⚛️ general relativity

Cosmological Expansion from Machian Phase Normalization by Horizon Constraints

This paper proposes that cosmological expansion is driven by Machian phase normalization of the gravitational path integral constrained by causal horizons, a framework that reinterprets the cosmological constant as an equilibrium phase density to explain effective dark energy behaviors and late-time tensions without introducing new degrees of freedom.

Original authors: Maurice H. P. M. van Putten

Published 2026-01-30

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CC BY 4.0

Original authors: Maurice H. P. M. van Putten

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. For decades, scientists have been trying to figure out why it's inflating and what force is pushing it. The standard explanation involves a mysterious, unchanging energy called "Dark Energy" (represented by the symbol $\Lambda$ ). However, this explanation has some cracks in it: it doesn't quite fit all the data, and it creates mathematical headaches for physicists.

This paper proposes a completely different way of thinking about the universe's expansion. Instead of a mysterious force pushing from the inside, the author suggests the expansion is actually a result of the universe "checking its own size" against its own boundaries.

Here is the breakdown using simple analogies:

1. The "Global Clock" Problem (Mach's Principle)

The Old Idea: Imagine you are in a room with no windows. You try to measure how fast you are spinning, but you have no reference point. In physics, this is like trying to define the "phase" (the timing or rhythm) of the universe without a reference.
The New Idea: The author says the universe does have a reference point: its own edge, called the Cosmological Horizon. Think of this horizon as the edge of a swimming pool. You can't define the water's movement without knowing where the pool walls are.
The paper argues that the universe's expansion is governed by a rule called Mach's Principle. In simple terms, this means "the whole universe tells the parts how to behave." The universe sets its own rhythm based on the total amount of matter inside its "pool."

2. The "Dial" vs. The "Engine" (The Conformal Factor)

The Problem: In standard physics, the "size" of the universe (the scale factor) is treated like an engine part that can vibrate, wiggle, and cause instability. This causes a famous mathematical problem called the "conformal-factor problem."
The Solution: This paper says, "Stop treating the size of the universe like an engine part." Instead, think of it as a global dial or a gauge.

Analogy: Imagine a thermostat in a house. The temperature (the universe's size) isn't a wild, chaotic thing; it's a setting fixed by the total heat in the house (the matter content). It doesn't "propagate" or wiggle on its own; it is simply set by the boundary conditions of the house.
Because it's just a setting and not a moving part, the scary mathematical problems disappear.

3. The "Thermostat" and the "Turning Point"

The paper suggests the universe acts like a system trying to reach thermal equilibrium (perfect balance), similar to a cup of coffee cooling down to room temperature.

The Turning Point: There are moments in the universe's history where the expansion pauses or changes direction (like a car reaching the top of a hill before rolling down). At these specific moments, the universe is perfectly balanced.
The Fix: At these balanced moments, the "Dark Energy" value ( $\Lambda$ ) is mathematically locked to the curvature of space. It's not a random number; it's a necessary value to keep the "thermodynamics" (the heat and entropy) of the universe's edge consistent.

4. Why the Universe is "Phantom" (The Variance Parameter $\beta$ )

Here is where it gets interesting for the present day.

The Old View: Dark Energy is a constant, unchanging force.
The New View: The universe is constantly adjusting its "thermostat" as it grows. However, it's not perfect. It takes a little time for the universe to adjust its settings to the new size.
The Analogy: Imagine driving a car with a slightly delayed cruise control. You press the gas, but the car takes a moment to speed up. This "lag" is what the paper calls the variance parameter ( $\beta$ ).
The Result: This lag makes the universe behave as if it has "Phantom Energy" (a type of dark energy that gets stronger over time), even though no new, exotic energy was added. It's just the universe catching up to its own growth. This explains why recent observations show the universe accelerating faster than the old models predicted.

5. The "Tension" Between Early and Late Universe

Scientists are currently confused because measurements of the early universe (from the Big Bang) and the late universe (today) don't match up. This is called the "Hubble Tension."

The Paper's Explanation: This isn't a mistake in our measurements. It's because the universe isn't in a perfect, stable state. The "lag" in the universe's adjustment means that the rules for the early universe (when it was in equilibrium) are slightly different from the rules for today (when it is adjusting).
The Conclusion: The universe is not settling into a permanent, stable state (like a perfect de Sitter space). It is constantly evolving, and the "tension" we see is actually a signature of this global adjustment process.

Summary

The paper argues that:

Expansion isn't driven by a mysterious constant force, but by the universe's need to keep its global "phase" (rhythm) consistent with its total size and matter.
The "size" of the universe is a fixed setting, not a wiggly, unstable field, which solves major mathematical problems.
The acceleration we see today is a natural result of the universe "lagging" behind its own growth, not a sign of exotic new physics.
The conflicts in data are actually proof that the universe is a dynamic, adjusting system, not a static one.

In short: The universe isn't being pushed by a ghost; it's just trying to keep its own internal clock synchronized with the size of its own horizon.

Technical Summary: Cosmological Expansion from Machian Phase Normalization by Horizon Constraints

Problem Statement
The paper addresses three interconnected conceptual issues in modern cosmology and quantum gravity:

The Conformal-Factor Problem: In Euclidean quantum gravity, the conformal mode of the metric possesses a wrong-sign kinetic term, leading to instabilities.
The Nature of the Cosmological Constant ( $\Lambda$ ): The physical meaning of $\Lambda$ remains ambiguous, particularly regarding whether it is a fundamental coupling or an integration constant, and its role as a propagating degree of freedom.
Late-Time Cosmological Tensions: Persistent discrepancies exist between early-universe and late-universe determinations of the Hubble parameter and other cosmological parameters within the standard $\Lambda$ CDM model.

The author argues that these problems stem from treating the cosmological constant as a fundamental local coupling rather than a global quantity fixed by causal boundary conditions.

Methodology
The paper proposes a reformulation of cosmological expansion based on the gravitational path integral constrained by Mach's principle. The core methodology involves:

Machian Phase Normalization: Instead of viewing Mach's principle as a philosophical postulate about inertia, it is elevated to a structural requirement for the gravitational path integral. The phase of the action must be defined relative to a global reference fixed by the total matter content within the cosmological horizon.
Horizon Thermodynamics as a Global Constraint: The cosmological horizon is used to delimit the causal phase space. Thermodynamic consistency (specifically the Clausius relation) is applied not to generate local field equations (as in Jacobson's approach) but to fix the global phase normalization of the path integral.
Reinterpretation of the Conformal Factor: The cosmological conformal factor is treated not as a propagating degree of freedom with a canonical momentum, but as a global gauge variable. Its value is fixed by the Hamiltonian constraint and boundary conditions at the horizon, rendering the conventional conformal-factor problem inapplicable.
Non-Equilibrium Dynamics: Departures from exact de Sitter equilibrium are modeled using a single variance parameter, $\beta$ , which encodes the delayed adjustment of the integrating factor to the evolving causal phase space.

Key Contributions and Results

Resolution of the Conformal-Factor Problem: By identifying the conformal factor as a global gauge variable fixed by the Hamiltonian constraint rather than a dynamical field, the paper argues that the instability associated with the wrong-sign kinetic term is an artifact of misinterpreting the variable's status. The conformal factor possesses no canonical conjugate momentum.
Derivation of $\Lambda$ as an Integrating Factor: The cosmological constant is derived as an integrating factor ( $\Lambda = R/6$ , where $R$ is the Ricci scalar) required to render the horizon Clausius relation exact at cosmological turning points (where the deceleration parameter $q = -1$ ). This elevates $\Lambda$ from a mere integration constant to a thermodynamically fixed quantity.
Emergence of Effective Phantom Behavior: The framework explains the emergence of effective $w$ $w$ CDM behavior, including phantom regimes ( $w < -1$ $w < - 1$ ), without introducing new degrees of freedom or exotic matter. This behavior arises naturally from the non-adiabatic evolution of the Machian phase normalization parameterized by $\beta$ $β$ .
- The parameter $\beta$ is constrained by the requirement that dark energy be negligible during the matter-dominated era ( $\beta \lesssim 1/12$ ).
- For $\beta$ near this critical value, the model predicts a crossing of the phantom divide ( $w = -1$ ) at a redshift $z_c \simeq 1.85$ , consistent with fits to Planck CMB data.
Duality between $\Lambda$ CDM and JCDM: The paper introduces a duality between the standard $\Lambda$ $Λ$ CDM model and a Machian counterpart, JCDM.
- $\Lambda$ CDM assumes a strictly constant $\Lambda$ , appropriate for early-time equilibrium but inconsistent with evolving causal phase spaces.
- JCDM identifies the dark energy sector with the Machian phase density itself, allowing $\Lambda$ to track the evolving causal phase space.
- Effective $w$ CDM is described as a coarse-grained interpolation between these limits, capturing the non-adiabatic response of the phase normalization.

Significance and Claims
The paper claims that the cosmological constant problem, the conformal-factor problem, and late-time cosmological tensions share a common origin: the treatment of an integrating factor as a fundamental constant rather than a causal, horizon-fixed quantity.

Reassessment of de Sitter Stability: The framework challenges the stability of de Sitter space in cosmology. In this formulation, the de Sitter state ( $q = -1$ ) is a thermodynamic turning point where Gibbs' principle applies, but it is not a stable fixed point of the global phase constraint. Consequently, analytic continuation away from this point generically permits crossings of $q = -1$ , implying that de Sitter space is not asymptotically stable once causal phase normalization is enforced.
Origin of Tensions: The tensions between early- and late-universe parameter determinations are presented not as anomalies to be eliminated, but as natural signatures of an underlying global constraint structure that precludes a stable de Sitter state.
Conceptual Clarity: The framework provides a natural origin for late-time cosmological tensions and clarifies the status of $\Lambda$ CDM by grounding cosmological expansion in global phase consistency rather than local stationarity of the action.

The paper concludes that general relativity remains valid for sub-horizon geometric deformations, while cosmological expansion and dark energy phenomenology arise from enforcing global phase consistency on a horizon-bounded spacetime.