An effective $\boldsymbol{\Lambda}$-Szekeres modelling of the local Universe with Cosmicflows-4

By modeling the local Universe as a superposition of inhomogeneous $\Lambda$-Szekeres patches calibrated to Cosmicflows-4 data, this study demonstrates that accounting for local cosmic structure increases the Hubble tension by shifting the best-fit value of $H_0$ by approximately $0.5\ \mathrm{km\,s^{-1}Mpc^{-1}}$.

The Gist

The Big Picture: The "Hubble Tension" Puzzle

Imagine you are trying to measure the speed of a car driving away from you.

• Method A (The Baby Picture): You look at a photo of the car taken when it was a baby (the early Universe, 13 billion years ago). Based on how the universe expanded back then, you calculate the car should be going 67 km/h.
• Method B (The Current Drive): You stand on the side of the road and watch the car right now (the local Universe, nearby). You measure its speed directly and find it's going 73 km/h.

This difference is called the Hubble Tension. It's a major mystery in physics. The two numbers don't match, and scientists are worried that maybe our understanding of the universe is broken.

The Hypothesis: "We Live in a Bumpy Neighborhood"

One possible explanation for this mismatch is that we are living in a weird, bumpy neighborhood.

• The Standard View: Scientists usually assume the universe is like a smooth, flat ocean. If you are anywhere in the ocean, the water looks the same in every direction.
• The "Bumpy" View: But what if we live in a valley surrounded by hills? If you are in a valley, the ground slopes down around you. If you are on a hill, the ground slopes up.

If our local neighborhood (the "Local Universe") is full of giant empty spaces (voids) and massive clumps of galaxies (clusters), it might trick us into measuring the expansion speed of the universe incorrectly. Maybe the "73 km/h" measurement is just a local illusion caused by our specific location.

What This Paper Did: The "Pizza Slice" Map

The authors of this paper wanted to test the "Bumpy Neighborhood" theory. They didn't just guess; they built a super-detailed 3D map of our cosmic neighborhood using data from the Cosmicflows-4 project (which tracks how galaxies are moving).

Here is how they modeled it:

1. The Pizza Analogy: Imagine the universe around us is a giant pizza. Instead of looking at the whole pizza at once, they sliced it into 256 triangular wedges (like pizza slices).
2. The Layers: Inside each slice, they looked at different layers (like the crust, the sauce, and the cheese) to see how dense the matter was at different distances.
3. The Physics Engine: They used a complex mathematical model called $\Lambda$-Szekeres. Think of this as a "physics engine" for the universe. Unlike the standard "smooth ocean" model, this engine allows the space to stretch and squeeze differently in different directions. It accounts for the fact that space near a giant galaxy cluster stretches differently than space inside a giant empty void.

They fed their real-world data (galaxy movements) into this engine to create a "quasi-local" map. This map shows exactly how much the universe is expanding in every single direction around us.

The Results: The Problem Got Worse

The team took this new, bumpy map and applied it to a list of Type Ia Supernovae (these are "standard candles"—exploding stars that act like perfect lightbulbs used to measure distance).

They asked: "If we account for all these bumps and dips in our local neighborhood, does the Hubble Tension go away?"

The Answer: No. It actually got worse.

• The Expectation: They hoped that once they corrected for the local "bumps," the local measurement would drop from 73 km/h down to 67 km/h, solving the puzzle.
• The Reality: When they included the local structure, the calculated speed of the universe increased slightly (by about 0.5 km/h).
• The Metaphor: Imagine you are trying to measure the speed of a river. You thought the water was flowing fast because you were standing in a narrow, fast-moving channel. You hoped that if you accounted for the channel, you'd realize the river is actually slow. Instead, you realized that because of the channel, the water is actually faster than you thought, making the difference between your measurement and the "baby picture" even bigger.

The Conclusion

The paper concludes that our local neighborhood is not the culprit for the Hubble Tension.

• We do live in a bumpy, messy neighborhood with giant voids and clusters.
• We have a very good map of it.
• But even when we perfectly account for all that messiness, the universe still looks like it's expanding faster locally than it did in the early days.

The Takeaway: The mystery of the Hubble Tension isn't because we are in a weird spot in the universe. The solution must lie elsewhere—perhaps in new physics we don't understand yet, or a fundamental flaw in our theory of gravity or dark energy. The "bumpy neighborhood" theory has been tested and ruled out as the solution.

Technical Summary

1. Problem Statement

The standard $\Lambda$CDM cosmological model relies on the Cosmological Principle (CP), assuming the Universe is statistically homogeneous and isotropic on large scales, described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. However, the local Universe ($z \lesssim 0.1$) is highly inhomogeneous and anisotropic, containing a complex network of voids, filaments, and clusters.

This discrepancy raises critical questions:

• Does the local inhomogeneous structure bias cosmological observables, specifically the Hubble constant ($H_0$)?
• Can the "Hubble Tension" (the discrepancy between early-Universe CMB measurements and late-Universe supernova measurements of $H_0$) be resolved by accounting for local structure effects (backreaction)?
• Existing models often rely on simplified assumptions (e.g., single voids or spherically symmetric LTB models) or perturbative approaches that may fail to capture strong non-linearities and anisotropies.

2. Methodology

The authors propose a general-relativistic, effective model of the local Universe using multi-structured $\Lambda$-Szekeres solutions. Unlike perturbative methods, Szekeres models are exact solutions to Einstein's field equations that allow for inhomogeneity and anisotropy without assuming spherical symmetry.

A. Data Source: Cosmicflows-4 (CF4)

• Input: The study utilizes the HAMLET reconstruction of the local density and peculiar velocity (PV) fields based on the Cosmicflows-4 (CF4) catalogue.
• Scope: Covers $\sim 56,000$ peculiar velocities out to $z \sim 0.1$ ($\sim 300$ Mpc/h), providing a 3D map of the local cosmic web.
• Preprocessing: To mitigate non-linear contamination and ensure validity within the quasi-linear regime, the authors apply a coarse-graining procedure (smoothing over $16 \times 16 \times 16$ voxels, $\approx 64$ Mpc/h). This reduces density contrasts to the range $-0.3 \leq \delta \leq 0.7$.

B. The Effective $\Lambda$-Szekeres Model

• Geometry: The local Universe is partitioned into a "patchwork" of spherical wedges (like pizza slices), each subdivided into radial shells.
• Formalism: Within each wedge, the metric is modeled as a superposition of monopole (isotropic) and dipole (anisotropic) components, described by the $\Lambda$-Szekeres metric.
  • The model assumes the large-scale background is FLRW, with the local environment described by exact deviations from this background.
  • The Quasilocal Formalism (Sussman & Bolejko) is employed to define volume-averaged scalars: density ($\rho_q$), expansion rate ($H_q$), and spatial curvature ($K_q$).
• Key Assumption: The observed peculiar velocities in CF4 are interpreted not as motions on top of an FLRW background, but as manifestations of an inhomogeneous and anisotropic expansion field ($H_{los}$) within an exact inhomogeneous spacetime.
• Fitting: The model fits the reconstructed density and PV fields to extract the line-of-sight dependent expansion rate $H_{los}(\theta, \phi, z)$.

C. Cosmological Inference

• Test Sample: Low-redshift Type Ia Supernovae (SNe Ia) from the Pantheon+ catalogue ($0.023 < z < 0.15$), the same sample used by the SH0ES collaboration to measure $H_0$.
• Distance Calculation: The authors compute the luminosity distance ($d_L$) corrections induced by the anisotropic expansion field using the quasilocal average $H_q$.
  $$d_L^{\Lambda-Sk}(z, \hat{n}) \approx (1+z) \int_0^z \frac{c \, dz'}{H_q(z', \hat{n})}$$
• Statistical Analysis: A Markov Chain Monte Carlo (MCMC) analysis is performed to constrain the cosmographic parameters ($H_0, q_0$).
  • Three covariance scenarios are tested: (1) Standard FLRW, (2) FLRW with standard PV corrections, and (3) FLRW with the new $\Lambda$-Szekeres corrections.
  • A bootstrapping approach (1,000 realizations) is used to propagate uncertainties from the CF4 reconstruction.

3. Key Contributions

1. First Multi-Structured Relativistic Model: This is the first application of a multi-structured $\Lambda$-Szekeres framework calibrated against a full 3D peculiar velocity field (CF4) to model the entire local cosmic web (voids, walls, clusters) simultaneously, rather than isolated structures.
2. Quasilocal Expansion Field Mapping: The study successfully maps the quasilocal expansion field ($H_q$) and spatial curvature ($\bar{\Omega}_{Kq}$) of the local Universe, revealing a non-trivial, dipole-like anisotropic structure.
3. Validation of Approximations: The authors demonstrate that for the local Universe, the quasilocal average ($H_q$) dominates the expansion field, while exact deviations ($D(H)$) are negligible. This simplifies the calculation of distance corrections while maintaining relativistic rigor.
4. Directional Corrections: The paper provides a sky map of distance corrections, showing they are coherent and directional (reflecting the cosmic web geometry) rather than random noise.

4. Key Results

• Local Expansion Variations: The reconstructed quasilocal expansion field shows variations in the Hubble parameter magnitude of up to 10% relative to the background $H_0$, with a distinct dipole-like feature in certain planes (SGX-SGY, SGX-SZ).
• Impact on $H_0$:
  • When accounting for the local inhomogeneous/anisotropic expansion, the best-fit value of the Hubble constant increases.
  • Standard FLRW fit: $H_0 \approx 72.46$.
  • With PV corrections: $H_0 \approx 73.19$.
  • With $\Lambda$-Szekeres corrections: $H_0 \approx 72.9$ (with a shift of $\Delta H_0 \approx 0.5$ km s$^{-1}$ Mpc$^{-1}$).
• Hubble Tension: Crucially, the inclusion of local structure effects exacerbates the Hubble tension rather than resolving it. The local environment appears to act as an overall overdensity (or a specific configuration of structures) that pushes the locally inferred $H_0$ higher, widening the gap with early-Universe (CMB) measurements ($H_0 \approx 67.4$).
• Nature of Corrections: The distance corrections are not random scatter; they exhibit a coherent angular pattern corresponding to the underlying distribution of walls and voids. However, the uncertainties in these corrections do not show a systematic redshift-dependent bias.

5. Significance and Conclusion

• Ruling Out Local Solutions: The study provides strong evidence that local structure effects alone cannot resolve the Hubble Tension. If the local Universe were an underdensity (a "Hubble bubble"), it would lower the inferred $H_0$, potentially easing the tension. However, the data suggests the opposite: the local structure increases $H_0$, making the tension worse.
• Relativistic Framework: The work validates the use of exact inhomogeneous solutions (Szekeres) as a viable tool for precision cosmology, moving beyond perturbative approximations to capture the true relativistic kinematics of the local Universe.
• Future Directions: The findings imply that the solution to the Hubble Tension likely lies in new physics beyond the standard model or in systematic errors in measurements, rather than in the gravitational influence of the local cosmic web. The methodology established here offers a robust framework for future tests of the Cosmological Principle using next-generation surveys.