[Paper Review] Starobinsky Inflation in the Swampland

Dieter Lüst, Joaquin Masias+2 more|arXiv (Cornell University)|Dec 20, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper argues that the Starobinsky inflation model, which includes an $R^2$ term in the gravitational action, likely originates from quantum corrections due to a tower of light species, implying the $R^2$ scale $M$ must be of order the species scale $\Lambda_{\text{s}}$. This identification places the model at the edge of effective field theory validity and provides strong evidence that Starobinsky inflation lies within the Swampland, challenging its UV consistency in quantum gravity.

ABSTRACT

We argue that the Starobinsky model of inflation, realised via an $R^2$ term in the Lagrangian, can originate from quantum effects due to a tower of light species. By means of two separate arguments, we show how this implies that the scale of the $R^2$ term must be of order of the species scale $Λ_s$, namely the energy at which gravity becomes strongly coupled. We discuss the implications and challenges of this scenario for inflation, inflationary reheating, and string theory embeddings. In this context, we collect strong evidence to conclude that Starobinsky inflation lies in the Swampland.

Motivation & Objective

• To assess whether the Starobinsky inflation model, with its $R^2$ term, can be consistently embedded in quantum gravity.
• To investigate whether the $R^2$ term can emerge from quantum corrections due to a tower of light species.
• To determine the relationship between the $R^2$ scale $M$ and the species scale $\Lambda_{\text{s}}$ in the context of effective field theory and quantum gravity.
• To evaluate cosmological and reheating implications of $M \sim \Lambda_{\text{s}}$ for inflationary observables and string theory embeddings.
• To provide evidence that Starobinsky inflation lies within the Swampland, based on consistency with quantum gravity constraints.

Proposed method

• Using a cosmological argument based on the quasi-de Sitter phase to relate the $R^2$ scale $M$ to the species scale $\Lambda_{\text{s}}$.
• Performing explicit graviton propagator calculations in the presence of a tower of light species and $R+R^2$ terms to derive the $M \sim \Lambda_{\text{s}}$ relation.
• Analyzing the effective field theory behavior of the graviton propagator in $R + \mathcal{O}(R^2)$ theories to fix the index structure and renormalization scale.
• Applying the species scale formula $\Lambda_{\text{s}} = M_P / N^{1/(d-2)}$ in $d=4$ dimensions to estimate the required number of light species.
• Evaluating the impact of scalar field-dependent $\Lambda_{\text{s}}$ on inflationary observables, particularly the spectral tilt.
• Examining string compactifications (Type IIB and heterotic) to derive the $R^2$ term scale $M$ and compare it with $\Lambda_{\text{s}}$.

Experimental results

Research questions

• RQ1 Can the $R^2$ term in the Starobinsky model arise from quantum corrections due to a tower of light species?
• RQ2 What is the precise relationship between the $R^2$ scale $M$ and the species scale $\Lambda_{\text{s}}$?
• RQ3 Does the condition $M \sim \Lambda_{\text{s}}$ lead to inconsistencies in the effective field theory description of inflation?
• RQ4 How does a scalar field-dependent $\Lambda_{\text{s}}$ affect the spectral tilt of primordial curvature perturbations?
• RQ5 Can string theory compactifications naturally produce the $R^2$ term with $M \sim \Lambda_{\text{s}}$, and what does this imply for the Swampland program?

Key findings

• The $R^2$ scale $M$ in the Starobinsky model is identified with the species scale $\Lambda_{\text{s}}$ up to an order-one factor, i.e., $M \simeq \Lambda_{\text{s}}$, based on cosmological and perturbative arguments.
• Inflationary Hubble scale satisfies $H \simeq \Lambda_{\text{s}}$, placing the model at the boundary of validity for weakly coupled effective field theories.
• The required number of light species $N \simeq 10^{10}$ to realize $M \simeq 10^{14}\,\text{GeV}$ implies a potential inconsistency in the 4D effective description.
• To match current CMB constraints on the spectral tilt, the species scale must vary very weakly with the inflaton field, satisfying $-0.004 \leq \Lambda_{\text{s}}' / \Lambda_{\text{s}} \leq 0.001$.
• In Type IIB string compactifications, the $R^2$ scale is found to be $M \sim g_s M_{P,4} / \mathcal{V}_s^{1/6}$, which can be matched to $\Lambda_{\text{s}}$ under specific stabilization conditions.
• In heterotic string compactifications, the $R^2$ scale is $M \sim M_s$, the string scale, which is consistent with $M \sim \Lambda_{\text{s}}$ only if the number of species is tuned accordingly.