Machian Phase Normalization in Gravitational Cosmology

• Machian phase normalization is a framework that imposes a global boundary condition on the gravitational path integral by referencing the causal horizon.
• It fixes the cosmological conformal factor via the Hamiltonian constraint, resulting in a horizon-integrated phase density and dynamically evolving dark energy.
• This approach reinterprets ΛCDM by linking the cosmological constant to horizon dynamics, offering insights into phantom regimes and late-time cosmological tensions.

Machian phase normalization is a global boundary condition in the gravitational path integral, implemented by referencing the phase to the entirety of the causal horizon rather than to local or asymptotic Minkowski spacetime. This formulation, rooted in Mach’s principle, fixes the cosmological conformal factor via the Hamiltonian constraint and causal horizon boundary data. The cosmological term $\Lambda$ emerges as a horizon-integrated phase density—specifically, $\Lambda=R/6$ at equilibrium—while non-equilibrium effects are parameterized by a single global variance $\beta$. This approach reinterprets the canonical status of $\Lambda$CDM, produces effective $w$CDM and even phantom regimes dynamically, and naturally explains late-time cosmological tensions by the global constraints that preclude a stable de Sitter state (Putten, 28 Jan 2026).

1. Gravitational Path–Integral Phase and Global Normalization

In the Lorentzian gravitational path integral,

$\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$

only phase differences $\Delta S$ are physically meaningful. Standard practice sets the "zero" of the phase at Minkowski null infinity; however, in cosmology the causal structure is bounded by the horizon ${\cal H}$. Machian phase normalization achieves a background-independent formulation by referencing the global phase to the total matter and geometry within this horizon. The path integral is therefore modified: $$e^{iS[g]/\hbar} \to e^{i(S[g] - S_0[\mathcal{H}])/\hbar}$$ where $S_0[\mathcal{H}]$ anchors the phase to the causal horizon's content. This prescription neither arises from local field equations nor propagates local degrees of freedom; instead, it imposes a global boundary condition (Machian phase normalization) on the path integral.

2. Status of the Cosmological Conformal Factor

The flat FLRW line element in conformal time $\Lambda=R/6$0 is

$\Lambda=R/6$1

Classically, the scale factor $\Lambda=R/6$2 might be construed as dynamical, with associated canonical momentum and possible instabilities ("conformal-factor problem"). In contrast, homogeneity and isotropy reduce gravitational constraints to the Hamiltonian constraint alone: $\Lambda=R/6$3 with $\Lambda=R/6$4. The absence of an independent momentum conjugate to $\Lambda=R/6$5 renders the scale factor a global gauge variable (or Lagrange multiplier) rather than a propagating field. Consequently, the conformal-factor instability is evaded; $\Lambda=R/6$6 enforces the Machian phase normalization through the Hamiltonian constraint.

3. Phase Density and Horizon Clausius Relation at Cosmological Turning Points

A turning point in cosmic expansion is identified by

$\Lambda=R/6$7

corresponding to a stationary horizon. At such points, Gibbs’ variational principle applies to the total horizon-enclosed energy, and the Clausius relation

$\Lambda=R/6$8

with

$\Lambda=R/6$9

(where $\beta$0 is the horizon area) requires that $\beta$1 be an exact variation. This is achieved if the effective phase density is proportional to the trace of the Schouten tensor: $\beta$2 yielding

$\beta$3

Thus, $\beta$4 is not an arbitrary constant, but emerges as the integrating factor rendering horizon flux variations exact in entropy.

4. Dynamics Away from Equilibrium: The Variance Parameter $\beta$5

Outside equilibrium (i.e.\ away from $\beta$6), the horizon-anchored integrating factor $\beta$7 evolves according to cosmographic invariants: $\beta$8 with the jerk parameter

$\beta$9

and $\Lambda$0 as the sole global “variance” parameter quantifying non-adiabatic adjustment. In the radiation-dominated era ($\Lambda$1), the correction vanishes. The non-adiabatic evolution of the dark-energy density is governed by

$\Lambda$2

where $\Lambda$3. Once $\Lambda$4 is fixed, the background evolution is completely specified.

5. Emergent $\Lambda$5CDM and Phantom Behavior without New Degrees of Freedom

The effective equation of state parameter for dark energy is

$\Lambda$6

Nonzero $\Lambda$7 enables $\Lambda$8 to track horizon phase-space evolution, yielding dynamically evolving $\Lambda$9. In the matter-dominated epoch ($w$0), $w$1 is required for $w$2. At late times $w$3 rises relative to $w$4, and $w$5 can cross below $w$6, exhibiting “phantom” behavior without introducing ghost fields or supplementary degrees of freedom. Fits with evolution codes (e.g.\ CAMB) display phantom crossing at $w$7 around $w$8 for $w$9 near the upper bound, arising strictly via horizon-anchored phase normalization.

6. Cosmological Tensions and the Fate of de Sitter

The conventional $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$0CDM framework enforces a constant $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$1, with $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$2 acting as an asymptotic attractor (stable de Sitter future). Under Machian phase normalization, $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$3 is only a thermodynamic turning point. Analytic continuation permits $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$4 to cross back through $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$5, eliminating guaranteed approach to stable de Sitter. As the phase density $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$6 must adjust to changes in the causal domain, calibration schemes based on early-universe (e.g. CMB) distances and late-universe (distance ladder) $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$7 determinations generally yield different results. The Hubble tension is interpreted as a signature of the evolving global phase normalization, not an anomaly; specifically, it reflects mismatch between the horizon scale at last scattering and the present, after integrating factor evolution.

7. Conceptual Implications and Framework Summary

Machian phase normalization transfers the cosmological term $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$8 from a static parameter to a dynamical, horizon-referenced integrating factor. The cosmological conformal factor is a global gauge variable fixed by the Hamiltonian constraint. All effective $\mathcal{Z} = \int \mathcal{D}g\,e^{iS[g]/\hbar}$9CDM phenomenology, including dynamically realized phantom regimes, is produced with a single global parameter $\Delta S$0, obviating the need for additional degrees of freedom. The absence of a stable de Sitter attractor and the natural explanation for late-time cosmological tensions arise from the inherent global constraint structure of the Machian framework (Putten, 28 Jan 2026).