Andrei Linde
An improved volume-weighted probability measure for eternal inflation is proposed. For the models studied in this paper it leads to simple and intuitively expected gauge-invariant results.
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We investigate whether the eternal chaotic inflation can be achieved when the weak gravity conjecture is taken into account. We show that even the assisted chaotic inflation with potential $λϕ^4$ or $m^2ϕ^2$ can not be eternal. The effective field theory description for the inflaton field breaks down before inflation reaches the eternal regime. We also find that the total number of e-folds is still bounded by the inflationary entropy for the assisted inflation.
The quantum behavior of noncommutative eternal inflation is quite different from the usual knowledge. Unlike the usual eternal inflation, the quantum fluctuation of noncommutative eternal inflation is suppressed by the Hubble parameter. Due to this, we need to reconsider many conceptions of eternal inflation. In this paper we study the Hawking-Moss tunneling in noncommutative eternal inflation using the stochastic approach. We obtain a brand-new form of the tunneling probability for this process and find that the Hawking-Moss tunneling is more unlikely to take place in the noncommutative case …
The discovery of the string theory landscape has recently brought attention to the eternal nature of inflation. In contrast to the common belief that eternal inflation may be a generic feature of most inflationary models, in this note we argue that the suppressed amplitude of perturbations due to adiabatic regularization, together with a fine-tuning constraint on the equation of state of the rare inflating pockets with large fluctuations, render eternal inflation expensive in energy and may make it unlikely to occur. The energy scales of the eternally inflating pockets have to be very close to…
In this paper we explore the relationship between the existence of eternal inflation and the initial conditions leading to inflation. We demonstrate that past and future completion of inflation is related, in that past-incomplete inflation can not be future eternal. Bubble universes nucleating close to the initial conditions hypersurface have the largest Lorentz boosts and experience the highest anisotropy. Consequently, their probability to collide upon formation is one. Thus instead of continuing eternally inflation ends soon after it starts. The difficulty in actualizing eternal inflation o…
We investigate the condition for eternal inflation to take place in the noncommutative spacetime. We find that the possibility for eternal inflation's happening is greatly suppressed in this case. If eternal inflation cannot happen in the low energy region where the noncommutativity is very weak (the UV region), it will never happen during the whole inflationary history. Based on these conclusions, we argue that an initial condition for eternal inflation is available from the property of spacetime noncommutativity.
We investigate how eternal inflation is affected by quantum gravity effects. We consider general features of quantum gravity, such as renormalizability, complementarity, minimal length, definition of observables, and weak gravity conjecture. We also consider phenomenological models such as ghost inflation, non-commutative inflation, brane inflation, k-inflation and resonant tunneling. We find that all these features and models do not support eternal inflation. These evidences show hints that eternal inflation is prohibited by quantum gravity.
The swampland criteria put significant constraints on inflation models. In this paper, I show that hilltop inflation on the brane may provide an initial condition for hilltop inflation and produce topological eternal inflation without violating the swampland distance and refined de Sitter criteria. This shows a way that string theory may not be incompatible with eternal inflation.
We propose a new measure for eternal inflation that includes both conditions, large field fluctuations and smooth homogeneous domains, in the self reproducing probability estimate. We show that due to the increasing inhomogeneities in the background spacetime fractal, self-reproduction stops within a finite time t_f, thus inflation can not be eternal.
We study various probability measures for eternal inflation by applying their regularization prescriptions to models where inflation is not eternal. For simplicity we work with a toy model describing inflation that can interpolate between eternal and non-eternal inflation by continuous variation of a parameter. We investigate whether the predictions of four different measures (proper time, scale factor cutoff, stationary and causal {diamond}) change continuously with the change of this parameter. We will show that {only} for the stationary measure the predictions change continuously. For the p…
Dynamics of eternal inflation on the landscape admits description in terms of the Martin-Siggia-Rose (MSR) effective field theory that is in one-to-one correspondence with vacuum dynamics equations. On those sectors of the landscape, where transport properties of the probability measure for eternal inflation are important, renormalization group fixed points of the MSR effective action determine late time behavior of the probability measure. I argue that these RG fixed points may be relevant for the solution of the gauge invariance problem for eternal inflation.
In the global description of eternal inflation, probabilities for vacua are notoriously ambiguous. The local point of view is preferred by holography and naturally picks out a simple probability measure. It is insensitive to large expansion factors or lifetimes, and so resolves a recently noted paradox. Any cosmological measure must be complemented with the probability for observers to emerge in a given vacuum. In lieu of anthropic criteria, I propose to estimate this by the entropy that can be produced in a local patch. This allows for prior-free predictions.
I investigate whether eternal inflation is possible when the inflaton originally travels upslope the scalar potential and inevitably reaches a turn-around point where its classical kinetic energy momentarily vanishes. This behaviour occurs regardless of the steepness of the potential slope. Such steep eternal inflation, if achieved, would satisfy the recent swampland conjecture and allow eternal inflation in the string landscape.
In this paper, we show that eternal inflation of the random walk type is generically absent in the brane inflationary scenario. Depending on how the brane inflationary universe originated, eternal inflation of the false vacuum type is still quite possible. Since the inflaton is the position of the D3-brane relative to the anti-D3-brane inside the compactified bulk with finite size, its value is bounded. In DBI inflation, the warped space also restricts the amplitude of the scalar fluctuation. These upper bounds impose strong constraints on the possibility of eternal inflation. We find that ete…
Recently, based on swampland considerations in string theory, the (no) eternal inflation principle has been put forward. The natural question arises whether similar conditions hold in other approaches to quantum gravity. In this article, the asymptotic safety hypothesis is considered in the context of eternal inflation. As exemplary inflationary models the SU(N) Yang-Mills in the Veneziano limit and various RG-improvements of the gravitational action are studied. The existence of UV fixed point generically flattens the potential and our findings suggest no tension between eternal inflation and…
We investigate the validity of the assertion that eternal inflation populates the landscape of string theory. We verify that bubble solutions do not satisfy the Klein Gordon equation for the landscape potential. Solutions to the landscape potential within the formalism of quantum cosmology are Anderson localized wavefunctions. Those are inconsistent with inflating bubble solutions. The physical reasons behind the failure of a relation between eternal inflation and the landscape are rooted in quantum phenomena such as interference between wavefunction concentrated around the various vacua in th…
We examine various entropic inflation scenarios, under the light of eternality. After describing the inflation realization and the normal condition for inflation to last at the background level, we investigate the conditions for eternal inflation with the effect of thermal fluctuations produced from standard radiation and from the holographic screen. Furthermore, we incorporate stochastic quantum fluctuations through a phenomenological, Langevin analysis, studying whether they can affect the inflation eternality. In single-holographic-screen scenarios eternality can be easily obtained, while i…
Non-perpetual Eternal Inflation and the Emergent de Sitter Swampland Conjecture
本文提出关联 CMB 参数与永恒暴胀所需 e-folds 的新方法 $n_s$-$ΔN$，用于检验永恒暴胀模型，并揭示德西特沼泽地猜想参数与宇宙景观永恒性之间的联系，支持多重宇宙图景。
We show that eternal inflation is compatible with holography. In particular, we emphasize that if a region is asymptotically de Sitter in the future, holographic arguments by themselves place no bound on the number of past e-foldings. We also comment briefly on holographic restrictions on the production of baby universes.
We construct analytic and numerical solutions of the Fokker-Planck equation that arises in the context of stochastic inflation. We use these solutions to derive necessary conditions for eternal inflation on the higher derivatives of the scalar field potential and examine the prospects for eternal inflation in a variety of popular models. We note similarities between the conditions needed to avoid eternal inflation and several recently-proposed Swampland criteria, which leads us to speculate on the possibility that the de Sitter Swampland conjectures should be viewed as approximate consequences…
Since the advent of inflation, several theorems have been proven suggesting that although inflation can (and generically does) continue eternally into the future, it cannot be extended eternally into the past to create a ``steady-state'' model with no initial time. Here we provide a construction that circumvents these theorems and allows a self-consistent, geodesically complete, and physically sensible steady-state eternally inflating universe, based on the flat slicing of de Sitter space. This construction could be used as the background space-time for creation events that form big-bang-like …
It has been widely claimed that inflation is generically eternal to the future, even in models where the inflaton potential monotonically increases away from its minimum. The idea is that quantum fluctuations allow the field to jump uphill, thereby continually revitalizing the inflationary process in some regions. In this paper we investigate a simple model of this process, pertaining to inflation with a quartic potential, in which analytic progress may be made. We calculate several quantities of interest, such as the expected number of inflationary efolds, first without and then with various …
We discuss eternal inflation in context of classical probability spaces defined by a triplet: sample space, $σ$-algebra and probability measure. We show that the measure problem is caused by the countable additivity axiom applied to the maximal $σ$-algebra of countably infinite sample spaces. This is a serious problem if the bulk space-time is treated as a sample space which is thought to be effectively countably infinite due to local quantum uncertainties. However, in semiclassical description of eternal inflation the physical space expands exponentially which makes the sample space of infini…
In recent literature on eternal inflation, a number of measures have been introduced which attempt to assign probabilities to different pocket universes by counting the number of each type of pocket according to a specific procedure. We give an overview of the existing measures, pointing out some interesting connections and generic predictions. For example, pairs of vacua that undergo fast transitions between themselves will be strongly favored. The resultant implications for making predictions in a generic potential landscape are discussed. We also raise a number of issues concerning the type…
We consider constraints on inflation driven by a single, minimally coupled scalar field in the light of the WMAP5 dataset, as well as ACBAR and the SuperNova Legacy Survey. We use the Slow Roll Reconstruction algorithm to derive optimal constraints on the inflationary parameter space. The scale dependence in the slope of the scalar spectrum permitted by WMAP5 is large enough to lead to viable models where the small scale perturbations have a substantial amplitude when extrapolated to the end of inflation. We find that excluding parameter values which would cause the overproduction of primordia…
Even if nothing but a light Higgs is observed at the LHC, suggesting that the Standard Model is unmodified up to scales far above the weak scale, Higgs physics can yield surprises of fundamental significance for cosmology. As has long been known, the Standard Model vacuum may be metastable for low enough Higgs mass, but a specific value of the decay rate holds special significance: for a very narrow window of parameters, our Universe has not yet decayed but the current inflationary period can not be future eternal. Determining whether we are in this window requires exquisite but achievable exp…
We investigate the global structure of inflationary universe both by analytical methods and by computer simulations of stochastic processes in the early Universe. We show that the global structure of the universe depends crucially on the mechanism of inflation. In the simplest models of chaotic inflation the Universe looks like a sea of thermalized phase surrounding permanently self-reproducing inflationary domains. In the theories where inflation occurs near a local extremum of the effective potential corresponding to a metastable state, the Universe looks like de Sitter space surrounding isl…
We demonstrate that there is no controlled description of stochastic eternal inflation consistent with the refined swampland de Sitter conjecture.
We recapitulate multiple arguments that Eternal Inflation, and the String Landscape are actually part of the Swampland: ideas in Effective Quantum Field Theory that do not have a counterpart in genuine models of Quantum Gravity.
I propose a new volume-weighted probability measure for cosmological "multiverse" scenarios involving eternal inflation. The "reheating-volume (RV) cutoff" calculates the distribution of observable quantities on a portion of the reheating hypersurface that is conditioned to be finite. The RV measure is gauge-invariant, does not suffer from the "youngness paradox," and is independent of initial conditions at the beginning of inflation. In slow-roll inflationary models with a scalar inflaton, the RV-regulated probability distributions can be obtained by solving nonlinear diffusion equations. I d…
We study if eternal inflation is realized while satisfying the recently proposed string Swampland criteria concerning the range of scalar field excursion, $|Δϕ| < \mathcal{D} \cdot M_{\rm P}$, and the potential gradient, $|\nabla V| > c \cdot V/M_{\rm P}$, where $\mathcal{D}$ and $c$ are constants of order unity, and $M_{\rm P}$ is the reduced Planck mass. We find that only the eternal inflation of chaotic type is possible for $c \sim {\cal O}(0.01)$ and $1/\mathcal{D} \sim {\cal O}(0.01)$, and that the Hubble parameter during the eternal inflation is parametrically close to the Planck scale, …
We consider the long standing puzzle of how to obtain meaningful probabilities in eternal inflation. We demonstrate a new algorithm to compute the probability distribution of pocket universe types, given a multivacua inflationary potential. The computed probability distribution is finite and manifestly gauge-independent. We argue that in some scenarios this technique can be applied to disfavor some eternally inflating potentials.
We use the stochastic approach to investigate the measure for slow roll eternal inflation. The probability for the universe of a given Hubble radius can be calculated in this framework. In a solvable model, it is shown that the probability for the universe to evolve from a state with a smaller Hubble radius to that of a larger Hubble radius is dominated by the classical probability without the stochastic source. While the probability for the universe to evolve from a larger Hubble radius to a smaller one is suppressed by $\exp(-ΔS)$, where the de Sitter entropy $S$ arises naturally in this sto…
We study the effects which the back-reaction of long wavelength fluctuations exert on stochastic inflation. In the cases of power-law and Starobinsky inflation these effects are too weak to terminate the stochastic growth of the inflaton field. However, in the case of the cyclic Ekpyrotic scenario, the back-reaction effects prevent the unlimited growth of the scalar field.
Higher-order alpha'-corrections are a generic feature of type IIB string compactifications. In KKLT-like models of moduli stabilization they provide a mechanism of breaking the no-scale structure of the volume modulus. We present a model of inflation driven by the volume modulus of flux compactifications of the type IIB superstring. Using the effects of gaugino condensation on D7-branes and perturbative alpha'-corrections the volume modulus can be stabilized in a scalar potential which simultaneously contains saddle points providing slow-roll inflation with about 130 e-foldings. We can accommo…
Defying eternal inflation in warm inflation with a negative running
本文将负谱指数跑动阻止永恒暴胀的结论推广至暖暴胀，发现其所需负跑动更小，且永恒暴胀的触发条件更严格。
As a result of discussions with Bousso and Vilenkin I want to return to the question of whether the multiverse is past-eternal or if there was a beginning. Not surprisingly, given three people, there were three answers. However, the discussions have led to some common ground. The multiverse being past-eternal, or at least extremely old has content and potential phenomenological implications. I will discuss how the oldness of the multiverse is connected with recent speculations of Douglas.
The possibility that the non-minimal coupling inflation could be eternal is investigated. We calculate the quantum fluctuation of the inflaton in a Hubble time and find that it has the same value as in the minimal case in the slow-roll limit. Armed with this result, we have studied some concrete non-minimal inflationary models including the chaotic inflation and the natural inflation while the inflaton is non-minimally coupled to the gravity and we find that these non-minimal inflations could be eternal in some parameter regions.
The usual theory of inflation breaks down in eternal inflation. We derive a dual description of eternal inflation in terms of a deformed Euclidean CFT located at the threshold of eternal inflation. The partition function gives the amplitude of different geometries of the threshold surface in the no-boundary state. Its local and global behavior in dual toy models shows that the amplitude is low for surfaces which are not nearly conformal to the round three-sphere and essentially zero for surfaces with negative curvature. Based on this we conjecture that the exit from eternal inflation does not …
Present treatments of eternal inflation regulate infinities by imposing a geometric cutoff. We point out that some matter systems reach the cutoff in finite time. This implies a nonzero probability for a novel type of catastrophe. According to the most successful measure proposals, our galaxy is likely to encounter the cutoff within the next 5 billion years.
Reheating surface volume renormalization and eternal inflation under broken scale invariance
本研究探讨标度不变性轻微破缺（$\varepsilon=n_s-1<0$）如何通过两种独立方法修改重加热表面体积重整化，进而缓和无永恒暴胀条件。
We calculate the probability distribution for the volume of the Universe after slow-roll inflation both in the eternal and in the non-eternal regime. Far from the eternal regime the probability distribution for the number of e-foldings, defined as one third of the logarithm of the volume, is sharply peaked around the number of e-foldings of the classical inflaton trajectory. At the transition to the eternal regime this probability is still peaked (with the width of order one e-folding) around the average, which gets twice larger at the transition point. As one enters the eternal regime the pro…
We prove that the light-cone time cut-off on the multiverse defines the same probabilities as a causal patch with initial conditions in the longest-lived metastable vacuum. This establishes the complete equivalence of two measures of eternal inflation which naively appear very different (though both are motivated by holography). The duality can be traced to an underlying geometric relation which we identify.
I apply recently proposed "Swampland" conjectures to eternal inflation in single-scalar field theories. Eternal inflation is a phase of infinite self-reproduction of a quasi-de Sitter universe which has been argued to be a generic consequence of cosmological inflation. The originally proposed de Sitter swampland conjectures were shown by Matsui and Takahashi and by Dimopoulos to be generically incompatible with eternal inflation. However, the more recently proposed "refined" swampland conjecture imposes a slightly weaker criterion on the scalar field potential in inflation, and is consistent w…
I address some recently raised issues regarding the time-parametrization dependence in stochastic descriptions of eternal inflation. To clarify the role of the choice of the time gauge, I show examples of gauge-dependent as well as gauge-independent statements about physical observables in eternally inflating spacetimes. In particular, the relative abundance of thermalized and inflating regions is highly gauge-dependent. The unbounded growth of the 3-volume of the inflating regions is found in certain time gauges, such as the proper time or the scale factor gauge. Yet in the same spacetimes th…
The Wheeler-DeWitt equation provides the probability distribution for the curvature perturbation, the gauge invariant quantum fluctuation of the inflaton. From this, we can find a tower of power spectra which is not found in a perturbative approach. Since the power spectrum for the modes that cross the horizon contributes to the uncertainty in the classical inflaton displacement, we obtain new conditions for eternal inflation. In the presence of the patch in the higher excitations, the bound on the slow-roll parameter allowing eternal inflation is given by at most $ε\lesssim (2n+1)(H/m_{\rm Pl…
We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential $V(ϕ)=V_{0}+δV(ϕ)$, where $|δV(ϕ)|\ll V_{0}$ is random. We find that the diffusion of the distribution function $ρ(ϕ,t)$ of the inflaton expectation value in different Hubble patches may be suppressed due to the effect analogous to the Anderson localization in disordered quantum systems. At $t \to \infty$ only the localized part of the distribution function $ρ(ϕ, t)$ survives which leads to dynamical selection principle on the landscape. The probability to measure any but a sma…
For slow-roll inflation we study the phase transition to the eternal regime. Starting from a finite inflationary volume, we consider the volume of the universe at reheating as order parameter. We show that there exists a critical value for the classical inflaton speed, \dotϕ^2/H^4 = 3/(2 π^2), where the probability distribution for the reheating volume undergoes a sharp transition. In particular, for sub-critical inflaton speeds all distribution moments become infinite. We show that at the same transition point the system develops a non-vanishing probability of having a strictly infinite rehea…
We review the paradigm of eternal inflation in the light of the recently proposed corpuscular picture of space-time. Comparing the strength of the average fluctuation of the field up its potential with that of quantum depletion, we show that the latter can be dominant. We then study the full respective distributions in order to show that the fraction of the space-time which has an increasing potential is always below the eternal-inflation threshold. We prove that for monomial potentials eternal inflaton is excluded. This is likely to hold for other models as well.
Cosmological inflation, if it occurred, radically alters the picture of the `big bang', which would merely point to reheating at the end of inflation. Moreover, this reheating may be only local, so that inflation continues elsewhere and forever, continually spawning big-bang-like regions. This chapter reviews this idea of `eternal inflation', then focuses on what this may mean for the ultimate beginning of the universe. In particular, I will argue that given eternal inflation, the universe may be free of a cosmological initial singularity, might be eternal (and eternally inflating) to the past…
It has been suggested that low energy effective field theories should satisfy given conditions in order to be successfully embedded into string theory. In the case of a single canonically normalized scalar field this translates into conditions on its potential and the derivatives thereof. In this Letter we revisit stochastic models of small field inflation and study the compatibility of the swampland constraints with entropy considerations. We show that stochastic inflation either violates entropy bounds or the swampland criterium on the slope of the scalar field potential. Furthermore, we ill…
I summarize the arguments that strongly suggest that our universe is the product of inflation. The mechanisms that lead to eternal inflation in both new and chaotic models are described. Although the infinity of pocket universes produced by eternal inflation are unobservable, it is argued that eternal inflation has real consequences in terms of the way that predictions are extracted from theoretical models. The ambiguities in defining probabilities in eternally inflating spacetimes are reviewed, with emphasis on the youngness paradox that results from a synchronous gauge regularization techniq…
When the slow-roll parameter $ε_H$ is smaller than $H^2/M_{\rm Pl}^2$, the quantum fluctuations of the inflaton after the horizon crossing are large enough to realize eternal inflation. Whereas they do not generate a sufficient amount of density fluctuation of the inflaton to produce the black hole in quasi-de Sitter space, they can also generate the sizeable density fluctuation of the radiation when the number of degrees of freedom increases rapidly in time, as predicted by the distance conjecture. We argue that the condition that the density fluctuation of the radiation is not large enough t…
Eternal inflation is studied in the context of warm inflation. We focus on different tools to analyze the effects of dissipation and the presence of a thermal radiation bath on the fluctuation-dominated regime, for which the self-reproduction of Hubble regions can take place. The tools we explore are the threshold inflaton field and threshold number of e-folds necessary to establish a self-reproduction regime and the counting of Hubble regions, using generalized conditions for the occurrence of a fluctuation-dominated regime. We obtain the functional dependence of these quantities on the dissi…
We consider landscape models that admit several regions where the conditions for eternal inflation hold. It is shown that one can use the no-boundary wave function to calculate small departures from homogeneity within our past light cone despite the possibility of much larger fluctuations on super horizon scales. The dominant contribution comes from the history exiting eternal inflation at the lowest value of the potential. In a class of landscape models this predicts a tensor to scalar ratio of about 10%. In this way the no-boundary wave function defines a measure for the prediction of local …
We consider the cosmological dynamics associated with volume weighted measures of eternal inflation, in the Bousso-Polchinski model of the string theory landscape. We find that this measure predicts that observers are most likely to find themselves in low energy vacua with one flux considerably larger than the rest. Furthermore, it allows for a satisfactory anthropic explanation of the cosmological constant problem by producing a smooth, and approximately constant, distribution of potentially observable values of Lambda. The low energy vacua selected by this measure are often short lived. If w…
An eternally inflating universe produces an infinite amount of spatial volume, so every possible event happens an infinite number of times, and it is impossible to define probabilities in terms of frequencies. This problem is usually addressed by means of a measure, which regulates the infinities and produces meaningful predictions. I argue that any measure should obey certain general axioms, but then give a simple toy model in which one can prove that no measure obeying the axioms exists. In certain cases of eternal inflation there are measures that obey the axioms, but all such measures appe…
Inflation creates large-scale cosmological density perturbations that are characterized by an isotropic, homogeneous, and Gaussian random distribution about a locally flat background. Even in a flat universe, the spatial curvature measured within one Hubble volume receives contributions from long wavelength perturbations, and will not in general be zero. These same perturbations determine the Cosmic Microwave Background (CMB) temperature fluctuations, which are O(10^-5). Consequently, the low-l multipole moments in the CMB temperature map predict the value of the measured spatial curvature Ω_k…
Eternal inflation near inflection points: a challenge to primordial black hole models
该研究指出，具有拐点势的膨胀模型中，永恒膨胀的条件为λ₁ ≤ 3，且在特定参数下会导致婴儿宇宙的形成，从而挑战原初黑洞模型与大尺度结构观测的兼容性。
We present an interpretation of the physics of space-times undergoing eternal inflation by repeated nucleation of bubbles. In many cases the physics can be interpreted in terms of the quantum mechanics of a system with a finite number of states. If this interpretation is correct, the conventional picture of these space-times is misleading.
Motivated by the lessons of black hole complementarity, we develop a causal patch description of eternal inflation. We argue that an observer cannot ascribe a semiclassical geometry to regions outside his horizon, because the large-scale metric is governed by the fluctuations of quantum fields. In order to identify what is within the horizon, it is necessary to understand the late time asymptotics. Any given worldline will eventually exit from eternal inflation into a terminal vacuum. If the cosmological constant is negative, the universe crunches. If it is zero, then we find that the observer…
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