Matter and Energy in de Sitter Space

An excellent question that gets to the heart of modern cosmology and quantum field theory in curved spacetime.

The short answer is: Yes, in an asymptotically pure de Sitter space, almost all forms of matter energy will be diluted away to effectively zero, except for a constant vacuum energy density. However, there are crucial subtleties.

The Classical Picture: Redshift to Nothing

De Sitter space is the maximally symmetric spacetime solution to Einstein's equations with a positive cosmological constant (Λ). It describes an exponentially expanding universe.

The scale factor grows as a(t) ∝ e^Ht, where H is the Hubble constant. Any freely propagating matter (particles, radiation) experiences an extreme form of cosmological redshift.

Massless particles (photons): Their wavelength is stretched by the expansion. Energy E ∝ 1/λ, so their energy redshifts exponentially to zero.

Massive particles: Their peculiar (kinetic) momentum also redshifts as p ∝ 1/a. A particle initially moving with high energy will eventually come to a near standstill relative to the cosmic expansion, its kinetic energy drained away. Its rest mass energy (E=mc²) remains, but that's not "most" of its energy if it started with high kinetic energy.

Classically, matter fields are diluted and redshifted into irrelevance. The energy density of matter (ρ_matter) decays as ρ ∝ a⁻³ (for dust) or a⁻⁴ (for radiation), while the vacuum energy density ρ_Λ remains constant. The universe becomes an almost pure de Sitter vacuum.

The Quantum Picture: A More Subtle Story

This is where it gets interesting. Quantum fields in de Sitter space don't just passively redshift.

Dilution vs. Particle Production

While classical field amplitudes are diluted, quantum mechanics introduces fluctuations. The exponential expansion can "pull" virtual particles out of the vacuum, a phenomenon akin to Hawking radiation from black holes.

The De Sitter Horizon & Temperature

De Sitter space has a cosmological event horizon at a distance ~c/H. An observer in de Sitter space perceives a thermal bath with a Hawking-Gibbons temperature:

T_dS = ħH / (2πk_B)

This is a quantum mechanical effect of the horizon.

The Fate of a Quantum Field

Consider a scalar field (e.g., the inflaton, or a matter field).

Super-horizon modes: Quantum fluctuations that get stretched beyond the horizon become "frozen" as classical field perturbations. They effectively give up their local kinetic energy to the gravitational field, contributing to a kind of large-scale structure (but in eternal de Sitter, this is subtle).

Equilibration?: There is a long-standing debate: will a quantum field in de Sitter space eventually thermalize with the horizon temperature T_dS? Many calculations suggest that over enormous timescales, an initially out-of-equilibrium field will relax to a thermal state at T_dS. In this steady state, the field has not "given up all its energy" but has reached an equilibrium with the horizon, where particle creation and annihilation balance.

The Critical Distinction: "Give Up" to Whom?

Energy in general relativity is not globally conserved, especially in an expanding universe. We must ask: according to which observer?

A comoving observer (moving with the expansion) sees particle energies redshift to zero.

An observer using static coordinates (hovering at a fixed distance from the origin in a patch of de Sitter) sees a different picture. They are surrounded by a thermal bath at T_dS. A particle they emit might fall toward the horizon, have its energy hugely redshifted from the comoving perspective, but from the static observer's view, the energy is transferred to the gravitational field when the particle crosses the horizon.

Summary & Key Points

Aspect	Fate in Asymptotic De Sitter Space
Classical Matter Density	Dilutes away exponentially: ρmatter → 0.
Kinetic Energy of Particles	Redshifts to zero for comoving observer.
Rest Mass Energy	Remains, but becomes a negligible component of the total energy budget.
Quantum Fields	Tend toward a thermal equilibrium state at the de Sitter temperature TdS, interacting with the horizon. They don't vanish but reach a steady state.
Vacuum Energy (Λ)	Constant. Ultimately dominates everything.
Gravitational Potential Energy	Increases (becomes more negative) to balance the "loss" of kinetic energy. This is the GR conservation law at work.

Conclusion

Matter does give up its local, usable, non-rest-mass energy from the perspective of a comoving observer in de Sitter space. Its kinetic energy is redshifted to zero, and its density is diluted to nothingness. However, quantum fields don't simply disappear; they interact with the cosmological horizon, potentially reaching a thermal equilibrium characterized by the de Sitter temperature.

The ultimate victor is the constant vacuum energy density ρ_Λ, which governs the asymptotic future.

Thus, for all practical purposes regarding structure, dynamics, and usable energy, yes, matter gives up almost everything, leaving behind a cold, near-empty, thermal quantum vacuum governed by Λ and its associated horizon.