Inflationary Dynamics and Perturbations in Fractal Cosmology

TL;DR

The paper investigates inflation in a fractal cosmology where the effective spacetime dimension is $D$, modifying the background Friedmann and continuity equations and altering slow-roll dynamics. It develops a fractal Mukhanov–Sasaki formalism with an effective momentum $k_{\text{eff}}$, leading to $D$-, $L$-dependent corrections to the scalar power spectrum and spectral index $n_s$, and analyzes cubic, Starobinsky, and Natural inflation potentials. Planck 2018 data constrain $D$ to approximately $2.7\lesssim D\lesssim 3$, with the fractional scale $L$ required to be near the Planck length to avoid degeneracies; notably, Natural Inflation can accommodate sub-Planckian decay constants for suitable $D$, while Starobinsky-like plateaus lose their unique dominance in this framework. Overall, the work demonstrates that fractal geometry can meaningfully modify inflationary predictions and opens new parameter space for reconciling inflation with observations in a scale-dependent spacetime.

Abstract

We study inflationary dynamics within the framework of fractal cosmology, where spacetime exhibits a non-integer effective dimension, sourced through a relaxation of the cosmological principle. Using Friedmann and continuity equations, modified by an effective fractal dimension $D$; we derive generalized slow-roll parameters and examine their evolution for cubic, Starobinsky and Natural inflationary potentials. We then formulate a fractal extension of the Mukhanov-Sasaki equation by introducing an effective momentum term $k_{\text{eff}}$, arising from the fractal decomposition of the spatial Laplacian, that captures the geometric influence of fractal cosmology on scalar perturbations. This leads to corrections in the power spectrum and a scalar spectral index $n_s$ that depends explicitly on both the fractal dimension $D$ and a fractional scale $L$, which controls the strength of the fractal deformation. Comparison with the Planck 2018 data ($n_s=0.9649\pm 0.0042$) constrains the allowed range of $D$ ($2.7\lesssim D\lesssim3$) depending on the cosmological and inflationary model assumed.

Figures (2)

• Figure 1: The above figure contains two plots. The left plot shows the evolution of $\epsilon_1$ with the number of e-folds, while the right plot shows the evolution of $\epsilon_2$ with the number of e-folds. Here we have chosen a linear potential $V=V_0(1+\frac{\phi}{10})$ and plotted for 4 cases; $D=1,2,3,4$. Here we have chosen the initial value of $\phi=10^{-2}$ and $\dot{\phi}=0$; corresponding to the scales expected for a constant offset monomial potential.
• Figure 2: The above figure contains two plots therein. The left plot shows the evolution of $\epsilon_1$ with the number of e-folds, while the right plot shows the evolution of $\epsilon_2$ with the number of e-folds. Here we have chosen a cubic potential $V=V_0(1+\frac{\phi^3}{10})$ and plotted for 4 cases; $D=1,2,3,4$. Here we have chosen the initial value of $\phi=10^{-2}$ and $\dot{\phi}=0$, corresponding to the scales expected for a constant-offset monomial potential.