Slow-roll Inflation with the Gauss-Bonnet and Chern-Simons Corrections

TL;DR

The paper develops a slow-roll inflation framework in the presence of Gauss-Bonnet and Chern-Simons corrections, deriving general expressions for $n_S$, $n_T$, $r$, and the tensor polarization $\Pi$. It shows that Gauss-Bonnet interactions violate the standard single-field consistency relation $r=-8n_T$, allowing blue or scale-invariant tensor spectra when the GB coupling derivative $\xi_{,\phi}$ is large (around $10^8/M_{ m Pl}$), and that the Chern-Simons term induces parity-violating circular polarization in gravitational waves. The authors reconstruct the potential and couplings from observables and present analytic and numeric examples illustrating viable blue and scale-invariant tensor spectra, highlighting how future observations could fix or constrain these couplings at CMB scales. They also discuss the Lyth bound within this modified gravity context, the possibility of inflation with steep potentials, and the observational implications of tensor polarization for testing high-energy gravity theories. Overall, the work provides a bridge between high-curvature corrections in gravity and testable cosmological signatures, guiding future data-driven constraints on quantum-gravity-inspired couplings.

Abstract

We study slow-roll inflation with the Gauss-Bonnet and Chern-Simons corrections. We obtain general formulas for the observables: spectral indices, tensor-to-scalar ratio and circular polarization of gravitational waves. The Gauss-Bonnet term violates the consistency relation r = -8n_T. Particularly, blue spectrum n_T > 0 and scale invariant spectrum |8n_T|/r << 1 of tensor modes are possible. These cases require the Gauss-Bonnet coupling function of ξ_{,φ} \sim 10^8/M_{Pl}. We use examples to show new-inflation-type potential with 10M_{Pl} symmetry breaking scale and potential with flat region in φ\gtrsim 10M_{Pl} lead to observationally consistent blue and scale invariant spectra, respectively. Hence, these interesting cases can actually be realized. The Chern-Simons term produce circularly polarized tensor modes. We show an observation of these signals supports existence of the Chern-Simons coupling function of ω_{,φ} \sim 10^8/M_{Pl}. Thus, with future observations, we can fix or constrain the value of these coupling functions, at the CMB scale.

Figures (5)

• Figure 1: The configuration of the potential $V$ and the coupling function $\xi$, which achieve a blue spectrum, is shown. We need a climbing-up situation to realize a blue spectrum. Therefore the following situation is required: At early stage, the effective potential $\xi$ makes $\phi$ climb up potential and a blue spectrum is realized. At late stage, $\phi$ rolls down potential as usual.
• Figure 2: The trajectories, in which $\theta$ is varying, of analytic blue models are shown on a $n_{\rm S}-r$ plane. Two contours roughly corresponds to the 68% and 95% confidence level of the WMAP 7-year result. We draw eight lines, which are corresponding to $\phi_0=9$ (red, bottom) to $16M_{\rm Pl}$ (blue, top) with an even interval. Thick dotted and thick solid lines denote the blue spectra of $0.003>n_{\rm T}>0$ and $n_{\rm T}>0.003$, respectively. Models with $\phi_0\sim 11M_{\rm Pl}$ are consistent with observations and realizes blue spectra.
• Figure 3: The trajectories, in which $b$ is varying, of some blue models are shown on a $n_{\rm S}-r$ plane. In the $b\rightarrow0$ limit, all trajectories converges to one point. Two contours roughly corresponds to the 68% and 95% confidence level of the WMAP 7-year result. We draw nine lines, which are corresponding to $a=0.01$ (red, left in left panels and top in right panels) to $1$ (blue, right in left panels and bottom in right panels) with an even logarithmic interval. Thick dotted and thick solid lines denote $0.003>n_{\rm T}>0$ and $n_{\rm T}>0.003$ blue spectra, respectively. The model with $\phi_0=15M_{\rm Pl}$ is appropriate to achieve blue spectrum. An exponential coupling makes large $\phi_0$ ($\phi_0=20M_{\rm Pl}$) model result in a blue spectrum, and small $\phi_0$ ($\phi_0=11.1M_{\rm Pl}$) model achieve an observable amount of $r$.
• Figure 4: The trajectories, in which $\theta$ is varying, of analytic flat models are shown on a $n_{\rm S}-r$ plane. Two contours roughly corresponds to the 68% and 95% confidence level of the WMAP 7-year result. We draw five lines, which are corresponding to $\phi_0=2$ (red, bottom) to $10M_{\rm Pl}$ (blue, top) with an even interval. Thick lines denote the regions, which realize $r/|8n_{\rm T}| > 100$. Small $\phi_0$ models, namely $\phi_0=2M_{\rm Pl}$ and $\phi_0=4M_{\rm Pl}$ models, realize observationally consistent flat spectra.
• Figure 5: The trajectories, in which $b$ is varying, of some flat models is shown on a $n_{\rm S}-r$ plane. In the $b\rightarrow0$ limit, all trajectories converges to one point. Two contours roughly corresponds to the 68% and 95% confidence level of the WMAP 7-year result. We draw nine lines, which are corresponding to $a=0.01$ (red, left in left panels and right in right panels) to $1$ (blue, right in left panels and left in right panels) with an even logarithmic interval. In this case, tensor spectrum is always red. Thick lines denote the regions of flat spectra, $r/8|n_{\rm T}|>100$. This figure shows that small $\phi_0$ leads to a flat spectrum. Note that in $s=-1$ models, trajectories approach to some points, in large $b$ limit.