Inflation and Primordial Perturbations in Fractal Cosmology

TL;DR

This paper explores how fractal space dimensions affect inflationary cosmology, modifying key equations and observational predictions, and constrains the fractal dimension using Planck data.

Contribution

It introduces a fractal extension of inflationary dynamics, deriving modified equations and spectra, and shows how fractality influences inflation models and observational constraints.

Findings

Fractal effects suppress slow roll parameters for D<3.

Corrections to the scalar power spectrum depend on D and length scale L.

Fractal corrections allow Natural Inflation with sub-Planckian axion decay constants.

Abstract

We study inflationary dynamics within the framework of fractal cosmology, where space is characterized by an effective non-integer dimension $D$. In our work, fractal effects are sourced through thermodynamic modifications at the cosmological horizon. Using the modified Friedmann and continuity equations, we then derive the modified slow roll parameter and their evolution for linear, cubic, Starobinsky ($R+R^2$) and Natural inflationary potentials, showing that the slow roll parameters get suppressed for $D<3$. We further derive a fractal extension of the Mukhanov-Sasaki equation by introducing an effective momentum $k_{\text{eff}}$, which captures the modification of spatial Laplacian due to fractality. This leads to explicit corrections to the scalar power spectrum and the spectral index $n_s$, depending on both $D$ and a fractional length scale $L$. Confrontation with Planck 2018 data constrains the effective dimension to a best-fit range of $2.7\lesssim D \lesssim 3$ for the Starobinsky model. Furthermore, in the case of Natural Inflation, fractal corrections relax the usual requirement of super-Planckian axion decay constants, opening a phenomenologically viable parameter space inaccessible in the standard $3+1$ dimensional cosmology.