[Paper Review] A new parameter in attractor single-field inflation

Jinn-Ouk Gong, Misao Sasaki|arXiv (Cornell University)|Feb 14, 2015

Cosmology and Gravitation Theories16 references3 citations

TL;DR

This paper introduces a new dynamical parameter, $ p $, to generalize slow-roll inflation in single-field attractor models where $ \dot{\phi} = \dot{\phi}(\phi) $, enabling non-slow-roll dynamics consistent with Planck data. The parameter $ p $, which vanishes in canonical models, becomes significant when the inflaton is not slowly rolling, and the authors show that $ \epsilon $, $ \eta $, $ s $, and $ p $ are all required to fully describe such inflationary dynamics, with two examples demonstrating observational viability.

ABSTRACT

We revisit the notion of slow-roll in the context of general single-field inflation. As a generalization of slow-roll dynamics, we consider an inflaton $ϕ$ in an attractor phase where the time derivative of $ϕ$ is determined by a function of $ϕ$, $\dotϕ=\dotϕ(ϕ)$. In other words, we consider the case when the number of $e$-folds $N$ counted backward in time from the end of inflation is solely a function of $ϕ$, $N=N(ϕ)$. In this case, it is found that we need a new independent parameter to properly describe the dynamics of the inflaton field in general, in addition to the standard parameters conventionally denoted by $ε$, $η$, $c_s^2$ and $s$. Two illustrative examples are presented to discuss the non-slow-roll dynamics of the inflaton field consistent with observations.

Motivation & Objective

• To generalize the slow-roll approximation in single-field inflation beyond canonical models by considering attractor dynamics where $ \dot{\phi} = \dot{\phi}(\phi) $.
• To identify a new independent parameter $ p $ that becomes non-negligible in non-slow-roll attractor inflation, distinct from standard parameters $ \epsilon $, $ \eta $, $ c_s^2 $, and $ s $.
• To demonstrate that the standard set of slow-roll parameters is insufficient for describing general attractor inflation, requiring the inclusion of $ p $ for full dynamical characterization.
• To construct and analyze two illustrative models of non-slow-roll attractor inflation that remain consistent with current CMB observations, particularly Planck constraints on $ n_{\cal R} $ and $ f_{\rm NL} $.

Proposed method

• Formulates a general $ P(X,\phi) $ theory of single-field inflation, where $ X = -g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi/2 $, to describe non-canonical kinetic terms.
• Defines the new parameter $ p $ as a measure of non-slow-roll dynamics, derived from the second-order term in the $ \delta N $ formalism, which vanishes in canonical models but can be of order unity in general attractor models.
• Uses the $ \delta N $ formalism to relate the curvature perturbation $ \cal R $ to the field fluctuation $ \delta\phi $, expanding $ \cal R $ in powers of $ \delta\phi $ to extract the parameter $ p $.
• Derives the spectral index $ n_{\cal R} - 1 = -2\epsilon - \eta - s $ and its running $ \alpha_{\cal R} $, showing that $ p $ modifies the evolution of $ \cal R $ beyond the standard slow-roll framework.
• Analyzes the intrinsic non-Gaussianity of $ \cal R $ via the cubic-order action of $ \delta\phi $, deriving the bispectrum and finding $ f_{\rm NL} = \frac{5}{6}\epsilon $, which contributes to the full consistency relation.
• Constructs two explicit models of attractor inflation where $ \dot{\phi} = \dot{\phi}(\phi) $, and shows that they satisfy observational constraints on $ n_{\cal R} \approx 0.968 $ and $ f_{\rm NL} \ll 1 $, even when $ \epsilon $, $ \eta $, or $ s $ are not small.

Experimental results

Research questions

• RQ1 Can the standard slow-roll parameters $ \epsilon $, $ \eta $, $ s $, and $ c_s^2 $ fully describe inflationary dynamics in non-canonical, attractor single-field models?
• RQ2 What new parameter is required to describe non-slow-roll dynamics in attractor inflation, and how does it differ from canonical slow-roll behavior?
• RQ3 Can non-slow-roll attractor inflation models still be consistent with Planck observations of the nearly scale-invariant power spectrum and low primordial non-Gaussianity?
• RQ4 How does the $ \delta N $ formalism need to be extended to include the new parameter $ p $, and what is its physical interpretation in terms of field evolution?
• RQ5 What is the role of intrinsic non-Gaussianity in the full consistency relation for $ f_{\rm NL} $, and how does it relate to the new parameter $ p $?

Key findings

• A new parameter $ p $ is introduced that characterizes non-slow-roll dynamics in attractor single-field inflation, which vanishes in canonical models but can be of order unity when the inflaton is not slowly rolling.
• The parameter $ p $ is derived from the second-order term in the $ \delta N $ expansion of the curvature perturbation $ \cal R $, and is related to the ratio of $ \ddot{\phi} $ to $ H\dot{\phi} $.
• The standard set of slow-roll parameters $ \epsilon $, $ \eta $, $ s $, and $ c_s^2 $ is insufficient to describe general attractor inflation; $ p $ must be included as an independent parameter.
• Two explicit models of non-slow-roll attractor inflation are constructed that satisfy Planck constraints: $ n_{\cal R} = 0.968 \pm 0.006 $ and low $ f_{\rm NL} $, demonstrating the viability of such models.
• The intrinsic non-Gaussianity of $ \cal R $ contributes $ f_{\rm NL} = \frac{5}{6}\epsilon $, which is half of the full consistency relation, with the other half coming from super-horizon evolution captured by the $ \delta N $ formalism.
• The paper shows that the moment of horizon crossing is not necessarily the key instant in general attractor inflation, as curvature perturbations can continue evolving on super-horizon scales.