Zeldovich Approximation in Cosmology

• Zeldovich approximation is a first-order Lagrangian method that maps initial positions to final mass distributions, capturing large-scale flows and the onset of nonlinear structure formation.
• Its analytic formulation, using Green's function analysis, provides closed-form expressions for n-point statistics and BAO damping, validated by N-body simulations.
• Extensions incorporating Lagrangian bias and iterative improvements make the approximation a robust tool for cosmic web reconstruction, RSD removal, and density field reconstruction.

The Zeldovich approximation is a first-order Lagrangian perturbative solution for the gravitational evolution of cosmic structure in cold dark matter cosmology. It provides a mapping from initial (Lagrangian) positions to final (Eulerian) positions of mass elements, resumming all linear bulk flows and capturing the onset of nonlinear structure formation—caustics, pancakes, filaments—across a wide range of cosmological and theoretical contexts. Due to its analytic tractability, minimal input requirements (initial density and linear growth factor), and close agreement with both simulations and general relativity on large scales, the Zeldovich approximation remains central to modern treatments of nonlinear cosmic structure, power spectra, cosmic web geometry, and reconstruction techniques.

1. Formalism and Mapping

The core of the Zeldovich approximation is the Lagrangian-to-Eulerian mapping for a fluid element labeled by its initial position q: $$\mathbf{x}(\mathbf{q}, t) = \mathbf{q} + \Psi(\mathbf{q}, t)$$ where the displacement field $\Psi(\mathbf{q}, t)$ is given, to linear order, by

$\Psi(\mathbf{q}, t) = D(t)\, \Psi_0(\mathbf{q})$

with $D(t)$ the linear growth factor and $\Psi_0(\mathbf{q}) = -\nabla\phi_0(\mathbf{q})$ related to the initial density contrast via the Poisson equation $\nabla^2\phi_0 = \delta_0$. In Fourier space, the solution is

$$\Psi(\mathbf{k}, t) = -i\,\mathbf{k}\,k^{-2} \delta(\mathbf{k}, t)$$

Mass conservation relates the Eulerian overdensity $\delta(\mathbf{x}, t)$ to the divergence of the displacement: $$\delta(\mathbf{q}, t) \simeq -D(t)\, \nabla\cdot\Psi_0(\mathbf{q})$$

2. Derivation, Validity, and Green's Function Analysis

The equations of motion for collisionless dark matter, combined with the continuity and Euler equations, lead (in the absence of shell crossing) to the separable ansatz for displacements above. A Green's function analysis (Bartelmann, 2014) reveals that, when the exact cosmological Hamiltonian is split into a free particle part plus time-dependent corrections, the effective gravitational potential acting to drive deviations from Zeldovich trajectories vanishes exactly at initial time and remains significant only during a brief, early epoch. Consequently, the Zeldovich approximation is exceptionally accurate in the quasi-linear regime, especially on scales larger than the typical shell-crossing scale. Iterative improvement schemes can generate "free" trajectories with even more rapid decay of residual perturbations, but the principal accuracy of the Zeldovich solution arises from its absorption of most time dependence into inertial propagation.

3. Statistical Predictions and Comparisons with Simulations

The Zeldovich approximation leads to explicit, analytical expressions for n-point statistics (power spectrum, correlation functions, bispectrum, ..., in both real and redshift space). The two-point correlation function in the ZA is

$$1 + \xi(r) = \int \frac{d^3q}{(2\pi)^{3/2} |A|^{1/2}} \exp\left[ -\frac{1}{2} (r - q)^T A^{-1}(q) (r - q) \right]$$

where the displacement correlation tensor $A_{ij}(q)$ is a positive-definite, kernel-integrated function of the linear power spectrum. Generalization to redshift space is achieved by transforming $\Psi \to \Psi^{(s)}$ under the plane-parallel or full angular RSD mapping, see (Tassev, 2013, White, 2014, Castorina et al., 2018). High-fidelity codes such as ZelCa implement these formulas, enabling BAO and higher-order correlation modeling directly.

Compared to standard perturbative (Eulerian) approaches, the ZA non-perturbatively resums large-scale displacements, leading to the correct exponential damping of baryon acoustic oscillation (BAO) features and the realistic broadening of correlation peaks, which are sharper and less physically accurate in tree-level SPT. The effectiveness of the Zeldovich approximation and its variants (e.g., the Halo-Zeldovich-PT model (Seljak et al., 2015)) is confirmed by percent-level agreement with N-body simulations out to $k\simeq 1\,\mathrm{h/Mpc}$ in P(k), and for correlation functions above $\sim5\,\mathrm{Mpc}/h$.

4. Relation to General Relativity and Large-Scale Structure

General relativistic gradient expansions show that the Zeldovich mapping is not merely a Newtonian artifact, but emerges (at linear order) from the fully relativistic solution to the Einstein-fluid system in synchronous/comoving gauge in $\Lambda$CDM (Rampf et al., 2012, Rampf et al., 2014, Fidler et al., 2015). On horizon and super-horizon scales, the displacement field computed from GR matches ZA; the only modification is that the mapping between the Newtonian potential and density becomes a Helmholtz rather than a Poisson equation, reflecting causal constraints. At second order, genuine GR corrections appear at the level of $(H/k)^2\times \mathrm{2LPT}$, but these are negligible except for extremely large simulation volumes or modes near the horizon.

In practice, setting up N-body initial conditions with standard ZA in the so-called "N-body gauge" includes all first-order relativistic corrections, provided radiation and anisotropic stress are negligible at the initial redshift (Fidler et al., 2015).

5. Cosmic Web Geometry and the Formation of Singularities

The Zeldovich approximation naturally predicts the formation of caustics (pancakes), filaments, and nodes via the mapping's singularities. The eigenvalues $\lambda_i(\mathbf{q})$ of the initial deformation tensor dictate collapse: caustics form where $1 - D(t) \lambda_i(\mathbf{q}) = 0$. Catastrophe theory (Arnold's A-D-E classification) structures the generic caustic network: A₂ (folds), A₃ (cusps/filaments), A₄ (swallowtails/branchings), and D₄ (umbilics/nodes) (Hidding et al., 2013, Hidding et al., 2016). The mapping of these singularities defines the evolving geometric "skeleton" or "spine" of the cosmic web. Lagrangian lines and ridges—A₃-lines—act as progenitors of filaments, with their intersections and hierarchies establishing the high connectivity and percolation observed in the web (Hidding et al., 2013).

Adhesion models build upon ZA by introducing infinitesimal viscosity (Burgers' equation), causing mass elements to coalesce sharply onto pancakes, filaments, and nodes—more closely representing the post-shell-crossing cosmic web. Application to local universe constraints utilizes these skeleton extractions for rapid and robust structure visualization (Hidding et al., 2016).

6. Density Field Reconstruction and BAO Sharpening

The Zeldovich approximation is fundamental to density-field reconstruction techniques for sharpened BAO measurement and redshift-space distortion mitigation. The standard reconstruction pipeline employs the ZA to estimate the shift field from the observed (possibly biased) galaxy density, after smoothing with a Gaussian of scale $R_s \sim 10\,h^{-1}\mathrm{Mpc}$. The observed galaxies are displaced by the computed field, and a synthetic control sample ("shifted field") is similarly moved. The reconstructed density is the difference between the displaced and shifted fields, restoring the sharpness of the BAO feature and reducing the impact of nonlinear evolution (White, 2015, Chen et al., 2019).

Recent work (Nadathur et al., 2018) demonstrates explicit use of the ZA for RSD-removal in void clustering analysis: applying an FFT-based solution to the RSD-modified Zeldovich equation, galaxies are shifted by the line-of-sight RSD component before void-finding. This recovers real-space void statistics and enables unbiased measurement of the growth rate, a feat not achievable from voids identified directly in redshift space.

Both bias and counterterms up to quadratic order can be incorporated in the ZA framework to reach sub-percent modeling accuracy over all relevant cosmological scales (Chen et al., 2019). The method is robust, provided parameters such as the growth rate $f$, bias $b$, and the smoothing scale $R_s$ are specified a priori and tested for each cosmological model. Computational cost arises primarily from repeating reconstruction for each parameter set under consideration.

7. Extensions, Limitations, and Current Applications

Extensions of the Zeldovich approximation include higher-order Lagrangian perturbation theory (2LPT, 3LPT), inclusion of Lagrangian bias (analytic up to third order), incorporation of wide-angle RSDs (Castorina et al., 2018), and explicit treatment of unequal-time correlators with scale-dependent damping (Chisari et al., 2019). The underlying physical assumption is that large-scale flows dominate on scales above $\gtrsim 10-20\,\mathrm{Mpc}/h$; shell crossing and multi-streaming break down ZA locally, but its predictions remain globally and statistically robust for bulk properties and BAO.

Core limitations include the neglect of multi-stream, fully nonlinear regions, absence of small-scale velocity dispersion (fingers-of-god), and accurate treatment of high-density cluster environments. Application to BEC dark matter and adhesion models generalizes the mapping to include quantum and pressure effects, demonstrating ZA's foundational role in a broader class of structure formation paradigms (Chavanis, 2011).

Through analytic insight, computational efficiency, and broad validation, the Zeldovich approximation underlies state-of-the-art large-scale structure analysis, initial condition generation, and geometric characterization of the cosmic web. Ongoing research focuses on deeper integration with effective field theory, precision modeling at higher orders, and real survey application via iterative or adaptive generalizations of the basic ZA framework.