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Bulk Viscosity from Geometry, Not Only Kinetic Theory

The idea that bulk viscosity 

could be an alternative to dark energy for a cosmological effective theory has been around for a while. For example, , 2011 Dark goo: bulk viscosity as an alternative to dark energy, or Hu, 2024 Viscous universe with cosmological constant,  or Khan, 2025 Spatial Phonons: A Phenomenological Viscous Dark Energy Model for DESI. 

Now, Paul, 2025 Origin of bulk viscosity in cosmology and its thermodynamic implications, takes a kinetic/thermodynamic slant. 

 

Paul found:

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

lead to

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

for an expanding FLRW universe with the apparent horizon treated as a thermodynamic boundary.

What entropy? 

  • $S_{BH}$ is the Bekenstein–Hawking entropy. It does not increase. 
  • $S_m$ is the coarse-grained, hydrodynamic entropy of the vacuum fluid, it is not a microscopic von Neumann entropy.   
  • $S_h$ is the entropy associated with the horizon degrees of freedom, the apparent-horizon entropy, and it is the entropy that absorbs the dissipation created by the bulk viscosity. 
  • Total entropy  $\dot S_{\text{tot}}$ is the Second Law for semiclassical horizons. It is NOT an increase in area. 

Our approach 

Here, we synthesise the kinetic, fluid-dynamical, EFT, thermodynamic, and horizon-fluid perspectives into a single, self-contained derivation, focusing on two main points:

  1. Exact de Sitter with bulk viscosity is not equilibrium; it is a non‑equilibrium steady state (NESS) with constant $(\rho, H)$ but non-zero entropy production $\nabla_\mu s^\mu>0$.

  2. The bulk viscosity $$\zeta = \frac{H c^2}{8\pi G}$$ is the simplest local closure (within a broad, natural class) that:

    • reproduces the observed de Sitter background,
    • matches a kinetic mean free path picture for relativistic "vacuum quanta",
    • and is consistent with the horizon-fluid viscosity/entropy scales.

1. Kinetic Derivation: Momentum Flux Across Layers

1.1 Setup

Following the layered kinetic construction of Paul (2025), consider three layers of "vacuum quanta" separated by a mean free path $\lambda_m$: $S_1$ (below), $S_2$ (observer), $S_3$ (above). The fluid has velocity, relative to a comoving observer,

$$v(d) = H d$$

where $H$ is the Hubble parameter and $d$ is the physical distance from the observer.

We assume that:

  • the "vacuum quanta" form a relativistic gas with characteristic rest energy $u = m_s c^2$ and average speed $\langle c\rangle \simeq c$,
  • they have a mean free path $\lambda_m$ determined by horizon-scale physics (specified below).

This is a different microscopic model from Paul's thermal matter fluid, but we reuse his geometric setup and velocity gradient.

1.2 Momentum from layers

Momentum transported from the layer below at $d - \lambda_m$:

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

and from the layer above at $d + \lambda_m$:

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

where $n$ is the number flux (particles per unit area per unit time) crossing $S_2$, and $m$ is the particle mass.

The net momentum flux (viscous pressure) is:

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

1.3 Number density and relativistic quanta

Let:

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

The number flux is

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

For relativistic "vacuum quanta", we assume

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

so

$$n = \frac{\rho}{m_s c^2} c.$$

Substituting:

$$p_\text{vis} = -2 \left(\frac{\rho}{m_s c^2} c\right) m_s \lambda_m H = -\frac{2 \rho \lambda_m H}{c}.$$

1.4 Bulk viscosity definition

By definition of bulk viscous pressure in an expanding isotropic fluid:

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

1.5 Mean free path and de Sitter parameters

We now make a horizon-scale assumption for the mean free path:

  • In kinetic theory, $\lambda = \bar v \tau$.
  • Take $\bar v \simeq c$,
  • Take $\tau$ to be of order the light-crossing time of the de Sitter horizon, say $$\tau = \frac{l_\Lambda}{2c},\qquad l_\Lambda \sim \frac{c}{H}.$$

Then

$$\lambda_m = \bar v \tau \simeq c\cdot\frac{l_\Lambda}{2c} = \frac{l_\Lambda}{2} \simeq \frac{c}{2H}.$$

For a de Sitter spacetime with Hubble parameter $H$, take:

$$\rho = \rho_\Lambda = \frac{3 H^2 c^2}{8 \pi G}.$$

Substituting into $\zeta$:

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

Thus, kinetically, under:

  • relativistic vacuum quanta with $u = m_s c^2$ and $\langle c\rangle = c$,
  • a horizon-scale mean free path $\lambda_m = c/(2H)$,

one is naturally led to:

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

2. Fluid-Dynamical and EFT Interpretation

In relativistic hydrodynamics, the effective pressure of a bulk-viscous fluid is:

$$p_{\rm eff} = p_{\rm eq} + \Pi,$$

with

$$\Pi = -\zeta \theta, \qquad \theta = \nabla_\mu u^\mu.$$

In spatially flat FLRW with comoving four-velocity $u^\mu=(1,0,0,0)$,

$$\theta = 3H.$$

For the vacuum fluid, we take the equilibrium pressure in the comoving frame to be negligible:

$$p_{\rm eq} \simeq 0,$$

so:

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

At late times, observations are consistent with an effective cosmological constant with

$$p_{\rm eff} = -\rho_\Lambda c^2,\qquad \rho_\Lambda = \frac{3H^2 c^2}{8\pi G}.$$

Matching:

$$-3\zeta H = -\rho_\Lambda c^2$$

gives

$$\zeta = \frac{\rho_\Lambda c^2}{3H} = \frac{H c^2}{8\pi G},$$

in agreement with the kinetic estimate.

Thus the viscous pressure:

$$p_{\rm vis} = -3\zeta H = -\frac{3H^2 c^2}{8\pi G} = -\rho_\Lambda c^2$$

exactly reproduces the cosmological constant equation of state at the background level.

The Raychaudhuri equation:

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

with $\rho = \rho_\Lambda$, $p_{\rm eff} = -\rho_\Lambda c^2$, implies:

$$\dot H = -\frac{4\pi G}{c^2}(\rho_\Lambda - \rho_\Lambda) = 0,$$

so $H = H_0$ is a fixed point. Linear perturbations $H = H_0 + \delta H$ satisfy:

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

showing that de Sitter is a stable attractor when supported by this viscous vacuum fluid.

3. Thermodynamics and Irreversibility: de Sitter as a NESS

For a bulk-viscous relativistic fluid, the entropy production is

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

where $T$ is the relevant temperature for the dissipative sector and $\theta=3H$.

If the dissipative degrees of freedom are effectively characterized by the de Sitter (Gibbons–Hawking) temperature:

$$T \approx T_{dS} = \frac{\hbar H}{2\pi k_B},$$

and $\zeta = H c^2/(8\pi G)$, then:

$$\nabla_\mu s^\mu = \frac{H c^2}{8\pi G} \cdot \frac{(3H)^2}{T_{dS}} = \frac{H c^2}{8\pi G} \cdot \frac{2\pi k_B}{\hbar H} \cdot 9H^2 = \frac{9 k_B c^2 H^2}{4 \hbar G} > 0.$$

Importantly:

  • The energy density $\rho$ and Hubble parameter $H$ are constant (de Sitter geometry),
  • Yet the entropy production $\nabla_\mu s^\mu$ is strictly positive.

Thus exact de Sitter supported by bulk viscosity is not a thermal equilibrium state; it is a non-equilibrium steady state (NESS) with:

  • constant macroscopic variables $H,\rho$,
  • ongoing entropy production in the bulk.

In the apparent-horizon thermodynamic picture of Paul (2025), a viscous cosmology satisfies:

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

and hence

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

The horizon entropy increases, the fluid entropy decreases, and the total entropy increases — an inherently irreversible evolution with the horizon acting as a thermodynamic sink for the entropy produced by bulk viscous dissipation.

4. Holographic/Horizon-Fluid Consistency

From the membrane paradigm, the horizon behaves as a 2D viscous fluid with shear viscosity per unit area:

$$\eta_{\rm area} = \frac{c^3}{16\pi G}.$$

For an apparent horizon of radius:

$$R_A = \frac{c}{H},$$

the area is:

$$A = 4\pi R_A^2 = 4\pi \frac{c^2}{H^2},$$

and the enclosed volume:

$$V = \frac{4\pi}{3} R_A^3 = \frac{4\pi}{3} \frac{c^3}{H^3}.$$

An effective shear viscosity per unit volume inside the horizon is:

$$\eta_{\rm vac} \sim \frac{\eta_{\rm area} A}{V} = \frac{c^3}{16\pi G} \cdot \frac{4\pi c^2/H^2}{(4\pi/3)c^3/H^3} = \frac{3 c^2 H}{16\pi G}.$$

The associated entropy density (smearing the Bekenstein–Hawking horizon entropy over the volume) is:

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

The ratio:

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

saturates the Kovtun–Son–Starinets (KSS) bound for strongly coupled quantum fluids. Note, we are not applying KSS to bulk viscosity, merely justify the scale of the viscous sector. 

With:

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

we have $\zeta$ of the same order as $\eta_{\rm vac}$, set by the same horizon-controlled scale. This makes $\zeta = H c^2/(8\pi G)$ a natural closure compatible with:

  • mean-free-path kinetic intuition,
  • FLRW + GR energy conservation with $p_{\rm eff}=-\rho_\Lambda c^2$,
  • and horizon-fluid viscosity and entropy saturating the KSS bound.

5. Cosmological Implications

Background expansion

With:

$$\zeta = \frac{\rho_\Lambda}{3H} = \frac{H c^2}{8\pi G},\quad p_{\rm eff} = -\rho_\Lambda c^2,$$

the background expansion history is identical to ΛCDM:

  • supernova luminosity distances,
  • BAO scales,
  • distance to last scattering,

are all unchanged at the background level.

Structure formation

Differences appear at the level of perturbations:

  • Bulk viscosity introduces non-adiabatic pressure and effective friction in the growth of density perturbations.
  • This tends to suppress late-time structure growth, potentially lowering $f\sigma_8$ at low redshift, while leaving early-universe observables nearly intact if viscosity is negligible there.
  • Phenomenological models by Gagnon (2011), Hu (2024), and Khan (2025) provide frameworks for confronting such viscous dark fluids with DESI, weak lensing, and other LSS data, and may help relieve the $S_8$ tension.

Thus, bulk viscous dark energy with $\zeta = Hc^2/(8\pi G)$ is observationally degenerate with Λ at the background level, but potentially distinguishable via structure formation.

6. Wrap

  • Bulk viscosity of the vacuum $$\boxed{\zeta = \frac{H c^2}{8\pi G}}$$ emerges from:

    • a kinetic mean-free-path picture for relativistic vacuum quanta with horizon-scale relaxation,
    • matching the effective viscous pressure to the observed de Sitter pressure,
    • and consistency with horizon-fluid viscosity and KSS-saturating entropy.

  • The effective negative pressure $$\boxed{p_{\rm vis} = -3\zeta H = -\rho_\Lambda c^2}$$ is dissipative: it encodes internal friction in a spacetime medium driven out of equilibrium by expansion.

  • The de Sitter phase is a non-equilibrium steady state:

    • $\rho$ and $H$ are constant,
    • but $$\nabla_\mu s^\mu \propto \zeta,\theta^2 > 0,\qquad \dot S_{\rm tot} > 0,$$ so entropy is continually produced.

  • Dark energy is thus reinterpreted not as a perfectly static cosmological constant, but as the hydrodynamic and thermodynamic response of the horizon-fluid or quantum vacuum (i.e. an IR effective medium shaped by horizon physics) to cosmic expansion, with the horizon storing the entropy generated by viscous dissipation.

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