We review the problem of dark energy, including a survey of theoretical models and some aspects of numerical studies

Dark Energy

Beyond the cosmological standard model

The fact that one can associate thermodynamic properties with horizons brings together principles of quantum theory, gravitation and thermodynamics and possibly offers a window to the nature of quantum geometry. This review discusses certain aspects of this topic, concentrating on new insights gained from some recent work. After a brief introduction of the overall perspective, sections 2 and 3 provide the pedagogical background on the geometrical features of bifurcation horizons, path integral derivation of horizon temperature, black hole evaporation, structure of Lanczos-Lovelock models, the concept of Noether charge and its relation to horizon entropy. Section 4 discusses several conceptual issues introduced by the existence of temperature and entropy of the horizons. In section 5 we take up the connection between horizon thermodynamics and gravitational dynamics and describe several peculiar features which have no simple interpretation in the conventional approach. The next two sections describe the recent progress achieved in an alternative perspective of gravity. In section 6 we provide a thermodynamic interpretation of the field equations of gravity in any diffeomorphism invariant theory and in section 7 we obtain the field equations of gravity from an entropy maximization principle. The last section provides a summary.

http://iopscience.iop.org/article/10.1088/0034-4885/73/4/046901/pdf

Thermodynamical aspects of gravity: new insights

We introduce a large class of scalar-tensor models with interactions containing the second derivatives of the scalar field but not leading to additional degrees of freedom. These models exhibit peculiar features, such as an essential mixing of scalar and tensor kinetic terms, which we have named kinetic braiding. This braiding causes the scalar stress tensor to deviate from the perfect-fluid form. Cosmology in these models possesses a rich phenomenology, even in the limit where the scalar is an exact Goldstone boson. Generically, there are attractor solutions where the scalar monitors the behaviour of external matter. Because of the kinetic braiding, the position of the attractor depends both on the form of the Lagrangian and on the external energy density. The late-time asymptotic of these cosmologies is a de Sitter state. The scalar can exhibit phantom behaviour and is able to cross the phantom divide with neither ghosts nor gradient instabilities. These features provide a new class of models for Dark Energy. As an example, we study in detail a simple one-parameter model. The possible observational signatures of this model include a sizeable Early Dark Energy and a specific equation of state evolving into the final de-Sitter state from a healthy phantom regime.

https://iopscience.iop.org/article/10.1088/1475-7516/2010/10/026/pdf

Imperfect Dark Energy from Kinetic Gravity Braiding

After a decade and a half of research motivated by the accelerating universe, theory and experiment have a reached a certain level of maturity. The development of theoretical models beyond \Lambda, or smooth dark energy, often called modified gravity, has led to broader insights into a path forward, and a host of observational and experimental tests have been developed. In this review we present the current state of the field and describe a framework for anticipating developments in the next decade. We identify the guiding principles for rigorous and consistent modifications of the standard model, and discuss the prospects for empirical tests. We begin by reviewing attempts to consistently modify Einstein gravity in the infrared, focusing on the notion that additional degrees of freedom introduced by the modification must screen themselves from local tests of gravity. We categorize screening mechanisms into three broad classes: mechanisms which become active in regions of high Newtonian potential, those in which first derivatives become important, and those for which second derivatives are important. Examples of the first class, such as f(R) gravity, employ the familiar chameleon or symmetron mechanisms, whereas examples of the last class are galileon and massive gravity theories, employing the Vainshtein mechanism. In each case, we describe the theories as effective theories. We describe experimental tests, summarizing laboratory and solar system tests and describing in some detail astrophysical and cosmological tests. We discuss future tests which will be sensitive to different signatures of new physics in the gravitational sector. Parts that are more relevant to theorists vs. observers/experimentalists are clearly indicated, in the hope that this will serve as a useful reference for both audiences, as well as helping those interested in bridging the gap between them.

Beyond the Cosmological Standard Model

This book, first published in 2007, is an introductory textbook on quantum field theory in gravitational backgrounds intended for undergraduate and beginning graduate students in the fields of theoretical astrophysics, cosmology, particle physics, and string theory. The book covers the basic (but essential) material of quantization of fields in an expanding universe and quantum fluctuations in inflationary spacetime. It also contains a detailed explanation of the Casimir, Unruh, and Hawking effects, and introduces the method of effective action used for calculating the back-reaction of quantum systems on a classical external gravitational field. The broad scope of the material covered will provide the reader with a thorough perspective of the subject. Every major result is derived from first principles and thoroughly explained. The book is self-contained and assumes only a basic knowledge of general relativity. Exercises with detailed solutions are provided throughout the book.

Introduction to Quantum Effects in Gravity

Inflationary cosmology leads to the picture of a 'multiverse', involving an infinite number of (spatially infinite) post-inflationary thermalized regions, called pocket universes. In the context of theories with many vacua, such as the landscape of string theory, the effective constants of Nature are randomized by quantum processes during inflation. We discuss an analytic estimate for the volume distribution of the constants within each pocket universe. This estimate is based on the conjecture that the field distribution is approximately ergodic in the diffusion regime, when the dynamics of the fields is dominated by quantum fluctuations (rather than by the classical drift). We then propose a method for determining the relative abundances of different kinds of pocket universes. Both ingredients are combined into an expression for the distribution of the constants in pocket universes of all types.

https://iopscience.iop.org/article/10.1088/1475-7516/2006/01/017/pdf

Probabilities in the inflationary multiverse

In generic models of cosmological inflation, quantum fluctuations strongly influence the spacetime metric and produce infinitely many regions where the end of inflation (reheating) is delayed until arbitrarily late times. The geometry of the resulting spacetime is highly inhomogeneous on scales of many Hubble sizes. The recently developed string-theoretic picture of the “landscape” presents a similar structure, where an infinite number of de Sitter, flat, and anti-de Sitter universes are nucleated via quantum tunneling. Since observers on the Earth have no information about their location within the eternally inflating universe, the main question in this context is to obtain statistical predictions for quantities observed at a random location. I describe the problems arising within this statistical framework, such as the need for a volume cutoff and the dependence of cutoff schemes on time slicing and on the initial conditions. After reviewing different approaches and mathematical techniques developed in the past two decades for studying these issues, I discuss the existing proposals for extracting predictions and give examples of their applications

Predictions in eternal inflation

Cosmological scenarios with $k$-essence are invoked in order to explain the observed late-time acceleration of the Universe. These scenarios avoid the need for fine-tuned initial conditions (the ``coincidence problem'') because of the attractorlike dynamics of the $k$-essence field $\ensuremath{\phi}$. It was recently shown that all $k$-essence scenarios with Lagrangians $p=L(X){\ensuremath{\phi}}^{\ensuremath{-}2}$, where $X\ensuremath{\equiv}\frac{1}{2}{\ensuremath{\phi}}_{,\ensuremath{\mu}}{\ensuremath{\phi}}^{,\ensuremath{\mu}}$, necessarily involve an epoch where perturbations of $\ensuremath{\phi}$ propagate faster than light (the ``no-go theorem''). We carry out a comprehensive study of attractorlike cosmological solutions (``trackers'') involving a $k$-essence scalar field $\ensuremath{\phi}$ and another matter component. The result of this study is a complete classification of $k$-essence Lagrangians that admit asymptotically stable tracking solutions, among all Lagrangians of the form $p=K(\ensuremath{\phi})L(X)$. Using this classification, we select the class of models that describe the late-time acceleration and avoid the coincidence problem through the tracking mechanism. An analogous ``no-go theorem'' still holds for this class of models, indicating the existence of a superluminal epoch. In the context of $k$-essence cosmology, the superluminal epoch does not lead to causality violations. We discuss the implications of superluminal signal propagation for possible causality violations in Lorentz-invariant field theories.

Attractor scenarios and superluminal signals in k-essence cosmology

Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples.

Sergei Winitzki

Papers

Introduction to Quantum Effects in Gravity

Probabilities in the inflationary multiverse

Predictions in eternal inflation

Attractor scenarios and superluminal signals in k-essence cosmology

Cosmological particle production and the precision of the WKB approximation