Higgs-$R^2$ inflation -- full slow-roll study at tree-level

Abstract

We consider Higgs inflation with an $\alpha R^2$ term. It adds a new scalar degree of freedom, which leads to a two-field model of inflation. We do a complete slow-roll analysis of the three-dimensional parameter space of the $R^2$ coefficient $\alpha$, the non-minimal coupling $\xi$ and the Higgs self-coupling $\lambda$. We find three classes of inflationary solutions, but only pure $R^2$ and attractor solutions fit observations. We find that pure Higgs inflation is impossible when the $R^2$ term is present regardless of how small $\alpha$ is. However, we can have Higgs-like inflation, where the amplitude of the perturbations does not depend on $\alpha$ and the predictions as a function of e-folds are the same as in Higgs inflation, although the inflationary trajectory is curved in field space. The spectral index is $0.939<n_R<0.967$, and constraining it to the observed range, the tensor-to-scalar ratio varies from $3.8\times 10^{-3}$ to the maximum allowed by observations, $0.079$. Observational constraints on isocurvature perturbations contribute to these limits, whereas non-Gaussianity is automatically in the range allowed by observations.

v2. Added references, corrected numerical analysis of inflationary trajectories. Range of values for the spectral index narrowed down. v3. Added references and clarified text, fixed typos. Published version. 18 pages, 5 figures

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