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

TL;DR

This paper demonstrates that fractional scalar-field cosmology can reconcile power-law inflation models with observational data by suppressing the tensor-to-scalar ratio while maintaining the scalar tilt, through non-local corrections introduced by fractional calculus.

Contribution

It introduces fractional calculus into scalar-field cosmology, providing a minimal extension that aligns power-law inflation with current observational constraints.

Findings

Fractional order $oldsymbol{eta eq 1}$ suppresses tensor-to-scalar ratio $oldsymbol{r}$.

Achieves $oldsymbol{n_s ext{ around } 0.965}$ and $oldsymbol{r extless 0.04}$ for fractional parameters.

Stable inflationary attractors are identified in the fractional power-law solutions.

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 $\alpha \neq 1$ 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 $\alpha(n,m)$ and recover the standard power-law limit as $\alpha \to 1$. For observationally favored values $\alpha \approx 0.8$-$0.9$ 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.