[Paper Review] On the tensor-to-scalar ratio in large single-field inflation models

Gabriela Barenboim, Wan-Il Park|arXiv (Cornell University)|Sep 23, 2015

Cosmology and Gravitation Theories3 citations

TL;DR

This paper demonstrates that in large single-field inflation models—where the inflaton field traverses a large fraction of the Planck scale—the tensor-to-scalar ratio $ r_T $ is generically bounded below by $ \mathcal{O}(10^{-3}) $ for spectral indices $ n_s $ consistent with Planck observations. Using slow-roll approximations and $ e $-folding constraints, the authors show this lower bound arises from the interplay between the required number of $ e $-foldings and the field excursion, making $ r_T \gtrsim 10^{-3} $ a robust prediction for realistic large-field models.

ABSTRACT

We show that generically the tensor-to-scalar ratio in large single-field inflation scenarios is bounded to be larger than $\mathcal{O}(10^{-3})$ for the spectral index in the range favored by observations.

Motivation & Objective

• To determine whether large single-field inflation models generically predict a lower bound on the tensor-to-scalar ratio $ r_T $, given observational constraints on the scalar spectral index $ n_s $.
• To investigate how the number of $ e $-foldings required to solve the horizon problem affects the lower bound of $ r_T $ in large-field inflation scenarios.
• To assess whether realistic models—such as chaotic, Hilltop, inverse-Hilltop, and Starobinsky-like potentials—can evade the $ r_T \gtrsim 10^{-3} $ bound when $ \mu \sim M_{\rm P} $ or larger.
• To evaluate the impact of reheating temperature and additional $ e $-folding contributions (e.g., from thermal inflation) on the lower bound of $ r_T $.

Proposed method

• Used slow-roll inflation formalism with standard definitions of slow-roll parameters $ \epsilon $, $ \eta $, and $ \xi^2 $, derived from the inflaton potential $ V(\phi) $.
• Applied the $ e $-folding formula $ N_{e,*}^{\rm th} \simeq -\frac{1}{M_{\rm P}^2} \int_{\phi_*}^{\phi_e} \frac{V}{V'} d\phi $ to relate the number of $ e $-foldings to the inflaton field excursion.
• Evaluated $ r_T = 16\epsilon $ using the slow-roll approximation, with $ \epsilon $ determined by the potential's shape and field value at horizon exit.
• Analyzed specific potential forms: chaotic monomial $ V \propto \phi^p $, Hilltop $ V \propto (1 - \phi^p/2)^2 $, inverse-Hilltop $ V \propto (1 - \phi^{-p}/2)^2 $, and Starobinsky-like $ V \propto (1 - e^{-\phi/\mu})^2 $.
• Imposed observational constraints: $ n_s \approx 0.96 $ from Planck (TT+lowP), $ r_T < 0.1 $, and $ V_*^{1/4} \lesssim 2 \times 10^{16}~\text{GeV} $, to fix $ \phi_* $ and $ V_* $.
• Performed numerical analysis across different $ p $-values and $ \mu $-scales, including $ \mu \sim M_{\rm P} $ and $ \mu > M_{\rm P} $, to determine viable parameter space.

Experimental results

Research questions

• RQ1 What is the minimum value of the tensor-to-scalar ratio $ r_T $ that can be achieved in large single-field inflation models with $ \mu \sim M_{\rm P} $ and $ p \leq 4 $, consistent with Planck $ n_s $ observations?
• RQ2 How do additional $ e $-foldings from thermal inflation or low reheating temperature affect the lower bound on $ r_T $?
• RQ3 Can models like inverse-Hilltop or Hilltop potentials with $ p \gg 4 $ evade the $ \mathcal{O}(10^{-3}) $ lower bound on $ r_T $, and what are the implications for field excursion?
• RQ4 To what extent does the choice of $ \mu $—the scale characterizing the end of inflation—affect the lower bound on $ r_T $ in realistic large-field models?
• RQ5 Is the $ \mathcal{O}(10^{-3}) $ lower bound on $ r_T $ robust across different potential forms (e.g., chaotic, Starobinsky-like) when $ e $-folding and spectral index constraints are applied?

Key findings

• For large single-field inflation models with $ \mu \sim M_{\rm P} $ and $ p \leq 4 $, the tensor-to-scalar ratio is generically bounded below by $ r_T \gtrsim \mathcal{O}(10^{-3}) $, given the observed spectral index $ n_s \approx 0.96 $.
• When $ \mu > M_{\rm P} $, the lower bound on $ r_T $ increases to $ \mathcal{O}(10^{-3}) $, even for $ p \leq 4 $, due to the required field excursion and $ e $-folding constraints.
• For $ p \gg 4 $, particularly in inverse-Hilltop and Hilltop potentials, $ r_T $ can be significantly reduced, but only if the field is far from the end of inflation, which may conflict with physical realizability.
• The inclusion of extra $ e $-foldings (e.g., from thermal inflation) or low reheating temperatures pushes the lower bound on $ r_T $ upward, reinforcing the $ \mathcal{O}(10^{-3}) $ bound.
• Starobinsky-like potentials with $ \mu = M_{\rm P} $ predict $ r_T \sim 10^{-3} $, consistent with the lower bound, and are within the 1-σ observational band.
• Models with $ p \leq 4 $ and $ \mu \sim M_{\rm P} $ either yield $ r_T \gtrsim \mathcal{O}(10^{-3}) $ or push $ n_s $ beyond the 3-σ observational bounds, indicating a fundamental lower limit in realistic scenarios.