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Anomalous Running Gravity

 

How running gravity, anomaly-driven vacuum energy, and quantum error correction combine to explain cosmic tensions, while preserving ΛCDM

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 ΛCDM fits the CMB, large-scale structure, and nucleosynthesis exceptionally well.

And yet:

  • The locally measured Hubble constant ($H_0$) is higher than the Planck CMB prediction
  • Weak lensing surveys find lower clustering amplitude ($S_8$) than ΛCDM predicts

These are small but persistent discrepancies.

Rather than discarding ΛCDM, what if these tensions are subtle signals about how vacuum energy and gravity behave dynamically?

1. Core Idea: Mildly Running Gravity

Standard ΛCDM

Vacuum energy is constant: $$\rho_\Lambda = \text{const.}$$

ARG 3.3: Running Gravity with Anomaly Source

In Anomalous Running Gravity (ARG 3.3), we promote vacuum energy and Newton's constant to dynamic quantities sourced by the trace anomaly of quantum fields and a Gauss–Bonnet term:

$$S \supset \int d^4x \sqrt{-g} \frac{b}{(4\pi)^2}\ln\left(\frac{\phi}{\phi_0}\right) E$$

where $E$ is the Euler density.

This naturally generates:

  • Early-time $G$-running near the MeV scale
  • Late-time effects near the dark-energy transition ($z\lesssim 2$)

Gravity compensates to preserve energy conservation:

$$\dot{\rho}_\Lambda + \rho_m \frac{\dot G}{G} \approx 0$$

Present-day variation remains tiny:

$$|\dot G/G|_0 \lesssim 10^{-13}~\mathrm{yr}^{-1}$$

This satisfies local and BBN bounds.

2. Dual-Phase Running

Early Phase

  • MeV-era Gauss–Bonnet source adjusts $G(a)$ near recombination
  • Triggered near matter-radiation equality as relativistic species decouple

Late Phase

  • Braiding and mass evolution tweak expansion at $z\lesssim 2$
  • Emerging from low-energy, IR vacuum fluctuations

Together, they allow $r_d$ and $H_0$ to shift without spoiling the CMB damping tail or BAO geometry.

3. Resolving the Tensions

3.1 Hubble Tension

The late-time evolution modifies effective expansion:

$$H^2 = \frac{8\pi G(a)}{3}(\rho_m+\rho_r+\rho_\Lambda) - H \frac{\dot G}{G}$$

ARG 3.3 produces: $$H_0 \sim 72.5-73.5~\mathrm{km/s/Mpc}$$

without violating the CMB angular scale.

3.2 $S_8$ Tension

Structure growth is suppressed by:

  1. Mildly weaker gravity: $G(a) < G_0$ at intermediate redshifts
  2. Braiding friction: Kinetic mixing between $\phi$ and metric perturbations slows growth

Growth equation: $$\ddot{\delta}_m + \left(2H + \frac{\dot G}{G}\right)\dot{\delta}_m - 4\pi G(a)\rho_m \delta_m = 0$$

Result: $$S_8 \sim 0.75-0.77$$

matching weak-lensing observations.

3.3 Soft ISW Prior: Low-$\ell$ CMB Power

ARG 3.3 enforces a soft prior on braiding vs. Planck mass evolution:

$$\alpha_{\rm B}(z)-\alpha_{\rm M}(z) \sim \mathcal{N}(0, \sigma_\epsilon^2),\quad \sigma_\epsilon \ll 1$$

Effects:

  • Nudges late ISW effect toward zero: $(\Phi+\Psi)' \approx 0$ at $z\lesssim 2$
  • Reduces low-$\ell$ CMB power
  • Generates mild gravitational slip:

$$\eta = \frac{\Phi}{\Psi} \sim 1.02-1.05 \quad \text{at } z \sim 0.5$$

This is a concrete, observable signature.

4. The QEC Connection

Spacetime as Quantum Error Correction

Recent advances suggest spacetime itself is a quantum code:

Key Ideas:

  • Logical qubits encode geometry
  • Expansion generates new degrees of freedom
  • Dark energy = thermodynamic cost of error correction

Landauer energy for erasing errors: $$\rho_{\rm vac}^{\rm QEC} \sim \frac{k_B T \ln 2 , N_{\rm qubits}(z)}{V_{\rm Hubble}(z)}$$

ARG 3.3 ↔ QEC Correspondence

ARG 3.3 QEC Interpretation
Dual-phase vacuum $\rho_{\rm vac}(z)$ Landauer cost of refreshing qubits at two epochs
Running $G(z)$ Feedback from maintaining coherence
Growth suppression Friction from computational energy cost
Late-time $H_0$ enhancement Faster local "refresh rate"

Emergent Viscoelasticity

From the QEC perspective:

Bulk viscosity from error correction: $$\zeta(z) \sim \frac{k_B T \ln 2}{\tau_{\rm QEC}} \frac{dN_{\rm qubits}}{dV}$$

Shear stress with finite relaxation: $$\pi_{\mu\nu}(z, \omega) = - \frac{2 \eta}{1 - i \omega \tau_{\rm QEC}} \sigma_{\mu\nu}$$

Stochastic stress from quantum fluctuations during syndrome readout produces effective local noise.

Topological Protection

QEC codes are inherently topological:

$$\Lambda_{\rm top} \sim \frac{\text{Code distance}}{V_{\rm Hubble}}$$

Key features:

  • ARG 3.3's dual-phase behavior naturally constrained by topological stability
  • Ensures small ~5–8% $G$ variation without fine-tuning
  • Large code distance → stable cosmological constant → $w \approx -1$
  • Berry phase holonomy suppresses quantum corrections

5. Microscopic Interpretation: Entropy Flow

ARG 3.3 can be interpreted as geometric information processing:

Hubble horizon entropy: $$S_H = \frac{\pi}{G H^2}$$

Thermodynamic relation: $$dE = T_H dS_H$$

QEC viewpoint:

  • Spacetime encodes geometric information non-locally
  • Expansion increases effective qubits
  • Maintaining coherence produces friction
  • Maps onto running vacuum energy and growth suppression

6. Unified Einstein Equation

$$\boxed{G_{\mu\nu} + \Lambda_{\rm top} g_{\mu\nu} = 8\pi G(z) T_{\mu\nu}^{\rm matter} + 8 \pi G(z) \xi_{\mu\nu}^{\rm QEC}(z)}$$

Components:

  • ARG 3.3 dual-phase vacuum ↔ $\rho_{\rm vac}(z)$
  • Running $G(z)$ ↔ computational feedback
  • Growth suppression ↔ local QEC overhead
  • Viscous effects ↔ emergent mesoscopic stress
  • Topological Berry phase ↔ global protection of $\Lambda$

7. ΛCDM as a Special Limit

ARG 3.3 reduces to ΛCDM when:

$$\alpha_{\rm B} = \alpha_{\rm M} = 0, \quad b = 0, \quad \Delta G(a) = 0$$

Key point: All local tests, CMB physics, and large-scale structure are preserved.

This is a conservative extension, not a radical alternative.

8. Observable Signatures

What We Predict

Hubble tension:

  • Slightly higher $H_0$ from SN Ia anchors
  • Local variations with cosmic web density

$S_8$ suppression:

  • Mild gravitational slip
  • Braiding friction slows growth

ISW effect:

  • Reduced low-$\ell$ power in CMB

Gravitational slip:

  • $\eta(z)\neq 1$ at intermediate redshifts
  • Testable with weak lensing

Gravitational waves:

  • Frequency-dependent damping from QEC shear viscosity

Dark energy stability:

  • Naturally protected by topological code distance

9. The Bottom Line

ARG 3.3 + QEC provides:

First-principles, anomaly-driven mechanism for mild vacuum-energy running

Newton's constant adjusts slightly to maintain conservation

Naturally produces:

  • Elevated $H_0$
  • Lower $S_8$
  • Mild gravitational slip
  • Reduced low-$\ell$ ISW power

ΛCDM preserved as the exact infrared fixed point

 Conservative extension with testable predictions

10. Wrap

Three Perspectives, One Framework

Macroscopic (ARG 3.3): Running vacuum energy + running gravity

Mesoscopic (Viscoelastic): Bulk and shear viscosity from quantum coarse-graining

Microscopic (QEC): Computational cost of maintaining spacetime coherence

All three are manifestations of the same underlying physics.

Cosmic tensions may simply be the first glimpses of vacuum energy doing what quantum field theory predicts:

  • Anomaly physics (trace anomaly, Gauss-Bonnet)
  • Geometric information flow (QEC, entropy)

Rather than a crisis for cosmology, these tensions are hints that:

  • Spacetime is computational
  • Gravity is emergent
  • Dark energy is the overhead of maintaining quantum coherence
  • ΛCDM is the noiseless, trivial-topology limit

The universe isn't broken. We're just seeing the computational microstructure for the first time.

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