We study the linear perturbations of multi-field inflationary models governed by a Lagrangian which is a general function of the scalar fields and of a global kinetic term combining their spacetime gradients with an arbitrary field space metric. Our analysis includes k-inflation, DBI inflation and its multi-field extensions which have been recently studied. For this general class of models, we calculate the action to second order in the linear perturbations. We decompose the perturbations into an (instantaneous) adiabatic mode, parallel to the background trajectory, and entropy modes. We show that all the entropy modes propagate with the speed of light whereas the adiabatic mode propagates with an effective speed of sound. We also identify the specific combination of entropy modes which sources the curvature perturbation on large scales. We then study in some detail the case of two scalar fields: we write explicitly the equations of motion for the adiabatic and entropy modes in a compact form and discuss their quantum fluctuations and primordial power spectra.
After a brief summary of general relativity and cosmology, we present the basic concepts underlying inflation, the currently best motivated models for the early Universe. We describe the simplest inflation models, based on a single scalar field, and discuss how primordial cosmological perturbations are generated. We then review some recent developments concerning multi-field inflation models, in particular multi-field Dirac-Born-Infeld inflation.
We study cosmological perturbations in two-field inflation, allowing for non-standard kinetic terms. We calculate analytically the spectra of curvature and isocurvature modes at Hubble crossing, up to first order in the slow-roll parameters. We also compute numerically the evolution of the curvature and isocurvature modes from well within the Hubble radius until the end of inflation. We show explicitly for a few examples, including the recently proposed model of `roulette' inflation, how isocurvature perturbations affect significantly the curvature perturbation between Hubble crossing and the …
We briefly review the standard derivation of the spectra of cosmological perturbations in the simplest models of inflation. We then consider models with several scalar fields, described by Lagrangians with an arbitrary dependence on the kinetic terms. We illustrate our general formalism with the case of multi-field DBI inflation.
The holographic entropy bound is used to estimate the quantum-gravitational discreteness of inflationary perturbations. In the context of scalar inflaton perturbations produced during standard slow-roll inflation, but assuming that horizon-scale perturbations ``freeze out'' in discrete steps separated by one bit of total observable entropy, it is shown that the Hilbert space of a typical horizon-scale inflaton perturbation is equivalent to that of about 10^5 binary spins-- approximately the inverse of the final scalar metric perturbation amplitude, independent of other parameters. Holography t…
We extend the theory of cosmological perturbations to the case when the ``matter'' Lagrangian is an arbitrary function of the scalar field and its first derivatives. In particular, this extension provides a unified description of known cases such as the usual scalar field and the hydrodynamical perfect fluid. In addition, it applies to the recently proposed k-inflation, which is driven by non-minimal kinetic terms in the Lagrangian. The spectrum of quantum fluctuations for slow-roll and power law k-inflation is calculated. We find, for instance, that the usual ``consistency relation'' between …
At energies higher than the brane tension, the dynamics of a scalar field rolling down a potential are modified relative to the predictions of General Relativity. The modifications imply, among other things, that steeper potentials can be used to drive an epoch of slow--roll inflation. We investigate the evolution of entropy and adiabatic modes during inflation driven by two scalar fields confined on the brane. We show that the amount of entropy perturbations produced during inflation is suppressed compared to the predictions made by General Relativity. As a consequence, the initial conditions…
It has been pointed out that the perturbation spectrum predicted by inflation may be sensitive to a natural ultraviolet cutoff, thus potentially providing an experimentally accessible window to aspects of Planck scale physics. A priori, a natural ultraviolet cutoff could take any form, but a fairly general classification of possible Planck scale cutoffs has been given. One of those categorized cutoffs, also appearing in various studies of quantum gravity and string theory, has recently been implemented into the standard inflationary scenario. Here, we continue this approach by investigating it…
Curvature perturbations generated after the end of inflation are estimated for a generic multi-field inflationary potential. The curvature perturbations are generated by conversion of inflationary isocurvature perturbations, similar to curvaton and inhomogeneous reheating scenarios. During inflation, the trajectory along which the fields are rolling down will fluctuate. When inflation ends, the fields enter the oscillatory phase along different directions in different parts of the universe. Different oscillatory trajectories can have different effective equations of state and reheating rates, …
We study the fourth order action of the comoving curvature perturbation in an inflationary universe in order to understand more systematically the de Sitter limit in nonlinear cosmological perturbation theory. We derive the action of the curvature perturbation to fourth order in the comoving gauge, and show that it vanishes sufficiently fast in the de Sitter limit. By studying the de Sitter limit, we then extrapolate to the n'th order action of the comoving curvature perturbation and discuss the slow-roll order of the n-point correlation function.
The formation of frozen classical perturbations from vacuum quantum fluctuations during inflation is described as a unitary quantum process with apparent "decoherence" caused by the expanding spacetime. It is argued that the maximum observable information content per comoving volume in classical modes is subject to the covariant entropy bound at the time those modes decohere, leading to a new quantitative bound on the information contained in frozen field modes in phase space. This bound implies holographic correlations of large-scale cosmological perturbations that may be observable.
In the present paper, we study the non-Gaussianity of multiple-field inflation model using the method of the field equation. We start from reviewing the background and the perturbation theory of multiple-field inflation, and then derive the Klein-Gorden equation for the perturbations at second order. Afterward, we calculate the tree-level bispectrum of the fields' perturbations and finally give the corresponding parameter $f_{NL}$ for the curvature perturbation $ζ$ in virtue of the $δN$ formalism. We also compare our result with the one already obtained from the Lagrangian formalism, and find …
We observe that when a local patch in a radiation filled Robertson-Walker universe inflates by some reason, outside perturbations can enter into the inflating region. Generally, the physical wavelengths of these perturbations become larger than the Hubble radius as they cross into the inflating space and their amplitudes freeze out immediately. It turns out that the corresponding power spectrum is not scale invariant. Although these perturbations cannot reach out to a distance inner observer shielded by a de Sitter horizon, they still indicate a curious boundary effect in local inflationary sc…
We use the dS/CFT correspondence and bulk gravity to predict the form of the renormalized holographic three-point correlation function of the operator which is dual to the inflaton field perturbation during single-field, slow-roll inflation. Using Maldcaena's formulation of the correspondence, this correlator can be related to the three-point function of the curvature perturbation generated during single-field inflation, and we find exact agreement with previous bulk QFT calculations. This provides a consistency check on existing derivations of the non-Gaussianity from single-field inflation a…
The slow-roll approximation to inflation is ultimately justified by the presence of inflationary attractors for the orbits of the solutions of the dynamical equations in phase space. There are many indications that the inflaton field couples nonminimally to the spacetime curvature: the existence of attractor points for inflation with nonminimal coupling is demonstrated, subject to a condition on the inflaton potential and the value of the coupling constant.
We explain why it is so difficult and perhaps even impossible to increase the cosmological tensor-to-scalar perturbation ratio during the post-inflationary evolution of the universe. Nevertheless, contrary to some recent claims, tensor perturbations can be relatively large in the simplest inflationary models which do not violate any rules of modern quantum field theory.
These lectures are intended to give a pedagogical introduction to the main current picture of the very early universe. After elementary reviews of general relativity and of the standard Big Bang model, the following subjects are discussed: inflation, the classical relativistic theory of cosmological perturbations and the generation of perturbations from scalar field quantum fluctuations during inflation.
We propose a simple realization of hybrid inflation in supergravity, which does not require any unnaturally small parameters. Inflation in this scenario is relatively short, and inflationary density perturbations fall down on large scales, which corresponds to a blue spectrum with n > 1. Superheavy strings which are formed after inflation in this scenario produce density perturbations of comparable magnitude. In this mixed perturbation (MP) model one may have microwave anisotropy produced by strings and galaxy formation due to inflationary perturbations.
We explore a new possibility that some inflaton fields in multi-field inflation models satisfy the observed value of the spectral index so that the curvature perturbation generated by them through post-inflationary dynamics may be relevant for the current observations. We illustrate this point using a simple and reasonable multi-field chaotic inflation model motivated from string theory. Possible non-Gaussianity and the tensor-to-scalar ratio are briefly addressed.
We construct a warm inflation model with inflaton field non-minimally coupled to induced gravity on a warped DGP brane. We incorporate possible modification of the induced gravity on the brane in the spirit of $f(R)$-gravity. We study cosmological perturbations in this setup. In the case of two field inflation such as warm inflation, usually entropy perturbations are generated. While it is expected that in the case of one field inflation these perturbations to be removed, we show that even in the absence of the radiation field, entropy perturbations are generated in our setup due to non-minima…
We review recent developments in the theory of inflation and cosmological perturbations produced from inflation. After a brief introduction of the standard, single-field slow-roll inflation, and the curvature and tensor perturbations produced from it, we discuss possible sources of nonlinear, non-Gaussian perturbations in other models of inflation. Then we describe the so-called $δN$ formalism, which is a powerful tool for evaluating nonlinear curvature perturbations on super Hubble scales.
On compactification from higher dimensions, a single free massive scalar field gives rise to a set of effective four-dimensional scalar fields, each with a different mass. These can cooperate to drive a period of inflation known as assisted inflation. We analyze the dynamics of the simplest implementation of this idea, paying particular attention to the decoupling of fields from the slow-roll regime as inflation proceeds. Unlike normal models of inflation, the dynamics does not become independent of the initial conditions at late times. In particular, we estimate the density perturbations obta…
We propose a new mechanism to generate density perturbations in inflationary models. Spatial fluctuations in the decay rate of the inflaton field to ordinary matter lead to fluctuations in the reheating temperature. We argue that in most realistic models of inflation the coupling of the inflaton to normal matter is determined by the vacuum expectation values of fields in the theory. If those fields are light during inflation (this is a generic situation in the minimal models of supersymmetric inflation) they will fluctuate leading to density perturbations through the proposed mechanism. We sho…
We present a class of very simple inflationary models of two scalar fields which leads to nongaussian isothermal perturbations with "blue" spectrum, n > 1. One of the models is inspired by supersymmetric theories where light scalar fields naturally acquire masses of the order of the Hubble constant H during inflation. Another model presumes that one of the fields has a nonminimal interaction with gravity. By a slight modification of parameters of these models one can obtain either gaussian isothermal perturbations, or nongaussian adiabatic perturbations with n > 1.
We study a scalar-tensor cosmological model where the Einstein tensor is non-minimally coupled to the free scalar field dynamics. Using FRW metric, we investigate the behavior of scale factor for vacuum, matter and dark energy dominated eras. Especially, we focus on the inflationary behavior at early universe. Moreover, we study the perturbation analysis of this model in order to confront the inflation under consideration with the observational results.
We show that the non-linear evolution of long wavelength perturbations may be important in a wide class of inflationary scenarios. We develop a solution for the evolution of such nonlinear perturbations which is exact to first order in a gradient expansion. As a first application, we demonstrate that in single field models of inflation there can be no parametric amplification of super-Hubble modes during reheating. We consider the implications of the solution for recent discussions of the back-reaction effect of long wavelength perturbations on the background geometry, give a new derivation of…
It is shown that non-adiabatic cosmological perturbations cannot arise during the period of reheating following inflation with a single scalar inflaton field.
There are many inflationary models in which inflaton field does not satisfy the slow-roll condition. However, in such models, it is always difficult to generate the curvature perturbation during inflation. Thus, to generate the curvature perturbation, one must introduce another component to the theory. To cite a case, curvatons may generate dominant part of the curvature perturbation after inflation. However, we have a question whether it is unrealistic to consider the generation of the curvature perturbation during inflation without slow-roll. Assuming multi-field inflation, we encounter the …
The dominant contribution to the primordial curvature perturbation may be generated at the end of inflation. Taking the end of inflation to be sudden, formulas are presented for the spectrum, spectral tilt and non-gaussianity. They are evaluated for a minimal extension of the original hybrid inflation model.
We study the dynamics of multiple scalar fields and a barotropic fluid in an FLRW-universe. The scalar potential is a sum of exponentials. All critical points are constructed and these include scaling and de Sitter solutions. A stability analysis of the critical points is performed for generalized assisted inflation, which is an extension of assisted inflation where the fields mutually interact. Effects in generalized assisted inflation which differ from assisted inflation are emphasized. One such a difference is that an (inflationary) attractor can exist if some of the exponential terms in th…
We study cosmological perturbations and observational aspects for mutated hilltop model of inflation. Employing mostly analytical treatment, we evaluate observable parameters during inflation as well as post-inflationary perturbations. This further leads to exploring observational aspects related to Cosmic Microwave Background (CMB) radiation. This semi-analytical treatment reduces complications related to numerical computation to some extent for studying the different phenomena related to CMB angular power spectrum for mutated hilltop inflation.
This review article aims at presenting the theory of inflation. We first describe the background spacetime behavior during the slow-roll phase and analyze how inflation ends and the Universe reheats. Then, we present the theory of cosmological perturbations with special emphasis on their behavior during inflation. In particular, we discuss the quantum-mechanical nature of the fluctuations and show how the uncertainty principle fixes the amplitude of the perturbations. In a next step, we calculate the inflationary power spectra in the slow-roll approximation and compare these theoretical predic…
This paper considers warm inflation as an interesting application of multi-field inflation. Delta-N formalism is used for the calculation of the evolution of the curvature perturbations during warm inflation. Although the perturbations considered in this paper are decaying after the horizon exit, the corrections to the curvature perturbations sourced by these perturbations can remain and dominate the curvature perturbations at large scales. In addition to the typical evolution of the curvature perturbations, inhomogeneous diffusion rate is considered for warm inflation, which may lead to signi…
Based on the stochastic gravity, we study the loop corrections to the scalar and tensor perturbations during inflation. Since the loop corrections to scalar perturbations suffer infrared (IR) divergence, we consider the IR regularization to obtain the finite value. We find that the loop corrections to the scalar perturbations are amplified by the e-folding; in other words there appear the logarithmic correction, just as discussed by M.Sloth et al. On the other hand, we find that the tensor perturbations do not suffer from infrared divergence.
Following our previous work in noncommutative braneworld inflation (arXiv:0911.4418), in this paper we use the smeared, coherent state picture of noncommutativity to study evolution of perturbations in a noncommutative braneworld scenario. We show that in this setup, the early stage of the universe evolution has a phantom evolution with imaginary effective sound speed. We show also that the amplitude of perturbations in the commutative regime decays faster than the noncommutative regime with the same parameter values, and as a result we need smaller number of e-folds in the noncommutative regi…
Isocurvature perturbations naturally occur in models of inflation consisting of more than one scalar field. In this paper we calculate the spectrum of isocurvature perturbations generated at the end of inflation for three different inflationary models consisting of two canonical scalar fields. The amount of non-adiabatic pressure present at the end of inflation can have observational consequences through the generation of vorticity and subsequently the sourcing of B-mode polarisation. We compare two different definitions of isocurvature perturbations and show how these quantities evolve in dif…
We review the theory of inflationary perturbations. Perturbations at both linear and nonlinear orders are reviewed. We also review a variety of inflation models, emphasizing their signatures on cosmic perturbations.
The effect of the initial conditions in inflation on scalar and tensor perturbations is investigated. Formulae for the power spectra of gravitational waves and curvature perturbations for any initial conditions in inflation are derived, and the ratio of scalar to tensor perturbations and spectral index are calculated. The formulation is applied to some simplified pre-inflationary cosmological models, and the differences of the ratio and spectral index are investigated with respect to two matching conditions. In addition, the present power spectrum of gravitational waves is derived. The propose…
We develop the gradient expansion formalism for shift-symmetric Galileon-type actions. We focus on backgrounds that undergo inflation, work in the synchronous gauge, and obtain a general solution up to second order without imposing extra conditions at first order. The solution simplifies during the late stages of inflation. We also define a curvature perturbation conserved up to first order.
The large-scale power deficit in the cosmic microwave background fluctuations might be relevant with the physics of pre-inflation, a bounce, or a superinflationary phase preceding slow-roll inflation, which can provide a singular-free realization of inflation. We investigate the primordial perturbations from such pre-inflationary evolutions, which generally may consist of multiple phases with different background dynamics, and give a universal formula for the power spectrum of primordial perturbations in terms of the recursive Bogoliubov coefficients. We also apply our formula to corresponding…
By making a suitable generalization of the Starobinsky stochastic inflation, we propose a classical phase space formulation of stochastic inflation which may be used for a quantitative study of decoherence of cosmological perturbations during inflation. The precise knowledge of how much cosmological perturbations have decohered is essential to the understanding of acoustic oscillations of cosmological microwave background (CMB) photons. In order to show how the method works, we provide the relevant equations for a self-interacting inflaton field. For pedagogical reasons and to provide a link t…
The scale--independence of the primordial curvature perturbation suggests that it comes from the vacuum fluctuation during inflation of a light scalar field. This field may be the inflaton, or a different `curvaton' field. The observation of primordial non--gaussianity would be a smoking gun for the curvaton model, while the observation of gravitational waves originating during slow--roll inflation would rule out the model.
We provide a concise review on multi-field inflation and cosmological perturbations. We discuss convenient and physically meaningful bases in terms of which perturbations can be systematically studied. We give formal accounts on the gauge fixing conditions and present the perturbation action in two gauges. We also briefly review non-linear perturbations.
We analyze the behavior of linear perturbations in vector inflation. In contrast to the scalar field inflation, the linearized theory with vector fields contains couplings between scalar, vector and tensor modes. The perturbations decouple only in the ultraviolet limit, which allows us to carry out the canonical quantization. Superhorizon perturbations can be approximately analyzed due to suppressed mixing between different modes in the small fields models. We find that the vector perturbations of the metric decay exponentially, but the scalar and tensor modes could remain weakly coupled throu…
The effective field theory (EFT) of inflation provides a natural framework to study the new physical effects on primordial perturbations. Recently a healthy extension of the EFT of inflation with high-order operators has been proposed, which avoids ghosts and meanwhile leads to a nonlinear dispersion relation of the scalar perturbations. This paper is devoted to studying the effects of these high-order operators by using the uniform asymptotic approximation method. In particular, we first construct the approximate analytical solution to the mode function of the scalar perturbations. Because of…
Even if non-adiabatic perturbations are generated in multi-field inflation, the perturbations will become adiabatic if the universe after inflation enters an era of local thermal equilibrium, with no non-zero conserved quantities, and will remain adiabatic as long as the wavelength is outside the horizon, even when local thermal equilibrium no longer applies. Small initial non-adiabatic perturbations associated with imperfect local thermal equilibrium remain small when baryons are created from out-of-equilibrium decay of massive particles, or when dark matter particles go out of local thermal …
We continue the analysis of perturbations in vector inflation. The dominant theme of this paper is the long wavelength limit of perturbations in small fields inflation and the controversial issue of its linear stability. We explain the nature of longitudinal modes, describe how they evolve, and show that they are not as harmful as it could seem at the first glance. On the other hand, the gravitational waves instability in large fields models is shown explicitly. It strongly limits a potential applicability of the recently proposed (delta N)-type approach to vector inflationary perturbations. F…
An inflationary universe scenario in the context of tachyon field is studied. This study is done from an ansantz for the effective potential of cosmological perturbations $U(η)$. We describe in great detail the analytical solutions of the scalar and tensor perturbations for two different ansantz in the effective potential of cosmological perturbations; the Easther's model and an effective potential similar to power law inflation. Also we find from the background equations that the effective tachyonic potential $V(φ)$, in both models satisfy the properties of a tachyonic potential. We consider …
In this paper we study the generation of primordial perturbations in a cosmological setting of bigravity during inflation. We consider a model of bigravity which can reproduce the $Λ$CDM background and large scale structure and a simple model of inflation with a single scalar field and a quadratic potential. Reheating is implemented with a toy-model in which the energy density of the inflaton is entirely dissipated into radiation. We present analytic and numerical results for the evolution of primordial perturbations in this cosmological setting. We find that the amplitude of tensor perturbati…
The purpose of these lectures is to give a pedagogical introduction to inflation and the production of primordial perturbations, as well as a review of some of the latest developments in this domain. After a short introduction, we review the main principles of the Hot Big Bang model, as well as its limitations. This motivates the study of cosmological inflation induced by a slow-rolling scalar field. We then turn to the analysis of cosmological perturbations, and explain how the vacuum quantum fluctuations are amplified during an inflationary phase. The next step consists in relating the pertu…
In a recent paper of Linde and Mukhanov, a hybrid inflationary model with nongaussian density perturbations and a ``blue'' spectral index n > 1 was presented. In this comment we point out that this model can be considered as a particular realization of the general framework for the spectrum and statistics of the CMBR we proposed in astro-ph/9611208, based on the hypothesis of conformal invariance. The implication of this hypothesis is that density perturbations are nongaussian with a spectral index n > 1.
Cosmological perturbations of massive higher-spin fields are generated during inflation, but they decay on scales larger than the Hubble radius as a consequence of the Higuchi bound. By introducing suitable couplings to the inflaton field, we show that one can obtain statistical correlators of massive higher-spin fields which remain constant or decay very slowly outside the Hubble radius. This opens up the possibility of new observational signatures from inflation.
We study the contribution of spectator massive scalar fields to the inflationary density perturbations through the universal gravitational coupling. We find that such contribution has several remarkable properties: it does not decrease as the mass of the spectator field increases; it has a significant size and cannot be turned off by any adjustable parameters; and it applies to all massive scalars existed during inflation, making the overall effect unexpectedly large. As a result, the primordial density perturbations are anomalously sensitive to the high energy physics.
The recently proposed trans-Planckian censorship conjecture (TCC) seems to require that the energy scale of inflation is significantly lower than the Planck scale $(H_\text{inf}<10^{-20} \Mpl)$. This, in turn, implies that the tensor-to-scalar ratio for inflation is negligibly small, \textit{independent} of assumptions of slow-roll or even of having a single scalar field, thus ruling out inflation if primordial tensor modes are ever observed. After demonstrating the robustness and generality of these bounds, we show that having an excited initial state for cosmological perturbations seems to b…
We study the cosmological perturbations created during the New Higgs inflationary phase. In the New Higgs Inflation, the Higgs boson is kinetically coupled to the Einstein tensor and only three perturbative degrees of freedom, a scalar and two tensorial (gravitational waves), propagate during Inflation. Scalar perturbations are found to match the latest WMAP-7yrs data within Standard Model Higgs parameters. Primordial gravitational waves also, although propagating with superluminal speed, are consistent with present data. Finally, we estimate the values of the parameter of the New Higgs Inflat…
We propose a new approach for calculating the curvature perturbations produced during inflation in the stochastic formalism. In our formalism, the fluctuations of the e-foldings are directly calculated without perturbatively expanding the inflaton field and they are connected to the curvature perturbations by the $δN$ formalism. The result automatically includes the contributions of the higher order perturbations because we solve the equation of motion non-perturbatively. In this paper, we analytically prove that our result (the power spectrum and the nonlinearity parameter) is consistent with…
We study generalized Galileons as a framework to develop the most general single-field inflation models ever, Generalized G-inflation, containing yet further generalization of G-inflation, as well as previous examples such as k-inflation, extended inflation, and new Higgs inflation as special cases. We investigate the background and perturbation evolution in this model, calculating the most general quadratic actions for tensor and scalar cosmological perturbations to give the stability criteria and the power spectra of primordial fluctuations. It is pointed out in the Appendix that the Horndes…
We compute the full bispectra, namely both auto- and cross- bispectra, of primordial curvature and tensor perturbations in the most general single-field inflation model whose scalar and gravitational equations of motion are of second order. The formulae in the limits of k-inflation and potential-driven inflation are also given. These expressions are useful for estimating the full bispectra of temperature and polarization anisotropies of the cosmic microwave background radiation.
