Bulk Viscosity from Geometry, Not Only Kinetic Theory

The idea that bulk viscosity 

could be an alternative to dark energy has been around for a while. For example, see the dark goo paper, or viscous universe, or spatial phonons.

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

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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.

 

 

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So, let us 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.

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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.

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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!

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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

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

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Step 4: Relativistic quanta

For vacuum quanta:

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

Then:

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

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Step 5: Bulk viscosity definition

By definition:

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

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

 

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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. Which means: \begin{equation}
    \zeta = \frac{\nu_{\text{vac}}}{c^2} Y_{\Lambda}
\end{equation} 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. 

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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:

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

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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!

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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!

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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!

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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 $\zeta > 0$.

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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!

 

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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

 

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Bulk Viscous Dark Fluid  

One can then whip up something like the following:

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Framework and Setup

In spatially flat FLRW with Hubble parameter $H=\dot a/a$ and expansion rate $\theta=3H$, a dissipative fluid has effective pressure $p_{\rm eff}=p+\Pi$ with bulk viscous pressure $\Pi=-\zeta \theta$, satisfying the energy conservation equation:

$$\dot\rho+3H(\rho+p+\Pi)=0$$

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Determining the Bulk Viscosity

From Kubo Formula and Relaxation Theory

Dimensional and Kubo/relaxation arguments in de Sitter space with Gibbons-Hawking temperature $T_H=\hbar H/2\pi k_B$ give:

$$\zeta \sim \chi_\Theta \tau_\Pi \propto H$$

where:

• $\chi_\Theta$ is the bulk susceptibility (trace channel)
• $\tau_\Pi \sim \alpha/H$ is the relaxation time
• $\alpha = \mathcal{O}(1)$ is a dimensionless coefficient

From Steady-State Energy Conservation

At late times with $\dot\rho \to 0$ and $p \simeq 0$, energy conservation fixes:

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

This is the unique de Sitter value, determined entirely by thermodynamic consistency.

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Thermodynamic Consistency

Entropy Production

The entropy production rate is:

$$\nabla_\mu s^\mu = \frac{\zeta}{T}\theta^2 = \frac{9 k_B c^2 H^2}{4\hbar G} > 0$$

This is positive-definite, satisfying the generalized second law and ensuring thermodynamic irreversibility.

Physical Interpretation

• Horizon entropy increases: $\dot S_h > 0$
• Matter entropy may decrease: $\dot S_m < 0$
• Total entropy always increases: $\dot S_{\rm tot} = \dot S_h + \dot S_m > 0$

The arrow of time is encoded in the bulk viscosity itself.

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Causal Structure: Israel–Stewart Theory

Evolution Equation

The Israel–Stewart evolution for the bulk pressure is:

$$\tau_\Pi \dot\Pi + \Pi = -\zeta \theta$$

This introduces a finite relaxation time, ensuring causality.

Bulk Mode Speed

The characteristic speed of bulk perturbations is:

$$\left(\frac{v_b}{c}\right)^2 = \frac{1}{3\tau_\Pi H}$$

Causality Constraint

Requiring $v_b \le c$ imposes:

$$\boxed{\tau_\Pi H \gtrsim \frac{1}{3}}$$

In practice, $\tau_\Pi H = \mathcal{O}(1)$ is needed for both causality and phenomenological viability.

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Microscopic Origin

Conformal Symmetry Breaking

The condition $\zeta \neq 0$ arises from conformal symmetry breaking via:

1. Spacetime curvature (non-zero Ricci scalar)
2. Trace anomalies (quantum corrections)
3. Mass terms (breaking scale invariance)

In de Sitter space, all three mechanisms are active.

Holographic Picture

From the holographic/membrane paradigm:

• Horizon degrees of freedom give shear viscosity $\eta \propto H$
• Bulk viscosity emerges as $\zeta \sim \mathcal{O}(\eta)$
• Bulk entanglement appears as a one-loop contribution

The horizon acts as a thermodynamic boundary encoding all bulk information.

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Observational Predictions

Background Evolution

At the background level, $\zeta=\rho/(3H)$ reproduces exact de Sitter expansion, making:

• SNe distances: Identical to ΛCDM
• BAO scales: Identical to ΛCDM
• CMB distance to last scattering: Identical to ΛCDM

The background expansion history is observationally indistinguishable from a cosmological constant.

Structure Formation

Finite $\tau_\Pi$ suppresses late-time growth through:

• Reduced growth rate: $f\sigma_8(z)$ suppressed at low $z$
• Minimal impact on CMB lensing (integrated effect)
• Potential resolution of $S_8$ tension

This is the key observable signature distinguishing bulk viscous dark energy from pure ΛCDM.

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Open Questions

Theoretical

1. Compute $\alpha$ from first principles using the retarded correlator $G^R_{\Theta\Theta}$
2. Quantify low-$\omega$ spectral weight in the trace channel
3. Derive bulk susceptibility $\chi_\Theta$ from horizon thermodynamics

Phenomenological

1. Implement $\zeta = \rho/(3H)$, $\tau_\Pi = \alpha/H$ in CLASS/CAMB
2. Test against $S_8$ tension in weak lensing surveys
3. Constrain $\alpha$ from growth-rate measurements ($f\sigma_8$)
4. Compare with upcoming data: Euclid, Vera Rubin, Roman

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Summary: Four Pillars, One Result

All independent theoretical frameworks converge on the same conclusion:

1. Response Theory (Kubo Formula)

$\zeta = \frac{1}{9}\lim_{\omega\to 0}\frac{1}{\omega}\operatorname{Im} G^{R}{\Theta\Theta}(\omega) \sim \chi\Theta \tau_\Pi \propto H$

2. Israel-Stewart Hydrodynamics

$$\tau_\Pi \dot\Pi + \Pi = -\zeta \theta, \quad v_b^2 = \frac{c^2}{3\tau_\Pi H}$$

3. Horizon Thermodynamics

$$\dot S_{\rm tot} = \frac{\zeta}{T}\theta^2 > 0, \quad E_{BY} = 2T_H S_H$$

4. Holography (Membrane Paradigm)

$$\eta/s = \hbar/(4\pi k_B), \quad \zeta \sim \mathcal{O}(\eta)$$

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The Final Result

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

with relaxation time:

$$\boxed{\tau_\Pi H = \mathcal{O}(1)}$$

This ensures:

• Causality ($v_b \le c$)
• Thermodynamic consistency ($\dot S_{\rm tot} > 0$)
• Late-time growth suppression (testable with $f\sigma_8$)
• No background deviation (consistent with SNe/BAO/CMB)

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Physical Interpretation

Dark energy is not a substance. It is the macroscopic manifestation of vacuum bulk viscosity—the irreversible dissipative stress generated by cosmic expansion itself.

The cosmological constant $\Lambda$ emerges as the thermodynamic equilibrium state of an expanding universe in contact with its own event horizon, where:

• The horizon stores entropy
• Expansion generates viscous stress
• Irreversibility enforces the arrow of time
• Quantum gravity (holography) sets the scale

This is not a modification of general relativity, but a completion of its thermodynamic description.