scispace - formally typeset
Search or ask a question

Showing papers on "Particle horizon published in 2004"


Journal ArticleDOI
TL;DR: The role of positive curvature is negligible at late times, but can be crucial in the early universe as mentioned in this paper, allowing for cosmologies that originate as Einstein static universes, and then inflate and later reheat to a hot big-bang era.
Abstract: Observations indicate that the universe is effectively flat, but they do not rule out a closed universe. The role of positive curvature is negligible at late times, but can be crucial in the early universe. In particular, positive curvature allows for cosmologies that originate as Einstein static universes, and then inflate and later reheat to a hot big-bang era. These cosmologies have no singularity, no 'beginning of time' and no horizon problem. If the initial radius is chosen to be above the Planck scale, then they also have no quantum gravity era, and are described by classical general relativity throughout their history.

444 citations


Journal ArticleDOI
TL;DR: In this article, the authors provide a realization of a singularity-free inflationary universe in the form of a simple cosmological model dominated at early times by a single minimally coupled scalar field with a physically based potential.
Abstract: We provide a realization of a singularity-free inflationary universe in the form of a simple cosmological model dominated at early times by a single minimally coupled scalar field with a physically based potential. The universe starts asymptotically from an initial Einstein static state, which may be large enough to avoid the quantum gravity regime. It enters an expanding phase that leads to inflation followed by reheating and a standard hot big bang evolution. We discuss the basic characteristics of this emergent model and show that none is at odds with current observations.

326 citations


Journal ArticleDOI
TL;DR: In this paper, the idea of relating the infrared and ultraviolet cutoffs is applied to the Brans-Dicke theory of gravitation and it is shown that the Hubble scale or the particle horizon as the infrared cutoff will not give accelerating expansion.
Abstract: The idea of relating the infrared and ultraviolet cutoffs is applied to the Brans-Dicke theory of gravitation. We find that the Hubble scale or the particle horizon as the infrared cutoff will not give accelerating expansion. The dynamical cosmological constant with the event horizon as the infrared cutoff is a viable dark energy model.

247 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study the space-time evolution of the fine structure constant, α, inside evolving spherical overdensities in a lambda-CDM Friedmann universe using the spherical infall model.

225 citations


Journal ArticleDOI
TL;DR: In this paper, the authors use standard general relativity to illustrate and clarify several common misconceptions about the expansion of the universe and explain why these misconceptions do not violate special relativity and link these concepts to observational tests.
Abstract: We use standard general relativity to illustrate and clarify several common misconceptions about the expansion of the universe. To show the abundance of these misconceptions we cite numerous misleading, or easily misinterpreted, statements in the literature. In the context of the new standard ΛCDM cosmology we point out confusions regarding the particle horizon, the event horizon, the ‘observable universe’ and the Hubble sphere (distance at which recession velocity = c). We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light. We explain why this does not violate special relativity and we link these concepts to observational tests. Attempts to restrict recession velocities to less than the speed of light require a special relativistic interpretation of cosmological redshifts. We analyze apparent magnitudes of supernovae and observationally rule out the special relativistic Doppler interpretation of cosmological redshifts at a confidence level of 23σ.

189 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the universe can experience many cycles with different vacua, which is a generic behavior independent of the details of the model and might provide a distinct dynamical approach to an anthropically favorable vacuum.
Abstract: Recently, the notion that the number of vacua is astronomical has received increased attentions, which may be regarded as a possible anthropical explanation to incredibly small cosmological constant. Further, a dynamical mechanism to implement this possibility is required. We show in an operable model of cyclic universe that the universe can experience many cycles with different vacua, which is a generic behavior independent of the details of the model. This might provide a distinct dynamical approach to an anthropically favorable vacuum.

89 citations


Journal ArticleDOI
TL;DR: In this paper, the authors constrain the characterizing parameters of three classes of dark energy cosmological models to see whether they are in agreement with this kind of data, based on time measurements rather than distance observations.
Abstract: An impressive amount of different astrophysical data converges towards the picture of a spatially flat Universe undergoing today a phase of accelerated expansion. The nature of the dark energy dominating the energy content of the Universe is still unknown, and a lot of different scenarios are viable candidates to explain cosmic acceleration. Most of the methods employed to test these cosmological models are essentially based on distance measurements to a particular class of objects. A different method, based on the lookback time to galaxy clusters and the age of the Universe, is used here. In particular, we constrain the characterizing parameters of three classes of dark energy cosmological models to see whether they are in agreement with this kind of data, based on time measurements rather than distance observations.

77 citations


Book
17 Nov 2004
TL;DR: In a previous work as discussed by the authors, we have discussed the fundamental principles of cosmoparticle physics and the basis of inflation in the modern universe and fine-tuning of microphysical parameters in the universe.
Abstract: 1. Principles of Cosmoparticle Physics.- 2. Basis of Inflation.- 3. Quantum Fluctuations during Inflation.- 4. Strong Primordial Inhomogeneities and Galaxy Formation.- 5. Baryon Asymmetrical Universe with Antimatter Regions.- 6. Antimatter in the Modern Universe.- 7. Astronomy of Ultra High Energy Cosmic Rays.- 8. High Density Regions from First-order Phase Transitions.- 9. Fine-Tuning of Microphysical Parameters in the Universe.- 10. Inflation: Additional Resources.- Epilogue.- References.

71 citations


Journal ArticleDOI
TL;DR: In this paper, the SO(1, 1) scalar field model of dark energy was proposed and the Lagrangian may be decomposed as that of the real quintessence model and the negative coupling energy term of Phi to a. The existence of the coupling term L-c leads to a wider range of w(Phi) and overcomes the problem of negative kinetic energy in the phantom universe model.
Abstract: We suggest the SO(1, 1) scalar field model of dark energy. In this model, the Lagrangian may be decomposed as that of the real quintessence model and the negative coupling energy term of Phi to a. The existence of the coupling term L-c leads to a wider range of w(Phi) and overcomes the problem of negative kinetic energy in the phantom universe model. We propose a power-law expansion kinetics model of the universe with time-dependent power, which can describe the universe transition from ordinary acceleration to super acceleration. We also give a simple discussion of the Big Rip, singularity, and point out the possibility that the universe driven by a phantom avoids it.

70 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the accelerated expansion of the universe can be described as an extrinsic geometrical property, as an alternative to dark energy, and the standard Friedmann universe embedded in a five-dimensional bulk with constant curvature was examined.

68 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a detailed study of the lensing configuration in the cluster Abell 2218, using a simplified maximum likelihood analysis, and they find 0 < Ω_M < 0.30 and w < -0.85 for a flat universe with dark energy.
Abstract: In this Letter we present a detailed study of the lensing configuration in the cluster Abell 2218. Four multiple-images systems with measured spectroscopic redshifts have been identified in this cluster. These multiple images are very useful to constrain accurately the mass distribution in the cluster core, but they are also sensitive to the value of the geometrical cosmological parameters of the Universe. Using a simplified maximum likelihood analysis we find 0 < Ω_M < 0.30 assuming a flat Universe, and 0 < Ω_M < 0.33 and w < -0.85 for a flat Universe with dark energy. Interestingly, an Einstein-de Sitter model is excluded at more than 4σ. These constraints are consistent with the current constraints derived with CMB anisotropies or supernovae studies. The proposed method constitutes an independent test of the geometrical cosmological parameters of the Universe and we discuss the limits of this method and this particular application to Abell 2218. Application of this method with more sophisticated tools and to a larger number of clusters or with more multiple images constraints, will put stringent constraints on the geometrical cosmological parameters.

Journal ArticleDOI
TL;DR: In this paper, the Hartle-Hawking wave function for de Sitter space has been generalized to string theory, and the authors give a phenomenological analysis of the probability of quantum tunneling to various stringy vacua, and they find that the preferred tunneling is to an inflationary universe (like our early universe), not to a universe with a very small cosmological constant (i.e., like today's universe) and to a 10-dimensional (or a higher dimensional supercritical) uncompactified deSitter universe.
Abstract: We study some gravitational instanton solutions that offer a natural realization of the spontaneous creation of inflationary universes in the brane world context in string theory. Decoherence due to couplings of higher (perturbative) modes of the metric as well as matter fields modifies the Hartle-Hawking wavefunction for de Sitter space. Generalizing this new wavefunction to be used in string theory, we propose a principle in string theory that hopefully will lead us to the particular vacuum we live in, thus avoiding the anthropic principle. As an illustration of this idea, we give a phenomenological analysis of the probability of quantum tunneling to various stringy vacua. We find that the preferred tunneling is to an inflationary universe (like our early universe), not to a universe with a very small cosmological constant (i.e., like today's universe) and not to a 10-dimensional (or a higher dimensional supercritical) uncompactified de Sitter universe. Some solutions are interesting as they offer a cosmological mechanism for the stabilization of extra dimensions during the inflationary epoch.

Journal ArticleDOI
TL;DR: In this article, the time dependence of G is presented and the mass distribution at large scales and the correlation function are explained and are natural consequences of the evaluated varying G. The results suggest a universe based on El Naschie's e(∞) Cantorian space time.
Abstract: In this paper the time dependence of G is presented. It is a simple consequence of the Virial Theorem and of the self-similarity and fractality of the Universe. The results suggest a Universe based on El Naschie’s e(∞) Cantorian space–time. Moreover, we show the importance of the Golden Mean in respect to the large scale structures. Thanks to this study the mass distribution at large scales and the correlation function are explained and are natural consequences of the evaluated varying G. We demonstrate the agreement between the present hypotheses of segregation with a size of astrophysical structures, by using a comparison between quantum quantities and astrophysical ones. It appears clear that the Universe has a memory of its quantum origin. This appears in the G dependence too. Moreover, we see that a G=G(t) in El Naschie’s e(∞) Cantorian space–time can imply an accelerated Universe.

Journal ArticleDOI
TL;DR: For a universe containing a cosmological constant together with uniform arrangements of magnetic fields, strings, or domain walls, exact solutions to the Einstein equations are shown to lead to a universe with ellipsoidal expansion as mentioned in this paper.
Abstract: For a universe containing a cosmological constant together with uniform arrangements of magnetic fields, strings, or domain walls, exact solutions to the Einstein equations are shown to lead to a universe with ellipsoidal expansion. The magnetic field case is the easiest to motivate and has the highest possibility of finding application in observational cosmology.

Journal ArticleDOI
TL;DR: In this article, the authors obtained the metric for a Reissner-Nordstrom black hole in the background of the Friedman-Robertson-Walker universe and verified it and discussed the influence of the evolution of the universe on the size of the black hole.

Posted Content
TL;DR: The Review of Particle Physics 2004 (aka the Particle Data Book) as mentioned in this paper provides a compact review of knowledge of the cosmological parameters as at the end of 2003.
Abstract: This is a new review article for The Review of Particle Physics 2004 (aka the Particle Data Book). It forms a compact review of knowledge of the cosmological parameters as at the end of 2003. Topics included are Parametrizing the Universe; Extensions to the standard model; Probes; Bringing observations together; Outlook for the future.

Journal ArticleDOI
TL;DR: In this article, the authors consider the early universe and show that the effective mass-squared of a scalar field with only Planck-suppressed couplings with light fields and whose true mass is less than the Hubble parameter H is of order ±H 2 during inflation and matterdomination, but much smaller during radiation domination.
Abstract: We consider the effective mass-squared in the early Universe, of a scalar field which has only Planck-suppressed couplings with light fields and whose true mass is less than the Hubble parameter H. A detailed investigation shows that the effective mass-squared generically is of order ±H2 during inflation and matter-domination, but much smaller during radiation domination. We consider the special circumstances under which the mass-squared may be much bigger or much smaller than the generic value.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the case of a symmetric collision of infinitely thin branes having a hyperbolic or flat spatial geometry and show that such a collision results in a collapsing universe on the final brane unless the preexisting expansion rate in the bulk just prior to the collision is sufficiently large in comparison to the momentum transfer in the fifth dimension.
Abstract: It is intriguing to consider the possibility that the big bang of the standard (3+1)-dimensional cosmology originated from the collision of two branes within a higher dimensional spacetime, leading to the production of a large amount of entropy. In this paper we study, subject to certain well-defined assumptions, under what conditions such a collision leads to an expanding universe. We assume the absence of novel physics, so that ordinary (4+1)-dimensional Einstein gravity remains a valid approximation. It is necessary that the fifth dimension not become degenerate at the moment of collision. First the case of a symmetric collision of infinitely thin branes having a hyperbolic or flat spatial geometry is considered. We find that a symmetric collision results in a collapsing universe on the final brane unless the preexisting expansion rate in the bulk just prior to the collision is sufficiently large in comparison to the momentum transfer in the fifth dimension. Such prior expansion may either result from negative spatial curvature or from a positive five-dimensional cosmological constant. The relevance of these findings to the colliding bubble braneworld universe scenario is discussed. Finally, results from a numerical study of colliding thick-wall branes is presented, which confirm the results of the thin-wall approximation.

Journal ArticleDOI
TL;DR: In this paper, the effects of space-time noncommutativity on the quantum fluctuations of an inflaton field and their contributions to the cosmic microwave background (CMB) were investigated.

Journal ArticleDOI
TL;DR: In this paper, the authors show that the imprints of the inflationary perturbations thermalize during the late acceleration of the universe and they “inflate away, just like the initial inhomogeneities during ordinary inflation, making our era a most opportune time to study cosmology.

Journal ArticleDOI
TL;DR: In this article, a stability criterion is derived in general relativity for self-similar solutions with a scalar field and those with a stiff fluid, which is a perfect fluid with the equation of state P = ρ.
Abstract: A stability criterion is derived in general relativity for self-similar solutions with a scalar field and those with a stiff fluid, which is a perfect fluid with the equation of state P = ρ. A wide class of self-similar solutions turns out to be unstable against kink mode perturbation. According to the criterion, the Evans–Coleman stiff-fluid solution is unstable and cannot be a critical solution for the spherical collapse of a stiff fluid if we allow sufficiently small discontinuity in the density gradient field in the initial data sets. The self-similar scalar-field solution, which was recently found numerically by Brady et al (2002 Class. Quantum Grav. 19 6359), is also unstable. Both the flat Friedmann universe with a scalar field and that with a stiff fluid suffer from kink instability at the particle horizon scale.

Journal ArticleDOI
TL;DR: In this paper, the evolution of the universe is divided into several stages, in which the universe contracts from a Λ-dominated vacuum, and after the bounce, the universe expands.
Abstract: Recent observations of Type Ia supernovae provide evidence for the acceleration of our universe, which leads to the possibility that the universe is entering an inflationary epoch. We simulate it under a "big bounce" model, which contains a time variable cosmological "constant" that is derived from a higher dimension and manifests itself in 4D spacetime as dark energy. By properly choosing the two arbitrary functions contained in the model, we obtain a simple exact solution in which the evolution of the universe is divided into several stages. Before the big bounce, the universe contracts from a Λ-dominated vacuum, and after the bounce, the universe expands. In the early time after the bounce, the expansion of the universe is decelerating. In the late time after the bounce, dark energy (i.e. the variable cosmological "constant") overtakes dark matter and baryons, and the expansion enters an accelerating stage. When time tends to infinity, the contribution of dark energy tends to two thirds of the total energy density of the universe, qualitatively in agreement with observations.

Journal Article
TL;DR: In this article, the physics of the CMBRanisotropies and their role in probing cosmological parameters, especially in the light of the latest observations from the WMAP satellite, are discussed.
Abstract: Inter University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India.(Dated: February 2, 2008)The observed structures in the universe are thought to have arisen from gravitational instability acting on smallfluctuations generated in the early universe. These spatial fluctuations are imprinted on the CMBR as angularanisotropies. The physics which connects initial fluctuati ons in the early universe to the observed anisotropies isfairly well understood, since for most part it involves linear perturbation theory. This makes CMBR anisotropiesone of the cleanest probes of the initial fluctuations, vario us cosmological parameters governing their evolutionand also the geometry of the universe. We review here in a fairly pedagogical manner the physics of the CMBRanisotropies and explain the role they play in probing cosmological parameters, especially in the light of thelatest observations from the WMAP satellite.I. INTRODUCTION

Journal ArticleDOI
TL;DR: In this article, the cosmological horizon is viewed as the boundary of a cavity and the holographic D-bound in a de Sitter space, and an expression of the upper limit of the total number of e-foldings of inflation is derived for a simple evolution model of inflation.
Abstract: If the source of the current accelerating expansion of the universe is a positive cosmological constant, Banks and Fischler argued that there exists an upper limit of the total number of e-foldings of inflation. We further elaborate on the upper limit in the sense of viewing the cosmological horizon as the boundary of a cavity and of the holographic D-bound in a de Sitter space. Assuming a simple evolution model of inflation, we obtain an expression of the upper limit in terms of the cosmological constant, the energy density of the inflaton when the inflation starts, the energy density as the inflation ends and reheating temperature. We discuss how the upper limit is modified in the different evolution models of the universe. The holographic D-bound gives a higher upper limit than the entropy threshold in the cavity. For the most extreme case where the initial energy density of inflation is as high as the Planck energy, and the reheating temperature is as low as the energy scale of nucleosynthesis, the D-bound gives the upper limit as 146 and the entropy threshold as 122. For a reasonable assumption in the simplest cosmology, the holographic D-bound leads to a value of about 85, while the cavity model gives a value of around 65 for the upper limit, which is close to the value needed to solve the flatness problem and the horizon problem in the hot big bang cosmology.

Journal ArticleDOI
TL;DR: In this article, a conformal invariant gravitational model is proposed, which is taken to hold at early universe and the conformal factor acts as a quintessence field that leads the universe to accelerate at late times.
Abstract: We investigate a conformal invariant gravitational model which is taken to hold at early universe. The conformal invariance allows us to make a dynamical distinction between the two unit systems (or conformal frames) usually used in cosmology and elementary particle physics. In this model we argue that when the universe suffers phase transition, the resulting mass scale introduced by particle physics should have a variable contribution to vacuum energy density. This variation is controlled by the conformal factor which is taken as a dynamical field. We then deal with the cosmological consequences of this model. In particular, we shall show that there is an inationary phase at early times. At late times, on the other hand, it provides a mechanism which makes a large effective cosmological constant relax to a sufficiently small value. Moreover, we shall show that the conformal factor acts as a quintessence field that leads the universe to accelerate at late times.

17 Mar 2004
TL;DR: In this paper, the shape of the primordial spectrum is very similar to that predicted by generic inflation models, and the angular scale at which the first acoustic peak appears is consistent with the flat universe predicted by inflation.
Abstract: Cosmologists have developed a phenomenally successful picture of structure in the universe based on the idea that the universe expanded exponentially in its earliest moments. There are three pieces of evidence for this exponential expansion--inflation--from observations of anisotropies in the cosmic microwave background. First, the shape of the primordial spectrum is very similar to that predicted by generic inflation models. Second, the angular scale at which the first acoustic peak appears is consistent with the flat universe predicted by inflation. Here the author describes the third piece of evidence, perhaps the most convincing of all: the phase coherence needed to account for the clear peak/trough structure observed by the WMAP satellite and its predecessors. The author also discusses alternatives to inflation that have been proposed recently and explain how they produce coherent phases.

Journal ArticleDOI
TL;DR: In this paper, the authors show that the universe has a toroidal topology with wound scalar fields around its cycles, which can contribute significantly to the total energy density of the universe.
Abstract: The observational limits on the present energy density of the Universe allow for a component that redshifts like 1/a2 and can contribute significantly to the total. We show that a possible origin for such a contribution is that the universe has a toroidal topology with ``wound" scalar fields around its cycles.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the observed red shift at the LAB system is due to a real shrinkage of the quantum world, due to the decrease in size of quantum particles determined by a decreasing Planck's "constant".
Abstract: We find five fundamental reasons demanding that any gravitational mass m, and the speed of light c, vary with cosmological time such that mc remains constant. This is required by the universal condition of conservation of momentum in a Universe with spatial homogeneity. We prove that this is consistent with Einstein's Theory of General Relativity. We call this effect a "MASS BOOM". At the LAB system no such time variations can be detected, unless we make comparisons with cosmological observations. We have to stress that the physical conditions implied by a time varying mass, together with a time varying speed of light, preserving the constancy of momentum, are compatible with Einstein's field equations. We then integrate his cosmological equations and find the solution for the cosmological scale factor a(t) proportional to the square of time, implying an apparent accelerated expansion for the Universe, as seen from the LAB frame. This is the interpretation given to recent observations obtained from the Supernova Type Ia. This determination of the scale factor a(t) is based upon a LAB interpretation and therefore is an apparent effect. On the other hand we note that the product ct being a constant determines the real Universe as a static one, of constant size. The observed red shift at the LAB system is due to a real shrinkage of the quantum world, due to the decrease in size of the quantum particles determined by a decreasing Planck's "constant".

01 Dec 2004
TL;DR: The cyclic model of the universe as mentioned in this paper is a radical alternative to standard big bang/inflationary theory in which space and time exist indefinitely, the rapidly accelerating inflationary phase is avoided, and the universe undergoes periodic epochs of expansion and contraction.
Abstract: The "cyclic model of the universe" is a radical alternative to standard big bang/inflationary theory in which space and time exist indefinitely, the rapidly accelerating inflationary phase is avoided, and the universe undergoes periodic epochs of expansion and contraction. THROUGHOUT HUMAN HISTORY, there have been three cosmic paradigms commonly invoked to explain the nature of the universe. Each has been embodied in a modern cosmological model based on Einstein's general theory of relativity. One is the notion of an Unchanging Universe, which underlies Einstein's static universe model (the first cosmological model proposed after the introduction of General Relativity) and the Steady State Model of Hoyle, Gold, and Bondi. This paradigm has been set aside because of Hubble's discovery that the universe is expanding and the discovery by Penzias and Wilson of the cosmic microwave background, direct evidence of an earlier hotter and denser epoch. The paradigm of a Created Universe, in which space and time spring from nothingness and steadily evolve, underlies today's standard model of cosmology, the big bang/inflationary picture (1-3). Although the creation process and the first instants are not well understood, the sequence of events following is well defined and the predictions of the model appear to be in exquisite agreement with current observations. Finally, we come to the Cyclic Universe, the notion that the universe undergoes periodic epochs of evolution followed by a sequence of events that brings the universe back, phoenix-like, to its initial state. The cyclic concept is at least as old as recorded history. For example, cyclic evolution is an essential part of ancient Hindu cosmology. In the 1920s and 1930s, a version was explored in the context of modern cosmology by Tolman (4) and others. That version entailed an overdense closed universe in which our three-dimensional world expands and contracts at regular intervals. The idea was set aside because, as Tolman first pointed out, the entropy produced during each cycle would be concentrated during the contracting phase, adding to the entropy produced in earlier phases. The entropy density at the end of each contraction causes the next expansion cycle to be longer than the one before. The cycles cannot be identical, as originally imagined. Instead, extrapolating backward from the present entropy, the universe would have undergone smaller and smaller cycles in the past converging rapidly toward zero duration. The total age of the universe would not be significantly different from a universe with no cycles. Since the main purpose of considering a cyclic universe is to push back the "beginning" indefinitely, the entropy problem was viewed as a serious impediment and discouraged further work. Today, we would also discard this idea because it requires an overdense universe, whereas observations inform us that the mass density is only about one-fourth of the critical value required to cycle. In the last year, however, there has been a return of the Cyclic Universe in a new twenty-first-century version based on recent ideas developed in fundamental (superstring) physics (5, 6). The new cyclic universe appears capable of reproducing all of the successful predictions of the standard big bang/inflationary model with the same exquisite precision, even though the key events that shape the large-scale structure of the universe occur at different epochs and temperatures and entail different physical processes. A motivation for considering an alternative is that today's standard model has become more complicated. The initial big bang model assumed a universe that emerges full of radiation and density, and uniformly cools and condenses into structure over time. However, after the big bang, the universe should have been turbulent, chaotic, and disordered, whereas the observed universe seems remarkably homogeneous and uniform on large scales. A period of inflationary expansion has been added, which stretches the universe at exponentially fast rates, smoothing out the initial inhomogeneities, warps, or curvature. …

Journal ArticleDOI
TL;DR: In this article, the authors proposed a cosmological model in which not only the temperature but also the curvature is limited by the mass scale of the Hagedorn temperature.
Abstract: We propose a new type of cosmological model in which it is postulated that not only the temperature but also the curvature is limited by the mass scale of the Hagedorn temperature. We find that the big bang of this universe is smoothly connected to the big crunch of the previous universe through a Hagedorn universe, in which the temperature and curvature remain very close to their limiting values. In this way, we obtain the picture of a cyclic universe. By estimating the entropy gained in each big crunch and big bang, we reach the conclusion that our universe has repeated this process about forty times after it was created at the Planck scale. We also show that the model gives a scale-invariant spectrum of curvature perturbations.