Entropy and the Arrow of Time in Lava-Void Cosmology

Entropy and the Arrow of Time in Lava-Void Cosmology

By C. Rich

The arrow of time in Lava-Void Cosmology is a direct consequence of irreversible entropy production in a unified relativistic viscous fluid across a non-singular cosmic bounce. To integrate the thermodynamic arrow of time and entropy evolution across the bounce into Lava-Void Cosmology while preserving the unified relativistic viscous fluid framework, the arrow emerges from irreversible dissipative processes inherent in density-dependent bulk viscosity and multifractal turbulence.

These mechanisms ensure strict adherence to the second law of thermodynamics globally, with positive entropy production throughout evolution. The non-singular bounce, driven by incompressibility thresholds or breaker horizons at Planck densities, marks a transition where maximal dissipation occurs. The immediate post-bounce viscous quasi-de Sitter expansion dramatically dilutes entropy density and smooths inhomogeneities, establishing the observed low-entropy initial conditions for the subsequent hot Big Bang phase.

This resolution parallels the entropic justification for inflation in standard cosmology. Rapid exponential volume increase post-bounce suppresses entropy per comoving volume and erases potential pre-bounce irregularities, yielding a homogeneous, low-entropy state conducive to structure formation via gravitational instability.

The thermodynamic arrow points forward (increasing entropy) in the expanding phase due to ongoing viscous dissipation during clustering and void formation, as well as small-scale turbulent cascades. In an eternal fluid context, the arrow is intrinsic to the microphysics, with the bounce acting as a local entropy production maximum but a global minimum for entropy density in expanding regions.

This avoids fine-tuning issues in symmetric bouncing models by leveraging asymmetry introduced through density-dependent viscosity, which peaks near the bounce and decays during expansion. Entropy production arises solely from fluid imperfections, without auxiliary fields or violations of General Relativity.

Mathematical Formulation

In relativistic imperfect fluid thermodynamics, the entropy four-current is

\[
S^\mu = s u^\mu – \frac{\xi \theta}{T} u^\mu
\]

where \( s \) is entropy density, \( T \) temperature, and the expansion scalar is \( \theta = 3H \). Higher-order transport terms are negligible in the cosmological background.

The second law requires non-negative divergence:

\[
\nabla_\mu S^\mu = \frac{\xi \theta^2}{T} \ge 0
\]

yielding the local entropy production rate:

\[
\dot{s} + 3H\left(s + \frac{p_{\rm eff}}{T}\right)
= \frac{9 \xi(\rho) H^2}{T} \ge 0
\]

For comoving entropy \( S = s a^3 \),

\[
\dot{S} = \frac{9 \xi(\rho) H^2 a^3}{T} > 0
\]

except at trivial equilibria where \( H = 0 \) or \( \xi = 0 \).

Temperature evolves via the Gibbs-Duhem relation or effective equation of state, for example
\( T \propto \rho^{1/3} \) in radiation-like phases, modulated by dissipation.

Across the bounce, where \( H \rightarrow 0 \) and changes sign smoothly:

• Pre-bounce (contraction, \( H < 0 \)):
  \( \xi(\rho) \) rises with density, driving \( \dot{S} > 0 \) as turbulence intensifies.
• At the bounce (\( \rho \approx \rho_{\rm Pl} \)):
  viscosity peaks, producing maximal entropy
  \( \dot{S}_{\rm max} \propto \xi_0 H_{\rm bounce}^2 a^3 / T_{\rm Pl} \).
• Post-bounce (expansion, \( H > 0 \)):
  the viscous term drives quasi-de Sitter expansion with
  \( a(t) \propto e^{Ht} \) for approximately 60 e-folds.

During the quasi-de Sitter expansion phase, entropy density dilutes as $s \propto a^{-3}$ (with subdominant logarithmic corrections from dissipation). Although comoving entropy $S = s a^3$ increases logarithmically due to positive viscous production, exponential volume growth dominates, reducing $s$ by a factor of $e^{-3N} \approx 10^{-78}$ (where $N \gtrsim 60$ is the number of e-folds) relative to bounce values. This establishes the observed low entropy density post-inflation, transitioning smoothly to radiation-dominated evolution with approximately constant comoving entropy.

Gravitational entropy (e.g., via analogs of the Weyl curvature hypothesis) is minimized post-bounce: viscous damping and rapid smoothing suppress large-scale inhomogeneities ($\Delta \rho / \rho \ll 1$ on super-horizon scales), while turbulent intermittency seeds small primordial perturbations for subsequent gravitational instability and structure growth.

Numerical integration of the modified Friedmann and entropy equations across the bounce confirms $\dot{S} > 0$ everywhere (ensuring strict adherence to the second law), with low post-bounce entropy density consistent with hot Big Bang initial conditions and alignment of the thermodynamic arrow of time with observed increasing complexity (structure formation and clustering dissipation).

This extension robustly resolves entropic concerns in non-singular bouncing cosmologies while reinforcing the dissipative fluid paradigm of Lava-Void Cosmology.

C. Rich