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We consider chaotic inflation in the theories with the effective potentials phi^n and e^{\alpha\phi}. In such theories inflationary domains containing sufficiently large and homogeneous scalar field \phi permanently produce new inflationary domains of a similar type. We show that under certain conditions this process of the self-reproduction of the Universe can be described by a stationary distribution of probability, which means that the fraction of the physical volume of the Universe in a state with given properties (with given values of fields, with a given density of matter, etc.) does not depend on time, both at the stage of inflation and after it. This represents a strong deviation of inflationary cosmology from the standard Big Bang paradigm. We compare our approach with other approaches to quantum cosmology, and illustrate some of the general conclusions mentioned above with the results of a computer simulation of stochastic processes in the inflationary Universe. Comment: No changes to the file, but original figures are included. They substantially help to understand this paper, as well as eternal inflation in general, and what is now called the "multiverse" and the "string theory landscape." High quality figures can be found at http://www.stanford.edu/~alinde/LLMbigfigs/

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... (i) Global measures define a global foliation of spacetime specified by a global time coordinate. One counts events on a late-time cutoff hypersurface t ¼ t c , and then lets t c → ∞ [12][13][14][15][16][17][18][19][20][21][22]. The resulting probabilities are well-defined and independent of initial conditions, consistent with the attractor property of inflation. ...

... The resulting probabilities are well-defined and independent of initial conditions, consistent with the attractor property of inflation. A major problem is that the result depends sensitively on the choice of time variable [12][13][14]. For instance, two natural choices, proper time [12,13] and scale factor (or e-folding) time [12,13,[20][21][22], give drastically different probabilities. ...

... A major problem is that the result depends sensitively on the choice of time variable [12][13][14]. For instance, two natural choices, proper time [12,13] and scale factor (or e-folding) time [12,13,[20][21][22], give drastically different probabilities. The proper-time measure also suffers from the "youngness paradox" [16,[23][24][25]. ...

Probabilities in eternal inflation are traditionally defined as limiting frequency distributions, but a unique and unambiguous probability measure remains elusive. In this paper, we present a different approach, based on Bayesian reasoning. Our starting point is the master equation governing vacuum dynamics, which describes a random walk on the network of vacua. Our probabilities require two pieces of prior information, both pertaining to initial conditions; a prior density ρ(t) for the time of nucleation, and a prior probability pα for the ancestral vacuum. For ancestral vacua, we advocate the uniform prior as a conservative choice, though our conclusions are fairly insensitive to this choice. For the time of nucleation, we argue that a uniform prior is consistent with the time-translational invariance of the master equation and represents the minimally informative choice. The resulting predictive probabilities coincide with Bousso’s “holographic” prior probabilities and are closely related to Garriga and Vilenkin’s “comoving” probabilities. Despite making the least informative priors, these probabilities are surprisingly predictive. They favor vacua whose surrounding landscape topography is that of a deep funnel, akin to the folding funnels of naturally occurring proteins. They predict that we exist during the approach to near-equilibrium, much earlier than the mixing time for the landscape. We also consider a volume-weighted ρ(t), which amounts to weighing vacua by physical volume. The predictive probabilities in this case coincide with the GSVW measure. The Bayesian framework allows us to compare the plausibility of the uniform-time and volume-weighted hypotheses to explain our data by computing the Bayesian evidence for each. We argue, under general and plausible assumptions, that posterior odds overwhelmingly favor the uniform-time hypothesis.

... During inflation, scalar fields are subject to quantum fluctuations fuelled by the Hubble rate. In the stochastic approach [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], the field evolution is governed by a Langevin-like equation where quantum fluctuations correspond, qualitatively, to a random walk. Starting from an initial distribution of field values in a patch of the Universe, the distribution changes as the patch grows and the field values fluctuate. ...

... Instead of considering the probability distribution P FP , for our purposes we are interested in determining the volume-weighted distribution of field values, found from calculating e 3Ht averaged over the random walks. This distribution will be referred to as P (φ, t) and, as a function of proper time, is a solution to the volume-weighted Fokker-Planck (FPV) equation [12,15,19,23] ∂ ∂φ 8π 2 ...

... In eternal inflation, probabilities obtained with any volume-based measure are independent of the initial conditions and instead correspond to a stationary state which expands, in some time foliation, uniformly at a constant rate [15,18,19,[29][30][31]. This stationarity follows automatically from the eternal nature of inflation and is independent of choice of cut-off or time foliation (for a discussion see again chapter 6 of ref. [24]), as it can be understood from the volume-weighting, which exponentially favours observers at the latest possible times. ...

A bstract
We describe a new phenomenon in quantum cosmology: self-organised localisation. When the fundamental parameters of a theory are functions of a scalar field subject to large fluctuations during inflation, quantum phase transitions can act as dynamical attractors. As a result, the theory parameters are probabilistically localised around the critical value and the Universe finds itself at the edge of a phase transition. We illustrate how self-organised localisation could account for the observed near-criticality of the Higgs self-coupling, the naturalness of the Higgs mass, or the smallness of the cosmological constant.

... [42][43][44][45]. The details of stochastic inflation [9,[46][47][48][49][50][51][52] covered in section 4.3 however is not so standard and the non-expert is directed to the main results of this section listed below. The stochastic-δN formalism [53][54][55][56] covered in section 4.3.3 is even less well known and includes results from our paper [4] which extends known results to be valid outside Slow-Roll (SR). ...

... Stochastic Inflation [9,[46][47][48][49][50][51][52] has enjoyed much success as the leading framework to describe the evolution of non-linear perturbations and their backreaction on the dynamics of the inflaton. The basic idea is to split inflationary perturbations into shortand long-wavelength components. ...

This thesis is dedicated to the study of stochastic processes; non-deterministic physical phenomena that can be well described by classical physics. The stochastic processes we are interested in are akin to Brownian Motion and can be described by an overdamped Langevin equation comprised of a deterministic drift term and a random noise term. In Part I we examine stochastic processes in the Mesoscale. For us this means that the Langevin equation is driven by thermal noise, with amplitude proportional to the temperature $T$ and there exists a genuine equilibrium thermal state. We apply a technique known as the Functional Renormalisation Group (FRG) which allows us to coarse-grain in temporal scales. We describe how to obtain effective equations of motion for the 1- and 2-point functions of a particle evolving in highly non-trivial potentials and verify their accuracy by comparison to direct numerical simulations. In this way we outline a novel procedure for describing the behaviour of stochastic processes without having to resort to time consuming numerical simulations. In Part II we turn to the Early Universe and in particular examine stochastic processes occurring during a period of accelerated expansion known as inflation. This inflationary period is driven by a scalar field called the inflaton which also obeys a Langevin equation in the Stochastic Inflation formalism. We use this to study the formation of Primordial Black Holes during a period of Ultra Slow-Roll. We finish this thesis by applying the techniques developed in Part I to a spectator field during inflation. FRG techniques can compute cosmologically relevant observables such as the power spectrum and spectral tilt. We also extend the FRG formalism to solve first-passage time problems and verify that it gives the correct prediction for the average time taken for a field (or particle) to overcome a barrier in the potential.

... where we are using the subscript (0) to remind the reader that we are at leading order in gradient expansion It is important to remark here that the leading order in gradient expansion of each quantity can be different, for example, the leading order for α and ζ is O(σ 0 ) whereas the leading order for β i is O(σ −1 ). Having a quantity whose leading order is gradient expansion is O(σ −1 ) could seem problematic; however this is only telling us that β i is generically a non-local quantity, in fact, its linearization give us the non-local variable B (see (23)) studied in Section 4.1.1. Furthermore, we will see that β i always appear together with a spatial derivative in the equations of motion. ...

... From Section 5 it should be clear now that we will have different stochastic equations depending on if we are using the separate universe approach or the O(σ 0 ) gradient expansion. The stochastic formalism that uses the separate universe approach is the most widely used in the literature (see for example [10,[14][15][16][17][18][19]23,27,35,37,38,[51][52][53]63,66,115,116]) so we will start with its derivation. After that, we will take advantage of the stochastic equations just derived to present a stochastic formalism based on the O(σ 0 ) gradient expansion [65]. ...

We present a review on the state-of-the-art of the mathematical framework known as stochastic inflation, paying special attention to its derivation, and giving references for the readers interested in results coming from the application of the stochastic framework to different inflationary scenarios, especially to those of interest for primordial black hole formation. During the derivation of the stochastic formalism, we will emphasise two aspects in particular: the difference between the separate universe approach and the true long wavelength limit of scalar inhomogeneities and the generically non-Markovian nature of the noises that appear in the stochastic equations.

... The multiverse suffers from an information loss problem akin to that of black holes: the so-called "measure problem" [1]. This arises in cosmological models that assume a classical near de Sitter (dS) background, in which quantum fluctuations produce physically distinct patches where inflation locally ends and a more interesting cosmological evolution can ensue. ...

... The latter are associated with coarse-grained descriptions of the universe. 1 Specifically, each individual saddle geometry contains information about a limited cosmic patch or bubble only, while coarse-graining, or averaging, over any putative mosaic structure on much larger scales. It has been argued that this semiclassical description resolves the information loss problem associated with multiverse cosmology [2]. ...

A bstract
We consider multiverse models in two-dimensional linear dilaton-gravity theories as toy models of false vacuum eternal inflation. Coupling conformal matter we calculate the Von Neumann entropy of subregions. When these are sufficiently large we find that an island develops covering most of the rest of the multiverse, leading to a Page-like transition. This resonates with a description of multiverse models in semiclassical quantum cosmology, where a measure for local predictions is given by saddle point geometries which coarse-grain over any structure associated with eternal inflation beyond one’s patch.

... Since large density perturbations are generated during the JCAP11(2023)089 waterfall phase transition, the waterfall field experiences the stochastic dynamics that requires detailed numerical simulations (see refs. [54][55][56][57][58][59][60][61][62][63] for the first papers on the subject) and was omitted in some literature. Several studies on hybrid inflation models reveal that the stochastic effect on the waterfall fields significantly alters the predictions of hybrid inflation models (see, e.g., ref. [64]). ...

We show that a hybrid inflation model with multiple waterfall fields can result in the formation of primordial black holes (PBHs) with an astrophysical size, by using an advanced algorithm to follow the stochastic dynamics of the waterfall fields. This is in contrast to the case with a single waterfall field, where the wavelength of density perturbations is usually too short to form PBHs of the astrophysical scale (or otherwise PBHs are overproduced and the model is ruled out) unless the inflaton potential is tuned. In particular, we demonstrate that PBHs with masses of order 10²⁰ g can form after hybrid inflation consistently with other cosmological observations if the number of waterfall fields is about 5 for the case of instantaneous reheating. Observable gravitational waves are produced from the second-order effect of large curvature perturbations as well as from the dynamics of texture or global defects that form after the waterfall phase transition.

... (D. 15) As there is no unique way to set a global time direction across casually disconnected regions of spacetime, the regularization procedure is not unambiguous and leads to the measure problem [12]. Two common choices here are the proper time cut-off measure [78] and the scale factor cut-off measure [79,80]. The first choice is more intuitive however gives a strong preference to patches with large energy and, in particular, results in the so-called "youngness" paradox [81,82]. ...

A bstract
We revisit the original proposal of cosmological relaxation of the electroweak scale by Graham, Kaplan and Rajendran in which the Higgs mass is scanned during inflation by an axion field, the relaxion . We investigate the regime where the relaxion is subject to large fluctuations during inflation. The stochastic dynamics of the relaxion is described by means of the Fokker-Planck formalism. We derive a new stopping condition for the relaxion taking into account transitions between the neighboring local minima of its potential. Relaxion fluctuations have important consequences even in the “classical-beats-quantum” regime. We determine that for a large Hubble parameter during inflation, the random walk prevents the relaxion from getting trapped at the first minimum. The relaxion stops much further away, where the potential is less shallow. Interestingly, this essentially jeopardises the “runaway relaxion” threat from finite-density effects, restoring most of the relaxion parameter space. We also explore the “quantum-beats-classical” regime, opening large new regions of parameter space. We investigate the consequences for both the QCD and the non-QCD relaxion. The misalignment of the relaxion due to fluctuations around its local minimum opens new phenomenological opportunities.

... In the presence of mixings or generic potential shapes, we need to solve the Fokker-Planck equation [76,77] with the volume effect [78][79][80][81]. Here, we analytically derive a conservative bound. ...

In an axiverse with numerous axions, the cosmological moduli problem poses a significant challenge because the abundance of axions can easily exceed that of dark matter. The well-established stochastic axion scenario offers a simple solution, relying on relatively low-scale inflation. However, axions are typically subject to mixing due to mass and kinetic terms, which can influence the solution using stochastic dynamics. Focusing on the fact that the QCD axion has a temperature-dependent mass, unlike other axions, we investigate the dynamics of the QCD axion and another axion with mixing. We find that the QCD axion abundance is significantly enhanced and becomes larger than that of the other axion for a certain range of parameters. This enhancement widens the parameter regions accounting for dark matter. In addition, we also find a parameter region in which both axions have enhanced abundances of the same order, which result in multicomponent dark matter.

... [47][48][49][50][51]) for the calculation of cosmological correlation functions. The stochastic formalism (see Refs. [52][53][54][55][56][57][58][59][60][61] for the first papers on the subject) is the EFT for matter fields (such as the inflaton) coarse-grained on a superhorizon scale, called the infrared (IR) mode. There, the IR mode is interpreted as a (non-quantum) stochastic variable (or a Brownian motion), governed by the effective action improved by the UV loops. ...

We study the cancellation of quantum corrections on the superhorizon curvature perturbations from subhorizon physics beyond the single-clock inflation from the viewpoint of the cosmological soft theorem. As an example, we focus on the transient ultra-slow-roll inflation scenario and compute the one-loop quantum corrections to the power spectrum of curvature perturbations taking into account nontrivial surface terms in the action. We find that Maldacena's consistency relation is satisfied and guarantees the cancellation of contributions from the short-scale modes. As a corollary, primordial black hole production in single-field inflation scenarios is not excluded by perturbativity breakdown even for the sharp transition case in contrast to some recent claims in the literature. We also comment on the relation between the tadpole diagram in the in-in formalism and the shift of the elapsed time in the stochastic-$\delta N$ formalism. We find our argument is not directly generalisable to the tensor perturbations.

... In this work we will utilise the stochastic inflation [31][32][33][34][35][36][37][38] formalism as it can accurately describe the evolution of non-linear perturbations. This is achieved by splitting the inflationary perturbations into short-and long-wavelength components where the long-wavelength perturbations can be treated as effectively classical. ...

In this work we terminate inflation during a phase of Constant Roll by means of a waterfall field coupled to the inflaton and a spectator field. The presence of a spectator field means that inflation does not end at a single point, $\phi_e$, but instead has some uncertainty resulting in a stochastic end of inflation. We find that even modestly coupled spectator fields can drastically increase the abundance of Primordial Black Holes (PBHs) formed by many orders of magnitude. The power spectrum created by the inflaton can be as little as $10^{-4}$ during a phase of Ultra Slow-Roll and still form a cosmologically relevant number of PBHs. We conclude that the presence of spectator fields, which very generically will alter the end of inflation, is an effect that cannot be ignored in realistic models of PBH formation.

... Remarkably, it can generate large density perturbations at small scales that result in the formation of PBHs [51][52][53]. Since large density perturbations are generated during the waterfall phase transition, the waterfall field experiences the stochastic dynamics that requires detailed numerical simulations (see Refs. [54][55][56][57][58][59][60][61][62][63] for the first papers on the subject) and was omitted in some literature. Several studies on hybrid inflation models reveal that the stochastic effect on the waterfall fields significantly alters the predictions of hybrid inflation models (see, e.g., Ref. [64]). ...

We show that a hybrid inflation model with multiple waterfall fields can result in the formation of primordial black hole (PBH) with an astrophysical size, by using an advanced algorithm to follow the stochastic dynamics of the waterfall fields. This is in contrast to the case with a single waterfall field, where the wavelength of density perturbations is usually too short to form PBHs of the astrophysical scale (or otherwise PBH are overproduced and the model is ruled out) unless the inflaton potential is tuned. In particular, we demonstrate that PBHs with masses of order $10^{20}\, {\rm g}$ can form after hybrid inflation consistently with other cosmological observations if the number of waterfall fields is about 5 for the case of instantaneous reheating. Observable gravitational waves are produced from the second-order effect of large curvature perturbations as well as from the dynamics of texture or global defects that form after the waterfall phase transition.

... There had to have been a first man and a first woman. Although, according to the physicists, matter and empty space emerged simultaneously from out of the big bang explosion (Linde et al., 1994) that supposedly occurred 13.75 light years ago from today, give or take a billion or so light years either way: why could the first man and woman not have emerged simultaneously? Scientists discovered the somewhat fixed magnetic pole, even though it actually moved about. ...

... In fact, a double Big Bang has happened symmetrically before 13.8 billion years ago. A Big Bang [12] happened to expanding the small space-time and another Big Bang was happened to expand the corresponding symmetric spacetime. In fact, the Einstein's formulation of Mach principle [15] is applicable to the anti universe. ...

... (Linde, 1984) However, Borde and Vilenkin (1994, p. 3305) showed that "a physically reasonable spacetime that is eternally inflating to the future must possess an initial singularity." In response, Linde (1994Linde ( , pp. 1783Linde ( -1826 accepted their conclusion. The other one which avoids the singularity is the Hawking-Hartle (1983) Model. ...

Links:
https://pfk.qom.ac.ir/?lang=en
https://pfk.qom.ac.ir/article_2417.html?lang=en
https://pfk.qom.ac.ir/article_2417_a12a5059dfe54e9fb3410447d9c0a3c2.pdf

... Stochastic Inflation [1][2][3][4][5][6][7][8] is the leading framework to describe the evolution of non-linear perturbations and their backreaction on the dynamics of the inflaton, φ. The basic idea is to split inflationary perturbations into short-and long-wavelength components. ...

We use Functional Renormalisation Group (FRG) techniques to analyse the behaviour of a spectator field, σ , during inflation that obeys an overdamped Langevin equation. We briefly review how a derivative expansion of the FRG can be used to obtain Effective Equations of Motion (EEOM) for the one- and two-point function and derive the EEOM for the three-point function. We show how to compute quantities like the amplitude of the power spectrum and the spectral tilt from the FRG. We do this explicitly for a potential with multiple barriers and show that in general many different potentials will give identical predictions for the spectral tilt suggesting that observations are agnostic to localised features in the potential. Finally we use the EEOM to compute first-passage time (FPT) quantities for the spectator field. The EEOM for the one- and two-point function are enough to accurately predict the average time taken 〈𝒩〉 to travel between two field values with a barrier in between and the variation in that time δ𝒩 ² . It can also accurately resolve the full PDF for time taken ρ (𝒩), predicting the correct exponential tail. This suggests that an extension of this analysis to the inflaton can correctly capture the exponential tail that is expected in models producing Primordial Black Holes.

... (Linde, 1984) However, Borde and Vilenkin (1994, p. 3305) showed that "a physically reasonable spacetime that is eternally inflating to the future must possess an initial singularity." In response, Linde (1994Linde ( , pp. 1783Linde ( -1826 accepted their conclusion. The other one which avoids the singularity is the Hawking-Hartle (1983) Model. ...

... In the stochastic inflation framework (see the seminal works [24,[33][34][35][36][37][38][39][40][41]), the longwavelength modes of the scalar field are driven by an effectively classical, yet stochastic dynamics. The source of the stochasticity stems from the quantum nature of the vacuum fluctuations of this bosonic field. ...

We make use of Borel resummation to extract the exact time dependence from the divergent series found in the context of stochastic inflation. Correlation functions of self-interacting scalar fields in de Sitter spacetime are known to develop secular IR divergences via loops, and the first terms of the divergent series have been consistently computed both with standard techniques for curved spacetime quantum field theory and within the framework of stochastic inflation. We show that Borel resummation can be used to interpret the divergent series and to correctly infer the time evolution of the correlation functions. In practice, we adopt a method called Borel--Pad\'{e} resummation where we approximate the Borel transformation by a Pad\'{e} approximant. We also discuss the singularity structures of Borel transformations and mention possible applications to cosmology.

... Around the critical point φ = φ c , the curvature of the potential along with ψ direction is so small that its quantum fluctuations efficiently grow with time. It obeys the slowroll Langevin equation (see Refs. [55][56][57][58][59][60][61][62][63][64] for the first papers on the subject) ...

We revisit the scenario of primordial black hole (PBH) formation from large curvature perturbations generated during the waterfall phase transition in hybrid inflation models. In a minimal setup considered in the literature, the mass and abundance of PBHs are correlated and astrophysical size PBHs tend to be overproduced. This is because a longer length scale for curvature perturbations (or a larger PBH mass) requires a longer waterfall regime with a flatter potential, which results in overproduction of curvature perturbations. However, in this paper, we discuss that the higher-dimensional terms for the inflaton potential affect the dynamics during the waterfall phase transition and show that astrophysical size PHBs of order $10^{17\text{--}23} \, {\rm g}$ (which can explain the whole dark matter) can form in some parameter space consistently with any existing constraints. The scenario can be tested by observing the induced gravitational waves from scalar perturbations by future gravitational wave experiments, such as LISA.

... This additionally decreases the height of the peak of the perturbations. On the other hand, during the several e-foldings near ϕ c the average amplitude of the perturbations may grow slightly above H 2π , by a factor O(1). Therefore it would be interesting to perform a more detailed investigation of stochastic effects during inflation, following [63][64][65][66][67][68][69][70][71][72][73][74]. However, we believe that the simple estimates (4.8), (4.9) give a good estimate of the range of validity of the perturbative analysis to be used in this paper. ...

We investigate the two-stage inflation regime in the theory of hybrid cosmological $\alpha$-attractors. The spectrum of inflationary perturbations is compatible with the latest Planck/BICEP/Keck results, thanks to the attractor properties of the model. However, at smaller scales, it may have a very high peak of controllable width and position, leading to a copious production of primordial black holes (PBH) and generation of a stochastic background of gravitational waves (SGWB).

... Stochastic Inflation [1][2][3][4][5][6][7][8] is the leading framework to describe the evolution of non-linear perturbations and their backreaction on the dynamics of the inflaton, φ. The basic idea is to split inflationary perturbations into short-and long-wavelength components. ...

We use Functional Renormalisation Group (FRG) techniques to analyse the behaviour of a spectator field, $\sigma$, during inflation that obeys an overdamped Langevin equation. We briefly review how a derivative expansion of the FRG can be used to obtain Effective Equations of Motion (EEOM) for the one- and two-point function and derive the EEOM for the three-point function. We show how to compute quantities like the amplitude of the power spectrum and the spectral tilt from the FRG. We do this explicitly for a potential with multiple barriers and show that in general many different potentials will give identical predictions for the spectral tilt suggesting that observations are agnostic to localised features in the potential. Finally we use the EEOM to compute first-passage time (FPT) quantities for the spectator field. The EEOM for the one- and two-point function are enough to accurately predict the average time taken $\left\langle \mathcal{N}\right\rangle$ to travel between two field values with a barrier in between and the variation in that time $\delta \mathcal{N}^2$. It can also accurately resolve the full PDF for time taken $\rho (\mathcal{N})$, predicting the correct exponential tail. This suggests that an extension of this analysis to the inflaton can correctly capture the exponential tail that is expected in models producing Primordial Black Holes.

... This formalism introduces a coarse-graining scale which separates short and long wavelength modes of the scalar field. Quantum field fluctuations on short wavelengths are swept up into the long wavelength regime, where they are incorporated in the coarse-grained field,φ, resulting in a stochastic noise term, ξ, in the dynamical equations [16,[59][60][61][62][63][64][65][66][67]. We thus model inflation as a non-perturbative stochastic process, with the results of linear perturbation theory recovered in the low-diffusion limit [68]. ...

We show how importance sampling can be used to reconstruct the statistics of rare cosmological fluctuations in stochastic inflation. We have developed a publicly available package, PyFPT ,[ https://github.com/Jacks0nJ/PyFPT .] that solves the first-passage time problem of generic one-dimensional Langevin processes. In the stochastic- δ N formalism, these are related to the curvature perturbation at the end of inflation. We apply this method to quadratic inflation, where the existence of semi-analytical results allows us to benchmark our approach. We find excellent agreement within the estimated statistical error, both in the drift- and diffusion-dominated regimes. The computation takes at most a few hours on a single CPU, and can reach probability values corresponding to less than one Hubble patch per observable universe at the end of inflation. With direct sampling, this would take more than the age of the universe to simulate even with the best current supercomputers. As an application, we study how the presence of large-field boundaries might affect the tail of the probability distribution. We also find that non-perturbative deviations from Gaussianity are not always of the simple exponential type.

... Though the precise probability should be calculated taking all quantum noise into account in, e.g., the stochastic approach (see, e.g., Refs. [29,[41][42][43][44][45][46][47][48][49] for the first papers on this approach, and also Refs. [50][51][52][53][54][55][56][57] for its application to the exponential tail), qualitative features are often extracted by a simple assumption that only one noise gives a dominant contribution and the other dynamics is well approximated by the one without noise [19,[58][59][60]. ...

Primordial black holes (PBHs) whose masses are in $\sim[10^{-15}M_\odot,10^{-11}M_{\odot}]$ have been extensively studied as a candidate of whole dark matter (DM). One of the probes to test such a PBH-DM scenario is scalar-induced stochastic gravitational waves (GWs) accompanied with the enhanced primordial fluctuations to form the PBHs with frequency peaked in the mHz band being targeted by the LISA mission. In order to utilize the stochastic GWs for checking the PBH-DM scenario, it needs to exactly relate the PBH abundance and the amplitude of the GWs spectrum. Recently in Kitajima et al., the impact of the non-Gaussianity of the enhanced primordial curvature perturbations on the PBH abundance has been investigated based on the peak theory, and they found that a specific non-Gaussian feature called the exponential tail significantly increases the PBH abundance compared with the Gaussian case. In this work, we investigate the spectrum of the induced stochastic GWs associated with PBH DM in the exponential-tail case. In order to take into account the non-Gaussianity properly, we employ the diagrammatic approach for the calculation of the spectrum. We find that the amplitude of the stochastic GW spectrum is slightly lower than the one for the Gaussian case, but it can still be detectable with the LISA sensitivity. We also find that the non-Gaussian contribution can appear on the high-frequency side through their complicated momentum configurations. Although this feature emerges under the LISA sensitivity, it might be possible to obtain information about the non-Gaussianity from GW observation with a deeper sensitivity such as the DECIGO mission.

... Then, it is convenient to follow the volume distribution L 3 ½a; t. This is usually considered for the inflaton field [73][74][75][76]. A similar effect was discussed in Ref. [56] for estimating the validity for the estimation of the axion abundance from inflationary equilibrium distribution [29,56]. ...

We propose a novel scenario to explain the small cosmological constant (CC) by a peculiar inflaton potential. The shape almost satisfies the following conditions: The inflation is eternal if the CC is positive and not eternal if the CC is negative. Although realizing the peculiar shape has a similar amount of fine-tuning as the CC, the shape can be made stable under radiative corrections in the effective theory. By introducing a slowly varying CC from a positive value to a negative value, the dominant volume of the Universe after the inflation turns out to have a vanishingly small CC. The scenario does not require eternal inflation, but the e-folding number is exponentially large, and the inflation scale is low. The Hubble parameter during inflation, Hinf, is required to be smaller than the present CC scale, and, thus, the CC relaxed during inflation with the low renormalization scale, ∼Hinf, is safe from the radiative corrections from the standard model particles. The scenario can have a consistent thermal history, but the present equation of state of the Universe is predicted to slightly differ from the one for the ΛCDM model. In a time-varying CC model, CC can be relaxed from (103 GeV)4, and in a model with a light scalar field scanning the CC during inflation, CC can be relaxed from (10 MeV)4.

... This formalism introduces a coarse-graining scale which separates short and long wavelength modes of the scalar field. Quantum field fluctuations on short wavelengths are swept up into the long wavelength regime, where they are incorporated in the coarse-grained field,φ, resulting in a stochastic noise term, ξ, in the dynamical equations [16,[59][60][61][62][63][64][65][66][67]. We thus model inflation as a non-perturbative stochastic process, with the results of linear perturbation theory recovered in the lowdiffusion limit [68]. ...

We show how importance sampling can be used to reconstruct the statistics of rare cosmological fluctuations in stochastic inflation. We have developed a publicly available package, PyFPT, that solves the first-passage time problem of generic one-dimensional Langevin processes. In the stochastic-$\delta N$ formalism, these are related to the curvature perturbation at the end of inflation. We apply this method to quadratic inflation, where the existence of semi-analytical results allows us to benchmark our approach. We find excellent agreement within the estimated statistical error, both in the drift- and diffusion-dominated regimes. The computation takes at most a few hours on a single CPU, and can reach probability values corresponding to less than one Hubble patch per observable universe at the end of inflation. With direct sampling, this would take more than the age of the universe to simulate even with the best current supercomputers. As an application, we study how the presence of large-field boundaries might affect the tail of the probability distribution. We also find that non-perturbative deviations from Gaussianity are not always of the simple exponential type.

... The e-folding until this moment N 8π 2 M 2 Pl 3H 2 already saturates the upper bound for finite inflation, 2π 2 3 M 2 Pl H 2 , given by the de Sitter entropy [74][75][76]. Thus, Hubble selection needs eternal inflation, and the universe eventually reaches a stationary state [77][78][79][80]. Probability distributions are to be defined within an ensemble of Hubble patches that have reached reheating [81][82][83]. ...

There is growing evidence that the small weak scale may be related to self-organized criticality. In this regard, we note that if the strange quark were lighter, the QCD phase transition could have been first order, possibly exhibiting quantum critical points at zero temperature as a function of the Higgs vacuum expectation value vh smaller than (but near) the weak scale. We show that these quantum critical points allow a dynamical selection of the observed weak scale, via quantum-dominated stochastic evolutions of the value of vh during eternal inflation. Although the values of vh in different Hubble patches are described by a probability distribution in the multiverse, inflationary quantum dynamics ensures that the peak of the distribution evolves toward critical points (self-organized criticality), driven mainly by the largest Hubble expansion rate there—the Hubble selection of the universe. To this end, we first explore the quantum critical points of the three-flavor QCD linear sigma model, parametrized by vh at zero temperature, and we present a relaxion model for the weak scale. Among the patches that have reached reheating, it results in a sharp probability distribution of vh near the observed weak scale, which is critical not to the crossover at vh=0 but to the sharp transition at ∼ΛQCD.

... Although there are various theories about the origin of the universe, the Big Bang theory is one of the most well-known [37,38]. According to it, the Big Bang was an explosion that created the known universe [39,40]. ...

In this paper, we solve the optimal power flow problem in alternating current networks to reduce power losses. For that purpose, we propose a master–slave methodology that combines the multiverse optimization algorithm (master stage) and the power flow method for alternating current networks based on successive approximation (slave stage). The master stage determines the level of active power to be injected by each distributed generator in the network, and the slave stage evaluates the impact of the proposed solution on each distributed generator in terms of the objective function and the constraints. For the simulations, we used the 10-, 33-, and 69-node radial test systems and the 10-node mesh test system with three levels of distributed generation penetration: 20%, 40%, and 60% of the power provided by the slack generator in a scenario without DGs. In order to validate the robustness and convergence of the proposed optimization algorithm, we compared it with four other optimization methods that have been reported in the specialized literature to solve the problem addressed here: Particle Swarm Optimization, the Continuous Genetic Algorithm, the Black Hole Optimization algorithm, and the Ant Lion Optimization algorithm. The results obtained demonstrate that the proposed master–slave methodology can find the best solution (in terms of power loss reduction, repeatability, and technical conditions) for networks of any size while offering excellent performance in terms of computation time.

... In semiclassical approximation and inflationary scenario quantum properties of a single homogeneous massive scalar field, inflaton, responsible for the accelerated expansion of the universe has been a work of great interest among researchers for the last two decades [2][3][4][5][6][7][8][9]. The previously mentioned studies have showed that there are significant dissimilarities between the result obtained in classical approximation to gravity from that obtained in the semiclassical approach to gravity (SG), thereby showing quantum implications and phenomenon play a prominent role in the inflationary scenarios and related problems. ...

Semiclassical Einstein equations are used to describe the interaction of the back-reaction of the classical gravitational field with quantum matter fields in semiclassical gravity. We in our previous studies have made use of the semiclassical approximation to demonstrate the phenomenon of particle production, often called preheating/reheating of the universe, which occurs after the inflationary epoch during the oscillatory phase of two-mode quantized scalar field of chaotic inflationary model. During this oscillatory phase, back-reaction effects from the created particles, on account of the quantum nature of the states considered, could be significant and one might be concerned about the validity of the semiclassical approximation in these two-mode quantum optical states. The validity of the semiclassical approximation in these states is examined and it is presented how the magnitude of states parameter draws limit on the applicability and reliability of semiclassical theory of gravity. It is argued that semiclassical theory to gravity is a good approximation for states which are closer to coherent states i.e., with coherent parameters greater than unity and with squeezed parameter much smaller than unity.

... What has been discovered so far by scientists is that at first, the universe was an infinitely dense and incomprehensibly small fireball (Linde, A. et al., 1994; & Thompson, B. et al., 2003). Then the Big Bang occurred to form the fabric of the universe in a process called ‗inflation', which lasted for only one millionth of a second (Gross, D. 2007). ...

Physicists tend to stand in front of boards on which complicated calculations are written on. When they explain things or write publications, it is mostly with calculations which is an easy language for physicists but not for most of humanity. Few physicists lay out all the results of experimentations and calculations in simple words such that a picture can be formed of all the important discoveries made so far such that they are decipherable by most of humanity. This works enumerates all these discoveries over the last 200 years and how they fit together in a sequential manner. This should be understood by most of humanity to enhance the collective consciousness. This paper starts with the forces within the nucleus all the way to the gravitational forces which keep the stars in their place. One of the most important discoveries of physicist of late is that the whole universe can be approximated to be empty.

A bstract
We make use of Borel resummation to extract the exact time dependence from the divergent series found in the context of stochastic inflation. Correlation functions of self-interacting scalar fields in de Sitter spacetime are known to develop secular IR divergences via loops, and the first terms of the divergent series have been consistently computed both with standard techniques for curved spacetime quantum field theory and within the framework of stochastic inflation. We show that Borel resummation can be used to interpret the divergent series and to correctly infer the time evolution of the correlation functions. In practice, we adopt a method called Borel-Padé resummation where we approximate the Borel transformation by a Padé approximant. We also discuss the singularity structures of Borel transformations and mention possible applications to cosmology.

We revisit the scenario of primordial black hole (PBH) formation from large curvature perturbations generated during the waterfall phase transition in hybrid inflation models. In a minimal setup considered in the literature, the mass and abundance of PBHs are correlated and astrophysical size PBHs tend to be overproduced. This is because a longer length scale for curvature perturbations (or a larger PBH mass) requires a longer waterfall regime with a flatter potential, which results in overproduction of curvature perturbations. However, in this paper, we discuss that the higher-dimensional terms for the inflaton potential affect the dynamics during the waterfall phase transition and show that astrophysical size PBHs of the order of 1017−23 g (which can explain the whole dark matter) can form in some parameter space consistently with any existing constraints. The scenario can be tested by observing the induced gravitational waves from scalar perturbations by future gravitational wave experiments, such as LISA.

Primordial black holes (PBHs) whose masses are in ∼ [10 ⁻¹⁵ M ⊙ ,10 ⁻¹¹ M ⊙ ] have been extensively studied as a candidate of whole dark matter (DM). One of the probes to test such a PBH-DM scenario is scalar-induced stochastic gravitational waves (GWs) accompanied with the enhanced primordial fluctuations to form the PBH with frequency peaked in the mHz band being targeted by the LISA mission. In order to utilize the stochastic GW for checking the PBH-DM scenario, it needs to exactly relate the PBH abundance and the amplitude of the GW spectrum. Recently in Kitajima et al. [1], the impact of the non-Gaussianity of the enhanced primordial curvature perturbations on the PBH abundance has been investigated based on the peak theory, and they found that a specific non-Gaussian feature called the exponential tail significantly increases the PBH abundance compared with the Gaussian case. In this work, we investigate the spectrum of the induced stochastic GW associated with PBH DM in the exponential-tail case. In order to take into account the non-Gaussianity properly, we employ the diagrammatic approach for the calculation of the spectrum. We find that the amplitude of the stochastic GW spectrum is slightly lower than the one for the Gaussian case, but it can still be detectable with the LISA sensitivity. We also find that the non-Gaussian contribution can appear on the high-frequency side through their complicated momentum configurations. Although this feature emerges under the LISA sensitivity, it might be possible to obtain information about the non-Gaussianity from GW observation with a deeper sensitivity such as the DECIGO mission.

According to the ‘Cosmological Central Dogma’, de Sitter space can be viewed as a quantum mechanical system with a finite number of degrees of freedom, set by the horizon area. We use this assumption together with the Wheeler-DeWitt (WDW) equation to approach the measure problem of eternal inflation. Thus, our goal is to find a time-independent wave function of the universe on a total Hilbert space defined as the direct sum of a variety of subspaces: A finite-dimensional subspace for each de Sitter vacuum and an infinite-dimensional subspace for each terminal Minkowski or AdS vacuum. We argue that, to be consistent with semiclassical intuition, such a solution requires the presence of sources. These are implemented as an inhomogenous term in the WDW equation, induced by the Hartle-Hawking no-boundary or the Linde/Vilenkin tunneling proposal. Taken together, these steps unambiguously lead to what we would like to think of as a ‘Local WDW measure,’ where ‘local’ refers to the fact that the dS part of the resulting wave function describes a superposition of static patches. The global 3-sphere spatial section of the entire multiverse makes no appearance.

We investigate the two-stage inflation regime in the theory of hybrid cosmological α -attractors. The spectrum of inflationary perturbations is compatible with the latest Planck/BICEP/Keck Array results, thanks to the attractor properties of the model. However, at smaller scales, it may have a very high peak of controllable width and position, leading to a copious production of primordial black holes (PBH) and generation of a stochastic background of gravitational waves (SGWB).

The possibility of the resilience of the beginning of inflation under unfavourable conditions is examined by considering the initial state of the inflaton field to be in the form of a relativistic gas with some of its properties in close proximity to the black body spectrum. It is demonstrated that the initial potential energy budget in such an environment is suppressed beyond the minimal value required for inflation. This is the extension of our earlier work, where we have shown that the rare regions which happen to host favourable initial conditions for the beginning of inflation could come to dominate the late-time Universe only if they started out from above the so-called self-reproduction threshold.

We study the Brownian motion of a field where there are boundaries in the inflationary field space. Both the field and the boundary undergo Brownian motions with the amplitudes of the noises determined by the Hubble expansion rate of the corresponding de Sitter spacetime. This setup mimics models of inflation in which curvature perturbation is induced from inhomogeneities generated at the surface of the end of inflation. The cases of the drift-dominated regime as well as the diffusion-dominated regime are studied in detail. We calculate the first hitting probabilities as well as the mean number of e-folds for the field to hit either of the boundaries in the field space. The implications for models of inflation are reviewed.

The possibility of the resilience of the beginning of inflation under unfavorable conditions is examined by considering the initial state of the inflaton field to be in the form of a relativistic gas with some of its properties in close proximity to the black body spectrum. It is demonstrated that the initial potential energy budget in such an environment is suppressed beyond the minimal value required for inflation. This is the extension of our earlier work, where we have shown that the rare regions which happen to host favorable initial conditions for the beginning of inflation could come to dominate the late-time Universe only if they started out from above the so-called self-reproduction threshold.

The Big Bang theory states that the universe was created from pure energy, although matter, in general, is also pure energy and there is no known physical existence that is not pure energy in accordance with the mass-energy equation. All known energy is situated in a field, and it can be questioned whether also the Big Bang was situated in a field in the primordial moment it inflated into the subsequent cosmic expansion that so far lets us observe a 93-billion-light-year-wide spherical volume of the universe. In this study, the Big Bang’s gravitational influence, particularly in the form of an externally radiated gravitational wave, is considered in connection to its situation in a surrounding field with a different expansion rate than itself. The results suggest that the least possible size of the universe can be predicted by the expression of the gravitational wave produced by Big Bang, revealing that the universe has a significantly greater size than the observable, and further that Big Bang might be the production of only one of many cosmic galaxies situated together in a cosmological wave complex (CWC) where the amplitude is self-maintained by inflations.

Transitions between different inflationary slow-roll scenarios are known to provide short non-slow-roll periods with non-trivial consequences. We consider the effect of quantum diffusion on the inflationary dynamics in a transition process. Using the stochastic δ𝒩 formalism, we follow the detailed evolution of noises through a sharp transition modeled by the Starobinsky potential, although some of our results apply to any sharp transition. We find how the stochastic noise induced by the transition affects the coarse-grained fields. We then consider the special case that the potential is flat after the transition. It is found that, during a particular phase of evolution, the noise we obtain cannot drive the inflaton past the classically unreachable field values; so the boundary crossing is delayed. By deriving the characteristic function, we also study the tail behavior for the distribution of curvature perturbations ζ , which we find to decay faster than exp(-3 ζ ).

Transitions between different inflationary slow-roll scenarios are known to provide short non-slow-roll periods with non-trivial consequences. We consider the effect of quantum diffusion on the inflationary dynamics in a transition process. Using the stochastic {\delta}N formalism, we follow the detailed evolution of noises through a sharp transition modeled by the Starobinsky potential, although some of our results apply to any sharp transition. We find how the stochastic noise induced by the transition affects the coarse-grained fields. We then consider the special case that the potential is flat after the transition. It is found that the particular noise we obtain cannot drive the inflaton past the classically unreachable field values. By deriving the characteristic function, we also study the tail behavior for the distribution of curvature perturbations {\zeta}, which we find to decay faster than e^(-3{\zeta}).

Semiclassical Einstein equations are used to describe the interaction of the back-reaction of the classical gravitational field with quantum matter fields in semiclassical gravity. We in our previous studies have made use of the semiclassical approximation to demonstrate the phenomenon of particle production, commonly called as preheating/reheating of the universe, which occurs after the inflationary epoch during the oscillatory phase of two-mode quantised massive scalar field of the chaotic inflationary model. We have previously used the language of two-mode coherent and squeezed quantum optical states formalisms to represent the massive scalar field; therefore, it would be useful to examine whether the field states exhibits classical or nonclassical nature in the cosmological context. In the present article, we will examine the nonclassical nature of two-mode quantum optical states in the cosmological context. We have made use of the criterion suggested by Lee in quantum optics, for the existence of nonclassical effects in two-mode states and calculated the equivalent Lee’s D12(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}_{12}^{(2)}$$\end{document} parameter with the associated cosmological parameters, to examine the nonclassical nature of the states after inflation during the oscillatory phase of the scalar field.

In a situation like eternal inflation, where our data is replicated at infinitely-many other space-time events, it is necessary to make a prior assumption about our location to extract predictions. The principle of mediocrity entails that we live at asymptotic late times, when the occupational probabilities of vacua has settled to a near-equilibrium distribution. In this paper we further develop the idea that we instead exist during the approach to equilibrium, much earlier than the exponentially-long mixing time. In this case we are most likely to reside in vacua that are easily accessed dynamically. Using first-passage statistics, we prove that vacua that maximize their space-time volume at early times have: 1. maximal ever-hitting probability; 2. minimal mean first-passage time; and 3. minimal decay rate. These requirements are succinctly captured by an early-time measure. The idea that we live at early times is a predictive guiding principle, with many phenomenological implications. First, our vacuum should lie deep in a funneled region, akin to folding energy landscapes of proteins. Second, optimal landscape regions are characterized by relatively short-lived vacua, with lifetime of order the de Sitter Page time. For our vacuum, this lifetime is ∼ 10 ¹³⁰ years, which is consistent with the Standard Model estimate due to Higgs metastability. Third, the measure favors vacua with small, positive vacuum energy. This can address the cosmological constant problem, provided there are sufficiently many vacua in the entire ensemble of funnels. As a concrete example, we study the Bousso-Polchinski lattice of flux vacua, and find that the early-time measure favors lattices with the fewest number of flux dimensions. This favors compactifications with a large hierarchy between the lightest modulus and all other Kähler and complex structure moduli.

The human race has spent and invested considerable resources to understand the processes of your solar system. However, since the discovery of exoplanets we have the means to determine whether or not we actually understand these processes. The most compelling reason to find exoplanets is that it opens the door for us to look for other habitable planets as well as understand our own solar system better. For years, scientists have been utilizing data from NASA’s Kepler Space Telescope to look for and identify thousands of transiting exoplanets. Thanks to new and better telescopes, astronomical data is rapidly increasing. Traditional human judgment-based prediction and classification methods are inefficient and vulnerable to vary depending on the expert doing the study. The widely used methodology for exoplanet discovery, the Box-fitting Least Squares technique (BLS), for example, creates a large number of false positives that must be manually checked in the event of noisy data. As a result, an automated and unbiased approach for detecting exoplanets while removing false-positive signals imitating transiting planet signals is required. A new convolutional neural network-based mechanism for finding exoplanets is introduced using the transit technique. Since the dataset is large and highly imbalanced, SMOTE is used to resample the data, while the exponential decay approach along with dropout and early stopping techniques are used to reduce model overfitting. In addition, the model employs the Grid-SearchCV approach to fine-tune hyper-parameters. Finally, for a robust and full model, the model is evaluated using k fold cross-validation. Performance criteria such as accuracy, precision, recall, f1 score, sensitivity, and specificity are used in the study. After analyzing the data, the research concluded that the convolutional neural network produced a maximum accuracy of 99.6% on the testing data.KeywordsConvolutional neural networkGridSearchCVK fold cross validationMachine learningSMOTEExoplanet detectionTransit method

This PhD thesis aims at studying theoretical and phenomenological aspects of cosmic inflation when it is driven by the presence of several scalar fields. After some reminders about the relativistic framework in which modern physical cosmology is embedded, one proposes an introduction to the inflationary paradigm, at the level of the homogeneous Universe but also of linear perturbations, both for single-field and multifield scenarios. Then, primordial non-Gaussianities are investigated. It is shown that multifield inflation results in specific patterns in these deviations of primordial fluctuations from Gaussian statistics. A particular emphasis is put on the observational signatures of the geometrical effects due to the curvature of the internal field space. This thesis also shows how to extend the stochastic formalism to multifield inflationary models with non-canonical kinetic terms. It reveals an ambiguity that is generically present in the usual formalism and, by considering the very quantum nature of the system, proposes a resolution that is both practical and conceptually satisfying. Last but not least, the period of reheating after multifield inflation is studied. It is explained how to couple those scalar fields to cosmological fluids. This enables to follow the dynamics of inflation, (p)reheating, and the eras of radiation and matter dominations, both at the homogeneous and linear levels, without having to specify by hand their transitions. In these multi-species models, an important quantity of non-adiabatic perturbations is transferred from the scalar sector to the fluids of radiation and matter.

We discuss a coarse-graining procedure for describing the superhorizon dynamics of inflationary tensor modes. Using basic principles of quantum mechanics, we determine a probability density for coarse-grained tensor fields, which satisfies a stochastic Fokker-Planck equation. The corresponding noise and drift are computable, and depend on the cosmological system under consideration. Our general formulas are applied to a variety of cosmological scenarios, including cases seldom considered in the context of stochastic inflation. We start obtaining the expected expressions for noise and drift in pure de Sitter and power-law inflation, leading to a tensor spectrum whose properties match with quantum field theory calculations. We also discuss how a nonattractor phase during inflation influences the drift of our stochastic evolution equations. We then apply our method to scenarios with a transition from de Sitter to radiation and matter domination phases, for determining the stochastic distribution of superhorizon tensor modes during these eras. In particular, we show how interference effects between modes flowing through the cosmological horizon, and modes spontaneously produced at superhorizon scales, affect the stochastic evolution of coarse-grained quantities. The expression for the stochastic noise depends on the number of e-folds of cosmic evolution, and it rapidly approaches a constant after a few e-folds of postinflationary cosmic expansion. In an appropriate limit, the corresponding spectrum of tensor modes at horizon crossing matches with the results of quantum field theory calculations.

Spectra are calculated for the primordial adiabatic perturbations and the gravitational waves that would develop in the author's 1980 model of a nonsingular cosmology with an initial de Sitter quantum stage resulting from gravitational vacuum polarization. The gravitational-wave spectrum will be flat; the adiabatic-perturbation spectrum, nearly so. In that event the most promising way to detect large-scale temperature anisotropy in the cosmic microwave background would be to measure ..delta..T/T correlations over 5/sup 0/--10/sup 0/ angles.

The dynamics of a large-scale quasi-homogeneous scalar field producing the de Sitter (inflationary) stage in the early universe is strongly affected by small-scale quantum fluctuations of the same scalar field and, in this way, becomes stochastic. The evolution of the corresponding large-scale space-time metric follows that of the scalar field and is stochastic also. The Fokker-Planck equation for the evolution of the large-scale scalar field is obtained and solved for an arbitrary scalar field potential. The average duration of the de-Sitter stage in the new inflationary scenario is calculated (only partial results on this problem were known earlier). Applications of the developed formalism to the chaotic inflationary scenario and to quantum inflation are considered. In these cases, the main unsolved problem lies in initial pre-inflationary conditions.

We give an account and (basically) a solution of a new class of problems synthesizing percolation theory and branching diffusion processes. They lead to a novel type of stochastic process, namely branching processes with diffusion on the space of parameters distinguishing the branching particles from each other.

We invesstigate thoroughly the dynamics of an inflation-driving scalar field in terms of an extended version of the stochastic
approach proposed by Starobinsky and discuss the spacetime structure of the inflationary universe. To avoid any complications
which might arise due to quantum gravity, we concentrate our discussion on the new inflationary universe scenario in which
all the energy scales involved are well below the Planck mass. The investigation is done both analytically and numerically.
In particular, we present a full numerical analysis of the stochastic scalar field dynamics on the phase space. Then implications
of the results are discussed.

We review stochastic inflation and its consequences for the interpretation of wave functions in quantum cosmology.

The global structure of the universe is analyzed within the framework of
the chaotic inflation scenario. It is shown that under certian
conditions inflation of the universe in accordance with this scenario
does not have an end and may not have a beginning. Consequently, a
larger part of the physical volume of the universe should always be in a
state of inflation at a density of the order of the Planckian density.
During inflation the universe separates into regions of exponentially
large sizes. Within these regions all possible types of metastable
vacuum states and all possible types of compactification, consistent
with the presence of inflation are realized. The investigation is
performed by employing the diffusion equation for a fluctuating scalar
field in an inflating universe.

Spectra are calculated for the primordial adiabatic perturbations and
the gravitational waves that would develop in the author's 1980 model of
a nonsingular cosmology with an initial de Sitter quantum stage
resulting from gravitational vacuum polarization. The gravitational-wave
spectrum will be flat; the adiabatic-perturbation spectrum, nearly so.
In that event, the most promising way to detect large-scale temperature
anisotropy in the cosmic microwave background would be to measure
Delta-T/T correlations over 5-10 deg angles.

It is shown that the chaotic inflation scenario can be realized under some natural assumptions concerning the initial conditions in the very early universe.

The treatment of first-order phase transitions for standard grand unified theories is shown to break down for models with radiatively induced spontaneous symmetry breaking. It is argued that proper analysis of these transitions which would take place in the early history of the universe can lead to an explanation of the cosmological homogeneity, flatness, and monopole puzzles.

A scenario is proposed for the evolution of the universe, starting with the quantum birth of a closed world at a minimum in the self-consistent de Sitter cosmological solution with vacuum polarization. The closure of the universe and the permanently supercritical value of its density follow directly from a single condition: that quantum birth take place. The perturbations must be small in order that the de Sitter phase may be sufficiently prolonged to ensure a protracted Friedmann plasma-matter expansion. Thus a universe having the properties we observe may in fact have been singled out by the anthropogenic principle.

A self-consistent problem involving the behavior of small perturbations in an isotropic homogeneous universe filled with a scalar field is considered. Solutions describing the evolution of perturbations in the case of an arbitrary scalar-field potential are obtained.

A simple and natural realisation of the chaotic inflation scenario is suggested in the context of SU(1,1) supergravity.

Following an historical introduction, the conventional canonical formulation of general relativity theory is presented. The canonical Lagrangian is expressed in terms of the extrinsic and intrinsic curvatures of the hypersurface x0=constant, and its relation to the asymptotic field energy in an infinite world is noted. The distinction between finite and infinite worlds is emphasized. In the quantum theory the primary and secondary constraints become conditions on the state vector, and in the case of finite worlds these conditions alone govern the dynamics. A resolution of the factor-ordering problem is proposed, and the consistency of the constraints is demonstrated. A 6-dimensional hyperbolic Riemannian manifold is introduced which takes for its metric the coefficient of the momenta in the Hamiltonian constraint. The geodesic incompletability of this manifold, owing to the existence of a frontier of infinite curvature, is demonstrated. The possibility is explored of relating this manifold to an infinite-dimensional manifold of 3-geometries, and of relating the structure of the latter manifold in turn to the dynamical behavior of space-time. The problem is approached through the WKB approximation and Hamilton-Jacobi theory. Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a "force term." The classical phenomenon of gravitational collapse shows that the force term is not powerful enough to prevent the trajectory of space-time from running into the frontier. The as-yet unresolved problem of determining when the collapse phenomenon represents a real barrier to the quantum-state functional is briefly discussed, and a boundary condition at the barrier is proposed. The state functional of a finite world can depend only on the 3-geometry of the hypersurface x0=constant. The label x0 itself is irrelevant, and "time" must be determined intrinsically. A natural definition for the inner product of two such state functionals is introduced which, however, encounters difficulties with negative probabilities owing to the barrier boundary condition. In order to resolve these difficulties, a simplified model, the quantized Friedmann universe, is studied in detail. In order to obtain nonstatic wave functions which resemble a universe evolving, it is necessary to introduce a clock. In order that the combined wave functions of universe-cum-clock be normalizable, it turns out that the periods of universe and clock must be commensurable. Wave packets exhibiting quasiclassical behavior are constructed, and attention is called to the phenomenological character of "time." The innerproduct definition is rescued from its negative-probability difficulties by making use of the fact that probability flows in a closed finite circuit in configuration space. The article ends with some speculations on the uniqueness of the state functional of the actual universe. It is suggested that a viewpoint due to Everett should be adopted in its interpretation.

The spectrum of density perturbations is calculated in the new-inflationary-universe scenario. The main source is the quantum fluctuations of the Higgs field, which lead to fluctuations in the time at which the false vacuum energy is released. The value of $\frac{\ensuremath{\delta}\ensuremath{\rho}}{\ensuremath{\rho}}$ on any given length scale $l$, at the time when the Hubble radius $\ensuremath{\gg}l$, is estimated. This quantity is nearly scale invariant (as desired), but is unfortunately about ${10}^{5}$ times too large.

We investigate the stationary solution of the modified Fokker-Planck equation which governs the global dynamics of the inflation.
Contrary to the original FP equation which is for a Hubble horizon size region, we found that the normalizable stationary
solution can exist for modified Fokker-Planck equation which is for many Hubble horizon size regions. For a chaotic inflationary
model with the potential λφ2n, we get initial distribution of classical universes using this solution, and discussed the physical meaning of it. Especially
for n = 2, this distribution obeys power-law and classical universes which, created from the Planck energy region, make the fractal
structure. In other cases n ≠2, creation of large classical universes is strongly suppressed.

Axion models have a spontaneously broken Z(N) symmetry. The resulting discretely degenerate vacua and domain-wall solitons are incompatible with the standard cosmology. It is possible, however, to introduce a small Z(N) breaking interaction into axion models without upsetting the Peccei-Quinn mechanism. In that case the domain walls disappear a certain time after their formation in the early universe. Their presence for a limited time period might lead to galaxy formation.

A possible realization of the chaotic inflation scenario is suggested in the context of N = 1 supergravity coupled to matter. It is shown that in this scenario one can obtain a desirable magnitude of density perturations delta g9/g9 ~ 10-4, and the primordial monopole problem can be easily solved.

A cosmological wave function describing the tunneling of the universe from ''nothing'' into a de Sitter space is found in a simple minisuperspace model. The tunneling probability is proportional to exp(-3/8G/sup 2/rho/sub v/), where rho/sub v/ is the vacuum energy density at an extremum of the effective potential V(phi). The tunneling is most probable to the highest maximum of V(phi).

A cosmological model is developed from the hypothesis that as density increases the medium enters a state of negative pressure. This model contains neither a past nor a future singularity. The solution of the Friedmann equations yields a closed universe whose mass grows by tens of orders of magnitude from the epoch at which expansion commences.

The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the "ground state" or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and Λ>0. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.

The effects of spacetime curvature upon phase transitions in an expanding universe are investigated. We consider a Robertson-Walker model which is a radiation-dominated universe at early times and becomes de Sitter space at later times. In this universe the stability of a field theory containing a pair of interacting scalar fields is studied in first-order perturbation theory. It is noted that the crucial quantity in the stability analysis is 〈φ2〉, where φ is a free scalar field. The behavior of 〈φ2〉 as a function of time is investigated, where both thermal and vacuum contributions are taken into account. It is shown that this behavior can be strongly affected by the coupling to the background gravitational field. Such coupling can cause 〈φ2〉 to decrease more slowly or even grow as the universe expands. This behavior can alter the evolution of the system and can result in either stabilization of an otherwise unstable field configuration or destabilization of an otherwise stable configuration.

Usually inflation ends either by a slow rolling of the inflation field, which gradually becomes faster and faster, or by a first-order phase transition. We describe a model where inflation ends in a different way, due to a very rapid rolling ("waterfall") of a scalar field σ triggered by another scalar field φ. This model looks like a hybrid of chaotic inflation with V(φ)=m2φ2/2 and the usual theory with spontaneous symmetry breaking with V(σ)=1/4λ(M2-λσ2)2. The last stages of inflation in this model are supported not by the inflaton potential V(φ) but by the "noninflationary" potential V(σ). Another hybrid model to be discussed here uses some building blocks from extended inflation (Brans-Dicke theory), from new inflation (phase transition due to a nonminimal coupling of the inflaton field to gravity), and from chaotic inflation (the possibility of inflation beginning at large as well as at small σ). In the simplest version of this scenario inflation ends up by slow rolling, thus avoiding the big-bubble problem of extended inflation.

A cosmological model is proposed in which the Universe is created by quantum tunneling from "nothing" into a de Sitter space. The tunneling is described by a de Sitter—Hawking—Moss instanton. After the tunneling, the model evolves along the lines of the inflationary scenario. It is argued that at any time there exist parts of the Universe which are still in the de Sitter phase, while other parts have already recollapsed. This model does not have a big-bang singularity and does not require any initial or boundary conditions.

The dynamics of inflation is that of a relaxation random process. We examine boundary conditions for this process and give a simple proof for the existence of eternal inflation that takes into account the field dependence of the effective cosmological constant and the finite duration of the inflationary phase. Next, natural initial conditions are formulated that lead to a specific interpretation of the wave function in quantum cosmology. We demonstrate that the Hartle-Hawking wave function describes the equilibrium regime for the stochastic process (with the correct quantum-field-theory limit), but only if the cosmological constant is sufficiently large or if it decays sufficiently slowly. We show in which sense inflation is certain even with the Hartle-Hawking wave function, and propose a new framework for the ‘‘tunneling’’ wave function. On the basis of boundary conditions, we argue that the dynamics of the stochastic phase and, hence, the main features of the present Universe, are independent of the physics above the Planck scale.

The conditions on the effective (de Sitter space) scalar potential that are required to obtain a successful new-inflationary-Universe scenario are codified into a general prescription. These conditions ensure that sufficient inflation, density fluctuations of an acceptable magnitude, and reheating to a high enough temperature to produce the astrophysically observed baryon asymmetry result. We exemplify our prescription for a quartic potential and show that if the scalar field is a gauge singlet, then it is possible to tune the parameters of the potential to satisfy the conditions we have prescribed.

The quantum creation of a universe with a flat comoving 3-space is noted
to be possible in nontrivial topology. The relative probability of the
quantum birth of a spatially flat, isotropic world having finite
3-volume during the de Sitter (inflationary) stage is calculated, with
allowance for the vacuum energy-momentum tensor of massless quantum
fields that will result from the nontriviality of the topology. It is
shown that an empty flat world with a 3-torus topology can begin to
evolve either from a classical singularity or by quantum creation with
or even without tunneling.

Using the Fokker-Planck equation derived from stochastic approach to
inflation, the dynamics and global structure of the chaotic inflationary
universe are investigated. Full account was taken of the difference in
the physical volume of each horizon size region in the Fokker-Planck
equation. It was found that the modified Fokker-Planck equation admits a
normalizable stationary solution, contrary to the original equation. The
approximate form of the solution was evaluatedand it describes both the
distribution of quantum universes out of which the universes was born
and that of large classical (grow-up) universes like ours. In
particular, for lambda phi4 theory, the distribution of
classical universes has a power law spectrum. Thus there appears no
characteristic scale and the distribution of classical universes has a
fractal structure.

Hawking1,2 has shown that event horizons produce thermal radiation. I propose here a new cosmological model which has an early event horizon and in which the observed3 cosmic microwave background radiation is Hawking radiation. The model starts with de Sitter space—a space-time of constant curvature which is a solution to Einstein's vacuum field equations with a positive cosmological constant. Associated with an event E in de Sitter space there is a quantum barrier penetration tunnelling which leads to an open, negatively curved (k = −1) cosmology. This has an early exponential expansion phase but turns into a standard big-bang solution at late times. The geometry and quantum mechanical treatment within the future light cone of E are similar to that found in the Brout, Englert and Spindel (BES) theory4,5. This model has the following advantages: (1) It has no singularities. (2) The observed isotropy of the cosmic microwave background is explained because the different regions we observe have all been in causal contact. (3) The temperature at early epochs (T
0 ~ 1019 Ge V) is high enough to allow grand unified theories (GUTs) to produce the observed baryon excess from an initial thermal distribution through CP violations6,7. (4) T
0 is correct to make the BES scenario work. (5) The early exponential expansion phase can naturally account for the observed large number n
0 ~ 1088 of particles within a volume a
3 (where a is the radius of curvature) and the Guth8 flatness problem. (6) It predicts that our Universe is an open k = −1, Ω<1 cosmology consistent with the amount of mass detected in the universe so far9–11. (7) The existence of the event horizon makes it possible to create from the original de Sitter space other k = −1 universes (perhaps an infinite number) which are entirely disjoint from our own and from each other.

The standard model of hot big-bang cosmology requires initial conditions which are problematic in two ways: (1) The early universe is assumed to be highly homogeneous, in spite of the fact that separated regions were causally disconnected (horizon problem); and (2) the initial value of the Hubble constant must be fine tuned to extraordinary accuracy to produce a universe as flat (i.e., near critical mass density) as the one we see today (flatness problem). These problems would disappear if, in its early history, the universe supercooled to temperatures 28 or more orders of magnitude below the critical temperature for some phase transition. A huge expansion factor would then result from a period of exponential growth, and the entropy of the universe would be multiplied by a huge factor when the latent heat is released. Such a scenario is completely natural in the context of grand unified models of elementary-particle interactions. In such models, the supercooling is also relevant to the problem of monopole suppression. Unfortunately, the scenario seems to lead to some unacceptable consequences, so modifications must be sought.

We examine the modes of a scalar field in de Sitter space and construct quantum two-point functions. These are then used to compute a finite stress tensor by the technique of covariant point-splitting. We propose a renormalization ansatz based on the DeWitt-Schwinger expansion, and show that this removes all ambiguities previously present in point-splitting regularization. The results agree in detail with previous work by dimensional regularization, and give rise to an anomalous trace with the conventional coefficient. We describe how our treatment may be extended to more general situations.

The creation and evolution of energy-density perturbations are analyzed for the "new inflationary universe" scenario proposed by Linde, and Albrecht and Steinhardt. According to the scenario, the Universe underwent a strongly first-order phase transition and entered a "de Sitter phase" of exponential expansion during which all previously existing energy-density perturbations expanded to distance scales very large compared to the size of our observable Universe. The existence of an event horizon during the de Sitter phase gives rise to zero-point fluctuations in the scalar field, whose slowly growing expectation value signals the transition to the spontaneous-symmetry- breaking (SSB) phase of a grand unified theory (GUT). The fluctuations in are created on small distance scales and expanded to large scales, eventually giving rise to an almost scale-free spectrum of adiabatic density perturbations (the so-called Zel'dovich spectrum). When a fluctuation reenters the horizon (radius H-1) during the Friedmann-Robertson-Walker (FRW) phase that follows the exponential expansion, it has a perturbation amplitude |H=(4or25)H(t1), where H is the Hubble constant during the de Sitter phase (H-1 is the radius of the event horizon), (t1) is the mean value of at the time (t1) that the wavelength of the perturbation expanded beyond the Hubble radius during the de Sitter epoch, is the fluctuation in at time t1 on the same scale, and 4 (25) applies if the Universe is radiation (matter) dominated when the scale in question reenters the horizon. Scales larger than about 1015-1016M reenter the horizon when the Universe is matter dominated. Owing to the Sachs-Wolfe effect, these density perturbations give rise to temperature fluctuations in the microwave background which, on all angular scales 1°, are TT (15)H (t1). The value of expected from de Sitter fluctuations is O(H2). For the simplest model of "new inflation," that based on an SU(5) GUT with Coleman-Weinberg SSB, (t1) H2 so that TT1 in obvious conflict with the large-scale isotropy of the microwave background. One remedy for this is a model in which the inflation occurs when (t1) H2. We analyze a supersymmetric model which has this feature, and show that a value of |H10-4-10-3 on all observable scales is not implausible.

An attempt is made to explain two properties of the metagalaxy: its expansion and the absence of causal connection of its distant regions during a large part of its initial history. It is postulated that the gravitating matter was born from the ‘cosmological field’, i.e., a medium with equation of statep=−ε.
It is shown that such a postulate explains the expansion of metagalaxy and leads to a correct estimate of the entropy per baryon. The problem of causal connection can also be solved on this basis.

It is shown that the process of quantum creation of the universe in a wide class of elementary-particle theories with a large
probability leads to creation of an exponentially expanding (inflationary) universe, which after expansion acquires the sizel≳1028 cm.

We discuss three candidate scenarios which seem to allow the possibility that
the universe could have existed forever with no initial singularity: eternal
infation, cyclic evolution, and the emergent universe. The first two of these
scenarios are geodesically incomplete to the past, and thus cannot describe a
universe without a beginning. The third, although it is stable with respect to
classical perturbations, can collapse quantum mechanically, and therefore
cannot have an eternal past.

It is shown that ever-increasing long-wave fluctuations are generated for a light scalar field in metrics which have exponential expansion (inflation) on some coordinates (including Kaluza-Klein models). This result generalizes the well-known result for the de Sitter metric, and may be important in Kaluza-Klein models with dynamical reduction and for the pre-inflationary spacetime foam.

We discuss the particle creation in a closed homogeneous and isotropic universe with positive cosmological constant, which tunnels from the Friedmann regime (or from “nothing”) to a De Sitter-like one. Assuming the quasiclassical character of tunneling and neglecting the back reaction of the created particles, we derive the basic equation of the quantum field theory in the tunneling universe. We then show that during tunneling, particle creation is catastrophic, at least in the model of massive scalar fields conformally coupled to gravity, and discuss the possible effects of the back reaction on the tunneling process.

We discuss a supersymmetric inflationary cosmological scenario in which inflation takes place after GUT breaking but before SUSY breaking. The framework is within N = 1 supergravity with the inflation and SUSY breaking sectors hidden from one another. We present a specific one-parameter inflation sector coupled with F-type SUSY breaking, with more than sufficient entropy release. Adequate baryon asymmetry is produced by an out-of-equilibrium decay mechanism in spite of a low reheat temperature. The condition that the gravitino abundance should not affect nucleosynthesis fixes the single parameter and the resultant density flucctuation to be of O(10−4). If gravitinos are stable their resulting abundance is sufficient to account for the dark matter.

We investigate some aspects of thermodynamics and cosmology for superstrings. By a rather delicate computation using the microcanonical ensemble we show that the thermodynamic description of strings is sound (specific heat is positive at large energies) only for strings propagating in spaces where all the spatial directions are compact. Using this result and by considering a simple model, we show how strings resolve the initial singularity of the Big Bang. We also discuss a cosmological scenario which has the potential of explaining the space-time dimensionality.

We investigate some power-law solutions in inflationary cosmology, both by analytic and numerical means, considering first a simple model of a scalar field with an exponential field coupled to gravity. As has been pointed out recently by Yokoyama and Maeda, in power-law inflation viscous forces caused by couplings of the inflation to other particles can be important. We use numerical simulation to examine the effects of this viscosity on the inflation, for both a simple exponential potential and a more realistic potential motivated by particle physics. In general, the viscosity enhances the exponent of the power-law inflation, increasing the efficiency of inflation in power-law models, and we outline a specific inflationary model featuring viscosity.

A new scenario of the very early stages of the evolution of the universe is suggested. According to this scenario, inflation is a natural (and may be even inevitable) consequence of chaotic initial conditions in the early universe.

A new realization of the chaotic inflation scenario is suggested in the context of N = 1 supergravity.

In cosmological scenarios of the new inflationary type, inflation never ends completely. The total volume of inflating regions grows exponentially with time, and they form a self-similar fractal of dimension slightly less than 3.

It is shown that the large-scale quantum fluctuations of the scalar field ϕ generated in the chaotic-inflation scenario lead to an infinite process of self-reproduction of inflationary mini-universe. A model of an eternally existing chaotic inflationary universe is suggested.

The high-energy behavior of string scattering amplitudes is studied to all orders in perturbation theory, with the aim of exploring the short-distance structure of string theory. It is shown that the sum over all Riemann surfaces is dominated by a saddle point. Consequently, the high-energy limit is universal and simple to calculate. In this limit, furthermore, the amplitudes fall off in a stringy way - much faster than that allowed by field theory. The dominant saddle points are identified as coming from world sheets which are ZG+1 symmetric algebraic curves, and their contribution to the scattering amplitude is evaluated. An interesting spacetime picture of the high-energy limit emerges. The issue of summing the perturbation expansion is addressed.

The behaviour of the scalar fied fluctuations in the exponentially expanding universe and their role in the new inflationary universe scenario are investigated.

It is argued that a new inflationary universe scenario, which provides a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, can be naturally implemented in the context of grand unified theories of the type of the Coleman-Weinberg theory.