V2.95: Non-Equilibrium First Law with Log-Corrected Entropy

Objective

Compute the internal entropy production d_iS at the cosmological horizon when the entropy includes the log correction from the trace anomaly, and determine whether this modifies the Friedmann equation enough to close the Lambda gap.

Background

Paper 4 derives the Friedmann equation from the Clausius relation using Wald (area-law) entropy: dQ = T * dS_Wald. Paper 1 uses the full quantum entropy S = alphaA + deltaln(R_H). When we include the log term, the Clausius relation picks up an extra term:

dQ = T * (dS_Wald + d_iS^{log})
where d_iS^{log}/dt = delta/(2*A_H) * dA_H/dt = -delta * Hdot / H

This modifies the effective Newton’s constant: G_eff = G_Wald / (1 + epsilon), where epsilon = delta/(2alphaA_H).

Method

Phase 1: Analytical Framework

Decompose the quantum entropy into Wald + log parts and derive the d_iS rate analytically.

Phase 2: LCDM Numerical Evolution

Evolve LCDM from a=0.01 to a=1.0 and compute all Clausius quantities at each epoch.

Phase 3: Modified Friedmann Equation

Evaluate the correction factor (1 + epsilon) for different SM field contents.

Phase 4: Impact on Self-Consistency

Compute the corrected R_eff = R_bare * (1 + epsilon) for each scenario.

Phase 5: Quadratic Suppression Check

Determine whether d_iS_log scales linearly or quadratically with the non-equilibrium parameter.

Results

The Non-Equilibrium Parameter

The correction is controlled by:

epsilon = delta / (2 * alpha * A_H)

Units	A_H	epsilon (photon)
Dimensionless (H_0=1)	4*pi ≈ 12.6	-0.575
Planck (physical)	~8.3 x 10^122	~8.7 x 10^{-123}

The dimensionless computation gives a misleadingly large epsilon because A_H = 4*pi in H_0 units. In physical Planck units, the cosmological horizon has area ~10^122 L_P^2, making epsilon negligibly small.

Key Epochs

| Epoch | a | d_iS_log/dt | dS_Wald/dt | |ratio| | |:—|:—|:—|:—|:—| | z=99 | 0.01 | -580 | 0.003 | 181000 | | z=2 | 0.33 | -2.9 | 0.55 | 5.3 | | z=1 | 0.5 | -1.5 | 0.79 | 1.8 | | Lambda=matter | 0.77 | -0.6 | 0.77 | 0.79 | | Today | 1.0 | -0.33 | 0.57 | 0.58 |

(All quantities in dimensionless H_0 units.)

Scaling

d_iS_log scales linearly in epsilon (not quadratically), because it comes from the first-order mismatch between using alpha*dA vs (alpha + delta/(2A))*dA in the entropy gradient. However, even linear-in-epsilon corrections are negligible when epsilon ~ 10^{-122}.

Physical Correction to Lambda

In Planck units, the correction to the self-consistency ratio is:

Delta_R / R ~ epsilon ~ 10^{-122}

This is 120 orders of magnitude too small to affect the Lambda gap.

Key Finding

The log correction does NOT close the Lambda gap. The internal entropy production d_iS from the trace anomaly log term is suppressed by the enormous area of the cosmological horizon (A_H ~ 10^{122} L_P^2). In physical units, epsilon ~ 10^{-122}, making this correction utterly negligible.

This is a useful null result: it eliminates one potential correction mechanism and confirms that the gap must be closed by field content (V2.93), the Omega_Lambda target correction (V2.94), or other physics.

Implications

The remaining paths to close the Lambda gap are:

1. Field content (V2.93): Full SM gives R=0.530, photon gives R=1.205
2. Omega_Lambda target (V2.94): Target is 0.685, not 1.0
3. Viscous corrections (V2.96): Bulk viscosity from trace anomaly
4. Some intermediate decoupling: Between full SM and photon-only

The non-equilibrium d_iS from the log correction is definitively ruled out as a contributor.

Runtime

< 0.1s (analytical + array operations)