[Paper Review] Inflationary scenarios in Starobinsky model with higher order corrections

Michał Artymowski, Zygmunt Lalak+1 more|arXiv (Cornell University)|Feb 4, 2015

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates higher-order corrections to the Starobinsky inflation model via two parameters, $\lambda_1$ and $\lambda_2$, in the Einstein frame, revealing a potential with a plateau, steep slope, and possible local minimum or maximum. It identifies three inflationary regimes—plateau, topological (at local maximum), and saddle-point inflation—and shows that $\lambda_1 \gtrsim 10^{-3}$ or $\lambda_2 \gtrsim 10^{-4}$ suppresses eternal inflation, while the false vacuum at $\phi_{\text{min}}$ is stable against quantum tunnelling and thermal effects.

ABSTRACT

We consider the Starobinsky inflation with a set of higher order corrections parametrised by two real coefficients $\\lambda_1, \\lambda_2$. In the Einstein frame we have found a potential with the Starobinsky plateau, steep slope and possibly with an additional minimum, local maximum or a saddle point. We have identified three types of inflationary behaviour that may be generated in this model: i) inflation on the plateau, ii) at the local maximum (topological inflation), iii) at the saddle point. We have found limits on parameters $\\lambda_i$ and initial conditions at the Planck scale which enable successful inflation and disable eternal inflation at the plateau. We have checked that the local minimum away from the GR vacuum is stable and that the field cannot leave it neither via quantum tunnelling nor via thermal corrections.

Motivation & Objective

• To analyze the impact of higher-order corrections ($\lambda_1$, $\lambda_2$) on the Starobinsky inflation model in the Einstein frame.
• To determine whether the modified potential supports successful inflation without eternal inflation, particularly on the plateau.
• To assess the stability of a false vacuum at $\phi_{\text{min}}$ against quantum tunnelling and thermal corrections.
• To identify and characterize distinct inflationary scenarios: plateau, topological (at local maximum), and saddle-point inflation.
• To establish parameter and initial condition constraints ensuring $N \geq 60$ e-folds and avoiding eternal inflation.

Proposed method

• Transform the Jordan frame action with $f(R) = R + R^2/(6M^2)$ to the Einstein frame using conformal transformation, deriving a modified potential $U(\phi) = U_S(1 + \lambda_1 U_S/M_p^4 + \lambda_2 U_S^2/M_p^8)$.
• Use numerical integration to solve the inflaton equation of motion and Friedmann equations under slow-roll approximation for various $\lambda_1$, $\lambda_2$.
• Apply the Coleman-De Luccia (CDL) instanton method to compute quantum tunnelling rates from $\phi_{\text{min}}$ to the GR vacuum, using Euclidean action $S_{\text{CDL}} = 4\pi^2 \int d\tau \left( r^3 V(\phi) - 3r \right)$.
• Model thermal corrections to the potential at finite temperature $T$, showing they do not destabilize $\phi_{\text{min}}$ even at $T \gg M_p$.
• Use the undershoot/overshoot method to numerically determine initial field values $\phi_0$ that lead to classical evolution ending at $\phi_{\text{min}}$.
• Identify attractor solutions in phase space separating plateau inflation from minimum-trapping evolution, with a critical separatrix at $\phi_{\text{max}}$ or $\phi_{\text{saddle}}$.

Experimental results

Research questions

• RQ1 Can higher-order corrections $\lambda_1$, $\lambda_2$ in the Starobinsky model generate a viable inflationary scenario without eternal inflation?
• RQ2 What are the conditions on $\lambda_1$, $\lambda_2$, and initial field values that allow $N \geq 60$ e-folds on the plateau or at a local maximum?
• RQ3 Is the false vacuum at $\phi_{\text{min}}$ stable against quantum tunnelling via Coleman-De Luccia or Hawking-Moss mechanisms?
• RQ4 How do finite-temperature corrections affect the stability of $\phi_{\text{min}}$ and the possibility of thermal escape?
• RQ5 Can the model support topological inflation at a local maximum or saddle point, and what are the constraints on $\lambda_1$, $\lambda_2$ for such behavior?

Key findings

• The Einstein frame potential exhibits a plateau, a steep slope, and may contain a local minimum at $\phi_{\text{min}}$, a local maximum at $\phi_{\text{max}}$, or a saddle point depending on $\lambda_1$, $\lambda_2$.
• For $\lambda_1 \gtrsim 10^{-3}$ or $\lambda_2 \gtrsim 10^{-4}$, the steep slope suppresses eternal inflation on the plateau, enabling $N \geq 60$ e-folds.
• The maximum number of e-folds $N_{\text{max}}$ occurs at the saddle point, with $N_{\text{max}}$ peaking for specific $\lambda_1$, $\lambda_2$ values.
• Thermal corrections do not destabilize $\phi_{\text{min}}$; the minimum deepens at finite $T$, and the field cannot be thermally excited to the GR vacuum.
• Quantum tunnelling from $\phi_{\text{min}}$ to the GR vacuum is suppressed: only the Hawking-Moss mechanism is relevant for $N_{\text{max}} \geq 60$, yielding lifetimes much longer than the age of the Universe.
• For $\lambda_1 \gg 10^6$, which is inconsistent with inflation, tunnelling via CDL instantons becomes possible but still results in a vacuum lifetime vastly exceeding the age of the Universe.