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Power-Law Inflation in n-Dimensional Fractional Scalar Field Cosmology: Observational Constraints and Dynamical Analysis

Daniel Oliveira, Seyed Rasouli, Joao Marto, Paulo Moniz

TL;DR

This work develops power-law inflation within an $n$-dimensional fractional scalar-field cosmology by introducing a fractional order $\alpha$ in the minisuperspace action, which induces memory terms that modify the background and perturbation dynamics. The authors derive an explicit mapping $\alpha(n,m)$ that relates the fractional parameter to the dimensionality and the inflationary exponent, reproduce the standard limit as $\alpha\to1$, and show that for $\alpha\approx0.8$–$0.9$ in four dimensions the model yields $n_s\approx0.965$ and $r\lesssim0.04$, aligning with Planck/BICEP constraints. A self-consistent exponential potential emerges from the dynamics, and a dynamical-systems analysis reveals the fractional power-law solutions are stable inflationary attractors within viable parameter ranges. The framework decouples the scalar tilt from the tensor amplitude through memory effects, providing a predictive, testable extension of canonical inflation with concrete targets for upcoming CMB polarization measurements, while noting the need for a graceful exit mechanism and reheating within an extended or running-$\alpha$ scenario.

Abstract

Power-law inflation with $a(t) \propto t^m$ is conceptually simple and predicts a scalar tilt $n_s = 1 - 2/m$ compatible with CMB data, but in four-dimensional Einstein gravity it typically yields a tensor-to-scalar ratio $r = 16/m$ that is too large to satisfy current bounds. We show that a minimal extension based on fractional scalar-field cosmology resolves this tension. Introducing a fractional order $α\neq 1$ generates non-local (memory) corrections in the Friedmann and Klein-Gordon dynamics that suppress $r$ while keeping $n_s$ essentially unchanged. We derive an explicit mapping $α(n,m)$ and recover the standard power-law limit as $α\to 1$. For observationally favored values $α\approx 0.8$-$0.9$ in four dimensions we obtain $n_s \approx 0.965$ and $r \lesssim 0.04$, bringing power-law inflation into agreement with data. The scalar potential follows self-consistently as an exponential, and a dynamical-systems analysis shows the fractional power-law solutions form stable inflationary attractors over the viable parameter range. These results establish fractional power-law inflation as a predictive and testable framework, with clear targets for forthcoming CMB polarization measurements.

Power-Law Inflation in n-Dimensional Fractional Scalar Field Cosmology: Observational Constraints and Dynamical Analysis

TL;DR

This work develops power-law inflation within an -dimensional fractional scalar-field cosmology by introducing a fractional order in the minisuperspace action, which induces memory terms that modify the background and perturbation dynamics. The authors derive an explicit mapping that relates the fractional parameter to the dimensionality and the inflationary exponent, reproduce the standard limit as , and show that for – in four dimensions the model yields and , aligning with Planck/BICEP constraints. A self-consistent exponential potential emerges from the dynamics, and a dynamical-systems analysis reveals the fractional power-law solutions are stable inflationary attractors within viable parameter ranges. The framework decouples the scalar tilt from the tensor amplitude through memory effects, providing a predictive, testable extension of canonical inflation with concrete targets for upcoming CMB polarization measurements, while noting the need for a graceful exit mechanism and reheating within an extended or running- scenario.

Abstract

Power-law inflation with is conceptually simple and predicts a scalar tilt compatible with CMB data, but in four-dimensional Einstein gravity it typically yields a tensor-to-scalar ratio that is too large to satisfy current bounds. We show that a minimal extension based on fractional scalar-field cosmology resolves this tension. Introducing a fractional order generates non-local (memory) corrections in the Friedmann and Klein-Gordon dynamics that suppress while keeping essentially unchanged. We derive an explicit mapping and recover the standard power-law limit as . For observationally favored values - in four dimensions we obtain and , bringing power-law inflation into agreement with data. The scalar potential follows self-consistently as an exponential, and a dynamical-systems analysis shows the fractional power-law solutions form stable inflationary attractors over the viable parameter range. These results establish fractional power-law inflation as a predictive and testable framework, with clear targets for forthcoming CMB polarization measurements.
Paper Structure (59 sections, 87 equations, 2 figures)

This paper contains 59 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fractional cosmological framework in $n$ dimensions
  3. Metric and Fractional Action
  4. Field Equations and Dynamical Consistency
  5. Conservation Laws and Non-Locality
  6. Thermodynamics of Horizons
  7. Standard Limit and Physical Interpretation
  8. The Standard Power-Law Inflation Model: Benchmark and Observational Tension
  9. Power-law ansatz and field equations
  10. Self-consistent solution
  11. Slow-roll parameters and observables
  12. Confrontation with observations
  13. Physical origin of the incompatibility
  14. Strategy for resolution
  15. Power Law Inflationary Solutions
  16. ...and 44 more sections

Figures (2)

  • Figure 1: The $(n_s,r)$ plane for fractional power-law inflation with $\alpha\in(0,1]$, compared with Planck 2018 constraints (green shaded regions). Markers represent a discrete sampling of the inflationary exponent $m$.
  • Figure 2: Local phase portrait around the fractional power-law fixed point in the physical (constrained) phase space $(x,u)$ for $n=4$ and $\alpha=0.9$. The variable $y$ is reconstructed from the constraint via $y^2=1+(1-\alpha)u-x^2$.