Local Universe Construction: 3D Cosmological Models

• Local Universe Construction is the process of generating detailed 3D models of our nearby cosmic environment by converting observational data into continuous density and velocity fields.
• It employs techniques such as Wiener Filter reconstruction and constrained realizations to correct biases and fill gaps in galaxy surveys.
• These reconstructions validate cosmological theories and simulate key structures like clusters, voids, and filaments for enhanced understanding of structure formation.

Local Universe Construction refers to the set of methodologies and frameworks employed to generate detailed three-dimensional models of the observed cosmological environment within a few hundred megaparsecs, matching the spatial and kinematic structures actually observed in the local cosmic web. This process involves converting redshift and peculiar-velocity measurements from galaxy surveys into continuous density and velocity fields, correcting for observational biases and incompleteness, statistically reconstructing matter and flow fields, and validating or ranking simulated realizations according to their fidelity to key observed structures. These constructed models serve as essential laboratories for testing cosmological theories of structure formation, galaxy evolution, and the influence of environmental and cosmographic features.

1. Observational Foundations and Preprocessing

The foundational datasets for Local Universe construction are large, nearly all-sky spectroscopic galaxy redshift catalogs, such as the V8k catalog (30,124 galaxies, $cz\leq8000\,\text{km\,s}^{-1}$, mapped in Supergalactic coordinates), and compilations of high-precision distance and peculiar velocity indicators, such as the Cosmicflows series (Cosmicflows-1, -2, -3) (Pomarede et al., 2012, Sorce et al., 2015, Pfeifer et al., 2023). Redshifts are first corrected to the Cosmic Microwave Background (CMB) frame, and, for nearby galaxies, the conversion to real-space distance involves flow models to account for peculiar motions (e.g., "numerical action" models for infall to Virgo, removal of virial motions in clusters).

Selection incompleteness due to flux limits is a key systematic: the galaxy luminosity function $\Phi(L)$ is used to estimate observational incompleteness, with each galaxy upweighted by $1/S(r)$, where $S(r)$ is the selection function given by

$$S(r) = \frac{\int_{L_\text{min}(r)}^\infty \Phi(L) dL}{\int_0^\infty \Phi(L) dL}.$$

These corrections enable faithful reconstruction of the underlying luminosity and number density fields as a function of position.

2. Density and Velocity Field Reconstruction

Constructing spatially continuous fields from discrete galaxy data involves statistical smoothing and bias correction. The luminosity density $\rho(\mathbf{r})$ is computed on a regular grid by summing the selection-corrected galaxy luminosities with a smoothing kernel, typically Gaussian with a characteristic scale $R$ (e.g., $R=500\,\text{km\,s}^{-1}\approx7\,\text{Mpc}$):

$$\rho(\mathbf{r}) = \sum_i \left[\frac{L_i}{S(r_i)}\right] W(|\mathbf{r} - \mathbf{r}_i|; R),$$

where

$W(d; R) = (2\pi R^2)^{-3/2} \exp(-d^2/2R^2).$

The line-of-sight peculiar velocity field is statistically sparse and noisy. The minimum-variance linear Wiener Filter (WF) estimator is used to reconstruct the full 3D velocity and underlying matter density field (Pomarede et al., 2012, Sorce et al., 2015):

$\Phi(L)$0

where $\Phi(L)$1 is the prior power spectrum, $\Phi(L)$2 is the velocity-density cross power spectrum, $\Phi(L)$3 the noise power, and $\Phi(L)$4 the transform of the observed data. In real space:

$\Phi(L)$5

with $\Phi(L)$6 the covariance between velocity and data.

Reverse Zeldovich Approximation (RZA) is employed to shift constraints from observed Eulerian positions to their Lagrangian progenitors for initial condition construction in $\Phi(L)$7-body simulations (Sorce et al., 2015, Pfeifer et al., 2023).

3. Simulation Pipelines and Constrained Realizations

The central methodology for numerical Local Universe construction is the combination of Wiener Filter and Hoffman-Ribak constrained realizations (WF+CR) (Sorce et al., 2015, Gottloeber et al., 2010, Yepes et al., 2013). The process is as follows:

• Observational constraints (line-of-sight velocities, cluster masses) are mapped into linear functional forms on the initial Gaussian random field, allowing direct computation of the minimum-variance reconstructed field.
• CRs augment the uncertainty in unconstrained modes by adding random power consistent with the prior, ensuring the ensemble of simulations captures cosmic variance on all scales not directly constrained by data.
• These reconstructed initial conditions are evolved via $\Phi(L)$8-body codes (e.g., GADGET-3, AREPO) from high redshift to $\Phi(L)$9, typically in periodic cubic volumes of $1/S(r)$0, with resolutions up to $1/S(r)$1 particles, and with softening lengths of $1/S(r)$2.

The output is an ensemble of simulated universes, reproducing the major local structures (clusters, filaments, voids) and dynamically consistent flow fields, with cosmic variance reduced by factors of $1/S(r)$3–$1/S(r)$4 compared to random-phase simulations on Mpc scales (Sorce et al., 2015, Yepes et al., 2013).

4. Cosmographic Structures and Environmental Diagnostics

The constructed Local Universe enables detailed analysis of its key cosmographic components (Pomarede et al., 2012, Sorce et al., 2015):

• Local Void: Identified as a pronounced underdensity with coherent outflows in WF-reconstructed flows.
• Virgo Cluster: The dominant nearby overdensity ($1/S(r)$5), with clear infall signatures.
• Supercluster Filaments/Walls: Structures such as the "Centaurus wall," the Perseus–Pisces filament, Southern Wall, and the Great Wall are resolved.
• Great Attractor Region: Secondary mass concentration shaping large-scale bulk motions.

The environment of each galaxy or halo in the simulation is classified via web-type algorithms, e.g., the eigenvalue analysis of the tidal tensor derived from the local gravitational potential. This enables precise mapping of volume and mass filling fractions of knots, filaments, sheets, and voids, as well as analysis of the galaxy morphology–environment relation (Nuza et al., 2014).

Volume- and mass-weighted statistics are compared across ensembles of random and constrained simulations to quantify the cosmic variance and determine the "fairness" of the local volume relative to $1/S(r)$6CDM expectations (Sorce et al., 2015, Nuza et al., 2014).

5. Selection and Ranking of Realizations: The Local Universe Model (LUM)

Quality assessment and ranking of constrained simulations are addressed with frameworks such as the Local Universe Model (LUM) (Pfeifer et al., 2023). LUM operates as follows:

• Observational cluster positions and masses (e.g., 11 rich clusters from Cosmicflows-3) are matched to simulated halo catalogs (halos with $1/S(r)$7).
• A null-hypothesis $1/S(r)$8-value quantifies the likelihood that a given simulated counterpart would occur by chance in a random-phase simulation; the lower the $1/S(r)$9-value, the more statistically significant the match.
• The overall merit of each simulation is given by aggregated scores (e.g., product or sum over clusters' $S(r)$0 values).
• The best realizations are those with the maximum number of high-significance matches ($S(r)$1 per cluster), and minimum overall $S(r)$2 score.

This approach provides rigorous statistical control, enabling both quantification of the fidelity to observables and principled selection of optimal simulation seeds for high-resolution, follow-up studies (Pfeifer et al., 2023).

6. Advanced Applications and Theoretical Validation

Local Universe constructions facilitate diverse cosmological investigations:

• Structure Formation with $S(r)$3: Semi-analytic models incorporating the cosmological constant in weak-field gravity, Vlasov–Poisson kinetics, and virial analyses predict critical scales (e.g., filament spacings, critical radii) observed in the local web. These approaches also interpret the "Hubble tension" as a natural outcome of density-dependent expansion rates in inhomogeneous environments (Gurzadyan, 5 Feb 2025).
• Testing Fundamental Laws: Empirical laws, such as the Hubble–Humason–Sandage linear redshift–distance law ($S(r)$4, $S(r)$5) and the Carpenter–Karachentsev–de Vaucouleurs density–radius power-law ($S(r)$6, $S(r)$7), are implemented in the construction to ensure consistency with local cosmological observations (Baryshev, 2016).
• Feedback into Theory and Simulation: Constrained Local Universe models are benchmarks for galaxy formation, cosmological hydrodynamics, and testing dark matter scenarios (e.g., predictions for subhalo abundances or velocity functions relevant for distinguishing cold vs. warm dark matter) (Yepes et al., 2013, Gottloeber et al., 2010, Sorce et al., 2015).
• Environmental Effects on Modified Gravity: The precise mapping of the local density and potential fields provides the basis for environmental screening analyses in modified gravity models (e.g., $S(r)$8 gravity), as demonstrated by the high-resolution screening maps from the LOCUSTS pipeline (Shao et al., 2019).

7. Visualization and Survey Enabling Instruments

The constructed Local Universe is rendered and explored through advanced visualization techniques and high-etendue integral-field spectroscopy (Prada et al., 2020, Pomarede et al., 2012):

• 3D visualization of galaxy distributions, colored by peculiar velocity.
• Transparent isosurfaces tracing density structures at biologically significant levels.
• Streamlines and velocity arrows tracing bulk flows and flow convergence/divergence zones.
• Interactive tools (e.g., SDvision) for real-time exploration of data cubes, planar slices, and dynamic density/velocity thresholds.

State-of-the-art instruments (such as the LUCA IFU-6000) are specifically designed to provide the spatially resolved spectroscopic data required for high-fidelity density and velocity reconstruction of the local galaxy population, supporting direct input into Local Universe construction pipelines (Prada et al., 2020).

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References:

• (Pomarede et al., 2012) Visualization of structures and cosmic flows in the Local Universe
• (Sorce et al., 2015) Cosmicflows Constrained Local UniversE Simulations
• (Pfeifer et al., 2023) A Local Universe model for constrained simulations
• (Yepes et al., 2013) Dark Matter in the Local Universe
• (Nuza et al., 2014) The cosmic web of the Local Universe: cosmic variance, matter content and its relation to galaxy morphology
• (Baryshev, 2016) Two fundamental cosmological laws of the Local Universe
• (Gurzadyan, 5 Feb 2025) Structure formation in the local Universe and the cosmological constant
• (Shao et al., 2019) Screening maps of the local Universe I -- Methodology
• (Prada et al., 2020) The Local Universe from Calar Alto (LUCA)
• (Gottloeber et al., 2010) Constrained Local UniversE Simulations (CLUES)