We study non-minimal Coleman-Weinberg inflation in the Palatini formulation of gravity in the presence of an $R^2$ term. The Planck scale is dynamically generated by the vacuum expectation value of the inflaton via its non-minimal coupling to the curvature scalar $R$. We show that the addition of the $R^2$ term in Palatini gravity makes non-minimal Coleman-Weinberg inflation again compatible with observational data.
We investigate in detail a family of quintessential inflation models in the context of $R+R^2$ Palatini modified gravity. We find that successful inflation and quintessence are obtained with an inflaton scalar potential that is approximately quadratic in inflation and inverse quartic in quintessence. We show that corrections for the kination period due to Palatini modified gravity are subdominant, while the setup does not challenge constraints on modified gravity from solar system observations and microscopic experiments, in contrast to the metric case. We obtain concrete predictions regarding…
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 study a model of quintessential inflation constructed in $R^2$ modified gravity with a non-minimally coupled scalar field, in the Palatini formalism. Our non-minimal inflaton field is characterised by a simple exponential potential. We find that successful quintessential inflation can be achieved with no fine-tuning on the model parameters. Predictions on the characteristics of dark energy will be tested by observations in the near future, while contrast with existing observations provides insights on the modified gravity background, such as the value of the non-minimal coupling and its run…
With a scalar field non-minimally coupled to curvature, the underlying geometry and variational principle of gravity - metric or Palatini - becomes important and makes a difference, as the field dynamics and observational predictions generally depend on this choice. In the present paper we describe a classification principle which encompasses both metric and Palatini models of inflation, employing the fact that inflationary observables can be neatly expressed in terms of certain quantities which remain invariant under conformal transformations and scalar field redefinitions. This allows us to …
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…
If the inflaton and the quintessence fields are identified, the background geometry evolves through a stiff epoch undershooting the expansion rate of a radiation-dominated plasma. For some classes of inflationary potentials this scenario is at odds with the current observational evidence since the corresponding tensor-to-scalar ratio is too large. Quintessential inflation is analyzed when the gravitational action is supplemented by a contribution quadratic in the Einstein-Hilbert term. In the Palatini formulation the addition such a term does not affect the scalar modes during the inflationary…
When considered in the Palatini formalism, the Starobinsky model does not provide us with a mechanism for inflation due to the absence of a propagating scalar degree of freedom. By (non)--minimally coupling scalar fields to the Starobinsky model in the Palatini formalism we can in principle describe the inflationary epoch. In this article, we focus on the minimally coupled quartic and natural inflation models. Both theories are excluded in their simplest realization since they predict values for the inflationary observables that are outside the limits set by the Planck data. However, with the …
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 study preheating in the Palatini formalism with a quadratic inflaton potential and an added $αR^2$ 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 $α\gtrsim 10^{13}$ and that the energy density of the fragmented field grows with the rate $Γ/H \approx 0.011 \times α^{0.31}$. The model extends the family of plateau models with similar preheating behaviour. Although it contains…
In this work, we show the effect of the non-minimal coupling $ξϕ^2 R$ on the inflationary parameters by considering the single-field inflation and present the inflationary predictions of the appealing potential for the particle physics viewpoint: Natural Inflation, an axion-like inflaton which has a cosine-type periodic potential and the inflaton naturally emerges as a pseudo-Nambu-Goldstone boson with a spontaneously broken global symmetry. We present the inflationary predictions for this potential, $n_s$, $r$, and $α=\mathrm{d}n_s/\mathrm{d}\ln k$. In addition, we assume standard thermal his…
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 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 construct a model of quintessential inflation in Palatini $R^2$ gravity employing a scalar field with a simple exponential potential and coupled to gravity with a running non-minimal coupling. At early times, the field acts as the inflaton, while later on it becomes the current dark energy. Combining the scalar sector with an ideal fluid, we study the cosmological evolution of the model from inflation all the way to dark energy domination. We interpret the results in the Einstein frame, where a coupling emerges between the fluid and the field, feeding energy from the former to the latter du…
It has been previously shown that the linear inflation appears naturally as a solution of Coleman-Weinberg inflation, provided that the inflaton has a non-minimal coupling to gravity and the Planck scale is dynamically generated. We revisit the previous study by improving the discussion of reheating and by comparing the results of the metric and the Palatini formulations of non-minimal gravity. We find that both formulations predict linear inflation but a different number of $e$-folds. If the non-minimal coupling is larger than one, future experimental sensitivity can discriminate between the …
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 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 scalar field inflation in $F(R)$ gravity in the Palatini formulation of general relativity. Unlike in the metric formulation, in the Palatini formulation $F(R)$ gravity does not introduce new degrees of freedom. However, it changes the relations between existing degrees of freedom, including the inflaton and spacetime curvature. Considering the case $F(R)=R+αR^2$, we find that the $R^2$ term decreases the height of the effective inflaton potential. By adjusting the value of $α$, this mechanism can be used to suppress the tensor-to-scalar ratio $r$ without limit in any scalar field mod…
We study preheating following Higgs inflation in the Palatini formulation of gravity. We numerically evolve perturbations of the radial mode of the Higgs field and that of three scalars modeling the gauge bosons. We compare the two non-perturbative mechanisms of growth of excitations -- parametric resonance and tachyonic instability -- and confirm that the latter plays the dominant role. Our results provide further evidence that preheating in Palatini Higgs inflation happens within a single oscillation of the Higgs field about the bottom of its potential, consistent with the approximation of a…
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 present two scale invariant models of inflation in which the addition of quadratic in curvature terms in the usual Einstein-Hilbert action, in the context of Palatini formulation of gravity, manages to reduce the value of the tensor-to-scalar ratio. In both models the Planck scale is dynamically generated via the vacuum expectation value of the scalar fields.
A new setup of cosmic inflation with a periodic inflaton potential and conformal factor is discussed in the metric and Palatini formulations of gravity. As a concrete example, we focus on a natural-inflation-like inflaton potential, and show that the inflationary predictions fall into the allowed region of cosmic microwave background observations in both formulations.
We study a model of quintessential inflation in the context of Palatini gravity. As a representative example, we consider the Peebles-Vilenkin model of quintessential inflation with a small non-minimal coupling to gravity, which is consistent with the most recent Planck measurements. At the end of inflation, the inflaton field passes through a tachyonic region and it leads to explosive particle production through the tachyonic preheating process. After preheating, the Universe becomes dominated by the kinetic energy of the inflaton and enters a period of kination. Eventually, the total energy …
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 …
We examine the generation of primordial perturbations during an inflationary epoch in generalised theories of gravity when the equations of motion are derived using the Palatini variational principle. Both f(R) and Scalar-Tensor theories are considered and we compare our results with those obtained under the conventional metric formalism. Non-linear generalisations of the action lead to different theories under the two variational choices and we obtain distinct results for scalar and tensor spectral indices and their ratio. We find the following general result; inflation driven solely by f(R) …
Marmara University（马尔马拉大学）；Ankara University（安卡拉大学）；Tallinn University of Technology（塔林理工大学）；National Institute of Chemical Physics and Biophysics（国家化学物理与生物物理学研究所）
This thesis presents research exploring aspects of inflation and cosmology in the context of inflation models in which an inflaton is non-minimally coupled to the Ricci scalar, or is considered in conjunction with a term quadratic in the Ricci scalar. We consider a $ϕ^{2}$ Palatini inflation model in $R^{2}$ gravity and investigate whether this model can overcome some of the problems of the original $ϕ^{2}$ chaotic inflation model. We investigate the compatibility of this model with the observed CMB when treated as an effective theory of inflation in quantum gravity by examining the constraint…
Attractor inflation is a particularly robust framework for developing inflationary models that are insensitive to the details of the potential. Such models are most often considered in the metric formulation of gravity. However, non-minimal models may not necessarily maintain their attractor nature in the Palatini formalism where the connection is independent of the metric. In this work, we employ the $β$-function formalism to classify the strong coupling limit of inflationary models in both the metric and the Palatini approaches. Furthermore, we determine the range of values for the non-minim…
We present a simple generalisation of the $Λ$CDM model which on the one hand reaches very good agreement with the present day experimental data and provides an internal inflationary mechanism on the other hand. It is based on Palatini modified gravity with quadratic Starobinsky term and generalized Chaplygin gas as a matter source providing, besides a current accelerated expansion, the epoch of endogenous inflation driven by type III freeze singularity. It follows from our statistical analysis that astronomical data favors negative value of the parameter coupling quadratic term into Einstein-H…
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: $ξ\lesssim 10^8$.…
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 single field slow-roll inflation in the presence of $F(R)$ gravity in the Palatini formulation. In contrast to metric $F(R)$, when rewritten in terms of an auxiliary field and moved to the Einstein frame, Palatini $F(R)$ does not develop a new dynamical degree of freedom. However, it is not possible to solve analytically the constraint equation of the auxiliary field for a general $F(R)$. We propose a method that allows us to circumvent this issue and compute the inflationary observables. We apply this method to test scenarios of the form $F(R) = R + αR^n$ and find that, as in the pre…
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 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…
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…
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…
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 models of quartic inflation where the inflaton field $ϕ$ is coupled non-minimally to gravity, $ξϕ^2 R$, and perform a study of quantum corrections in curved space-time at one-loop level. We specifically focus on comparing results between the metric and Palatini theories of gravity. Transformation from the Jordan to the Einstein frame gives different results for the two formulations and by using an effective field theory expansion we derive the appropriate $β$-functions and the renormalisation group improved effective potentials in curved space for both cases in the Einstein frame. In …
In the context of the Palatini formalism of gravity with an $R^{2}$ term, a $ϕ^{2}$ potential can be consistent with the observed bound on $r$ whilst retaining the successful prediction for $n_{s}$. Here we show that the Palatini $ϕ^{2} R^2$ inflation model can also solve the super-Planckian inflaton problem of $ϕ^{2}$ chaotic inflation, and that the model can be consistent with Planck scale-suppressed potential corrections. If $α\gtrsim 10^{12}$, where $α$ is the coefficient of the $R^2$ term, the inflaton in the Einstein frame, $σ$, remains sub-Planckian throughout inflation. In addition, if…
Slow-roll inflation is analyzed in the context of modified gravity within the Palatini formalism. As shown in the literature, inflation in this framework requires the presence of non-traceless matter, otherwise it does not occur just as a consequence of the non-linear gravitational terms of the action. Nevertheless, by including a single scalar field that plays the role of the inflaton, slow-roll inflation can be performed in these theories, where the equations lead to an effective potential that modifies the dynamics. We obtain the general slow-roll parameters and analyze a simple model to il…
A modified model of gravity with additional positive and negative powers of the scalar curvature, $R$, in the gravitational action is studied. This is done using the Palatini variational principle. It is demonstrated that using such a model might prove useful to explain both the early time inflation and the late time cosmic acceleration without the need for any form of dark energy.
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 study how higher-order gravity affects Higgs inflation in the Palatini formulation. We first review the metric and Palatini formulations in comparative manner and discuss their differences. Next cosmic inflation driven by a scalar field and inflationary observables are discussed. After this we review the Higgs inflation and compute the inflationary observables both in the metric and Palatini formulations. We then consider adding higher-order terms of the curvature to the action. We derive the equations of motion for the most general action quadratic in the curvature that does not violate pa…
Large-scale magnetogenesis is analyzed within the Palatini approach when the gravitational action is supplemented by a contribution that is nonlinear in the Einstein-Hilbert term. While the addition of the nonlinear terms does not affect the scalar modes of the geometry during the inflationary phase, the tensor-to-scalar ratio is nonetheless suppressed. In this context it is plausible to have a stiff phase following the standard inflationary stage provided the potential has a quintessential form. The large-scale magnetic fields can even be a fraction of the nG over typical length scales of the…
NMDC-Palatini cosmology in slow-roll regime is of our interests. We present flat FLRW cosmological NMDC-Palatini field equations and acceleration condition. Late time trajectory is approximated. Chaotic inflation potential is considered here as it is viable in a range of positive coupling constant as constrained by CMB results. Phase portraits show that the NMDC-Palatini gives new saddle-point solutions. In $V \propto ϕ^2$ potential, the NMDC-Palatini effect enlarges the acceleration-allowed region from that of GR case.
In this article, we intend to find a specific model which can satisfy the further refining dS swampland conjecture and scalar weak gravity conjecture (SWGC) simultaneously, in particular, Higgs inflation model and its two extensions: Higgs-Dilaton model and Palatini Higgs inflation. We find that although Higgs inflation model and Higgs-Dilaton model could satisfy the further refining dS swampland conjecture, the two models cannot satisfy SWGC and its strong version. While Palatini Higgs inflation could satisfy these conjectures simultaneously. Therefore Palatini Higgs inflation could be a "rea…
The inflationary dynamics and preheating in a model with a non-minimally coupled inflaton field in the metric and Palatini formalisms are studied in this paper. We find that in both formalisms, irrespective of the initial conditions, our Universe will evolve into a slow-roll inflationary era and then the scalar field rolls into an oscillating phase. The value of the scalar field at the end of the inflation in the Palatini formalism is always larger than that in the metric one, which becomes more and more obvious with the increase of the absolute value of the coupling parameter $|ξ|$. During th…
We discuss how the non-minimal coupling $ξϕ^2 R$ between the inflaton and the Ricci scalar affects predictions of single field inflation models in Palatini formalism. To transition radiation dominated era, the inflaton field $ϕ$ must interact to matter fields at the end of inflation. Interactions of the inflaton with other fields lead to radiative corrections to the inflationary potential. These radiative corrections can be explained at leading order by Coleman-Weinberg (CW) one-loop corrections. In this work, using two different prescriptions debated in the literature, the effect of radiative…
We perform an analysis of models of chaotic inflation where the inflaton field $ϕ$ is coupled non-minimally to gravity via $ξϕ^n g^{μν}R_{μν}(Γ), n>0$. We focus on the Palatini theory of gravity, i.e. the case where the assumptions of the General Relativity are relaxed (that of the connection being the Levi-Civita one) and the gravitational degrees of freedom are encoded not only in the metric but also the connection $Γ$, which is treated as an independent variable. We show that in this case the famous attractor behaviour of simple non-minimally coupled models of inflation is lost. Therefore t…
The theory of General Relativity was established on a spacetime manifold equipped with a metric tensor, $(\mathcal{M}_4,\text{g})$, and the connection on $\mathcal{M}_4$ identified with the Levi-Civita one. Even though there are valid reasons to assume a torsionless manifold that preserves the metric, it was shown that dealing away with these assumptions the Levi-Civita condition can be reproduced at the level of equations of motion of GR. It was not long before the equivalence of General Relativity between the two descriptions, known as the Palatini or first-order formalism in which the conne…
We study inflation with the most general non-degenerate gravitational action that depends on the symmetric part of the Ricci tensor coupled to a scalar field in the Palatini formulation of gravity. We use field redefinitions to shift the effect of the Ricci terms from gravity to the scalar field, and apply the result to slow-roll inflation. As examples, we consider actions quadratic and cubic in the Ricci tensor. In the quadratic case the results are similar to the case $R+αR^2$ that has been studied earlier: the tensor-to-scalar ratio $r$ can be suppressed by an arbitrary amount, while the sc…
