What question did this study set out to answer?

The aim is to summarize key results of the Triptyque Conceptuel, focusing on the modified relations in inflationary cosmology.

April 4, 2026Open Access

Self-consistent inflationary trajectory in the Triptyque Conceptuel: nₛ=0. 9649, spectral closure, and modified tensor-to-scalar consistency relation

Key Points

The aim is to summarize key results of the Triptyque Conceptuel, focusing on the modified relations in inflationary cosmology.
Analyzed the inflationary background equation of motion with various parameters.
Established the slow-roll attractor's properties and calculated n_s at specific e-folds.
Derived constraints on physical parameters based on Planck measurements.
Found n_s = 0.9649 compatible with Planck measurements.
Determined the unique model parameters from the constraint P_s = 2.1×10⁻⁹.
Outlined a modified tensor-to-scalar relation differing from standard predictions.

Abstract

DESCRIPTION — NOTE V10 This companion note provides a concise self-contained summary of the three central results of Triptyque Conceptuel version 10 (DOI: 10. 5281/zenodo. 19389437). (i) The inflationary background equation of motion σNN + (3−ε) ·σN + V’/H² = 0, where V (σ) = Ω² (σ) ·TQ + m²σ², admits a universal slow-roll attractor independent of the initial condition σ₀. On this attractor, nₛ = 1 − 6εV + 2ηV = 0. 9649 at N = 57. 1 e-folds before the P10 threshold (Λ = 4), compatible with the Planck 2018 measurement (0. 9649 ± 0. 0042). (ii) The constraint Pₛ = 2. 1×10⁻⁹ (Planck 2018) fixes the unique free parameter of the model: mₚhys = 1. 585×10⁻³ MPl and TP10 = 7. 924×10⁻⁴ MPl. The model is thereby reduced to zero free parameters once Pₛ is imposed. (iii) A modified tensor-to-scalar consistency relation is established: r = κₚhys = 3·m²ₚhys = 7. 5×10⁻⁶, differing from the standard slow-roll result r = 16·εV = 0. 140. The Triptyque prediction is compatible with current bounds (r < 0. 034, 2025 datasets), whereas the standard value is not. The modified relation r ≠ −8·nₜ (with nₜ = −0. 0175) constitutes a falsifiable prediction in principle testable by future CMB polarisation experiments. The central open question for version 11 is the formal derivation of the canonical perturbative variable z in the effective geometry gₑff = Ω² (σ) ·η.

Self-consistent inflationary trajectory in the Triptyque Conceptuel: nₛ=0. 9649, spectral closure, and modified tensor-to-scalar consistency relation

Key Points

Abstract

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