Connection of Vacuum Energy with Inflation: The Elastic Potential, Initial Conditions, and the Origin of Primordial Fluctuations

This paper — the sixth in a programme of seventeen investigating the hypothesis that quantum vacuum energy gravitates locally — develops the connection between the elastic vacuum framework and inflationary cosmology.

Three results deferred from the companion paper (Paper 5, "The Elastic Vacuum") are delivered. First, the scalar field potential V(ψ) is derived from the elastic properties of spacetime rather than postulated: the Einstein–Hilbert action supplemented by its leading quantum correction (R²) yields, upon conformal transformation, the Starobinsky potential V(ψ) = V₀(1 − exp(−√(2/3) ψ/M_Pl))². This potential — currently the inflationary model most favored by Planck data — is identified as the leading-order nonlinear elastic potential of the vacuum within the R² truncation of the effective action: quadratic near equilibrium (Hooke's law), saturating at large deformation (the yield plateau). Inflation is reinterpreted as the elastic recoil of spacetime geometry from a state of extreme initial deformation.

Second, the flatness feedback parameter β > 0 is derived from the action in the quasi-static limit, completing the proof that spatial flatness is a stable late-time attractor of the elastic vacuum. The sign of β follows from the convexity of V(ψ) at its minimum — a structural result independent of specific parameter values.

Third, the paper argues on quantum-mechanical, thermodynamic, and structural grounds that the initial state of the universe cannot be exactly homogeneous. The Heisenberg uncertainty principle forbids any finite-energy field configuration from having zero spatial variance. Inflation need not create homogeneity from a homogeneous state; it creates approximate homogeneity from an inherently inhomogeneous initial state. A quantitative estimate shows that for total inflation lasting more than approximately 69 e-folds, even Planck-amplitude pre-inflationary perturbations are subdominant on CMB scales.

The paper also introduces a three-dimensional elastic analogy — injection of a substance into a transparent elastic medium — that resolves the principal limitations of the two-dimensional membrane model developed in Paper 5, and identifies the 10⁵⁵ mass hierarchy between the inflationary scalaron and the late-time screening field as the programme's most significant unsolved problem, distinguishing interpretive unification (achieved) from dynamical unification (not yet achieved).

The framework uses only general relativity with standard quantum corrections. The screening scalar field ψ is a new degree of freedom — acknowledged as such — but functions as a collective mode of the elastic vacuum rather than a new matter particle.

Recommended for submission to the Journal of Cosmology and Astroparticle Physics (JCAP).

Files: Connection_of_Vacuum_Energy_with_Inflation.pdf (compiled manuscript, 38 pages) Connection_of_Vacuum_Energy_with_Inflation.tex (LaTeX source) generate_figures.py (Python code for all figures) figures/ (directory containing PDF figures)

Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO) General Relativity and Quantum Cosmology (gr-qc) High Energy Physics - Theory (hep-th)

Authors: Boris Kriger Affiliation 1: Information Physics Institute, Gosport, Hampshire, United Kingdom Affiliation 2: Institute of Integrative and Interdisciplinary Research, Toronto, Canada ORCID: 0009-0001-0034-2903

Resource type: Preprint

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Language: English

Keywords: vacuum energy, cosmic inflation, elastic vacuum, Starobinsky potential, primordial fluctuations, flatness problem, initial conditions, cosmological constant problem, R-squared gravity, dark energy, general relativity, quantum corrections to gravity

Related identifiers: Is part of the research programme: "What If the Vacuum Gravitates Locally?" (Paper 6 of 17) Is supplement to: "The Elastic Vacuum: A Sequestering Mechanism for Vacuum Energy and the Complete Dynamical History of Cosmic Expansion" (Paper 5)

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