[Paper Review] Is power-law inflation really attractive?

César A. Terrero–Escalante|arXiv (Cornell University)|Apr 3, 2002

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper challenges the assumption that power-law inflation is a generic attractor in inflationary dynamics, showing that higher-order corrections in the spectral index calculations break the attractor behavior. Using next-to-next-to-leading-order expressions for scalar and tensor spectral indices, it demonstrates that power-law inflation is not universally favored, undermining its perceived theoretical robustness for cosmological observables.

ABSTRACT

It is argued that the order of the analytic expressions for the calculation of the primordial perturbations from inflation exerts a strong influence upon the results of the analysis of observables dynamics based on these expressions and, therefore, upon some predictions sometimes taken for granted as generic for the inflationary scenario.

Motivation & Objective

• To assess whether power-law inflation is a generic attractor in inflationary dynamics, challenging its assumed theoretical dominance.
• To investigate how the order of analytic approximations affects predictions of primordial perturbation spectra.
• To evaluate the reliability of inflaton potential reconstruction programs based on spectral index calculations.
• To determine whether power-law inflation remains a robust attractor when higher-order corrections in horizon flow functions are included.
• To analyze the dynamical behavior of the slow-roll parameter ε₁ beyond leading-order approximations using full next-to-next-to-leading-order equations.

Proposed method

• Derives next-to-next-to-leading-order expressions for scalar (Δ) and tensor (δ) spectral indices using the horizon flow formalism.
• Applies the horizon flow functions εₘ, with ε₁ = dln d_H / dN, to model inflationary dynamics beyond leading order.
• Uses the full dynamical system of differential equations (15) and (16) for ε₁, incorporating third-order derivatives and nonlinear terms.
• Analyzes the reduced phase space dynamics by fixing Δ or δ as constants to study fixed points and stability (e.g., saddle points).
• Evaluates the eigenvalues of the Jacobian matrix in the reduced phase space to determine the existence and nature of fixed points.
• Compares the results with previous works (e.g., Ayon-Beato et al.) to assess the validity of attractor behavior in power-law inflation.

Experimental results

Research questions

• RQ1 Is power-law inflation truly an attractor in inflationary dynamics when higher-order corrections are included?
• RQ2 How does the order of analytic approximation affect the predicted spectral indices and their scale dependence?
• RQ3 What is the dynamical behavior of the slow-roll parameter ε₁ beyond leading-order approximations?
• RQ4 Do the next-to-next-to-leading-order corrections in the spectral index expressions preserve the attractor nature of power-law inflation?
• RQ5 Can the assumption of power-law inflation as a generic outcome be justified in models with non-slow-roll potentials?

Key findings

• Power-law inflation is not a generic attractor when next-to-next-to-leading-order corrections are included in the spectral index calculations.
• The dynamical system for ε₁ at next-to-next-to-leading order exhibits a saddle point in the phase space for δ < 0, indicating no universal attractor behavior.
• The eigenvalues of the Jacobian matrix show opposite signs for δ < 0, confirming the presence of a saddle point, which implies unstable dynamics for most trajectories.
• The attractor-like behavior of power-law inflation is only observed for trajectories very close to unstable separatrices, limiting its generality.
• Higher-order corrections break the constraint that previously forced dynamics to remain close to power-law inflation, reducing its theoretical appeal.
• The results imply that claims about generic predictions in inflation must account for the order of approximation used in spectral index calculations.