization used in the question this can be rearranged asThe cosmological Lagrangian displacement obeys $\dot{\mathbf{\psi }}=\mathbf{v}/a$. Combining the last two relations and choosing the growing displacement gives the Zeldovich approximation

DefineThen $\mathbf{x}={\mathbf{x}}_i-b\nabla {\Phi }_i$ and $\mathbf{v}=-a\dot{b}\nabla {\Phi }_i$. Keeping the lowest nonvanishing order in displacement in the halo angular momentum givesThe angular momentum vanishes for a spherical Lagrangian patch, and also whenever the patch's inertia principal axes align with the local tidal-field principal axes. It likewise vanishes in a locally isotropic tidal field.

Taylor-expandThe constant-gradient term vanishes by the barycentre definition. Withone obtains the tidal torque theory result$I$ is the protohalo inertia tensor and $T$ is the local tidal, or gravitational Hessian, tensor. Their eigenframe misalignment produces the torque.

In an Einstein-de Sitter universe, $D\propto a\propto t^{2/3}$. Hence $\dot{b}\propto \dot{a}\propto t^{-1/3}$ andThis linear estimate is normally stopped near turnaround. Simulated haloes subsequently gain angular momentum through nonlinear torques, anisotropic accretion, and mergers, including the orbital angular momentum of infalling subhaloes, so the early linear estimate underpredicts the final value.

Write proper position as $\mathbf{r}=a(t)\mathbf{x}$ and proper velocity asSubstitute this decomposition into the proper-coordinate Euler equations for an inviscid fluid, use ${\nabla }_r=a^{-1}{\nabla }_x$, and subtract the homogeneous-background acceleration. With $\rho =\bar{\rho }(1+\delta )$, one obtains the comoving peculiar-velocity equationThe peculiar gravitational potential $\Phi$ is the total Newtonian potential with the potential of the exactly homogeneous expanding background subtracted. Its gradient therefore generates only accelerations relative to the Hubble flow.

For a pressureless fluid, linearization discards the quadratic advection term, leavingIn the early Einstein-de Sitter universe, the growing mode has $\Phi (\mathbf{x},t)=[D(a)/a]{\Phi }_i(\mathbf{x})$. Integration gives

The linear growth factor obeysSince $\bar{\rho }={\bar{\rho }}_i/a^3$ after choosing $a_i=1$, this is equivalent toIt follows that $a\mathbf{v}=-a^2\dot{D}\nabla {\Phi }_i/(4\pi G{\bar{\rho }}_i)$. Since $\dot{\mathbf{x}}=\mathbf{v}/a$, one final integration gives the Zeldovich approximationwhere an additive initial displacement has been absorbed into ${\mathbf{x}}_i$.

Let $\bar{\rho }(t)$ be the homogeneous density. A Lagrangian coordinate volume $d^{3}q$ contains the same mass as its image $d^{3}x$ under the Zeldovich approximation, so mass conservation givesSince $1+\delta =\rho /\bar{\rho }$, the density contrast is

For $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor isEquality of mixed partial derivatives makes it a real symmetric matrix. The real spectral theorem therefore supplies an orthonormal eigenbasis and real eigenvalues, which we denote by $-{\lambda }_1,-{\lambda }_2,-{\lambda }_3$. In that basis,and henceBefore the first crossing all factors are positive, allowing the absolute values to be omitted.

When $1-D{\lambda }_i=0$, the map loses rank in the corresponding principal direction. Its Jacobian determinant vanishes, trajectories meet, and shell crossing creates a cosmological caustic. The single-stream pressureless density formally diverges and the approximation no longer describes the subsequent multistream dynamics. If only one ${\lambda }_i$ is positive, one axis first collapses while the other two remain extended, producing a sheet or pancake. Collapse along a second and then a third principal axis produces filaments and nodes. Spatial variation of the eigenvalues joins these objects into the cosmic web around underdense voids.

The approximation succeeds because it reproduces linear growing-mode evolution exactly, preserves the initial tidal displacement and anisotropic collapse, and follows matter along nearly inertial comoving trajectories instead of expanding only the density at a fixed point. Large scales remain weakly nonlinear and are insensitive to the detailed dynamics after crossing. Its limitations begin at shell crossing: it permits streams to pass through one another, cannot produce virialized halos, and omits velocity dispersion, vorticity, gas pressure, shocks, feedback, and strongly nonlinear self-gravity.

For the one-dimensional displacement,so mass conservation givesThe earliest crossing occurs where the cosine is maximal:for integers $n$. The denominator vanishes at these isolated points.

Write $u=q-q_n$ at $t=t_c$. The Taylor series givesThus $|u|\sim (6|x-x_n|/k^2{)}^{1/3}$ andThe first caustic is therefore a cubic cusp with

For $\mathbf{r}=a[\mathbf{q}+b(t)(p(q_1),0,0)]$, put $J=1+b{p}^{\prime }$. Before shell crossing, mass conservation gives $\rho =\bar{\rho }/J$. The dust background equation $3\ddot{a}/a=-4\pi G\bar{\rho }$ and the growing-mode equation $\ddot{b}+2H\dot{b}=4\pi G\bar{\rho }b$ imply ${\nabla }_r\cdot \ddot{\mathbf{r}}=-4\pi G\bar{\rho }+4\pi G\bar{\rho }b{p}^{\prime }/J=-4\pi G\rho$. Thus planar Zeldovich approximation trajectories satisfy the full nonlinear dust force-divergence equation until the map loses invertibility. All three physical coordinates are included in the divergence; the two unperturbed directions still contribute background acceleration.

For the Zeldovich approximation $\mathbf{x}=\mathbf{q}-b{\nabla }_q\psi$, let ${\lambda }_i$ be the eigenvalues of the potential Hessian, with this sign convention. Before shell crossing, mass conservation gives the displayed density. The first positive eigenvalue reaches $b{\lambda }_i=1$ and collapses one direction, creating a sheetlike cosmological caustic. Extrapolating straight trajectories after crossing lets particles pass through without self-consistent gravitational trapping, so the predicted multistream sheet spreads excessively. Collisionless gravitational evolution keeps matter more tightly bound; the linear displacement extrapolation cannot determine its late thickness reliably.