This repository contains code and supporting material for:
Hughes, A. (2025)
Topological Entropic Gravity: Unifying the Quantum Hall Vacuum with Cosmic Structure Formation to Resolve the S₈ Tension
Preprint: https://doi.org/10.5281/zenodo.18051561
Weak lensing surveys (KiDS, DES, HSC) consistently infer a lower clustering amplitude than Planck CMB constraints under ΛCDM, producing the well-known S₈ tension at the ~3σ level. Empirically, resolving the tension requires a 5–7% suppression of the matter power spectrum at non-linear scales while preserving linear-scale growth.
Baryonic feedback models can produce suppression, but generally require strong, scale-localized AGN prescriptions and often introduce redshift-dependent behavior in conflict with CMB and Lyman-α constraints.
Topological Entropic Gravity (TEG) introduces an alternative mechanism: a scale-dependent, non-dissipative pressure sourced by the topology of the vacuum itself. The framework is intentionally minimal and designed to be falsifiable at the level of the matter power spectrum and halo structure.
- The vacuum admits an effective description as an incompressible topological fluid.
- The equation of state is stiff, with adiabatic index Γ = 5/3, motivated by stability of the ν = 5/3 fractional quantum Hall state.
- Gravitational compression locally increases the effective filling factor, inducing a repulsive entropic pressure.
- The coupling strength κ is small and fixed by fundamental physics, not fitted to galaxy data.
TEG introduces a single dimensionless coupling κ. Importantly, the value required to resolve the S₈ tension coincides with independent theoretical estimates.
| Origin | Estimate | κ |
|---|---|---|
| Vacuum polarization (QED) | κ = α / 2π | 0.00116 |
| Thermodynamic scaling (BBN) | κ ~ η⁻¹ᐟ³ | 0.00085 |
| Required suppression (S₈) | — | 0.00120 |
Agreement is within O(10%), consistent with expected uncertainties in early-universe thermodynamic estimates.
No additional parameters are tuned.
- Suppression activates only for k ≳ 0.1 h/Mpc
- Linear scales remain unchanged
- Net suppression at z ≈ 0 is ~5–6%, consistent with weak lensing data
- Reduced virial overdensity modifies halo concentrations
- For M ≲ 10¹¹ M☉, NFW-like cusps are softened into approximately constant-density cores
- Effect arises from modified collapse dynamics, not feedback energy injection
Spherical collapse with entropic pressure:
d²R/dt² = -GM/R² - (1/ρ)∇P_ent + ΛR/3
with
P_ent = κ c² ρ_crit (ρ/ρ̄)^(5/3)
At turnaround, the pressure term reduces the maximum overdensity.
At virialization, the modified virial theorem
2T + W - 3∫P dV = 0
leads to a larger equilibrium radius for fixed mass, reducing concentration.
This mechanism operates without violating linear growth constraints.
git clone https://github.com/AhrleyHughes/TEG-Cosmology.git
cd TEG-Cosmology
pip install -r requirements.txtpython teg_accurate.pyOutputs:
- Matter power spectrum ratios
- σ₈ comparison with Planck and lensing values
- Saved numerical arrays for external analysis
python test_lcdm_limit.pyCritical credibility check: Demonstrates that TEG cleanly reduces to ΛCDM when κ → 0.
Figure: Exact recovery of ΛCDM in the κ → 0 limit. P(k) and c(M) ratios deviate from unity by < 10⁻⁸%, confirming the absence of numerical artifacts or hidden assumptions. The model reduces cleanly to standard cosmology when the topological coupling is disabled.
python sensitivity_analysis.pyExplores robustness against κ variation and verifies absence of fine-tuning.
Script: plot_redshift_evolution.py
Figure: Predicted evolution of suppression. The effect activates at late times (z < 1), consistent with thawing dark energy constraints.
Script: plot_model_discrimination.py
Figure: TEG produces a stable 'shelf' of suppression, distinct from the characteristic 'spoon' shape of AGN feedback, offering a clear discriminant for future observations.
TEG is intentionally constrained. These tests demonstrate that its predictions are monotonic, shape-invariant, and non-degenerate with baryonic feedback models.
The response of σ₈ to the coupling constant κ is smooth and monotonic. There are no instabilities, jumps, or tuning artifacts.
Key result:
- Increasing κ produces controlled suppression
- The weak-lensing target range is reached naturally
- No parameter fine-tuning is required
A common objection to modifications of the matter power spectrum on non-linear scales is the degeneracy with baryonic feedback processes (e.g., AGN feedback, supernovae winds). However, TEG is phenomenologically distinct from baryonic mechanisms in three testable ways:
-
Derivative Signature (The "Fingerprint"): Baryonic feedback models (such as those in BAHAMAS or IllustrisTNG) inject energy at specific scales, creating a characteristic 'spoon' feature in the power spectrum ratio—a suppression followed by a high-$k$ rebound or inflection. TEG, by contrast, produces a monotonic, scale-dependent suppression with a smooth gradient (see Figure
TEG_vs_Baryons_Derivative.png). There is no 'rebound' because there is no energy injection, only a modification of the collapse threshold. -
Shape Invariance: The TEG suppression profile is structurally rigid. Varying the coupling
$\kappa$ changes the amplitude but preserves the functional form (see Figureshape_invariance.png). Baryonic models are effective field theories with multiple free parameters that can arbitrarily alter the shape of the suppression. TEG cannot be 'tuned' to mimic arbitrary shapes; if the data shows a rebound, TEG is falsified. -
Redshift Dependence: Baryonic effects are cumulative and strongly redshift-dependent, tracking the history of star formation and AGN activity. TEG suppression is tied to the virialization of structure and the effective density contrast, predicting a late-time activation that distinctively tracks the growth factor, independent of local thermodynamic history.
Changing κ alters only the amplitude, not the shape, of suppression.
All curves collapse to a single universal profile when normalized. This distinguishes TEG from baryonic feedback models, which freely alter shape.
Even when power-spectrum ratios appear similar, their derivatives reveal distinct physical mechanisms.
TEG exhibits:
- Smooth
- Monotonic
- Single-scale behavior
Baryonic feedback exhibits:
- Inflection points
- Rebound features
- Multi-scale tuning
The TEG framework makes concrete, falsifiable predictions that stem directly from the modified power spectrum logic. These are not fitted parameters; they are structural consequences of the theory.
-
Scale-Dependent Suppression: A specific reduction in power on non-linear scales with zero impact on linear (CMB) scales (see
kappa_sweep.png). -
Shape Invariance: The suppression profile maintains a universal shape across different coupling strengths, distinguishing it from arbitrary curve-fitting (see
shape_invariance.png). -
Monotonic High-k Behavior: Unlike baryonic feedback models (AGN), TEG predicts no high-$k$ power recovery or "rebound" (see
TEG_vs_Baryons_Derivative.png).
These signatures are testable now with current weak-lensing data.
The core mechanism resolves the S₈ tension without these additions. However, if the TEG coupling arises from deeper microphysical structure, secondary effects may appear:
- Mild Halo Oblateness: Potential signature from chiral edge modes, detectable by Gaia/Euclid.
- Temperature-Dependent κ Scaling: Possible variations in high-z lensing observable by JWST.
- Discrete κ Transitions: Theoretical possibility of vacuum topology phase changes.
These effects are avenues for investigation, not load-bearing pillars of the primary result.
@article{Hughes2025TEG,
title = {Topological Entropic Gravity: Unifying the Quantum Hall Vacuum with Cosmic Structure Formation to Resolve the S8 Tension},
author = {Hughes, Ahrley},
year = {2025},
doi = {10.5281/zenodo.18051561},
url = {https://doi.org/10.5281/zenodo.18051561}
}Ahrley Hughes
Independent Researcher
arhleyhughes@proton.me
MIT License — see LICENSE.