Squashed Holography with Scalar Condensates

We evaluate the partition function of the free and interacting O(N) vector model on a two-parameter family of squashed three spheres in the presence of a scalar deformation. We also find everywhere regular solutions of Einstein gravity coupled to a scalar field in AdS and in dS with the same double squashed boundary geometry. Remarkably, the thermodynamic properties of the AdS solutions qualitatively agree with the behavior predicted by the free O(N) model with a real mass deformation. The dS bulk solutions specify the semiclassical `no-boundary' measure over anisotropic deformations of inflationary, asymptotic de Sitter space. Through dS/CFT the partition function of the interacting O(N) model yields a holographic toy model of the no-boundary measure. We find this yields a qualitatively similar probability distribution which is normalizable and globally peaked at the round three sphere, with a low amplitude for strong anisotropies.Comment: 35 pages, version 2: presentation changed and comments adde

Comments on Squashed-sphere Partition Functions

We study the partition function of odd-dimensional conformal field theories placed on spheres with a squashed metric. We establish that the round sphere provides a local extremum for the free energy which, in general, is not a global extremum. In addition, we show that the leading quadratic correction to the free energy around this extremum is proportional to the coefficient, $C_T$, determining the two-point function of the energy-momentum tensor in the CFT. For three-dimensional CFTs, we compute explicitly this proportionality constant for a class of squashing deformations which preserve an $SU(2)\times U(1)$ isometry group on the sphere. In addition, we evaluate the free energy as a function of the squashing parameter for theories of free bosons, free fermions, as well as CFTs holographically dual to Einstein gravity with a negative cosmological constant. We observe that, after suitable normalization, the dependence of the free energy on the squashing parameter for all these theories is nearly universal for a large region of parameter space and is well approximated by a simple quadratic function arising from holography. We generalize our results to five-dimensional CFTs and, in this context, we also study theories holographically dual to six-dimensional Gauss-Bonnet gravity.Comment: 40 pages, 7 figures, 1 table; v2: additional comments and clarifications added, updated bibliograph

Lorentzian Condition in Holographic Cosmology

We derive a sufficient set of conditions on the Euclidean boundary theory in dS/CFT for it to predict classical, Lorentzian bulk evolution at large spatial volumes. Our derivation makes use of a canonical transformation to express the bulk wave function at large volume in terms of the sources of the dual partition function. This enables a sharper formulation of dS/CFT. The conditions under which the boundary theory predicts classical bulk evolution are stronger than the criteria usually employed in quantum cosmology. We illustrate this in a homogeneous isotropic minisuperspace model of gravity coupled to a scalar field in which we identify the ensemble of classical histories explicitly.Comment: 34 pages, 6 figures, revtex

Quantum Transitions Through Cosmological Singularities

In a quantum theory of cosmology spacetime behaves classically only in limited patches of the configuration space on which the wave function of the universe is defined. Quantum transitions can connect classical evolution in different patches. Working in the saddle point approximation and in minisuperspace we compute quantum transitions connecting inflationary histories across a de Sitter like throat or a singularity. This supplies probabilities for how an inflating universe, when evolved backwards, transitions and branches into an ensemble of histories on the opposite side of a quantum bounce. Generalising our analysis to scalar potentials with negative regions we identify saddle points describing a quantum transition between a classically contracting, crunching ekpyrotic phase and an inflationary universe.Comment: 32 pages, 27 figure

Beyond Classical Cosmology with the No-Boundary Wave Function

The goal of quantum cosmology is to find a quantum state that describes the entire evolution of our universe: from the fuzzy quantum dynamics dominating the universe’s evolution at early times to the classical cosmological evolution in our spacetime neighbourhood. Classical cosmology emerges in quantum cosmology under certain conditions only. In recent years significant progress has been made to understand the “classical realm” predicted by the No-Boundary Wave Function (NBWF), which will be the main focus of this thesis. However, it has not been understood how to go beyond this and learn something about the quantum realm of the universe as predicted by the NBWF. In this thesis we take a number of steps in this direction. The NBWF relates on a semiclassical level, Lorentzian de Sitter (dS) solutions to Euclidean Anti-de Sitter (AdS) solutions, which in their turn correspond, using the AdS/CFT conjecture, to a dual field theory defined on their boundary. This allows for a holographic formulation of the NBWF in which the relative weighting of different cosmological histories is given by the partition functions of (Euclidean) AdS/CFT duals. In this thesis we develop this novel holographic form of the NBWF in several directions. In the first part of this thesis we investigate the emergence of classical cosmological evolution from the boundary field theory and derive a sufficient set of conditions to obtain classical, Lorentzian bulk evolution at large spatial volumes. This derivation is based on the construction of a new wave function in terms of asymptotic variables, which are related to the sources of the dual field theory. With this new wave function it is possible to define new classicality conditions using the vacuum expectation values (vevs) from the dual boundary theory. In the second part of this thesis we look at the physics of eternal inflation, a regime where the dynamics of the theory is governed by large quantum fluctuations that get produced together with their backreaction on the geometry, meaning that the background does not evolve classically any more and that we therefore cannot get information about the global structure of the universe using the available techniques of the NBWF. With the use of the holographic NBWF proposal, it is possible to have an alternative calculation of the no-boundary measure, which is not plagued by the absence of a classical background. We show that it is possible to deduce some properties of the global structure of eternal inflation, by considering as a toy model a field theory living on a double squashed three-sphere. Both the squashed spheres and eternal inflation have highly curved regions and a high overall anisotropy. We start by calculating Euclidean AdS solutions that have the boundary of a squashed sphere and compare these with the free O(N) model. We find that the free energies of the two theories are remarkably similar, if we do not consider scalar excitations. We also comment on a universal property for CFTs on a squashed sphere. Namely, the field theory free energy has a local maximum in terms of the squashing parameter at zero squashing. Properties like this can be translated, using the holographic no-boundary conjecture, to cosmological spaces, with the result that the measure is peaked around isotropic universes, suggesting that holography predicts a smooth exit from eternal inflation. This is verified by the explicit calculation of the interacting O(N) model on the squashed sphere, which gives a distribution function that is globally peaked at the round sphere with zero scalar deformation with a low amplitude for geometries with negative scalar curvature. In the last part of this thesis we track the classical histories predicted by the NBWF back in time to the moment that the classicality conditions are not satisfied any more. Here, quantum-mechanical effects should be taken into account, making it possible that a classically forbidden transition happens between classical patches of cosmological evolution. We study these transitions by constructing complex saddle points that connect two classically evolving regions. The probabilities for transitions are then found to be the actions of these saddle points. We observe that universes at large values of the potential prefer a symmetric transition, while universes with a small value for the potential have a higher likelihood to transition to universes with a larger value of the potential.nrpages: 202status: publishe

The NUTs and Bolts of Squashed Holography

We evaluate the partition function of the free O(N ) model on a two-parameter family of squashed three spheres. We also find new solutions of general relativity with negative cosmological constant and the same double squashed boundary geometry and analyse their thermodynamic properties. Remarkably, both systems exhibit a qualitatively similar behaviour over the entire configuration space of boundary geometries. Recent formulations of dS/CFT enable one to interpret the field theory partition function as a function of the two squashing parameters as the Hartle-Hawking wave function in a minisuperspace model of anisotropic deformations of de Sitter space. The resulting probability distribution is normalisable and globally peaked at the round three sphere, with a low amplitude of boundary geometries with negative scalar curvature.41 pages, 19 figuresstatus: publishe

Lorentzian Condition in Holographic Cosmology

We derive a sufficient set of conditions on the Euclidean boundary theory in dS/CFT for it to predict classical, Lorentzian bulk evolution at large spatial volumes. Our derivation makes use of a canonical transformation to express the bulk wave function at large volume in terms of the sources of the dual partition function. This enables a sharper formulation of dS/CFT. The conditions under which the boundary theory predicts classical bulk evolution are stronger than the criteria usually employed in quantum cosmology. We illustrate this in a homogeneous isotropic minisuperspace model of gravity coupled to a scalar field in which we identify the ensemble of classical histories explicitly.34 pages, 6 figures, revtex4status: publishe

Quantum Transitions Through Cosmological Singularities

In a quantum theory of cosmology spacetime behaves classically only in limited patches of the configuration space on which the wave function of the universe is defined. Quantum transitions can connect classical evolution in different patches. Working in the saddle point approximation and in minisuperspace we compute quantum transitions connecting inflationary histories across a de Sitter like throat or a singularity. This supplies probabilities for how an inflating universe, when evolved backwards, transitions and branches into an ensemble of histories on the opposite side of a quantum bounce. Generalising our analysis to scalar potentials with negative regions we identify saddle points describing a quantum transition between a classically contracting, crunching ekpyrotic phase and an inflationary universe.32 pages, 27 figuresstatus: publishe