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Bulk Viscosity from Geometry, Not Only Kinetic Theory

The paper, Origin of bulk viscosity in cosmology and its thermodynamic implications, is interesting.  

The paper establishes four facts:

$$p_{\text{vis}}=-3\zeta H, \qquad \dot S_h>0, \qquad \dot S_m<0, \qquad T_m\neq T_h$$

and hence

$$\boxed{\dot S_{\text{tot}}>0 ;\text{always}}$$

These are consequences of:

  1. Hubble expansion producing velocity gradients,
  2. The apparent horizon behaving as a thermodynamic boundary,
  3. The fluid inside the horizon being an open system.

Generated image 

 

A complete, self-contained derivation synthesising the kinetic, fluid dynamics, EFT, and thermodynamic/irreversibility perspectives on bulk viscosity of the vacuum in an expanding FLRW universe.

1. Kinetic Derivation: Momentum Flux Across Layers

Step 1: Setup

  • As per the paper, consider three layers of "vacuum quanta" separated by mean free path $\lambda_m$: $S_1$ (below), $S_2$ (observer), $S_3$ (above).
  • Fluid velocity relative to the comoving observer:

$$v(d) = H d$$

where $H$ is the Hubble parameter.

Step 2: Momentum from layers

  • Momentum from layer below at $d - \lambda_m$:

$$P_\text{below} = n m (d - \lambda_m) H$$

  • Momentum from layer above at $d + \lambda_m$:

$$P_\text{above} = n m (d + \lambda_m) H$$

  • Net momentum flux (viscous pressure):

$$p_\text{vis} = P_\text{below} - P_\text{above} = -2 n m \lambda_m H$$

 Dimensionally consistent!

Step 3: Substitute number density

Assume (as per the paper)

$$n = \frac{\rho}{u} \langle c \rangle$$

where:

  • $\rho$ = energy density (J/m^3)
  • $u$ = energy per particle
  • $\langle c \rangle$ = average particle speed
  • $m$ = particle mass

$$p_\text{vis} = -2 \left( \frac{\rho}{u} \langle c \rangle \right) m \lambda_m H$$

Step 4: Relativistic quanta

For vacuum quanta:

$$u = m_s c^2, \quad \langle c \rangle = c, \quad m = m_s$$

Then:

$$p_\text{vis} = -2 \frac{\rho c}{m_s c^2} m_s \lambda_m H = -\frac{2 \rho \lambda_m H}{c}$$

Step 5: Bulk viscosity definition

By definition:

$$p_\text{vis} = -3 \zeta H \implies \zeta = \frac{2 \rho \lambda_m}{3 c}$$

Step 6: Vacuum parameters from our framework

  • Mean free path: In kinetic theory,  $\lambda = \bar v T $ and we have previously found that the average speed is $\bar v=c$ and relaxation time $T=l_{\Lambda}/2c$ so: $\lambda_m = \frac{c}{2 H}$, for relativistic quanta in de Sitter spacetime
  • Energy density: $\rho = p_\Lambda\ c^2 = \rho_\Lambda = \frac{3 H^2 c^2}{8 \pi G}$

$$\zeta = \frac{2 \rho}{3 c} \cdot \frac{c}{2 H} = \frac{\rho}{3 H} = \frac{1}{3 H} \cdot \frac{3 H^2 c^2}{8 \pi G} = \frac{H c^2}{8 \pi G}$$

Matches SI units: $[H c^2 / G] = \mathrm{kg/(m \cdot s)}$.

 

2. Fluid Dynamics Derivation

Step 1: Bulk viscous pressure

In fluid dynamics, the total pressure of a dissipative fluid is $P_{eff} = P_{static} + \Pi$, where $\Pi = -3 \zeta H$ is the bulk viscous pressure.

If we calculate the bulk viscosity $\zeta$ for your vacuum fluid, an interesting symmetry emerges. If we assume the Stokes condition for a monatomic-like fluid (where the second viscosity coefficient is zero), we get the relation:

$$\zeta = \frac{2}{3} \eta_{vac}$$

Plugging this into the dissipative pressure equation:

$$\Pi = -3 \left( \frac{2}{3} \eta_{vac} \right) H = -2 \eta_{vac} H = -2 \left( \frac{3 c^2 H}{16 \pi G} \right) H = -\frac{3 H^2 c^2}{8 \pi G}$$

This is exactly the negative pressure required to simulate the Cosmological Constant energy density ($\rho_{\Lambda} c^2$). Or equivalently as we previously discussed, the quantum Youngs Modulus of vacuum. In this framework, the "Dark Energy" driving the acceleration of the universe is simply the internal friction (bulk viscosity) of the spacetime fluid as it expands. A ratio of $\zeta=(2/3)\eta$  implies the vacuum fluid has internal degrees of freedom (microstructure). The vacuum "molecules" (spacetime quanta) are not simple points; they have structure that absorbs energy during expansion. 

3. EFT / Cosmological Derivation

Step 1: Viscous stress tensor

  • Relativistic bulk viscosity:

$T^{\mu\nu}{\rm visc} = -\zeta \theta \Delta^{\mu\nu}, \quad \theta = \nabla\mu u^\mu$

  • FLRW comoving frame: $u^\mu = (1,0,0,0) \implies \theta = 3 H$

  • Effective pressure:

$$p_{\rm eff} = p_{\rm eq} - \zeta \theta = p_{\rm eq} - 3 H \zeta$$

Step 2: Match dark energy

  • Assume $p_{\rm eq} = 0$ (vacuum quanta).
  • Observed dark energy: $p_{\rm eff} = - \rho_\Lambda c^2$

$$-3 H \zeta = -\frac{3 H^2 c^2}{8 \pi G} \implies \zeta = \frac{H c^2}{8 \pi G}$$

Same result as kinetic derivation!

Step 3: Dynamical stability

  • Raychaudhuri equation:

$$\dot H = -\frac{4 \pi G}{c^2} (\rho + p_{\rm eff}) = -\frac{4 \pi G}{c^2} (\rho_\Lambda - \rho_\Lambda) = 0$$

  • Small perturbation $H = H_0 + \delta H$:

$$\dot{\delta H} \approx -3 H_0 \delta H \implies \delta H(t) \propto e^{-3 H_0 t}$$

de Sitter solution is stable!

4. Thermodynamic / Irreversibility Derivation

Step 1: Entropy production

  • Entropy production for a viscous fluid:

$$\nabla_\mu s^\mu = \frac{\zeta}{T} \theta^2$$

  • Use Gibbons-Hawking temperature for de Sitter horizon:

$$T_{dS} = \frac{\hbar H}{2 \pi k_B}, \quad \theta = 3 H$$

$$\nabla_\mu s^\mu = \frac{\zeta}{\hbar H / 2 \pi k_B} (3 H)^2 = \frac{9 k_B c^2 H^2}{4 \hbar G} > 0$$

Positive-definite, satisfies the generalized second law!

Step 2: Total entropy

  • Horizon entropy increases: $\dot S_h > 0$
  • Matter entropy can decrease: $\dot S_m < 0$
  • Total entropy always increases:

$$\dot S_{\rm tot} = \dot S_h + \dot S_m > 0$$

  • Irreversibility principle fixes $\zeta > 0$.

5. Holographic Consistency (KSS Bound)

  • As we previously discussed, shear viscosity from membrane paradigm:

$$\eta_{\rm area} = \frac{c^3}{16 \pi G} \quad \text{per horizon area}$$

  • Convert to volume density using $l_\Lambda = c / H$:

$$\eta_{\rm vac} = \frac{3 c^2 H}{16 \pi G}$$

  • Entropy density:

$$s = \frac{3 k_B c^2 H}{4 \hbar G}$$

  • Ratio:

$$\frac{\eta}{s} = \frac{\hbar}{4 \pi k_B}$$

Exactly saturates KSS bound, consistent with holography!

 

Summary  

Bulk viscosity

  • Formula: $\zeta = \frac{H c^2}{8 \pi G}$
  • Interpretation: Vacuum behaves as a viscous fluid

Viscous pressure

  • Formula: $p_{\rm vis} = -3 H \zeta = - \rho_\Lambda c^2$
  • Interpretation: Negative pressure drives cosmic acceleration

Entropy production

  • Formula: $\nabla_\mu s^\mu = \frac{9 k_B c^2 H^2}{4 \hbar G} > 0$
  • Interpretation: Second law satisfied

Stability

  • Formula: $\delta H \propto e^{-3 H_0 t}$
  • Interpretation: de Sitter solution is a stable attractor

Holographic limit

  • Formula: $\eta / s = \frac{\hbar}{4 \pi k_B}$
  • Interpretation: Saturates KSS bound

Physical Picture

Dark energy = irreversible work of vacuum expansion

Expansion generates viscous stress, horizon stores entropy

Key Insights

  1. Kinetic derivation: microscopic momentum flux → $\zeta$
  2. EFT/cosmology: viscous stress tensor in FLRW → $\zeta$
  3. Thermodynamics / irreversibility: $\dot S_{\rm tot} > 0$ → $\zeta > 0$ and fixes its magnitude
  4. All approaches converge numerically:

$$\boxed{\zeta = \frac{H c^2}{8 \pi G}}$$

  1. Dark energy emerges as a macroscopic manifestation of vacuum bulk viscosity, not as an independent substance.

Conceptual Unity

All derivation methods—kinetic theory, fluid dynamics, effective field theory, and thermodynamic irreversibility—converge on the same fundamental result. This convergence is not a coincidence but reflects deep connections between:

  • Microscopic momentum transport (kinetic theory)
  • Macroscopic stress-energy dynamics (EFT)
  • Thermodynamic arrow of time (irreversibility)
  • Quantum gravity constraints (holography)

The bulk viscosity of the vacuum is not an ad hoc addition to cosmology, but an inevitable consequence of treating the expanding universe as a dissipative quantum system in contact with a thermodynamic horizon.


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