[Abridged] In contemporary literature, the calculation of modifications to the inflationary scalar power spectrum due to the loops from the higher order interaction terms in the Hamiltonian have led to a discussion regarding the validity of perturbation theory. Recently, there have been efforts to examine the contributions to the scalar power spectrum due to the loops arising from the cubic order terms in the action describing the perturbations, specifically in inflationary scenarios that permit an epoch of ultra slow roll (USR). A phase of USR inflation leads to significant observational consequences, such as the copious production of primordial black holes. In this work, we study the loop contributions to the scalar power spectrum in a scenario of USR inflation arising due to the quartic order terms in the action describing the scalar perturbations. We compute the loop contributions to the scalar power spectrum due to the dominant term in the action at the quartic order. We consider a scenario wherein a phase of USR is sandwiched between two stages of slow roll inflation and analyze the behavior of the loop contributions in terms of the parameters involved. We examine the late, intermediate and early epochs of USR during inflation. In the inflationary scenario involving a late phase of USR, for reasonable choices of the parameters, we show that the loop corrections are negligible for the entire range of wave numbers. In the intermediate case, the contributions from the loops prove to be scale invariant over large scales, and we find that these contributions can amount to 30% of the leading order power spectrum. In the case wherein USR sets in early, we find that the loop contributions could be negative and can dominate the power spectrum at the leading order, which indicates a breakdown of the perturbative expansion. We conclude with a brief summary and outlook.
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 …
For the constant-roll tachyon inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts and the tensor to scalar ratio up to the first order by using the method of Bessel function approximation. The derived $n_s-r$ results for the constant-roll inflation are also compared with the observations, we find that only one constant-roll inflation is consistent with the observations and observations constrain the constant-roll inflation to be slow-roll inflation. The tachyon potential is also reconstructed for the constant-roll inflat…
We investigate inflationary dynamics in the framework of the Einstein-Gauss-Bonnet gravity. In the model under consideration, the inflaton field is non-minimally coupled to the Gauss-Bonnet curvature invariant, so that the latter appears to be dynamically important. We consider a quartic potential for the inflaton field, in particular the one asymptotically connected to the Higgs inflation, and a wider class of coupling functions not considered in the earlier work. Keeping in mind the observational bounds on the parameters - the amplitude of scalar perturbations $A_s$, spectral index $n_s$ and…
In this paper, I present an analytic solution to the equation of motion of the inflaton field in a model of rapid roll inflation with a quartic potential by assuming that the Hubble parameter is a constant during inflation. The result is obtained by using Jacobi elliptic functions. The spectral index $n_s$ and the running spectral index $n_s^\prime$ is obtained without using rapid roll or extended slow roll approximations. The cosmological consequences of the model is discussed.
Quartic hilltop inflation remains one of the most successful inflationary models. Yet, the expectations of early treatments of hilltop inflation would contradict the observations and render the model excluded. However, recent numerical treatment has demonstrated that quartic hilltop inflation actually fares well with observations. In this work, a fully analytic treatment of the model aims to dispel the mystery surrounding the behaviour of quartic hilltop inflation. The results obtained are in excellent agreement with numerical works on the subject, yet offer simple analytic formulas to calcula…
By using the relations between the slow-roll parameters and the power spectrum for the single field slow-roll inflation, we derive the scalar spectral tilt $n_s$ and the tensor to scalar ratio $r$ for the constant slow-roll inflation and obtain the constraint on the slow-roll parameter $η$ from the Planck 2015 results. The inflationary potential for the constant slow-roll inflation is then reconstructed in the framework of both general relativity and scalar-tensor theory of gravity, and compared with the recently reconstructed E model potential. In the strong coupling limit, we show that the $…
The primordial power spectra of scalar and tensor perturbations during slow-roll inflation are usually calculated with the method of Bessel function approximation. For constant-roll or ultra slow-roll inflation, the method of Bessel function approximation may be invalid. We compare the numerical results with the analytical results derived from the Bessel function approximation, and we find that they differ significantly on super-horizon scales if the constant slow-roll parameter $η_H$ is not small. More accurate method is needed for calculating the primordial power spectrum for constant-roll i…
Loop corrections to observables in slow-roll inflation are found to diverge no worse than powers of the log of the scale factor, extending Weinberg's theorem to quasi-single field inflation models. Demanding perturbation theory be valid during primordial inflation leads to constraints on the effective lagrangian. This leads to some interesting constraints and coincidences on the landscape of inflationary vacua.
In this work, we investigate how a smooth transition from a constant-roll to a slow-roll inflationary era may be realized, in the context of a canonical scalar field theory. We study in some detail the dynamical evolution of the cosmological system, and we investigate whether a stable attractor exists, both numerically and analytically. We also investigate the slow-roll era and as we demonstrate, partially compatibility of the resulting scalar theory, may be achieved, with the potential of the latter belonging to a class of modular inflationary potentials. The novel features of the constant-ro…
We consider an inflationary scenario where the rate of inflaton roll defined by $\ddotϕ/H\dot ϕ$ remains constant. The rate of roll is small for slow-roll inflation, while a generic rate of roll leads to the interesting case of `constant-roll' inflation. We find a general exact solution for the inflaton potential required for such inflaton behaviour. In this model, due to non-slow evolution of background, the would-be decaying mode of linear scalar (curvature) perturbations may not be neglected. It can even grow for some values of the model parameter, while the other mode always remains consta…
In this work we generalize the constant-roll condition for minimally coupled canonical scalar field inflation. Particularly, we shall assume that the scalar field satisfies the condition $\ddotϕ=α(ϕ) V'(ϕ)$, and we derive the field equations under this assumption. We call the framework extended constant-roll framework. Accordingly we calculate the inflationary indices and the corresponding observational indices of inflation. In order to demonstrate the inflationary viability, we choose three potentials that are problematic in the context of slow-roll dynamics, namely chaotic, linear power-law …
In this paper we investigate the implications of having a varying second slow-roll index on the canonical scalar field inflationary dynamics. We shall be interested in cases that the second slow-roll can take small values and correspondingly large values, for limiting cases of the function that quantifies the variation of the second slow-roll index. As we demonstrate, this can naturally introduce a smooth transition between slow-roll and constant-roll eras. We discuss the theoretical implications of the mechanism we introduce and we use various illustrative examples in order to better understa…
The previously proposed class of phenomenological inflationary models in which the assumption of inflaton slow-roll is replaced by the more general, constant-roll condition is compared with the most recent cosmological observational data, mainly the Planck ones. Models in this two-parametric class which remain viable appear to be close to the slow-roll ones, and their inflaton potentials are close to (but still different from) that of the natural inflation model. Permitted regions for the two model parameters are presented.
We generalize the notion of constant-roll inflation earlier introduced in General Relativity (GR) and $f(R)$ gravity to inflationary models in more general scalar-tensor gravity. A number of novel exact analytic solutions for a FLRW spatially flat cosmological background is found for this case. All forms of the scalar field potential and its coupling to gravity producing the exact de Sitter solution, while the scalar field is varying, are presented. In the particular cases of induced gravity and GR with a non-minimally coupled scalar field, all constant-roll inflationary solutions are found. I…
We discuss the constant-roll inflation with constant $ε_2$ and constant $\barη$. By using the method of Bessel function approximation, the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor to scalar ratio are derived up to the first order of $ε_1$. The model with constant $ε_2$ is ruled out by the observations at the $3σ$ confidence level, and the model with constant $\barη$ is consistent with the observations at the $1σ$ confidence level. The potential for the model with constant $\barη$ is also obtained from the Hamilton-Jaco…
We present constraints on how far single field inflation may depart from the familiar slow-roll paradigm. Considering a fast-roll regime while requiring a (near)-scale-invariant power spectrum introduces large self-interactions for the field and consequently large and scale-dependent non-Gaussianities. Employing this signal, we use the requirement of weak-coupling together with WMAP constraints to derive bounds on generic $P(X,ϕ)$ theories of single field inflation.
The conventional slow-roll approximation is broken in the so-called "ultra slow-roll" models of inflation, for which the inflaton potential is exactly (or extremely) flat. The interesting nature of (canonical) ultra slow-roll inflation is that the curvature perturbation grows on superhorizon scales, but has a scale-invariant power spectrum. We study the ultra slow-roll inflationary dynamics in the presence of non-canonical kinetic terms of the scalar field, namely ultra slow-roll G-inflation. We compute the evolution of the curvature perturbation and show that the primordial power spectrum fol…
Warm inflationary universe models in the context of generalized chaplygin gas, modified chaplygin gas, generalized cosmic chaplygin gas are being studied. The dissipative coefficient of the form $Γ\propto T$, weak and strong dissipative regimes are being considered. We use quartic potential $\frac{λ_{*}ϕ^{4}}{4}$, which is ruled out by current data in cold inflation but in our models it is analyzed that it is in agreement with the WMAP$9$ and latest Planck data. In these scenarios, the power spectrum, spectral index, and tensor to scalar ratio are being examined under the slow roll approximati…
We present a complete formulation of the scalar bispectrum in the unified effective field theory (EFT) of inflation, which includes the Horndeski and beyond-Horndeski Gleyzes-Langlois-Piazza-Vernizzi classes, in terms of a set of simple one-dimensional integrals. These generalized slow-roll expressions remain valid even when slow-roll is transiently violated and encompass all configurations of the bispectrum. We show analytically that our expressions explicitly preserve the squeezed-limit consistency relation beyond slow-roll. As an example application of our results, we compute the scalar bis…
We show that if the inflaton has a non-minimal coupling to gravity and the Planck scale is dynamically generated, the results of Coleman-Weinberg inflation are confined in between two attractor solutions: quadratic inflation, which is ruled out by the recent measurements, and linear inflation which, instead, is in the experimental allowed region. The minimal scenario has only one free parameter -- the inflaton's non-minimal coupling to gravity -- that determines all physical parameters such as the tensor-to-scalar ratio and the reheating temperature of the Universe. Should the more precise fut…
In this paper we investigate the inflationary phenomenology of an Einstein-Gauss-Bonnet theory compatible with the GW170817 event, by imposing the constant-roll evolution on the scalar field. We develop the constant-roll GW170817-compatible Einstein-Gauss-Bonnet formalism, and we calculate the slow-roll indices and the observational indices of inflation, for several models of interest. As we demonstrate, the phenomenological viability of the models we study is achieved for a wide range of the free parameters. In addition, for the same values of the free parameters that guarantee the inflationa…
The previously introduced class of two-parametric phenomenological inflationary models in General Relativity in which the slow-roll assumption is replaced by the more general, constant-roll condition is generalized to the case of $f(R)$ gravity. A simple constant-roll condition is defined in the original Jordan frame, and exact expressions for a scalaron potential in the Einstein frame, for a function $f(R)$ (in the parametric form) and for inflationary dynamics are obtained. The region of the model parameters permitted by the latest observational constraints on the scalar spectral index and t…
We discuss the cosmological inflation in the context of $f(Q)$ non-metric gravity, where $Q$ is the non-metric scalar. After introducing conformal transformations for $f(Q)$ gravity, we first focus on the potential-slow-roll inflation by studying the corresponding potentials for different forms of the function $f(Q)$ in the Einstein frame. Secondly, we investigate the Hubble-slow-roll inflation for three classes of inflationary potentials considered for the specific form $f(Q)\propto Q^{2}$, in the Jordan frame. We compare results in both approaches with observations coming from Planck and BIC…
We explore the possibility of a blue-tilted gravitational wave spectrum from potential-driven slow-roll inflation in the Horndeski theory. In Kamada et al. (2012), it was claimed that a blue gravitational wave spectrum cannot be obtained from stable potential-driven slow-roll inflation within the Horndeski framework. However, it has been demonstrated that the spectrum of primordial gravitational waves can be blue in inflation with the Gauss-Bonnet term, where the potential term is dominant and slow-roll conditions as well as the stability conditions are satisfied. To fill in this gap, we clari…
It has recently been shown that in single field slow-roll inflation the total volume cannot grow by a factor larger than e^(S_dS/2) without becoming infinite. The bound is saturated exactly at the phase transition to eternal inflation where the probability to produce infinite volume becomes non zero. We show that the bound holds sharply also in any space-time dimensions, when arbitrary higher-dimensional operators are included and in the multi-field inflationary case. The relation with the entropy of de Sitter and the universality of the bound strengthen the case for a deeper holographic inter…
We, in the first part, contemplate a massless minimally coupled scalar which is Yukawa-coupled to a massless Dirac fermion in a locally de Sitter background of an inflating spacetime. We compute the scalar's quantum corrected mode function, power spectrum, spectral index and the running of the spectral index at one-loop order. We find that the spectrum is slightly blue-tilted; hence, the amplitudes of fluctuations grow slightly toward the smaller scales. Then, in the second part, we apply the computation method used in the first part to a massless minimally coupled scalar with a quartic self-i…
We study constant-roll inflation in the presence of a gauge field coupled to an inflaton. By imposing the constant anisotropy condition, we find new exact anisotropic constant-roll inflationary solutions which include anisotropic power-law inflation as a special case. We also numerically show that the new anisotropic solutions are attractors in the phase space.
A Remedy of the Trans-Planckian Censorship Problem with Smooth Slow-roll to Power-law Inflation Transitions in Scalar Field Theory
Institute of Space Sciences (ICE, CSIC)（空间科学研究所）；ICREA（加泰罗尼亚高级研究学院）；L.N. Gumilyov Eurasian National University - Astana（欧亚国立大学）；Department of Physics, Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学物理系）
We show that a corpuscular description of gravity can lead to an inflationary scenario similar to Starobinsky's model without requiring the introduction of the inflaton field. All relevant properties are determined by the number of gravitons in the cosmological condensate or, equivalently, by their Compton length. In particular, the relation between the Hubble parameter $H$ and its time derivative $\dot H$ required by CMB observations at the end of inflation, as well as the (minimum) initial value of the slow-roll parameter, are naturally obtained from the Compton size of the condensate.
From the assumption that the slow roll parameter $ε$ has a Lorentzian form as a function of the e-folds number $N$, a successful model of a quintessential inflation is obtained, as succinctly studied in \cite{Benisty:2020xqm}. The form corresponds to the vacuum energy both in the inflationary and in the dark energy epochs and satisfies the condition to climb from small values of $ε$ to $1$ at the end of the inflationary epoch. We find the corresponding scalar Quintessential Inflationary potential with two flat regions. Moreover, a reheating mechanism is suggested with numerical estimation for …
We propose the study of constant-roll inflation in $F(R)$ gravity. We use two different approaches, one that relates an $F(R)$ gravity to well known scalar models of constant-roll and a second that examines directly the constant-roll condition in $F(R)$ gravity. With regards to the first approach, by using well known techniques, we find the $F(R)$ gravity which realizes a given constant-roll evolution in the scalar-tensor theory. We also perform a conformal transformation in the resulting $F(R)$ gravity and we find the Einstein frame counterpart theory. As we demonstrate, the resulting scalar …
We develop a nonperturbative method through the Hartree factorization to examine the quantum fluctuation effects on the single-field inflationary models in a spatially flat FRW cosmological space-time. Apart from the background field equation as well as the Friedmann equation with the corrections of quantum field fluctuations, the modified Mukhanov-Sasaki equations for the mode functions of the quantum scalar field are also derived by introducing the nonzero $Δ_B$ term. We consider the Universe undergoing the slow roll (SR)-ultra slow roll (USR) -slow roll (SR) inflation where in particular th…
In loop quantum cosmology (LQC) the big bang is replaced by a quantum bounce which is followed by a robust phase of super-inflation. Rather than growing unboundedly in the past, the Hubble parameter \emph{vanishes} at the bounce and attains a \emph{finite universal maximum} at the end of super-inflation. These novel features lead to an unforeseen implication: in presence of suitable potentials all LQC dynamical trajectories are funneled to conditions which virtually guarantee slow roll inflation with more than 68 e-foldings, {without any input from the pre-big bang regime}. This is in striking…
We propose a new approach to constant-roll warm inflation as a generalization of constant-roll inflation. Based on the $β$-function formalism, it is shown that constant-roll warm inflation models with a natural end fall into universality classes defined by three different types of $β$-functions, under the assumption that radiation energy density is quasi-stable. Given that warm inflation is completely specified by the $β$-function and dissipation coefficient ratio, we investigate whether or not the inflation can physically be realized for enough number of e-foldings at the background level for…
It is often claimed that the ultra-slow-roll regime of inflation, where the dynamics of the inflaton field are friction dominated, is a non-attractor and/or transient. In this work we carry out a phase-space analysis of ultra-slow roll in an arbitrary potential, $V(ϕ)$. We show that while standard slow roll is always a dynamical attractor whenever it is a self-consistent approximation, ultra-slow roll is stable for an inflaton field rolling down a convex potential with $M_{\scriptscriptstyle{\mathrm{Pl}}} V''>|V'|$ (or for a field rolling up a concave potential with $M_{\scriptscriptstyle{\mat…
In this work we shall investigate the phenomenological implications of the constant-roll condition on a $k$-Inflation theory of gravity. The latter theories are particularly promising, since these remained robust to the results of GW170817, since these have a gravitational wave speed $c_T=1$ in natural units. We shall mainly focus on the phenomenology of the $k$-Inflation models, with the only assumption being the slow-roll condition imposed on the first and fourth slow-roll parameters, and the constant-roll condition for the evolution of the scalar field. We present in detail the final form o…
Slow roll solutions to inflationary potentials have been widely believed to be the only universal attractor. Over the last few years there has been growing interest in a new class of inflationary models known as Constant-Roll Inflation. Constant roll solutions are a generalization of "ultra-slow roll" dynamics, where the first slow roll parameter is small, but the second slow roll parameter $η$ is larger than unity. In Ultra-slow Roll Inflation, the large-$η$ solution is a dynamical transient, relaxing exponentially to the attractor de Sitter solution. In the constant roll generalization, rece…
We numerically calculate the evolution of second order cosmological perturbations for an inflationary scalar field without resorting to the slow-roll approximation or assuming large scales. In contrast to previous approaches we therefore use the full non-slow-roll source term for the second order Klein-Gordon equation which is valid on all scales. The numerical results are consistent with the ones obtained previously where slow-roll is a good approximation. We investigate the effect of localised features in the scalar field potential which break slow-roll for some portion of the evolution. The…
The effective field theory (EFT) of inflation provides an essential picture to explore the effects of the unknown high energy physics in the single scalar field inflation models. For a generic EFT of inflation, possible high energy corrections to simple slow-roll inflation can modify both the propagating speed and dispersion relations of the cosmological scalar and tensor perturbations. With the arrival of the era of precision cosmology, it is expected that these high energy corrections become more important and have to be taken into account in the analysis with future precise observational da…
In this work, we studied the slow-roll approximation of cosmic inflation within the context of $f(R,T)$ gravity, where $R$ is the scalar curvature, and $T$ is the trace of the energy-momentum tensor. By choosing a minimal coupling between matter and gravity, we obtained the modified slow-roll parameters, the scalar spectral index ($n_s$), the tensor spectral index ($n_{\textrm{T}}$), and the tensor-to-scalar ratio ($r$). We computed these quantities for a general power-law potential, Natural & Quartic Hilltop inflation, and the Starobinsky model, plotting the trajectories on the $(n_s,r)$ plan…
We introduce warm quintessential inflation and study it in the weak dissipative regime. We consider the original quintessential inflation model, which approximates quartic chaotic inflation at early times and thawing quartic inverse-power-law quintessence at present. We find that the model successfully accounts for both inflation and dark energy observations, while it naturally reheats the Universe, thereby overcoming a major problem of quintessential inflation model-building.
We consider the effect of quantum diffusion on the dynamics of the inflaton during a period of ultra-slow-roll inflation. We extend the stochastic-$δ\mathcal{N}$ formalism to the ultra-slow-roll regime and show how this system can be solved analytically in both the classical-drift and quantum-diffusion dominated limits. By deriving the characteristic function, we are able to construct the full probability distribution function for the primordial density field. In the diffusion-dominated limit, we recover an exponential tail for the probability distribution, as found previously in slow-roll inf…
Primordial perturbations from ultra-slow-roll single-field inflation with quantum loop effects
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Physics, National Dong Hwa University（国立东华大学物理系）；Institute of Astronomy and Astrophysics, Academia Sinica（中央研究院天文及天文物理研究所）
[Abridged] The primordial scalar power spectrum is well constrained on large scales, primarily by the observations of the anisotropies in the cosmic microwave background (CMB). Over the last few years, it has been recognized that a sharp rise in power on small scales will lead to enhanced formation of primordial black holes (PBHs) and also generate secondary gravitational waves (GWs) of higher and, possibly, detectable amplitudes. It is well understood that scalar power spectra with COBE normalized amplitude on the CMB scales and enhanced amplitudes on smaller scales can be generated due to de…
After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter $ε_1$ if and only if $ε_2^2\ll 6/ε_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $ΔN$ formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.
There has been considerable recent interest in a new class of non-slow roll inflationary solutions known as \textit{constant roll} inflation. Constant roll solutions are a generalization of the ultra-slow roll (USR) solution, where the first Hubble slow roll parameter $ε$ is small, but the second Hubble slow roll parameter $η$ is not. While it is known that the USR solutions represent dynamical transients, there has been some disagreement in literature about whether or not large-$η$ constant roll solutions are attractors or are also a class of transient solutions. In this paper we show that th…
We implement a procedure by which the parameters present in the potential of Quartic Hilltop Inflation (QHI) are eliminated in favor of the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. By doing this it is posible to obtain in a straightforward and simple way the equations of a previous analysis where an analytical treatment of QHI in the large field limit is given. This procedure also allows a more precise discussion of general properties of the model. Also, using a constraint from the reheating epoch it is possible to find bounds for the parameters of the model as well as f…
Constant-roll inflation was recently introduced by Motohashi, Starobinsky and Yokoyama as a phenomenological way to parametrize deviations from the slow-roll scenarios. In this paper, we investigate the dynamics of both the background and the perturbations in this model, without making any slow-roll assumptions. The perturbation spectra are computed with an efficient and accurate novel method that allowed us to quickly scan the parameter space of constant-roll inflation. We derive the constraints on the model parameters from the cosmic microwave background anisotropy measurements provided by t…
