Cosmological Expansion from Machian Phase Normalization by Horizon Constraints

We argue that cosmological expansion is governed by Machian phase normalization of the gravitational path integral, fixed by causal horizon boundary conditions rather than by local dynamics. In this formulation, the cosmological conformal factor is not a propagating degree of freedom but a global gauge variable fixed by the Hamiltonian constraint, rendering the conventional conformal-factor problem inapplicable. Thermal equilibrium at cosmological turning points uniquely fixes the equilibrium phase density ($Λ=R/6$) as the integrating factor that renders the horizon Clausius relation exact. Controlled departures from equilibrium are encoded by a single variance parameter governing non-adiabatic background evolution. The resulting framework clarifies the conceptual status of $Λ$CDM, explains the emergence of effective $w$CDM behavior–including phantom regimes without new degrees of freedom–and provides a natural origin for late-time cosmological tensions, arising from global constraints that preclude a stable de Sitter state.

💡 Research Summary

The paper proposes that cosmic expansion is not driven by local dynamical equations but by a global phase‑normalization condition imposed on the gravitational path integral. By invoking Mach’s principle at the level of the path integral, the authors argue that the causal horizon supplies a universal reference for the phase of the action; only phase differences are physical, and causality restricts correlations to lie within the horizon. Consequently, the overall phase of the gravitational action must be fixed relative to the total matter content inside the horizon.

In this framework the cosmological conformal factor a(t) is not a propagating degree of freedom. Instead it appears as a global gauge variable whose value is fixed by the Hamiltonian constraint (the Friedmann equation). Because it carries no canonical momentum, the notorious “conformal‑factor problem” (a wrong‑sign kinetic term in Euclidean quantum gravity) is rendered moot. The conformal factor simply rescales the whole spacetime to satisfy the global phase condition.

The authors then turn to horizon thermodynamics. By demanding that the Clausius relation δQ = T dS be exact for the horizon, they identify a unique integrating factor that makes the entropy variation an exact differential. This factor is Λ = R/6, equivalently the trace of the Schouten tensor J. At the turning point q = –1 (the de Sitter‑like moment when the horizon is momentarily stationary), Gibbs’ variational principle applies, fixing Λ to this equilibrium value. Thus Λ is not a fundamental coupling constant but an “integrating factor” that enforces global phase consistency.

Non‑adiabatic evolution is encoded in a single variance parameter β. When the universe departs from the equilibrium turning point, the phase reference cannot adjust instantaneously; the delay is statistically described by β. Expanding Λ(a) in cosmographic variables yields
 Λ(a) = J(a) + β (j – 3) + …,
where j is the jerk parameter. In the radiation era j = 3, so the β‑term vanishes. For β below a critical value β_crit ≈ 1/12 the effective dark‑energy density is negligible during matter domination, but as the universe ages the β‑term grows, causing Λ to increase relative to matter. This naturally drives the effective equation‑of‑state w across the phantom divide (w < –1) at redshift z ≈ 1.85, reproducing the phenomenology of wCDM without invoking exotic fields or instabilities.

The paper contrasts two limiting models: standard ΛCDM, which treats Λ as a fixed constant (thus consistent with early‑time adiabatic evolution but incompatible with the Machian phase condition later), and “JCDM”, where Λ is identified with the equilibrium phase density Λ = J = R/6 and evolves with the causal phase space. Observationally, the data favor an intermediate regime 0 < β ≲ β_crit, i.e., an effective wCDM that interpolates between these limits.

A crucial implication is that the de Sitter state (q = –1) is not a stable attractor in this Machian picture; it is merely a thermodynamic turning point where the horizon entropy variation is exact. Analytic continuation away from this point permits crossing q = –1, so the universe does not asymptotically settle into a stable de Sitter phase. This provides a natural explanation for the persistent H₀ and S₈ tensions: they are signatures of the underlying global phase constraint rather than anomalies to be eliminated.

In summary, the authors unify three long‑standing puzzles—the cosmological‑constant problem, the conformal‑factor problem, and late‑time cosmological tensions—by reinterpreting Λ as an integrating factor fixed by horizon‑based Machian phase normalization. Local General Relativity remains valid on sub‑horizon scales, while cosmic expansion and dark‑energy phenomenology emerge from enforcing global phase consistency on a horizon‑bounded spacetime. This perspective challenges the conventional view of de Sitter stability and suggests new avenues for confronting cosmological data with fundamental quantum‑gravitational principles.