Massless graviton in de Sitter as second sound in two-fluid hydrodynamics

Imagine the universe not as an empty void, but as a giant, invisible ocean. For a long time, physicists have been trying to figure out what happens when you drop a stone into this ocean. Does it make a ripple? Is that ripple a particle of gravity (a "graviton")? And if so, does it have weight (mass), or is it weightless?

This paper by G.E. Volovik tackles these deep questions by using a very clever trick: comparing the universe to a cup of super-cooled liquid helium.

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The Universe as a Two-Fluid Soup

In the 1940s, a physicist named Lev Landau discovered that super-cold liquid helium behaves like two different fluids mixed into one.

Fluid A (The Superfluid): This part is perfectly smooth, frictionless, and doesn't carry any heat. It's like a ghostly, silent flow.
Fluid B (The Normal Fluid): This part is "messy." It carries heat, has friction, and acts like a normal liquid.

Volovik suggests that our De Sitter universe (a universe expanding at an accelerating rate, like ours) works the same way. It's a cosmic soup made of two ingredients:

The "Superfluid" (Dark Energy): This is the vacuum energy (the $\Lambda$ term). It's the smooth, background fabric of space. It doesn't move or carry heat; it just is.
The "Normal Fluid" (Gravity's Heat): This is the part that comes from the expansion of the universe itself. Because the universe is expanding, it has a temperature (like a warm bath). This "heat" is carried by the gravitational field.

2. The "Second Sound" Analogy

In normal liquids, if you shake them, you get a sound wave (pressure moving through the liquid). This is First Sound.

But in superfluids, there is a weird, magical phenomenon called Second Sound.

First Sound is a wave of pressure (squeezing the liquid).
Second Sound is a wave of temperature (a ripple of heat moving through the liquid).

Imagine a room where the air pressure stays the same, but a wave of "hotness" and "coldness" ripples through the room without the air actually moving. That is second sound.

Volovik argues that in our cosmic ocean, this "Second Sound" exists too. It's a wave of entropy (disorder/heat) traveling through the "normal" gravitational part of the universe.

3. The Big Discovery: The Massless Graviton

The paper does the math on this cosmic "Second Sound." Here is the surprising result:

In a normal liquid, second sound moves slower than the speed of sound.
In this cosmic liquid, the "Second Sound" moves at exactly the speed of light.

Furthermore, the math shows this wave has zero mass.

The Metaphor:
Think of the universe as a giant drum. Usually, we think of gravity as the drumhead vibrating. Volovik is saying that the "Second Sound" is a specific type of vibration on that drum. Because it moves at the speed of light and has no mass, this vibration is the "graviton" (the particle that carries gravity).

The paper suggests that what we call a "massless graviton" might actually be this wave of heat/entropy moving through the cosmic fluid.

4. Why This Solves a Mystery

For decades, physicists have been confused about whether gravitons have mass in an expanding universe.

In a static universe (Minkowski space), it's clear: gravitons are massless.
In an expanding universe (De Sitter space), the math gets messy, and it's hard to tell if they have mass or not.

Volovik's "Two-Fluid" approach cuts through the confusion. By treating the universe like a thermodynamic fluid, he shows that this specific wave must be massless and travel at light speed. It's a natural consequence of the universe being a "hot" fluid of gravity.

5. What About the "Atoms" of the Vacuum?

The paper also touches on a famous problem: Why is the energy of empty space so small (or zero) when quantum physics says it should be huge?

The Analogy: Imagine a superfluid helium tank. The atoms inside are vibrating wildly (high energy), but because they are in a perfect, self-sustaining state, the net pressure is zero.
Volovik argues the universe works the same way. The "atoms of the vacuum" (whatever they are) have huge internal energy, but because the universe is in a perfect thermodynamic balance, all that energy cancels itself out. The "Second Sound" is just a ripple on top of this perfectly balanced state.

Summary: The Takeaway

This paper proposes a new way to see gravity:

The Universe is a Fluid: It has a smooth "superfluid" part (Dark Energy) and a "hot" part (Gravity).
Gravity is a Wave of Heat: The "Second Sound" in this fluid is a ripple of entropy.
The Graviton is Massless: This ripple moves at the speed of light and has no mass, identifying it as the graviton.
No Mystery Left: This approach explains why the vacuum energy is stable and why the graviton behaves the way it does, using the simple, everyday laws of thermodynamics rather than complex quantum math.

In short: Gravity might just be the "sound" of the universe's heat flowing through the cosmic ocean.

Here is a detailed technical summary of the paper "Massless graviton in de Sitter as second sound in two-fluid hydrodynamics" by G.E. Volovik.

1. Problem Statement

The nature of gravitons and their mass in de Sitter (dS) spacetime remains ambiguous compared to the well-defined case in Minkowski spacetime. Specifically, there is ongoing debate regarding the mass spectrum of gravitational modes in a universe with a positive cosmological constant ( $\Lambda$ ). While standard General Relativity predicts massless spin-2 gravitons, the dS background introduces subtleties regarding whether these modes acquire mass or if new collective modes (such as scalar or helicity-0 modes) emerge. The paper seeks to resolve this ambiguity by applying macroscopic thermodynamic principles to the quantum vacuum of the de Sitter state.

2. Methodology

The author employs a two-fluid hydrodynamic approach, originally developed by Landau for superfluid $^4$ He, and adapts it to the relativistic quantum vacuum of the de Sitter universe. The methodology involves:

Thermodynamic Analogy: Treating the de Sitter vacuum as a two-component fluid consisting of:
1. A "Superfluid" component: Representing the vacuum energy (the $\Lambda$ -term), which is coherent and has zero entropy.
2. A "Normal" component: Representing thermal excitations and gravitational degrees of freedom, which carry entropy.
Thermodynamic Relations: Utilizing the Gibbs-Duhem relation and the equation of state for both components. The vacuum energy is modeled using a 4-form field ( $q$ -theory) or scalar curvature ( $R$ ) to define the "atoms" of the vacuum.
Entropy and Temperature: Leveraging the observation that an observer in de Sitter space perceives a heat bath with temperature $T = H/\pi$ (where $H$ is the Hubble parameter). This thermal bath constitutes the normal component.
Hydrodynamic Equations: Applying the standard Landau two-fluid equations to derive the dispersion relation for collective modes, specifically focusing on the "second sound" (entropy waves).

3. Key Contributions and Theoretical Framework

A. Two-Fluid Decomposition of de Sitter Space

The paper establishes a rigorous thermodynamic mapping:

Superfluid Component ( $s$ ): Corresponds to the cosmological constant $\Lambda$ $Λ$ .
- Equation of state: $w_s = -1$ ( $P_s = -\epsilon_s$ ).
- Energy density: $\epsilon_s = \Lambda$ .
- This component is analogous to the ground state of a superfluid with zero temperature.
Normal Component ( $n$ ): Corresponds to gravitational degrees of freedom (scalar curvature $R$ $R$ ) and thermal excitations.
- Equation of state: $w_n = 1$ (stiff matter, $P_n = \epsilon_n$ ).
- The entropy density $S$ is derived from the Gibbons-Hawking entropy of the cosmological horizon: $S = 3\pi T / 4G$ .
- Crucially, the paper identifies the scalar curvature $R$ as the thermodynamic variable conjugate to the gravitational coupling, emerging via the "Kronecker anomaly" or Larkin-Pikin effect.

B. Thermodynamic Balance

The author demonstrates that in the absence of external pressure, the vacuum energy is self-sustained. The large micro-physical contributions to energy density ( $\epsilon$ ) are exactly compensated by the chemical potential ( $\mu$ ) terms, resulting in a net vacuum energy of zero in the Minkowski limit ( $T=0$ ). In the de Sitter limit ( $T \neq 0$ ), the balance shifts, creating the normal component.

4. Key Results

A. Existence of Second Sound

By applying the Landau formula for the velocity of second sound ( $s_2$ ) to the cosmological two-fluid system:
$s_2^2 = \frac{T S^2}{C_V \rho} \frac{\rho_s}{\rho_n}$
Where $C_V$ is the specific heat and $\rho$ is the total density.

Using the equation of state for the normal component ( $w_n = 1$ , stiff matter), the specific heat equals the entropy density ( $C_V = S$ ).
The densities of the superfluid and normal components are found to be equal ( $\rho_s = \rho_n$ ).
Result: The calculation yields $s_2 = c$ . The second sound mode propagates exactly at the speed of light.

B. Identification as a Massless Graviton

The paper argues that this propagating entropy wave (second sound) is the physical manifestation of a massless graviton in de Sitter spacetime.

Nature of the Mode: Unlike the standard transverse-traceless spin-2 graviton, this mode is a collective excitation of the scalar curvature $R$ (the normal component).
Masslessness: The mode is gapless (massless) and propagates at $c$ .
Helicity/Spin: The author suggests this mode may correspond to a spin-0 or helicity-0 graviton, potentially acting as a Goldstone boson arising from the spontaneous breaking of translational symmetry in de Sitter space (specifically, the symmetry under combined space-time translations).

C. Absence of First Sound

The paper notes that "first sound" (pressure waves) is absent in this specific cosmological model because the superfluid energy density ( $\Lambda$ ) is treated as a rigid constant. If the vacuum energy were dynamical (as in $q$ -theory or Starobinsky inflation), a Higgs-like mode would appear instead.

5. Significance and Implications

Resolution of the Cosmological Constant Problem: The framework provides a thermodynamic explanation for why the vacuum energy is zero in the ground state (Minkowski) and non-zero only when the system is perturbed (de Sitter), resolving the discrepancy between quantum field theory predictions and observed vacuum energy.
New Perspective on Gravitons: It challenges the conventional view that gravitons in de Sitter space are strictly spin-2. It proposes that the de Sitter background supports a massless scalar (or helicity-0) graviton mode that behaves as a second sound wave.
Universality of Hydrodynamics: The work demonstrates that Landau's hydrodynamic theory is robust enough to describe relativistic quantum vacua, bridging condensed matter physics and high-energy cosmology.
Goldstone Mechanism: It links the massless nature of this graviton to the breaking of asymptotic symmetries in de Sitter space, suggesting that the "soft graviton" is a Goldstone boson of the broken symmetry.

Conclusion

Volovik concludes that the de Sitter quantum vacuum behaves as a two-fluid system where the "normal" component (gravitational degrees of freedom) supports a massless collective mode propagating at the speed of light. This mode, identified as second sound, offers a compelling candidate for the massless graviton in de Sitter spacetime, potentially possessing spin-0 or helicity-0 characteristics. This approach unifies thermodynamic laws with gravitational dynamics, offering a novel pathway to understanding the spectrum of gravitational waves in an expanding universe.