Extrapolated Linear Density Contrast

• Extrapolated Linear Density Contrast is the linearized density fluctuation extracted from nonlinear fields to emulate primordial Gaussian modes.
• It employs Lagrangian perturbation theory and log-transform techniques to invert nonlinear mappings, enhancing fidelity to the initial density field.
• This method underpins BAO reconstruction and inverse problems in elasticity, improving cosmological parameter estimation and material characterization.

The extrapolated linear density contrast is a central concept in cosmological large-scale structure analysis, inverse problems in PDEs, and density field modeling. It refers to the linear-theory component of the total density fluctuation, reconstructed from nonlinear or observed fields using analytical transformations or perturbative expansions, so that this linearized field can be extrapolated, compared, or evolved as if it were an initial Gaussian mode. This approach underpins a variety of reconstruction schemes in cosmology and mathematical physics for recovering primordial fluctuations, correcting for nonlinear evolution, or solving coefficient inverse problems.

1. Theoretical Foundations and Definitions

The density contrast at Eulerian position $x$ and redshift $z$ is defined as

$$\delta(x, z) \equiv \frac{\rho(x, z) - \bar{\rho}(z)}{\bar{\rho}(z)},$$

where $\rho(x, z)$ is the comoving matter density and $\bar{\rho}(z)$ the mean. The extrapolated linear density contrast, denoted $\delta_L(x, z)$, is constructed such that, evolved with linear theory from initial redshift $z_{\mathrm{ini}}$, it would reproduce the observed full nonlinear field at redshift $z$ (Kitaura et al., 2011).

Within Lagrangian perturbation theory (LPT), the fluid element starting at Lagrangian coordinate $q$ is mapped to the Eulerian position $x = q + s(q, z)$, where $s(q, z)$ is the Lagrangian displacement field. The goal of linearization is to invert the mapping $q \to x$ or, equivalently, to reconstruct the linear field $\delta_L$ from the nonlinear $\delta$.

In mathematical inverse problems (e.g., elasticity), analogous constructs appear in the linearization of operator maps around a reference or engineered background, yielding a first-order approximation to the density contrast relevant for recovery from boundary data (Diao et al., 16 Jan 2026).

2. Linear and Logarithmic Density-Displacement Relations

From the continuity equation in comoving coordinates and to first order, the standard relation between the Eulerian overdensity and displacement divergence is

$\delta = -\nabla \cdot \psi,$

where $\psi$ is the displacement field (as in the Zel’dovich approximation) (Falck et al., 2011).

On quasi-linear or nonlinear scales, the linear relation loses fidelity due to the breakdown of first-order expansion. Retaining the full continuity equation and integrating yields a generalized, logarithmic relation:

$$\nabla \cdot \psi = -\ln(1+\delta) + \langle \ln(1+\delta) \rangle,$$

with the mean subtracted to ensure a zero mean for the displacement divergence. This log-transform "straightens" the relation between density and displacement, extending the regime of validity to higher contrasts ($\delta \sim 1-10$) and non-Gaussian fields. Empirically, the correlation between $-\nabla \cdot \psi$ and $\ln(1+\delta) - \langle \ln(1+\delta) \rangle$ is substantially higher than with $\delta$, with residual scatter $\sigma_{\log} \approx 0.22$ versus $\sigma_{\rm lin} \approx 0.50$ at $z=0$ for $1.6\,h^{-1}\,\mathrm{Mpc}$ CIC smoothing, and Pearson coefficients $r_{\log} \approx 0.92$ versus $r_{\rm lin} \approx 0.70$ (Falck et al., 2011).

3. Linearization via Lagrangian Perturbation Theory

In LPT, the displacement is expanded perturbatively:

$$s(q, z) = D_1(z) \nabla_q \phi^{(1)}(q) + D_2(z) \nabla_q \phi^{(2)}(q) + \mathcal{O}(D_3),$$

with $D_1$ the linear, $D_2$ the second-order growth factors, and $\phi^{(1)}$, $\phi^{(2)}$ the corresponding Lagrangian potentials determined by Poisson-like equations.

The log-transform admits a systematic expansion:

$$\ln[1+\delta] = -\nabla \cdot s + \tfrac{1}{2}\big[(\nabla \cdot s)^2 - s_{i,j}s_{j,i}\big] + \dots = \delta^{(1)}(x) + \delta_{\rm NL}(x),$$

where $\delta^{(1)}$ is the (linearly) extrapolated density contrast and $\delta_{\rm NL}$ encodes higher order corrections. To leading order, the linear component is isolated as

$$\delta_L(x) \simeq \alpha \ln[1+\delta(x)] + \beta\,\mathrm{Tr}\,T(x),$$

where $T_{ij}(x)$ is the tidal tensor of the linear field, $\alpha \simeq 1$, and $\beta \simeq D_2/D_1^2 \simeq -3/7$ for Einstein–de Sitter-like growth (Kitaura et al., 2011).

An iterative scheme in LPT inverts the nonlinear density to reconstruct $\delta_L$ via the Jacobian of the Lagrangian-to-Eulerian mapping, using both the log-transform and the tidal term to optimize Gaussianity and fidelity.

4. Extrapolation and Redshift Scaling

Once $\delta_L(x, z)$ is recovered at some redshift $z$, extrapolation to any other redshift $z'$ under linear theory uses the growth factor $D(z)$:

$$\delta_L(x, z') = \frac{D(z')}{D(z)}\,\delta_L(x, z),$$

with the growth factor determined by the background cosmology (Kitaura et al., 2011). This relabels the quasi-linearized field at any epoch, enabling primordial or future state reconstruction, backward or forward in cosmological time.

For the elastic Calderón problem, the first-order linearization of the Neumann-to-Dirichlet (N–D) map around an engineered background density yields an analogous linear functional of the density contrast, with explicit inversion possible in the Fourier domain via complex geometric optics (CGO) probing (Diao et al., 16 Jan 2026).

5. Numerical Methods and Practical Reconstruction

Various estimators are used for practical field measurement and reconstruction. In cosmological $N$-body simulations, density and displacement divergence are estimated using:

• Cloud-In-Cell (CIC) & Fourier divergence: Particles are interpolated onto a regular grid, and Fourier methods compute $\nabla \cdot \psi$. The grid sets the smoothing scale (e.g., $1.6\,h^{-1}\,\mathrm{Mpc}$ for $128^3$ cells).
• Adaptive SPH-style smoothing: At each grid point, a kernel encompasses a prescribed number of nearest particles ($M_{\min}=32$), yielding locally adaptive smoothing.
• Delaunay Tessellation Field Estimator (DTFE): Uses the unstructured mesh generated from the particle distribution and computes local densities from surrounding simplex volumes, applying Gauss’s theorem for divergence (Falck et al., 2011).

For the linearization-in-LPT approach, the recommended procedure includes: smoothing $\delta(x, z)$ on $r \gtrsim 5\,h^{-1}\,\mathrm{Mpc}$, computing the log-transformed field, extracting the tidal tensor via Poisson’s equation, and iteratively correcting for nonlinear tidal contributions. The iterative log-tidal scheme sharpens the Gaussianity of the recovered $\delta_L$ and enhances cross-correlation with the original primordial field by $\Delta G \sim 0.1$ on large scales.

In elasticity, explicit CGO boundary data and measurement of the perturbed N–D map enable recovery of Fourier coefficients of the density fluctuation, with controllable error by tuning the inclusion parameters and resonant frequency (Diao et al., 16 Jan 2026).

6. Accuracy, Regimes of Validity, and Limitations

The accuracy of extrapolated linear density contrast recovery depends on the chosen method and scale. In LPT-based cosmological schemes, on scales $k \lesssim 0.2\,h\,\mathrm{Mpc}^{-1}$ ($r \gtrsim 5\,h^{-1}\,\mathrm{Mpc}$), the reconstructed $\delta_L$ cross-correlates with the true initial field at $r > 0.9$. The skewness and kurtosis of $\delta_L$ drop by orders of magnitude relative to the raw nonlinear field (e.g., $S \rightarrow 0.3$ and $K \rightarrow 0.1$, vs. $S \sim 4$, $K \sim 200$ for the original) (Kitaura et al., 2011). At smaller scales or in multi-streaming (post-shell-crossing), the accuracy degrades due to LPT breakdown.

In log-transform approaches, the method’s regime of validity extends to $\delta \sim 1-10$, significantly beyond the linear regime. Best results arise when grid-based Eulerian smoothing of $2\,h^{-1}\,\mathrm{Mpc}$ or larger is employed; under-sampled cells require regularization to mitigate shot noise. Redshift-space distortions (“Fingers-of-God”) and bias affect mapping from galaxy to mass density but can be controlled in baryon acoustic oscillation (BAO) and similar analyses (Falck et al., 2011).

For inverse elastic problems, the error in density reconstruction can be made arbitrarily small by tuning the inclusion size, number, and the effective negative background shift, with explicit error terms:

$$O(\mathcal{P}^{-4} + \mathcal{P}^{-\iota} + a^\alpha \mathcal{P}^6),$$

ensuring reliable signal recovery as long as $\mathcal{P}$ is large and $a$ small (Diao et al., 16 Jan 2026).

7. Applications and Broader Significance

Extrapolated linear density contrasts are foundational in baryon acoustic oscillation reconstruction, primordial power spectrum recovery, velocity field estimation, constrained simulations, and improved cosmological parameter inference. The methodology supports practical workflows for both simulation-based and observational data, allowing the removal of nonlinear mode-coupling and restoration of lost information content, particularly at low redshift and on quasi-linear scales.

In mathematical inverse problems, the linearization and explicit inversion around engineered backgrounds provide a paradigm for global coefficient recovery, with explicit connections to metamaterial design and the use of resonant inclusions to optimize the informativeness of boundary data (Diao et al., 16 Jan 2026).

These techniques are continuously refined and generalized, e.g., to accommodate redshift-space systematics, non-Gaussian initial fields, higher-order LPT treatments, and the design of adaptive experimental probes for inverse problems. The ability to reconstruct a statistically linear or Gaussian field, even in the quasi-nonlinear regime, underpins a wide spectrum of research in physical cosmology, applied mathematics, and inverse problems.