[Paper Review] Scale independent $R^2$ inflation

Pedro G. Ferreira, Christopher T. Hill+2 more|Oxford University Research Archive (ORA) (University of Oxford)|Jun 8, 2019

Economic theories and models4 citations

TL;DR

This paper proposes a scale-invariant $R^2$ gravity model that generates mass scales via spontaneous Weyl symmetry breaking, leading to single-field inflation with predictions nearly identical to standard $R^2$ inflation. It derives analytic expressions for inflationary observables and predicts a tensor-to-scalar ratio of $0.0026 < r < 0.0033$, testable by upcoming CMB experiments.

ABSTRACT

Weyl (scale) invariant theories of scalars and gravity can generate all mass scales spontaneously. In this paper we study a particularly simple version -- scale invariant $R^2$ gravity -- and show that, during an inflationary period, it leads to fluctuations which, for a particular parameter choice, are almost indistinguishable from normal $R^2$ inflation. Current observations place tight constraints on the primary coupling constant of this theory and predict a tensor to scalar ratio, $0.0033 &gt; r &gt; 0.0026$, which is testable with the next generation of cosmic microwave background experiments.

Motivation & Objective

• To construct a fully Weyl-invariant $R^2$ gravity model that generates all mass scales spontaneously through spontaneous symmetry breaking.
• To show that this model leads to single-field inflation with observables nearly indistinguishable from standard $R^2$ inflation under specific parameter choices.
• To derive analytic expressions for inflationary observables—scalar spectral index $n_s$, tensor-to-scalar ratio $r$, and running—using a frame-independent approach.
• To demonstrate that the dilaton and Weyl photon decouple from the inflaton sector, avoiding fifth-force constraints and ensuring consistency with observations.

Proposed method

• Formulates a globally Weyl-invariant action with a Jordan-Brans-Dicke scalar field $\phi$ and an $R^2$ term, preserving scale invariance at the Lagrangian level.
• Applies the frame-independent formalism to compute inflationary observables directly in the Jordan frame, where scale invariance is manifest.
• Identifies the dilaton as the Goldstone boson of spontaneously broken Weyl symmetry and shows it decouples from the inflaton sector due to inertial spontaneous breaking.
• Demonstrates that the massive Weyl photon mode also decouples in the locally Weyl-invariant version, leading to a single scalar inflaton effective theory.
• Uses canonical normalization and field redefinitions to map the two-scalar system to an effective single-field scalar-tensor theory with frame-invariant quantities.
• Derives analytic expressions for $n_s$, $r$, and $\alpha_s$ by solving the slow-roll conditions in the limit of large $\alpha_2$, corresponding to the $R^2$-dominated regime.

Experimental results

Research questions

• RQ1 Can a fully scale-invariant $R^2$ gravity model generate a viable period of inflation without introducing explicit mass scales?
• RQ2 How does spontaneous Weyl symmetry breaking affect the dynamics of the inflaton and the decoupling of auxiliary fields like the dilaton and Weyl photon?
• RQ3 What are the analytic predictions for inflationary observables—specifically $n_s$, $r$, and $\alpha_s$—in this scale-invariant $R^2$ model?
• RQ4 How do the predictions of this model compare quantitatively to the original Starobinsky $R+R^2$ model and current CMB constraints?
• RQ5 Can the model avoid fifth-force constraints by ensuring the dilaton decouples from the inflaton sector?

Key findings

• The model predicts a tensor-to-scalar ratio in the range $0.0026 < r < 0.0033$, which is within reach of next-generation CMB experiments.
• For a specific parameter choice, the inflationary predictions of the scale-invariant $R^2$ model are nearly indistinguishable from those of the original Starobinsky $R+R^2$ model.
• The dilaton and Weyl photon modes decouple from the inflaton sector due to inertial spontaneous Weyl symmetry breaking, avoiding fifth-force constraints.
• The effective single-field description of inflation is derived via field redefinition, with the frame-invariant potential $\hat{U} = \hat{W}/\hat{f}^2$ and field-space metric $\hat{G} = 12\bar{K}/(\hat{\phi}^2 - 2\bar{K})^2$.
• The analytic expressions for $n_s$, $r$, and $\alpha_s$ are derived and shown to match the $|\alpha_2| \to \infty$ limit of a two-scalar model, confirming consistency.
• The model is consistent with current CMB observations and offers a precise, testable prediction for $r$ with a forecasted uncertainty of $\sigma(r) \approx 0.0005$.