[Paper Review] Non-canonical Conformal Attractors for Single Field Inflation

Tony Pinhero, Supratik Pal|arXiv (Cornell University)|Mar 21, 2017

Cosmology and Gravitation Theories4 citations

TL;DR

This paper introduces a new class of non-canonical conformal attractors in single-field inflation by extending conformal invariance to non-canonical kinetic terms via two approaches: $χ$-supergravity and superconformal theory. It derives a unified Lagrangian that reproduces canonical conformal attractors in the limit of canonical kinetic terms and shows that the model predicts $1-n_s = 2/N_e$ and $r = 12/N_e^2$ in the leading-order approximation, consistent with Planck and BICEP/Keck data.

ABSTRACT

We extend the idea of conformal attractors in inflation to non-canonical sectors by developing a non-canonical conformally invariant theory from two different approaches. In the first approach, namely, ${\cal N}=1$ supergravity, the construction is more or less phenomenological, where the non-canonical kinetic sector is derived from a particular form of the Kähler potential respecting shift symmetry. In the second approach i.e., superconformal theory, we derive the form of the Lagrangian from a superconformal action and it turns out to be exactly of the same form as in the first approach. Conformal breaking of these theories results in a new class of non-canonical models which can govern inflation with modulated shape of the T-models. We further employ this framework to explore inflationary phenomenology with a representative example and show how the form of the Kähler potential can possibly be constrained in non-canonical models using the latest confidence contour in the $n_s-r$ plane given by recent Planck and BICEP/Keck results.

Motivation & Objective

• To extend the conformal attractor mechanism to non-canonical kinetic sectors in single-field inflation.
• To develop a superconformal framework for non-canonical models that preserves conformal invariance before spontaneous breaking.
• To constrain the Kähler potential in non-canonical models using latest Planck 2018 and BICEP/Keck $n_s$-$r$ confidence contours.
• To demonstrate that canonical conformal attractors emerge as a limiting case of the proposed non-canonical framework.
• To explore inflationary phenomenology in non-canonical models with scale-dependent power spectra, relevant to recent observational evidence of spectral tilt.

Proposed method

• Construct a non-canonical conformally invariant theory in $χ$-supergravity using a shift-symmetric Kähler potential and phenomenological superpotential.
• Derive the same Lagrangian from a superconformal action, confirming consistency between the two approaches.
• Perform a conformal transformation to the Einstein frame, yielding a non-canonical kinetic term and a potential depending on a hyperbolic function of the canonically normalized inflaton field.
• Use the resulting effective action to compute slow-roll parameters and inflationary observables such as $n_s$ and $r$ in terms of $N_e$, the number of e-foldings.
• Apply the model to a representative example with a potential $V(\psi) \propto \tanh^{2n'}(\cdots)$, and analyze its behavior in the large-$\psi$ limit.
• Compare predictions with Planck 2018 and BICEP/Keck 2020 data, showing consistency with the $n_s$-$r$ confidence contours.

Experimental results

Research questions

• RQ1 Can the conformal attractor mechanism be generalized to non-canonical kinetic terms in single-field inflation?
• RQ2 Does a superconformal realization of non-canonical conformal invariance yield the same effective Lagrangian as the phenomenological $χ$-supergravity approach?
• RQ3 How do non-canonical kinetic terms affect the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$ in the attractor limit?
• RQ4 Can the model reproduce canonical conformal attractors in a specific limit of the parameters?
• RQ5 To what extent can the Kähler potential be constrained by current $n_s$-$r$ observational data in non-canonical models?

Key findings

• The non-canonical conformal attractor model yields the same leading-order attractor predictions: $1 - n_s = 2/N_e$ and $r = 12/N_e^2$, matching canonical conformal attractors in the limit of canonical kinetic terms.
• The effective Lagrangian in the Einstein frame features a non-canonical kinetic term proportional to $\left[1 + \beta(1 + 4e^{-\sqrt{2/3}\psi})\right]\partial_\mu\psi\partial^\mu\psi$, with $\beta$ encoding non-canonical corrections.
• The slow-roll parameter is found to be $\epsilon = \frac{3\alpha}{4N_e^2}$, consistent with the canonical attractor result, confirming the robustness of the attractor behavior.
• The model's predictions for $n_s$ and $r$ are compatible with the 95% confidence contour from Planck 2018 and BICEP/Keck 2020 data, validating its phenomenological viability.
• In the large-field limit ($\psi \gg 1$), the potential approaches a constant, and the kinetic term asymptotes to a form that supports slow-roll inflation with $\epsilon \propto 1/N_e^2$, confirming the attractor nature.
• The framework allows for scale-dependent power spectra, making it particularly relevant in light of Planck 2015 and 2018 results indicating a 5-$\sigma$ scale dependence in the primordial power spectrum.