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Register a new userMinimal Higgs inflation with an $R^2$ term in Palatini gravity
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Authors
Tommi Tenkanen
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It has recently been suggested that the Standard Model Higgs boson could act as the inflaton while minimally coupled to gravity - given that the gravity sector is extended with an $alpha R^2$ term and the underlying theory of gravity is of Palatini, rather than metric, type. In this paper, we revisit the idea and correct some shortcomings in earlier studies. We find that in this setup the Higgs can indeed act as the inflaton and that the tree-level predictions of the model for the spectral index and the tensor-to-scalar ratio are $n_ssimeq 0.941$, $rsimeq 0.3/(1+10^{-8}alpha)$, respectively, for a typical number of e-folds, $N=50$, between horizon exit of the pivot scale $k=0.05, {rm Mpc}^{-1}$ and the end of inflation. Even though the tensor-to-scalar ratio is suppressed compared to the usual minimally coupled case and can be made compatible with data for large enough $alpha$, the result for $n_s$ is in severe tension with the Planck results. We briefly discuss extensions of the model.
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We study preheating in the Palatini formalism with a quadratic inflaton potential and an added $alpha R^2$ term. In such models, the oscillating inflaton field repeatedly returns to the plateau of the Einstein frame potential, on which the tachyonic instability fragments the inflaton condensate within less than an e-fold. We find that tachyonic preheating takes place when $alpha gtrsim 10^{13}$ and that the energy density of the fragmented field grows with the rate $Gamma/H approx 0.011 times alpha^{0.31}$. The model extends the family of plateau models with similar preheating behaviour. Although it contains non-canonical quartic kinetic terms in the Einstein frame, we show that, in the first approximation, these can be neglected during both preheating and inflation.
We present an introduction to cosmic inflation in the context of Palatini gravity, which is an interesting alternative to the usual metric theory of gravity. In the latter case only the metric $g_{mu u}$ determines the geometry of space-time, whereas in the former case both the metric and the space-time connection $Gamma^lambda_{mu u}$ are a priori independent variables - a choice which can lead to a theory of gravity different from the metric one. In scenarios where the field(s) responsible for cosmic inflation are coupled non-minimally to gravity or the gravitational sector is otherwise extended, assumptions of the underlying gravitational degrees of freedom can have a big impact on the observational consequences of inflation. We demonstrate this explicitly by reviewing several interesting and well-motivated scenarios including Higgs inflation, $R^2$ inflation, and $xi$-attractor models. We also discuss some prospects for future research and argue why $r=10^{-3}$ is a particularly important goal for future missions that search for signatures of primordial gravitational waves.
In the framework of classical scale invariance, we consider quadratic gravity in the Palatini formalism and investigate the inflationary predictions of the theory. Our model corresponds to a two-field scalar-tensor theory, that involves the Higgs field and an extra scalar field stemming from a gauge $U(1)_X$ extension of the Standard Model, which contains an extra gauge boson and three right-handed neutrinos. Both scalar fields couple nonminimally to gravity and induce the Planck scale dynamically, once they develop vacuum expectation values. By means of the Gildener-Weinberg approach, we describe the inflationary dynamics in terms of a single scalar degree of freedom along the flat direction of the tree-level potential. The one-loop effective potential in the Einstein frame exhibits plateaus on both sides of the minimum and thus the model can accommodate both small and large field inflation. The inflationary predictions of the model are found to comply with the latest bounds set by the Planck collaboration for a wide range of parameters and the effect of the quadratic in curvature terms is to reduce the value of the tensor-to-scalar ratio.
In the context of the Palatini formalism of gravity with an $R^{2}$ term, a $phi^{2}$ potential can be consistent with the observed bound on $r$ whilst retaining the successful prediction for $n_{s}$. Here we show that the Palatini $phi^{2} R^2$ inflation model can also solve the super-Planckian inflaton problem of $phi^{2}$ chaotic inflation, and that the model can be consistent with Planck scale-suppressed potential corrections. If $alpha gtrsim 10^{12}$, where $alpha$ is the coefficient of the $R^2$ term, the inflaton in the Einstein frame, $sigma$, remains sub-Planckian throughout inflation. In addition, if $alpha gtrsim 10^{20}$ then the predictions of the model are unaffected by Planck-suppressed potential corrections in the case where there is a broken shift symmetry, and if $alpha gtrsim 10^{32}$ then the predictions are unaffected by Planck-suppressed potential corrections in general. The value of $r$ is generally small, with $r lesssim 10^{-5}$ for $alpha gtrsim 10^{12}$. We calculate the maximum possible reheating temperature, $T_{R;max}$, corresponding to instantaneous reheating. For $alpha approx 10^{32}$, $T_{R; max}$ is approximately $10^{10}$ GeV, with larger values of $T_{R;max}$ for smaller $alpha$. For the case of instantaneous reheating, we show that $n_{s}$ is in agreement with the 2018 Planck results to within 1-$sigma$, with the exception of the $alpha approx 10^{32}$ case, which is close to the 2-$sigma$ lower bound. Following inflation, the inflaton condensate is likely to rapidly fragment and form oscillons. Reheating via inflaton decays to right-handed neutrinos can easily result in instantaneous reheating. We determine the scale of unitarity violation and show that, in general, unitarity is conserved during inflation.
The predictions of standard Higgs inflation in the framework of the metric formalism yield a tensor-to-scalar ratio $r sim 10^{-3}$ which lies well within the expected accuracy of near-future experiments $ sim 10^{-4}$. When the Palatini formalism is employed, the predicted values of $r$ get highly-suppressed $rsim 10^{-12}$ and consequently a possible non-detection of primordial tensor fluctuations will rule out only the metric variant of the model. On the other hand, the extremely small values predicted for $r$ by the Palatini approach constitute contact with observations a hopeless task for the foreseeable future. In this work, we propose a way to remedy this issue by extending the action with the inclusion of a generalized non-minimal derivative coupling term between the inflaton and the Einstein tensor of the form $m^{-2}(phi) G_{mu u} abla^{mu}phi abla^{ u}phi$. We find that with such a modification, the Palatini predictions can become comparable with the ones obtained in the metric formalism, thus providing ample room for the model to be in contact with observations in the near future.
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