In this work, we study inflation with the non-minimally coupled quadratic, Standard Model (SM) Higgs and hilltop potentials through $ξϕ^2R$ term in the Palatini gravity. We first analyze observational parameters of Palatini quadratic potential as functions of $ξ$ for high-$N$ scenario and low-$N$ scenario. In addition to this, taking into account inflaton $ϕ$ has a non-zero vacuum expectation value $v$ afterwards inflation, we display observational parameters of well-known symmetry-breaking potentials type of Higgs potential and its generalizations which are hilltop potentials in the Palatini formalism for high-$N$ scenario and low-$N$ scenario. We calculate inflationary parameters of Palatini Higgs potential as functions of $v$ for different $ξ$ values where inflaton values both $ϕ>v$ and $ϕ<v$ during inflation as well as we show that observational parameters of Palatini Higgs potential in the induced gravity limit for high-$N$ scenario. On the other hand, we illustrate different from the Higgs potential the effect of $ξ$ on hilltop potentials which can agree with the observations for inflaton value solely $ϕ<v$ and $ξ$, $v\ll1$ for both two scenarios, which we mentioned above. For each considered potentials, we also display $n_s-r$ values fit the current data given by the Keck Array/BICEP2 and Planck collaborations.
We study inflation with the non-minimally coupled Standard Model Higgs in the case when quantum corrections generate a hilltop in the potential. We consider both the metric and the Palatini formulation of general relativity. We investigate hilltop inflation in different parts of the Higgs potential and calculate predictions for CMB observables. We run the renormalization group equations up from the electroweak scale and down from the hilltop, adding a jump in-between to account for unknown corrections in the intermediate regime. Within our approximation, no viable hilltop inflation is possible…
In the Higgs inflation scenario the Higgs field is strongly coupled to the Ricci scalar in order to drive primordial inflation. However, in its original form in pure metric formulation of gravity, the ultraviolet (UV) cutoff of the Higgs interactions and the Hubble rate are of the same magnitude, and this makes the whole inflationary evolution dependent of the unknown UV completion of the Higgs sector. This problem, the unitarity violation, plagues the Higgs inflation scenario. In this letter we show that, in the Palatini formulation of gravitation, Higgs inflation does not suffer from unitari…
We study the details of preheating in Palatini Higgs inflation. We show, that contrary to what happens in the metric formulation of the model, the Universe does not reheat through the creation of gauge bosons only, but also through the tachyonic production of Higgs excitations. The latest entropy production channel turns out to be very efficient and leads to an almost instantaneous onset of radiation domination after the end of inflation. As compared to the metric case, this reduces the number of e-folds needed to solve the usual hot big bang problems while leading to a smaller spectral index …
We investigate the UV-completion of the Higgs inflation in the metric and the Palatini formalisms. It is known that the cutoff scales for the perturbative unitarity of these inflation models become much smaller than the Planck scale to be consistent with observations. Expecting that the low cutoff scale originates in the curvature of a field-space spanned by the Higgs fields, we consider embedding the curved field-space into a higher dimensional flat space and apply this procedure to the metric-Higgs and the Palatini-Higgs scenarios. The new field introduced in this way successfully flattens t…
We investigate the sensitivity of Higgs(-like) inflation to higher dimensional operators in the nonminimal couplings and in the potential, both in the metric and Palatini formalisms. We find that, while inflationary predictions are relatively stable against the higher dimensional operators around the attractor point in the metric formalism, they are extremely sensitive in the Palatini one: for the latter, inflationary predictions are spoiled by $|ξ_4| \gtrsim 10^{-6}$ in the nonminimal couplings $(ξ_2 ϕ^2 + ξ_4 ϕ^4 + \cdots)R$, or by $|λ_6| \gtrsim 10^{-16}$ in the Jordan-frame potential $λ_4 …
We present the impact of non-minimal coupling $ξϕ^2 R$ on the inflationary parameters by taking into account the models of single-field inflation with the inflaton that has a non-zero vacuum expectation value ($v$) after the period of inflation in Palatini gravity. We discuss the well-known symmetry breaking type potentials, namely the Higgs potential and Coleman-Weinberg potential. We show the inflationary predictions of these potentials, for both $ϕ>v$ and $ϕ<v$ inflation, the regions in the $v-ξ$ plane for which the values of $n_s$ and $r$ are in agreement with the recent measurements. We a…
We calculate the tree-level amplitudes for electrically neutral $2\to2$ scattering for the Standard Model Higgs doublet non-minimally coupled to the Ricci scalar. We consider both the metric and the Palatini formulation of gravity. We find the partial wave unitarity limit for a general background field value. Our results are in agreement with previous work. In the electroweak vacuum, tree-level unitarity is violated at $\sim M_\text{Planck}/ξ$ in the metric formulation, and at $\sim M_\text{Planck}/\sqrtξ$ in the Palatini formulation. In the inflationary large field background, the unitarity l…
In this paper, we consider high scale (100 TeV) supersymmetry (SUSY) breaking and realize the idea of hilltop supernatural inflation in concrete particle physics models based on flipped-SU(5) and Pati-Salam models in the framework of supersymmetric grand unified theories (SUSY GUTs). The inflaton can be a flat direction including right-handed sneutrino and the waterfall field is a GUT Higgs. The spectral index is $n_s=0.96$ which fits very well with recent data by PLANCK satellite. There is no both thermal and non-thermal gravitino problems. Non-thermal leptogenesis can be resulted from the de…
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 $r\sim 10^{-12}$ 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 …
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 $αR^2$ 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 $n_s\simeq 0.941$, $r\simeq 0.3/(1+10^{-8}α)$, respectively, for a ty…
We compare Higgs inflation in the metric and Palatini formulations of general relativity, with loop corrections are treated in a simple approximation. We consider Higgs inflation on the plateau, at a critical point, at a hilltop and in a false vacuum. In the last case there are only minor differences. Otherwise we find that in the Palatini formulation the tensor-to-scalar ratio is consistently suppressed, spanning the range $1\times10^{-13}<r<7\times10^{-5}$, compared to the metric case result $2\times10^{-5}<r<0.2$. Even when the values of $n_s$ and $r$ overlap, the running and running of the…
We consider scalar field inflation in the Palatini formulation of general relativity. The covariant derivative of the metric is then non-zero. From the effective theory point of view it should couple to other fields. We write down the most general couplings between it and a scalar field that are quadratic in derivatives. We consider both the case when the torsion is determined by the field equations and the case when it is assumed to be zero a priori. We find the metric derivative terms can significantly modify inflationary predictions. We specialise to Higgs inflation and terms of only up to …
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: $ξ\lesssim 10^8$.…
We study Higgs inflation in the Palatini formulation with the renormalisation group improved potential in the case when loop corrections generate a feature similar to an inflection point. Assuming that there is a threshold correction for the Higgs quartic coupling $λ$ and the top Yukawa coupling $y_t$, we scan the three-dimensional parameter space formed by the two jumps and the non-minimal coupling $ξ$. The spectral index $n_s$ can take any value in the observationally allowed range. The lower limit for the running is $α_s>-3.5\times10^{-3}$, and $α_s$ can be as large as the observational upp…
This article is an addendum to [1]. We extend our computation of the on-shell scattering amplitudes in an arbitrary inflaton background $\barϕ_1$. Although the effective Einstein frame cutoff for $\barϕ_1>>M_P/\sqrtξ$ turns out to be $\barϕ_1$ or $ξ\barϕ_1$ for the $U(1)$ model, this is not the case for the realistic doublet Higgs model where the effective Einstein frame cutoff turns out to be the standard $M_P/\sqrtξ$ for both the Palatini and metric formulations. Then, as it has been pointed out in [1] the background $\barϕ_1$ is the effective Jordan frame cutoff for both the Palatini and me…
We consider $R^2$ inflation in the Palatini gravity assuming the existence of scalar fields, coupled to gravity in the most general manner. These theories, in the Einstein frame, and for one scalar field $h$, share common features with $K$ - inflation models. We apply this formalism for the study of popular inflationary models, whose potentials are monomials, $ V \sim h^{n} $, with $ n $ a positive even integer. We also study the Higgs model non-minimally coupled to gravity. Although these have been recently studied, in the framework of the Palatini approach, we show that the scalar power spec…
We study preheating in plateau inflation in the Palatini formulation of general relativity, in a special case that resembles Higgs inflation. It was previously shown that the oscillating inflaton field returns to the plateau repeatedly in this model, and this leads to tachyonic production of inflaton particles. We show that a minimally coupled spectator scalar field can be produced even more efficiently by a similar mechanism. The mechanism is purely gravitational, and the scalar field mass can be of order $10^{13}$ GeV, larger than the Hubble scale by many orders of magnitude, making this a c…
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 $g_{μν}$ determines the geometry of space-time, whereas in the former case both the metric and the space-time connection $Γ^λ_{μν}$ 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, assumpt…
The Starobinsky model, considered in the framework of the Palatini formalism, in contrast to the metric formulation, does not provide us with a model for inflation, due to the absence of a propagating scalar degree of freedom that can play the role of the inflaton. In the present article we study the Palatini formulation of the Starobinsky model coupled, in general nonminimally, to scalar fields and analyze its inflationary behavior. We consider scalars, minimally or nonminimally coupled to the Starobinsky model, such as a quadratic model, the induced gravity model or the standard Higgs-like i…
We consider ${\cal{R}}^2$-inflation in Palatini gravity, in the presence of scalar fields coupled to gravity. These theories, in the Einstein frame, and for one scalar field $h$, share common features with $K$ - inflation models. We apply this formalism for the study of single-field inflationary models, whose potentials are monomials, $ V \sim h^{n} $, with $ n $ a positive even integer. We also study the Higgs model non-minimally coupled to gravity. With ${\cal{R}}^2$-terms coupled to gravity as $\sim α{\cal{R}}^2 $, with $α$ constant, the instantaneous reheating temperature $T_{ins}$, is bou…
We present a comparative study of inflation in two theories of quadratic gravity with {\it gauged} scale symmetry: 1) the original Weyl quadratic gravity and 2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ($w_μ$) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric orig…
We study the production of primordial black hole (PBH) dark matter in the case when the Standard Model Higgs coupled non-minimally to gravity is the inflaton. PBHs can be produced if the Higgs potential has a near-critical point due to quantum corrections. In this case the slow-roll approximation may be broken, so we calculate the power spectrum numerically. We consider both the metric and the Palatini formulation of general relativity. Combining observational constraints on PBHs and on the CMB spectrum we find that PBHs can constitute all of the dark matter only if they evaporate early and le…
We consider a formulation of gauge field theory where the gauge field $A_α$ and the field strength $F_{αβ}$ are independent variables, as in the Palatini formulation of gravity. For the simplest gauge field action, this is known to be equivalent to the usual formulation. We add non-minimal couplings between $F_{αβ}$ and a scalar field, solve for $F_{αβ}$ and insert it back into the action. This leads to modified gauge field and scalar field terms. We consider slow-roll inflation and show that because of the modifications to the scalar sector, adding higher order terms to the inflaton potential…
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 desc…
We study the phenomenology of the Higgs-Dilaton model in the context of Einstein-Cartan gravity, focusing on the separate impact of the Holst and Nieh-Yan terms on the inflationary observables. Using analytical and numerical techniques, we show the predictions of these scenarios to display an attractor-like behaviour intrinsically related to the curvature of the field-space manifold in the metric formulation of the theory. Beyond that, the analysis of the Nieh-Yan case reveals the existence of an additional attractor solution induced by a cubic pole in the inflaton kinetic term that becomes re…
We consider models of chaotic inflation driven by the real parts of a conjugate pair of Higgs superfields involved in the spontaneous breaking of a grand unification symmetry at a scale assuming its Supersymmetric value. Employing Kaehler potentials with a prominent shift-symmetric part proportional to c- and a tiny violation, proportional to c+, included in a logarithm we show that the inflationary observables provide an excellent match to the recent Planck and Bicep2/Keck Array results setting, e.g., 0.0064<= c+/c-<1/N where N=2 or 3 is the prefactor of the logarithm. Deviations of these pre…
We investigate how inflationary predictions are affected by the difference in the number of $e$-folds between the Jordan and Einstein frames. We study several test models in relation to a Jordan frame defined by the common Higgs-inflation-like non-minimal coupling to gravity and consider two different formulations of gravity: metric and Palatini. We find that the difference is quite contained in case of a metric Jordan frame while can be quite remarkable in case of a Palatini Jordan frame. We also discuss a way to overcome the discrepancy by introducing a frame invariant physical distance and …
We study quadratic gravity $R^2+R_{[μν]}^2$ in the Palatini formalism where the connection and the metric are independent. This action has a {\it gauged} scale symmetry (also known as Weyl gauge symmetry) of Weyl gauge field $v_μ= (\tildeΓ_μ-Γ_μ)/2$, with $\tildeΓ_μ$ ($Γ_μ$) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $R_{[μν]}^2$ term acting as a gauge kinetic term for $v_μ$. We show that this theory has an elegant spontaneous breaking of gauged scale symmetry and mass generation in the absence of matter, where the necessa…
In the framework of mixed Higgs-Starobinsky inflation, we consider the generation of Abelian gauge fields due to their nonminimal coupling to gravity (in two different formulations of gravity -- metric and Palatini). We couple the gauge-field invariants $F_{μν}F^{μν}$ and $F_{μν}\tilde{F}^{μν}$ to an integer power of the scalar curvature $R^n$ in Jordan frame and, treating these interactions perturbatively, switch to the Einstein frame where they lead to effective kinetic and axial couplings between gauge fields and inflaton. We determine the power spectra, energy densities, correlation length…
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 Ein…
There are many single field inflationary models that are consistent with the recent Planck 2018 measurements of the spectral index $n_s$ and tensor-to-scalar ratio $r$. Despite good agreement with observational data some of these models suffer from having unregularized potentials which would produce a collapsing universe shortly after the end of inflation. In this paper we show that how one chooses to correct the behaviour potential towards the end of inflation can have a significant effect on the inflationary predictions of the model, specifically in the case of quartic hilltop and radiativel…
In order to accommodate the QCD axion as the dark matter (DM) in a model in which the Peccei-Quinn (PQ) symmetry is broken before the end of inflation, a relatively low scale of inflation has to be invoked in order to avoid bounds from DM isocurvature fluctuations, $H_* \lesssim \mathcal{O}(10^9)\,$GeV. We construct a simple model in which the Standard Model Higgs field is non-minimally coupled to Palatini gravity and acts as the inflaton, leading to a scale of inflation $H_* \sim 10^8\,$GeV. When the energy scale at which the PQ symmetry breaks is much larger than the scale of inflation, we f…
We study models of inflation where the scalar field $ϕ$ that drives inflation is coupled non-minimally to gravity via $ξϕ^2 R$, or where the gravity sector is enlarged by an $R^2$ term. We consider the original Higgs inflation, Starobinsky inflation, and two different versions of a scenario where the inflaton is a scalar field other than the Higgs, and discuss if they can be distinguished from each other by measuring the tensor-to-scalar ratio and runnings of the spectral index of primordial curvature perturbations, on top of the amplitude and spectral index of the perturbations. We consider b…
Inflation occurring at energy densities less than (10$^{14}$ GeV)$^4$ produces tensor perturbations too small to be measured by cosmological surveys. However, we show that it is possible to probe low scale inflation by measuring the mass of the inflaton at low energy experiments. Detection prospects and cosmological constraints are determined for low scale quartic hilltop models of inflation paired with a curvaton field, which imprints the spectrum of scalar perturbations observed in large scale structure and on the cosmic microwave background. With cosmological constraints applied, low scale …
What new physics can we extract from inflation using the ACT DR6 and DESI DR2 Observations?
SGT University（SGT大学）；Institute of Theoretical Physics, Faculty of Physics, University of Warsaw（华沙大学物理学院理论物理研究所）；Center for Theoretical Physics, L.N. Gumilyov Eurasian National University（L.N. 古米廖夫欧亚国立大学理论物理中心）；B.M.S. College of Engineering（B.M.S.工程学院）
We propose a model for combining the Standard Model (SM) with gravity. It relies on a non-minimal coupling of the Higgs field to the Ricci scalar and on the Palatini formulation of gravity. Without introducing any new degrees of freedom in addition to those of the SM and the graviton, this scenario achieves two goals. First, it generates the electroweak symmetry breaking by a non-perturbative gravitational effect. In this way, it does not only address the hierarchy problem but opens up the possibility to calculate the Higgs mass. Second, the model incorporates inflation at energies below the o…
We study initial conditions for inflation in scenarios where the inflaton potential has a plateau shape. Such models are those most favored by Planck data and can be obtained in a large number of model classes. As a representative example, we consider Higgs inflation with and without an $R^2$ term in the context of Palatini gravity. We show that inflation with a large number of e-folds generically occurs in a large part of the parameter space without any fine-tuning of parameters even when the scale of inflation and the inflaton field value during inflation are much smaller than the Planck sca…
We study inflation in the Brans-Dicke gravity as a special model of the scalar-tensor gravity. We obtain the inflationary observables containing the scalar spectral index, the tensor-to-scalar ratio, the running of the scalar spectral index and the equilateral non-Gaussianity parameter in terms of the general form of the potential in the Jordan frame. Then, we compare the results for various inflationary potentials in light of the Planck 2015 data. Our study shows that in the Brans-Dicke gravity, the power-law, inverse power-law and exponential potentials are ruled out by the Planck 2015 data.…
We compute the parametrized post-Newtonian parameter $γ$ in the case of a static point source for multiscalar-tensor gravity with completely general nonderivative couplings and potential in the Jordan frame. Similarly to the single massive field case $γ$ depends exponentially on the distance from the source and is determined by the length of a vector of non-minimal coupling in the space of scalar fields and its orientation relative to the mass eigenvectors. Using data from the Cassini tracking experiment, we estimate bounds on a general theory with two scalar fields. Our formalism can be utili…
If cosmic inflation was driven by an electrically neutral scalar field stable on cosmological time scales, the field necessarily constitutes all or part of dark matter (DM). We study this possibility in a scenario where the inflaton field $s$ resides in a hidden sector, which is coupled to the Standard Model sector through the Higgs portal $λ_{hs} s^2\mathcal{H}^\dagger\mathcal{H}$ and non-minimally to gravity via $ξ_s s^2 R$. We study scenarios where the field $s$ first drives inflation, then reheats the Universe, and later constitutes all DM. We consider two benchmark scenarios where the DM …
