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We review the recent progress in Higgs inflation focusing on Higgs-$R^2$ inflation, primordial black hole production and the $R^3$ term.
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We study quantum effects in Higgs inflation in the Palatini formulation of gravity, in which the metric and connection are treated as independent variables. We exploit the fact that the cutoff, above which perturbation theory breaks down, is higher than the scale of inflation. Unless new physics above the cutoff leads to unnaturally large corrections, we can directly connect low-energy physics and inflation. On the one hand, the lower bound on the top Yukawa coupling due to collider experiments leads to an upper bound on the non-minimal coupling of the Higgs field to gravity: $xi lesssim 10^8$. On the other hand, the Higgs potential can only support successful inflation if $xi gtrsim 10^6$. This leads to a fairly strict upper bound on the top Yukawa coupling of $0.925$ (defined in the $overline{text{MS}}$-scheme at the energy scale $173.2,text{GeV}$) and constrains the inflationary prediction for the tensor-to-scalar ratio. Additionally, we compare our findings to metric Higgs inflation.
We study inflation driven by the Higgs field in the Einstein-Cartan formulation of gravity. In this theory, the presence of the Holst and Nieh-Yan terms with the Higgs field non-minimally coupled to them leads to three additional coupling constants. For a broad range of parameters, we find that inflation is both possible and consistent with observations. In most cases, the spectral index is given by $n_s=1-2/N_star$ (with $N_star$ the number of e-foldings) whereas the tensor-to-scalar ratio $r$ can vary between about $10^{-10}$ and $1$. Thus, there are scenarios of Higgs inflation in the Einstein-Cartan framework for which the detection of gravitational waves from inflation is possible in the near future. In certain limits, the known models of Higgs inflation in the metric and Palatini formulations of gravity are reproduced. Finally, we discuss the robustness of inflationary dynamics against quantum corrections due to the scalar and fermion fields.
In this work, we revisit the non-minimally coupled Higgs Inflation scenario and investigate its observational viability in light of the current Cosmic Microwave Background, Baryon Acoustic Oscillation and type Ia Supernovae data. We explore the effects of the Coleman-Weinberg approximation to the Higgs potential in the primordial universe, connecting the predictions for the Lagrangian parameters at inflationary scales to the electroweak observables through Renormalization Group methods at two-loop order. As the main result, we find that observations on the electroweak scale are in disagreement with the constraints obtained from the cosmological data sets used in the analysis. Specifically, an $approx 8sigma$-discrepancy between the inflationary parameters and the electroweak value of the top quark mass is found, which suggests that a significant deviation from the scenario analysed is required by the cosmological data.
Higgs inflation and $R^2$-inflation (Starobinsky model) are two limits of the same quantum model, hereafter called Starobinsky-Higgs. We analyse the two-loop action of the Higgs-like scalar $phi$ in the presence of: 1) non-minimal coupling ($xi$) and 2) quadratic curvature terms. The latter are generated at the quantum level with $phi$-dependent couplings ($tildealpha$) even if their tree-level couplings ($alpha$) are tuned to zero. Therefore, the potential always depends on both Higgs field $phi$ and scalaron $rho$, hence multi-field inflation is a quantum consequence. The effects of the quantum (one- and two-loop) corrections on the potential $hat W(phi,rho)$ and on the spectral index are discussed, showing that the Starobinsky-Higgs model is in general stable in their presence. Two special cases are also considered: first, for a large $xi$ in the quantum action one can integrate $phi$ and generate a refined Starobinsky model which contains additional terms $xi^2 R^2ln^p (xi vert Rvert/mu^2)$, $p=1,2$ ($mu$ is the subtraction scale). These generate corrections linear in the scalaron to the usual Starobinsky potential and a running scalaron mass. Second, for a small fixed Higgs field $phi^2 ll M_p^2/xi$ and a vanishing classical coefficient of the $R^2$-term, we show that the usual Starobinsky inflation is generated by the quantum corrections alone, for a suitable non-minimal coupling ($xi$).
The measured Standard Model parameters lie in a range such that the Higgs potential, once extrapolated up to high scales, develops a minimum of negative energy density. This has important cosmological implications. In particular, during inflation, quantum fluctuations could have pushed the Higgs field beyond its potential barrier, triggering the formation of anti-de Sitter regions, with fatal consequences for our universe. By requiring that this did not happen, one can in principle connect (and constrain) Standard Model parameters with the energy scale of inflation. In this context, we highlight the sensitivity of the fate of our vacuum to seemingly irrelevant physics. In particular, the departure of inflation from an exact de Sitter phase, as well as Planck-suppressed derivative operators, can, already and surprisingly, play a decisive role in (de)stabilizing the Higgs during inflation. Furthermore, in the stochastic dynamics, we quantify the impact of the amplitude of the noise differing from the one of a massless field, as well as of going beyond the slow-roll approximation by using a phase-space approach. On a general ground, our analysis shows that relating the period of inflation to precision particle physics requires a knowledge of these irrelevant effects.
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