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# Anthropic distribution for cosmological constant and primordial density perturbations

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## Abstract

The Anthropic Principle has been proposed as an explanation for the observed value of the cosmological constant. Here we revisit this proposal by allowing for variation between universes in the amplitude of the scale-invariant primordial cosmological density perturbations. We derive a priori probability distributions for this amplitude from toy inflationary models in which the parameter of the inflaton potential is smoothly distributed over possible universes. We find that for such probability distributions, the likelihood that we live in a typical, anthropically-allowed universe is generally quite small.
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... Putting everything together we find t 0 ∼ t G ∼ t Λ and the coincidence problem seems to be resolved. It is clear that some assumptions must be made about the origin and evolution of intelligent life and these have drawn some criticism [143,144]. It is typically argued that, although anthropic constraints can select values of the cosmological constant which are close to the observed value, these values necessarily carry large uncertainties due to the lack of understanding of the conditions required for intelligent life to appear. ...
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The subject of this thesis is cosmological implications of string compactifications understood in a broad sense. In the first half of the thesis, we will begin by reviewing the four-dimensional description of the tree-level perturbative type IIB action. We will then review a number of open questions in cosmology and their relevance with regards to the remainder of the thesis. We will first explore some of these cosmological questions from the perspective of effective field theories motivated by supergravity. From the naturalness of dark energy and how to obtain a naturally light dark energy field in terms of the clockwork mechanism and the Dvali-Kaloper-Sorbo four-form mixing. We also discuss the coincidence problem for dynamical models of dark energy consistent with a quintessence field slowly rolling down a potential slope, of the type one would expect from the asymptotics of moduli space. In the second half of the thesis, we introduce the effects of perturbative and non-perturbative corrections to the tree-level type IIB action. We then focus on obtaining a viable model of quintessence from the type IIB effective field theory. However, we are able to show that such a model must have a non-supersymmetric Minkowski vacuum at leading order. When we consider the effects of quantum fluctuations during the early Universe, we see that such models must have extremely fine-tuned initial conditions to describe a slow-rolling scalar field at present times. We conclude that quintessence faces more challenges than a true cosmological constant, to the point that quintessence is very unattractive for model building modulo a ruling out of the cosmological constant by observations. Following this line of reasoning, we consider whether other perturbative corrections can generate de Sitter solutions in an appropriate setting.
... This scenario -without a matter dominated era -is an extreme version of cases where the vacuum energy density is too large relative to the primordial fluctuation amplitude to allow for structure formation (see Section 4 and Refs. [16,206,223,224,226,238,372,383,429,545]). ...
Preprint
(abridged) Both fundamental constants that describe the laws of physics and cosmological parameters that determine the cosmic properties must fall within a range of values in order for the universe to develop astrophysical structures and ultimately support life. This paper reviews current constraints on these quantities. The standard model of particle physics contains both coupling constants and particle masses, and the allowed ranges of these parameters are discussed first. We then consider cosmological parameters, including the total energy density, the vacuum energy density, the baryon-to-photon ratio, the dark matter contribution, and the amplitude of primordial density fluctuations. These quantities are constrained by the requirements that the universe lives for a long time, emerges from the BBN epoch with an acceptable chemical composition, and can successfully produce galaxies. On smaller scales, stars and planets must be able to form and function. The stars must have sufficiently long lifetimes and hot surface temperatures. The planets must be massive enough to maintain an atmosphere, small enough to remain non-degenerate, and contain enough particles to support a complex biosphere. These requirements place constraints on the gravitational constant, the fine structure constant, and composite parameters that specify nuclear reaction rates. We consider specific instances of possible fine-tuning in stars, including the triple alpha reaction that produces carbon, as well as the effects of unstable deuterium and stable diprotons. For all of these issues, viable universes exist over a range of parameter space, which is delineated herein. Finally, for universes with significantly different parameters, new types of astrophysical processes can generate energy and support habitability.
Chapter
Throughout this thesis we will concern ourselves with the consequences of a number of important open questions in modern physics and more particularly, its implications for cosmology.
Article
The subject of this thesis is cosmological implications of string compactifications understood in a broad sense. In the first half of the thesis, we will begin by reviewing the four-dimensional description of the tree-level perturbative type IIB action. We will then introduce a number of open questions in cosmology and their relevance with regards to the remainder of the thesis. We will first explore some of these questions from the perspective of effective field theories motivated by supergravity. In particular, we provide a description of a naturally light dark energy field in terms of the clockwork mechanism and the Dvali-Kaloper-Sorbo four-form mixing. We study its possible UV completion and show a no-go for its embedding within perturbative type IIA supergravity. We also discuss the coincidence problem for dynamical models of dark energy consistent with a quintessence field slowly rolling down a potential slope, of the type one would expect from the asymptotics of moduli space. As it rolls, a tower of heavy states will generically descend, triggering a phase transition in the low energy cosmological dynamics after at most a few hundred Hubble times. As a result, dark energy domination cannot continue indefinitely and there is at least a percentage chance that we find ourselves in the first Hubble epoch. In the second half of the thesis, we introduce the effects of perturbative and nonperturbative corrections to the tree-level type IIB action. We then focus on obtaining a viable model of quintessence from the type IIB effective field theory. However, we are able to show that such a model must have a non-supersymmetric Minkowski vacuum at leading order. Furthermore, it must necessarily take the form of axion hilltop quintessence. When we consider the effects of quantum fluctuations during the early Universe, we see that such models must have extremely fine-tuned initial conditions to describe a slow-rolling scalar field at present times. We conclude that quintessence faces more challenges than a true cosmological constant, to the point that quintessence is very unattractive for model building modulo a ruling out of the cosmological constant by observations. Following this line of reasoning, we consider whether other perturbative corrections can generate de Sitter solutions in an appropriate setting. In particular, we consider the effects of higher curvature corrections in the Gauss-Bonnet term. Remarkably, we are able to show that, for the particular setting of a fluxed runaway potential motivated by heterotic supergravity, the curvature corrections reduce the space of solutions.
Article
Both the fundamental constants that describe the laws of physics and the cosmological parameters that determine the properties of our universe must fall within a range of values in order for the cosmos to develop astrophysical structures and ultimately support life. This paper reviews the current constraints on these quantities. The discussion starts with an assessment of the parameters that are allowed to vary. The standard model of particle physics contains both coupling constants (α,α s ,α w ) and particle masses (m u ,m d ,m e ), and the allowed ranges of these parameters are discussed first. We then consider cosmological parameters, including the total energy density of the universe (Ω), the contribution from vacuum energy (ρ Λ ), the baryon-to-photon ratio (η), the dark matter contribution (δ), and the amplitude of primordial density fluctuations (Q). These quantities are constrained by the requirements that the universe lives for a sufficiently long time, emerges from the epoch of Big Bang Nucleosynthesis with an acceptable chemical composition, and can successfully produce large scale structures such as galaxies. On smaller scales, stars and planets must be able to form and function. The stars must be sufficiently long-lived, have high enough surface temperatures, and have smaller masses than their host galaxies. The planets must be massive enough to hold onto an atmosphere, yet small enough to remain non-degenerate, and contain enough particles to support a biosphere of sufficient complexity. These requirements place constraints on the gravitational structure constant (α G ), the fine structure constant (α), and composite parameters (C ⋆ ) that specify nuclear reaction rates. We then consider specific instances of possible fine-tuning in stellar nucleosynthesis, including the triple alpha reaction that produces carbon, the case of unstable deuterium, and the possibility of stable diprotons. For all of the issues outlined above, viable universes exist over a range of parameter space, which is delineated herein. Finally, for universes with significantly different parameters, new types of astrophysical processes can generate energy and thereby support habitability.
Chapter
After introducing some basic aspects of strings and branes in Chap. 12, we move to cosmological models arising in or motivated by the theory, concentrating on those based upon $$\mathbb{K}\mathrm{LT}$$ and large-volume uplifting scenarios [1–28] embedded in the string landscape [29–59] (reviews are [60–64]). Uplifting scenarios realize a de Sitter spacetime in mechanisms of moduli stabilization in M-theory and string theory, with two main consequences:
Chapter
A complete background-independent quantum theory of gravity is a hard goal to achieve and the formal discussion of any of the proposals in this direction lies beyond the scope of this book. However, in order to understand certain techniques it is often useful to specialize to a simple background and study the quantum system thereon (Sect. 10.1). Symmetry reduction is done not only for didactic purposes but also as a (hopefully temporary) necessity in active research. For instance, it is natural to consider the big-bang and cosmological constant problems in an expanding cosmological background. In such a setting, one does not employ the full machinery of general theories of quantum gravity and a number of simplifications take place. We apply the procedure of symmetry reduction to canonical gravity and quantum cosmological scenarios, first in the Wheeler–DeWitt approach in ADM variables (Sect. 10.2) and then within loop quantum cosmology (Sect. 10.3).
Article
Deriving the Einstein field equations (EFE) with matter fluid from the action principle is not straightforward, because mass conservation must be added as an additional constraint to make rest-frame mass density variable in reaction to metric variation. This can be avoided by introducing a constraint delta(root-g) = 0 to metric variations delta g(mu nu), and then the cosmological constant A emerges as an integration constant. This is a removal of one of the four constraints on initial conditions forced by EFE at the birth of the universe, and it may imply that EFE are unnecessarily restrictive about initial conditions. I then adopt a principle that the theory of gravity should be able to solve time evolution starting from arbitrary inhomogeneous initial conditions about spacetime and matter. The equations of gravitational fields satisfying this principle are obtained, by setting four auxiliary constraints on delta g(mu nu) to extract six degrees of freedom for gravity. The cost of achieving this is a loss of general covariance, but these equations constitute a consistent theory if they hold in the special coordinate systems that can be uniquely specified with respect to the initial space like hypersurface when the universe was born. This theory predicts that gravity is described by EFE with non-zero A in a homogeneous patch of the universe created by inflation, but A changes continuously across different patches. Then both the smallness and coincidence problems of the cosmological constant are solved by the anthropic argument. This is just a result of inhomogeneous initial conditions, not requiring any change of the fundamental physical laws in different patches.
Article
Deriving the Einstein field equations (EFE) with matter fluid from the action principle is not straightforward, because mass conservation must be added as an additional constraint to make rest-frame mass density variable in reaction to metric variation. This can be avoided by introducing a constraint 0δ(√−g) = to metric variations δ gμν, and then the cosmological constant Λ emerges as an integration constant. This is a removal of one of the four constraints on initial conditions forced by EFE at the birth of the universe, and it may imply that EFE are unnecessarily restrictive about initial conditions. I then adopt a principle that the theory of gravity should be able to solve time evolution starting from arbitrary inhomogeneous initial conditions about spacetime and matter. The equations of gravitational fields satisfying this principle are obtained, by setting four auxiliary constraints on δ gμν to extract six degrees of freedom for gravity. The cost of achieving this is a loss of general covariance, but these equations constitute a consistent theory if they hold in the special coordinate systems that can be uniquely specified with respect to the initial space-like hypersurface when the universe was born. This theory predicts that gravity is described by EFE with non-zero Λ in a homogeneous patch of the universe created by inflation, but Λ changes continuously across different patches. Then both the smallness and coincidence problems of the cosmological constant are solved by the anthropic argument. This is just a result of inhomogeneous initial conditions, not requiring any change of the fundamental physical laws in different patches.
Conference Paper
We discuss probability distributions for the cosmological constant A and the amplitude of primordial density fluctuations Q in models where they both are anthropic variables. With mild assumptions about the prior probabilities, the distribution P(Lambda, Q) factorizes into two independent distributions for the variables Q and y alpha Lambda/Q(3). The distribution for y is largely model-independent and is in a good agreement with the observed value of y. The form of P(Q) depends on the origin of density perturbations. If the perturbations are due to quantum fluctuations of the inflaton, then P(Q) tends to have an exponential dependence on Q, due to the fact that in such models Q is correlated with the amount of inflationary expansion. For simple models with a power-law potential, P(Q) is peaked at very small values of Q, far smaller than the observed value of 10(-5). This problem does not arise in curvaton-type models, where the inflationary expansion factor is not correlated with Q.
Book
The Early Universe has become the standard reference on forefront topics in cosmology, particularly to the early history of the Universe. Subjects covered include primordial nubleosynthesis, baryogenesis, phases transitions, inflation, dark matter, and galaxy formation, relics such as axions, neutrinos and monopoles, and speculations about the Universe at the Planck time. The book includes more than ninety figures as well as a five-page update discussing recent developments such as the COBE results.
Article
A general Friedmann big-bang cosmology can be specified by fixing a half-dozen cosmological parameters such as the photon-to-baryon ratio ηγ, the cosmological constant Λ, the curvature scale R, and the amplitude Q of (assumed scale-invariant) primordial density fluctuations. There is currently no established theory as to why these parameters take the particular values we deduce from observations. This has led to proposed “anthropic” explanations for the observed value of each parameter as the only value capable of generating a universe that can host intelligent life. In this paper, I explicitly show that the requirement that the universe generates Sun-like stars with planets does not fix these parameters, by developing a class of cosmologies (based on the classical “cold big-bang” model) in which some or all of the cosmological parameters differ by orders of magnitude from the values they assume in the standard hot big-bang cosmology, without precluding in any obvious way the existence of intelligent life. I also give a careful discussion of the structure and context of anthropic arguments in cosmology, and point out some implications of the cold big-bang model’s existence for anthropic arguments concerning specific parameters.
Article
We study cosmology from the point of view of quantum gravity. Some light is thrown on the nature of time, and it is suggested that the cosmological arrow of time is generated by a spontaneous breakdown of TCP. Conventional cosmological models in which quantum fields interact with a time-dependent gravitational field are shown to describe an approximation to the quantum gravitational wave function which is valid in the long-wavelength limit. Two problems with initial conditions are resolved in models in which a negative bare cosmological constant is cancelled by the classical excitation of a Bose field η with a very flat potential. These models can also give a natural explanation for the observed value of the cosmological constant.
Article
The wave function of an interacting ‘family’ of one large ‘parent’ and many Planck-sized ‘baby’ universes is computed in a semiclassical approximation using an adaptation of Hartle-Hawking initial conditions. A recently discovered gravitational instanton which exists for general relativity coupled to axions is employed. The outcome of a single experiment in the parent universe is in general described by a mixed state, even if the initial state is pure. However, a sequence of measurements rapidly collapses the wave function of the family of universes into one of an infinite number of ‘coherent’ states for which quantum incoherence is not observed in the parent universe. This provides a concrete illustration of an unexpected phenomena whose existence has been argued for on quite general grounds by Coleman: quantum incoherence due to information loss to baby universes is not experimentally observable. We further argue that all coupling constants governing dynamics in the parent universe depend on the parameters describing the particular coherent state into which the family wave function collapses. In particular, generically terms that violate any global symmetries will be induced in the effective action for the parent universe. These last results have much broader applicability that our specific model.