The form of the inflationary potential is severely restricted if one requires that it be natural in the technical sense, i.e. terms of unrelated origin are not required to be correlated. We determine the constraints on observables that are implied in such natural inflationary models, in particular on $r$, the ratio of tensor to scalar perturbations. We find that the naturalness constraint does not require $r$ to be lare enough to be detectable by the forthcoming searches for B-mode polarisation in CMB maps. We show also that the value of $r$ is a sensitive discriminator between inflationary models.
We suggest that the cosmological constant has been relaxed to its present, very small value during the inflationary stage of the evolution of the Universe. This requires relatively low scale, very long duration and unconventional source of inflation. We present a concrete mechanism of the cosmological constant relaxation at the inflationary epoch.
Formulae are derived for the spectra of scalar curvature perturbations and gravitational waves produced during inflation, special cases of which include power law inflation, natural inflation in the small angle approximation and inflation in the slow roll approximation.
A new idea of the cosmological constant is proposed in this paper. Due to the horizon is limited, the quantum fluctuation of the inflaton field is not zero, a nonzero vacuum energy is remained as a residual inflationary energy of an unusual potential, however the true stable vacuum energy is zero fortunately. A unified model of the cosmological constant and the chaotic inflation is proposed, which satisfies almost all cosmological phenomenology and will can be tested by data of the cosmic large scale structure.
From a general ansatz for the effective potential of cosmological perturbations we find new, exact solutions in single-scalar-field inflation: a three parameter family of exact inflationary solutions that encompasses all exact solutions that have been known previously (power-law inflation, Easther's model, and a generalised version of Starobinsky's solution). The main feature of this new family is that the spectral indices are scale dependent.
We calculate power spectra of cosmological perturbations at high accuracy for two classes of inflation models. We classify the models according to the behaviour of the Hubble distance during inflation. Our approximation works if the Hubble distance can be approximated either to be a constant or to grow linearly with cosmic time. Many popular inflationary models can be described in this way, e.g., chaotic inflation with a monomial potential, power-law inflation and inflation at a maximum. Our scheme of approximation does not rely on a slow-roll expansion. Thus we can make accurate predictions f…
Cosmological inflation provides the simplest and most promising mechanism for generating fluctuations for structure formation. Using powerful Hamilton-Jacobi methods, I will describe (1) how to compute density fluctuations and cosmic microwave anisotropies arising from inflation, and (2) improvements of the Zel'dovich approximation describing gravitational collapse. I compare these results with the latest cosmological observations.
Scalar and tensorial cosmological perturbations generated in warm inflationary scenarios whose matter-radiation fluid is endowed with a viscous pressure are considered. Recent observational data from the WMAP experiment are employed to restrict the parameters of the model. Although the effect of this pressure on the matter power spectrum is of the order of a few percent, it may be detected in future experiments.
The hypothesis that the Universe underwent a period of exponential expansion at very early times has become the most popular theory of the early Universe. Not only does it solve some of the problems of standard big bang cosmology, but it also provides a causal theory for the origin of primordial density fluctuations which may explain the observed density inhomogeneities and cosmic microwave fluctuations in the Universe. In these lectures, a review of the basic principles of inflationary cosmology is given, focusing on the theory of the origin and growth of cosmological perturbations. The lectu…
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.
A quantum field theory warm inflation model is presented that solves the horizon/flatness problems. The model obtains, from the elementary dynamics of particle physics, cosmological scale factor trajectories that begin in a radiation dominated regime, enter an inflationary regime and then smoothly exit back into a radiation dominated regime, with nonnegligible radiation throughout the evolution.
We consider approaches to cosmological parameter estimation in the inflationary cosmology, focussing on the required accuracy of the initial power spectra. Parametrizing the spectra, for example by power-laws, is well suited to testing the inflationary paradigm but will only correctly estimate cosmological parameters if the parametrization is sufficiently accurate, and we investigate conditions under which this is achieved both for present data and for upcoming satellite data. If inflation is favoured, reliable estimation of its physical parameters requires an alternative approach adopting its…
We derive the implications for anisotropies in the cosmic microwave background following from a model of inflation in which a bare cosmological constant is gradually screened by an infrared process in quantum gravity. The model predicts that the amplitude of scalar perturbations is $A_S = (2.0 \pm .2) \times 10^{-5}$, that the tensor-to-scalar ratio is $r \approx 1.7 \times 10^{-3}$, and that the scalar and tensor spectral indices are $n \approx .97$ and $n_T \approx -2.8 \times 10^{-4}$, respectively. By comparing the model's power spectrum with the COBE 4-year RMS quadrupole, the mass scale …
We study the evolution of regions of space with various initial field values for a simple theory that can support hybrid inflation. Only very narrow domains within the range of initial field values below the Planck scale lead to the onset of inflation. This implies a severe fine tuning for the initial configuration that will produce inflation.
We investigate cosmological moduli problem in the gauge-mediated supersymmetry breaking (GMSB). A mini-inflation (oscillating inflation) takes place when a scalar field corresponding to the flat direction in GMSB oscillates along the logarithmic potential induced by the gauge-mediation mechanism. It is shown that this oscillating inflation can sufficiently dilute the relic abundance of the string moduli for some ranges of parameters in the GMSB models.
We extend the idea of assisted inflation to the case of power-law potentials and demonstrate the simultaneous resolution of two major problems that plague chaotic inflation. The implementation of the same idea in the framework of a 5-dimensional, scalar field theory leads to a model of chaotic inflation free of fine-tuning.
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the power-law inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to ninth-order of the phase-integral approximation. We show that, the phase-integral approximation exactly reproduces the shape of the power spectra for scalar and tensor perturbations as well as the spectral indices. We compare the accuracy of the phase-integral approximation with the results for the power spectrum obtained with the slow-roll and unifor…
In spite of its great phenomenological success, current models of scalar field-driven inflation suffer from important unresolved conceptual issues. New fundamental physics will be required to address these questions. String theory is a candidate for a unified quantum theory of all four forces of nature. As will be shown, string theory may lead to a cosmological background quite different from an inflationary cosmology, and may admit a new stringy mechanism for the origin of a roughly scale-invariant spectrum of cosmological fluctuations.
We compute the spectrum of cosmological perturbations in a scenario in which inflation is driven by radiation in a non-commutative space-time. In this scenario, the non-commutativity of space and time leads to a modified dispersion relation for radiation with two branches, which allows for inflation. The initial conditions for the cosmological fluctuations are thermal. This is to be contrasted with the situation in models of inflation in which the accelerated expansion of space is driven by the potential energy of a scalar field, and in which the fluctuations are of quantum vacuum type. We fin…
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.
In the framework of Inflationary theory, the assumption that the quantum state of the perturbations is a non-vacuum state leads to a difficulty: non-vacuum initial states imply, in general, a large energy density of inflaton field quanta, not of a cosmological term type, that could prevent the inflationary phase. In this short note, we discuss in detail why this is so, keeping an eye on possible non-Gaussian features due to considering generic non-vacuum initial states.
We propose a paradigm for the inflation and the vanishing cosmological constant in a unified way with the self-tuning solutions of the cosmological constant problem. Here, we consider a time-varying cosmological constant in self-tuning models of the cosmological constant. As a specific example, we demonstrate it with a 3-form field in 5D.
In this article we review the theory of cosmological inflation with a particular focus on the beautiful connection it provides between the physics of the very small and observations of the very large. We explain how quantum mechanical fluctuations during the inflationary era become macroscopic density fluctuations which leave distinct imprints in the cosmic microwave background (CMB). We describe the physics of anisotropies in the CMB temperature and polarization and discuss how CMB observations can be used to probe the primordial universe.
We show a model-independent fine-tuning issue in the DBI inflationary mechanism. DBI inflation requires a warp factor h small enough to sufficiently slow down the inflaton. On the other hand, the Einstein equation in extra dimensions under the inflationary background deforms the warp space in the IR side. Generically these two locations coincide with each other, spoiling the DBI inflation. The origin and tuning of this ``h-problem'' is closely related, through the AdS/CFT duality, to those of the well-known ``eta-problem'' in the slow-roll inflationary mechanism.
Many models of supersymmetry breaking involve particles with weak scale mass and Planck mass suppressed couplings. Coherent production of such particles in the early universe destroys the successful predictions of nucleosynthesis. We show that this problem may be solved by a brief period of weak scale inflation. Furthermore the inflaton potential for such an inflation naturally arises from the same assumptions which lead to the cosmological problem. Successful baryogenesis and preservation of density fluctuations for large scale structure formation are also possible in this scenario.
Problem of cosmological singularity is discussed in the framework of gauge theories of gravitation. Generalizing cosmological Friedmann equations (GCFE) for homogeneous isotropic models including scalar fields and usual gravitating matter are introduced. It is shown that by certain restrictions on equation of state of gravitating matter and indefinite parameter of GCFE generic feature of inflationary cosmological models of flat, open and closed type is their regular bouncing character.
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…
Hybrid inflation faces two well-known problems: the blue spectrum of the non-supersymmetric version of the model and the fine-tuning of the initial conditions of the fields leading to sufficient inflation to account for the standard cosmological problems. They are investigated by studying the exact two-fields dynamics instead of assuming slow-roll. When the field values are restricted to be less than the reduced Planck mass, a non-negligible part of the initial condition space (around 15% depending on potential parameters) leads to successful inflation. Most of it is located outside the usual …
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 cosmological constant problem is how one chooses, without fine-tuning, one singular point $Λ_{eff}=0$ for the 4D cosmological constant. We argue that some recently discovered {\it weak self-tuning} solutions can be viewed as blowing-up this one point into a band of some parameter. These weak self-tuning solutions may have a virtue that only de Sitter space solutions are allowed outside this band, allowing an inflationary period. We adopt the hybrid inflation at the brane to exit from this inflationary phase and to enter into the standard Big Bang cosmology.
An introductory account is given of the inflationary cosmology, which postulates a period of accelerated expansion during the Universe's earliest stages. The historical motivation is briefly outlined, and the modelling of the inflationary epoch explained. The most important aspect of inflation is that it provides a possible model for the origin of structure in the Universe, and key results are reviewed, along with a discussion of the current observational situation and outlook.
We investigate the cosmological perturbation of two-scalar field model during the reheating phase after inflation. Using the exact solution of the perturbation in long-wavelength limit, which is expressed in terms of the background quantities, we analyze the behavior of the metric perturbation. The oscillating inflaton field gives rise to the parametric resonance of the massless scalar field and this leads to the amplification of the iso-curvature mode of the metric perturbations.
I give a brief overview of brane-antibrane inflation, with emphasis on the problems of tuning to get a flat potential in the KKLMMT framework, and recent work on the nature of superpotential corrections in that model.
A new idea of deriving a cosmological term from an underlying theory has been proposed in order to explain the expansion history of the universe. We obtain the scale factor with this derived cosmological term and demonstrate that it reflects all the characteristics of the expanding universe in different era so as to result in a transition from inflation to late acceleration through intermediate decelerating phases by this single entity. We further discuss certain observational aspects of this paradigm.
The motivation for studying the rolling tachyon and non-commutative inflation comes from string theory. In the tachyon inflation scenario, metric perturbations are created by tachyon field fluctuations during inflation. We drive the exact mode equation for scalar perturbation of the metric and investigate the cosmological perturbations in the commutative and non-commutative inflationary spacetime driven by the tachyon field which have a Born-Infeld Lagrangian.
We find exact exponentially expanding and contracting de Sitter solutions of the spatially homogeneous TeVeS cosmological equations of motion in the vacuum TeVeS model and a power law accelerated expanding solution in thepresence of an additional ideal fluid with equation of state parameter $-5/3 < ω< -1$. A preliminary stability analysis shows that the expanding vacuum solution is stable, while in the ideal fluid case stability depends on model parameter values. These solutions might provide a basis for incorporating early-time inflation or late-time accelerated expansion in TeVeS cosmology.
We develop a general formalism for analyzing linear perturbations in multiple-field cosmological inflation based on the gauge-ready approach. Our inflationary model consists of an arbitrary number of scalar fields with non-minimal kinetic terms. We solve the equations for scalar- and tensor-type perturbations during inflation to the first order in slow-roll, and then obtain the super-horizon solutions for adiabatic and isocurvature perturbations after inflation. Analytic expressions for power-spectra and spectral indices arising from multiple-field inflation are presented.
Previously defined covariant and gauge-invariant perturbation variables, representing, e.g., the fractional spatial energy density gradient on hypersurfaces of constant expansion, are used to simplify the linear perturbation analysis of a classical scalar field. With the help of conserved quantities on large scales we establish an exact first-order relation between comoving fluid energy density perturbations at `reentry' into the horizon and corresponding scalar field energy density perturbations at the first Hubble scale crossing during an early de Sitter phase of a standard inflationary scen…
We present the first computation of the cosmological perturbations generated during inflation up to second order in deviations from the homogeneous background solution. Our results, which fully account for the inflaton self-interactions as well as for the second-order fluctuations of the background metric, provide the exact expression for the gauge-invariant curvature perturbation bispectrum produced during inflation in terms of the slow-roll parameters or, alternatively, in terms of the scalar spectral $n_S$ and and the tensor to adiabatic scalar amplitude ratio $r$. The bispectrum represents…
We revisit the problem of constraining steps in the inflationary potential with cosmological data. We argue that a step in the inflationary potential produces qualitatively similar oscillations in the primordial power spectrum, independently of the details of the inflationary model. We propose a phenomenological description of these oscillations and constrain these features using a selection of cosmological data including the baryonic peak data from the correlation function of luminous red galaxies in the Sloan Digital Sky Survey. Our results show that degeneracies of the oscillation with stan…
It is commonly believed that our Universe has experienced two different stages of accelerated expansion. The early stage is known as inflation and the current acceleration is driven by dark energy. Observing inflation and dark energy dynamics are among the most important aspects of the current cosmological research. In this paper we make a first detailed probe of possible degeneracies between dynamical inflation and dark energy in light of current cosmological observations. We find the inclusion of inflation and dark energy parameters together make the parameter spaces broader, but interesting…
We study the presence of chaotic behavior in phase space in the pre-inflationary stage of hybrid inflation models. This is closely related to the problem of initial conditions associated to these inflationary type of models. We then show how an expected dissipative dynamics of fields just before the onset of inflation can solve or ease considerably the problem of initial conditions, driving naturally the system towards inflation. The chaotic behavior of the corresponding dynamical system is studied by the computation of the fractal dimension of the boundary, in phase space, separating inflatio…
We study the renormalized energy-momentum tensor (EMT) of cosmological scalar fluctuations during the slow-rollover regime for chaotic inflation with a quadratic potential and find that it is characterized by a negative energy density which grows during slow-rollover. We also approach the back-reaction problem as a second-order calculation in perturbation theory finding no evidence that the back-reaction of cosmological fluctuations is a gauge artifact. In agreement with the results on the EMT, the average expansion rate is decreased by the back-reaction of cosmological fluctuations.
We have investigated the generality of inflation (the probability of inflation in other words) in closed FRW models for a wide class of quintessence potentials. It is shown that inflation is not suppressed for most of them and for a wide enough range of their parameters. It allows us to decide inflation is common enough even in the case of closed Universe.
The dependence of cosmological and inflationary parameters on time during the last 60 e-folds in inflation is investigated using a slow-roll inflation model. The time dependence of the inflaton field is calculated for the case of chaotic inflation by a numerical method rather than the familiar approximations. It is found that the Hubble parameter and the spectral index decrease in the last 60 e-folds. The dependence of the power spectrum of the curvature perturbation on the size perturbation k is calculated numerically, and it is shown that the overall constant value of the power spectrum thus…
The inflationary potential and its derivatives determine the spectrum of scalar and tensor metric perturbations that arise from quantum fluctuations during inflation. The CBR anisotropy offers a promising means of determining the spectra of metric perturbations and thereby a means of constraining the inflationary potential. The relation between the metric perturbations and CBR anisotropy depends upon cosmological parameters -- most notably the possibility of a cosmological constant. Motivated by some observational evidence for a cosmological constant (large-scale structure, cluster-baryon frac…
Cosmological equations for homogeneous isotropic models filled by scalar fields and ultrarelativistic matter are investigated in the framework of gauge theories of gravity. Regular inflationary cosmological models of flat, closed and open type with dominating ultrarelativistic matter at a bounce are discussed. It is shown that essential part of inflationary cosmological models has bouncing character.
We study the possibility of canceling the cosmological constant in supergravity string models. We show that with a suitable choice of superpotential the vacuum energy may vanish with the dilaton field at its minimum and supersymmetry broken with a large hierarchy. We derive the condition for which the introduction of a chiral potential, e.g. the inflaton potential, does not destabilize the dilaton field even in the region where the scalar potential takes positive values. This allows for an inflationary potential with the dilaton frozen at its minimum.
We present a study of open inflation cosmological scenarios in which cosmic strings form betwen the two inflationary epochs. It is shown that in these models strings are stretched outside the horizon due to the inflationary expansion but must necessarily re-enter the horizon before the epoch of equal matter and radiation densities. We determine the power spectrum of cold dark matter perturbations in these hybrid models, finding good agreement with observations for values of $Γ=Ω_0h\sim0.3$ and comparable contributions from the active and passive sources to the CMB. Finally, we briefly discuss …
We study the renormalized energy-momentum tensor of cosmological scalar fluctuations during the slow-rollover regime for power-law inflation and find that it is characterized by a negative energy density at the leading order, with the same time behaviour as the background energy. The average expansion rate appears decreased by the back-reaction of the effective energy of cosmological fluctuations, but this value is comparable with the energy of background only if inflation starts at a Planckian energy. We also find that, for this particular model, the first and second order inflaton fluctuatio…
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 …
In this paper we investigate a model of an inflationary universe in Kaluza-Klein theory, which is a four-dimensional de Sitter space plus a one-dimensional compactified internal space. We find that the energy scale for inflation can be predicted from the fine-structure constant in a self-consistent solution of the semi-classical Einstein equations including the Casimir effect. From the observed value of the fine-structure constant, we obtain an energy scale for inflation of $ε=1.84\times 10^{16}g_*^{1/4}$ Gev, where $g_*$ is a dimensionless number depending on the spin and number of matter fie…
In this paper we provide approximate analytical analysis of stability of nonsingular inflationary chaotic-type cosmological models. Initial conditions for nonsingular solutions at the bounce correspond to dominance of potential part of the energy density of the scalar field over its kinetic part both within general relativity and gauge theories of gravity. Moreover, scalar field at the bounce exceeds the planckian value and on expansion stage these models correspond to chaotic inflation. Such solutions can be well approximated by explicitly solvable model with constant effective potential (cos…
We analyze the cosmological consequences of locked inflation, a model recently proposed by Dvali and Kachru that can produce significant amounts of inflation without requiring slow-roll. We pay particular attention to the end of inflation in this model, showing that a secondary phase of saddle inflation can follow the locked inflationary era. However, this subsequent period of inflation results in a strongly scale dependent spectrum that can lead to massive black hole formation in the primordial universe. Avoiding this disastrous outcome puts strong constraints on the parameter space open to m…
The pre-big-bang cosmology inspired by superstring theories has been suggested as an alternative to slow-roll inflation. We analyze, in both the Jordan and Einstein frames, the effect of spatial curvature on this scenario and show that too much curvature --- of either sign --- reduces the duration of the inflationary era to such an extent that the flatness and horizon problems are not solved. Hence, a fine-tuning of initial conditions is required to obtain enough inflation to solve the cosmological problems.
Cosmological inflation is studied in the case where the inflaton is the overall modulus $T$ for an orbifold. General forms of the (non-perturbative) superpotential are considered to ensure that $G=K+{\rm ln}|W|^2$ is modular invariant. We find generically that these models do not produce a potential flat enough for slow roll to a supersymmetric minimum, although we do find a model which produces up to 20 e-folds of inflation to a non-supersymmetric minimum.
Many pedagogical introductions of cosmological inflation are effective due to the simplicity of the relevant equations. Here an analytic solution of the cosmological equations is presented and used as an example to discuss fundamental aspects of the inflationary paradigm.
We describe the universe as a local, inhomogeneous spherical bubble embedded in a flat matter dominated FLRW universe. Generalized exact Friedmann equations describe the expansion of the universe and an early universe inflationary de Sitter solution is obtained. A non-perturbative expression for the deceleration parameter q is derived that can possibly describe the acceleration of the universe without dark energy, due to the effects associated with very long wave length super-horizon inflationary perturbations. The suggestion by Kolbe et al. [9] that long wave length super-horizon inflationary…
We analyze cosmological perturbations to the linear order in the context of inflation with an arbitrary number of scalar fields. The fields take values on a non-trivial manifold with a positive-definite metric and are non-minimally coupled to Einstein gravity. The perturbations are decomposed into three different types. The scalar-type perturbations are presented in a gauge-ready form without fixing the temporal gauge condition, as well as in terms of gauge-invariant variables. The gauge-ready method enables us to impose different gauge conditions which are most suitable to the problem at hand…
An anthropic understanding of the cosmological constant requires that the vacuum energy at late time scans from one patch of the universe to another. If the vacuum energy during inflation also scans, the various patches of the universe acquire exponentially differing volumes. In a generic landscape with slow-roll inflation, we find that this gives a steeply varying probability distribution for the normalization of the primordial density perturbations, resulting in an exponentially small fraction of observers measuring the COBE value of 10^-5. Inflationary landscapes should avoid this "σproblem…
We investigate how the uncertainty of noncommutative spacetime could explain the WMAP data. For this purpose, the spectrum is divided into the IR and UV region. We introduce a noncommutative parameter of $γ_0$ in the IR region and a noncommutative parameter of $μ_0$ in the UV region. We calculate cosmological parameters using the slow-roll expansion in the UV region and a perturbation method in the IR region. The power-law inflation is chosen to obtain explicit forms for the power spectrum, spectral index, and running spectral index. Further, these are used to fit the data.
