Supercurvature Mode Research Articles

Bianchi spacetimes as supercurvature modes around isotropic cosmologies

A powerful result in theoretical cosmology states that a subset of anisotropic Bianchi models can be seen as the homogeneous limit of (standard) linear cosmological perturbations. Such models are precisely those leading to Friedmann spacetimes in the limit of zero anisotropy. Building on previous works, we give a comprehensive exposition of this result, and perform the detailed identification between anisotropic degrees of freedom and their corresponding scalar, vector, and tensor perturbations of standard perturbation theory. In particular, we find that anisotropic models very close to open (i.e., negatively curved) Friedmann spaces correspond to some type of super-curvature perturbations. As a consequence, provided anisotropy is mild, its effects on all types of cosmological observables can always be computed as simple extensions of the standard techniques used in relativistic perturbation theory around Friedmann models. This fact opens the possibility to consistently constrain, for all cosmological observables, the presence of large scale anisotropies on the top of the stochastic fluctuations.

Large-scale inhomogeneity of dark energy produced in the ancestor vacuum

We investigate large-scale inhomogeneity of dark energy in the bubble nucleation scenario of the universe. In this scenario, the present universe was created by a bubble nucleation due to quantum tunneling from a metastable ancestor vacuum, followed by a primordial inflationary era. During the bubble nucleation, supercurvature modes of some kind of a scalar field are produced, and remain until present without decaying; thus they can play a role of the dark energy, if the mass of the scalar field is sufficiently light in the present universe. The supercurvature modes fluctuate at a very large spatial scale, much longer than the Hubble length in the present universe. Thus they create large-scale inhomogeneities of the dark energy, and generate large-scale anisotropies in the cosmic microwave background (CMB) fluctuations. This is a notable feature of this scenario, where quantum fluctuations of a scalar field are responsible for the dark energy. In this paper, we calculate imprints of the scenario on the CMB anisotropies through the integrated Sachs-Wolfe (ISW) effect, and give observational constraints on the curvature parameter $\Omega_K$ and on an additional parameter $\epsilon$ describing some properties of the ancestor vacuum.

Curvature and topology dependency of the cosmological spectra

In this article we investigate dependency of the adiabatic and entropy spectral indices of the cosmological perturbations on the geometry and topology of the background universe. Our discussion includes the post-inflationary universe i.e. radiation-dust mixture era. For this purpose, we first extract an explicit equation describing evolution of the comoving curvature perturbation in the FLRW universe with arbitrary spatial sectional curvature. We may percept when $K\neq 0$, curvature scale would be as significant as the perturbations scales to recognize the behavior of the spectral indices. We also focus on the entropy perturbation in order to extract behavior of the isocurvature spectral index in terms of the curvature index and time. Our analysis shows that spectra of curvature and entropy perturbations in sub-horizon scales could be function of topology. Moreover, an accurate analysis makes clear that time-average of isocurvature index in case $K=0$ is about zero,so that imprint of entropy perturbation in time duration may be negligible. We also consider evolution of the cosmological indices for super-curvature modes in the case $K=-1$. In the most results dependency to curvature, initial conditions and scale modes are thoroughly vivid.

Vacuum fluctuations in an ancestor vacuum: A possible dark energy candidate

We consider an open universe created by bubble nucleation, and study possible effects of an "ancestor vacuum" (de Sitter space in which bubble nucleation occurred) on the present universe. We compute vacuum expectation values of energy-momentum tensor for a minimally coupled scalar field, carefully taking into account the effect of the ancestor vacuum by the Euclidean prescription. In the study of the time evolution, an important role is played by the so-called supercurvature mode, which is non-normalizable on a spatial slice of open universe and decays in time most slowly. We point out that vacuum energy of a quantum field can be regarded as dark energy if mass of the field is of order the present Hubble parameter or smaller. We obtain preliminary results for the dark energy equation of state w(z) as a function of the redshift.

Vacuum state of the Dirac field in de Sitter space and entanglement entropy

We compute the entanglement entropy of a free massive Dirac field between two causally disconnected open charts in de Sitter space. We first derive the Bunch-Davies vacuum mode functions of the Dirac field. We find there exists no supercurvature mode for the Dirac field. We then give the Bogoliubov transformation between the Bunch-Davies vacuum and the open chart vacua that makes the reduced density matrix diagonal. We find that the Dirac field becomes more entangled than a scalar field as m2/H2 becomes small, and the difference is maximal in the massless limit.

Strongly scale-dependent CMB dipolar asymmetry from super-curvature fluctuations

We reconsider the observed CMB dipolar asymmetry in the context of open inflation, where a supercurvature mode might survive the bubble nucleation. If such a supercurvature mode modulates the amplitude of the curvature power spectrum, it would easily produce an asymmetry in the power spectrum. We show that current observational data can be accommodated in a three-field model, with simple quadratic potentials and a non-trivial field-space metric. Despite the presence of three fields, we believe this model is so far the simplest that can match current observations. We are able to match the observed strong scale dependence of the dipolar asymmetry, without a fine tuning of initial conditions, breaking slow roll or adding a feature to the evolution of any field.

Is there supercurvature mode of massive vector field in openinflation?

We investigate the Euclidean vacuum mode functions of a massive vectorfield in a spatially open chart of de Sitter spacetime. In theone-bubble open inflationary scenario that naturally predicts a negative spatial curvature after a quantum tunneling, it is known that a lightscalar field has the so-called supercurvature mode, i.e. an additionaldiscrete mode which describes fluctuations over scales larger than thespatial curvature scale. If such supercurvature modes exist for a vectorfield with a sufficiently light mass, then they would decay slower and easily survive the inflationary era. However, the existence ofsupercurvature mode strongly depends on details of the system. Toclarify whether a massive vector field has supercurvature modes, weconsider a U(1) gauge field with gauge and conformal invariances spontaneously broken through the Higgs mechanism, and present explicit expressions for the Euclidean vacuum mode functions. We findthat, for any values of the vector field mass, there is nosupercurvature mode. In the massless limit, the absence ofsupercurvature modes in the scalar sector stems from the gaugesymmetry.

Mode spectrum of the electromagnetic field in open universe models

We examine the mode functions of the electromagnetic field on spherically symmetric backgrounds with special attention to the subclass which allows for the foliation as open Friedmann-Lemaitre (FL) spacetime. It is well-known that in certain scalar field theories on open FL background there can exist so-called supercurvature modes, their existence depending on parameters of the theory. Looking at specific open universe models, such as open inflation and the Milne universe, we find that no supercurvature modes are present in the spectrum of the electromagnetic field. This excludes the possibility for superadiabatic evolution of cosmological magnetic fields within these models without relying on new physics or breaking the conformal invariance of electromagnetism.

Cosmological perturbations in a Friedmann-Robertson-Walker model with a scalar field and false vacuum decay

The unconstrained reduced action corresponding to the dynamics of scalar fluctuations about the Friedmann-Robertson-Walker (FRW) background is derived using Dirac's method of description of singular Lagrangian systems. The results are applied to so-called negative mode problem in description of tunneling transitions with gravity. With our special choice of physical variable, the kinetic term of the reduced action has a conventional signature for a wide class of models. In this representation, the existence of a negative mode justifying the false vacuum decay picture turns out to be manifest. We also explain how the present result becomes consistent with the previously proved ``no negative mode (supercritical supercurvature mode) theorem''.

The no-negative mode theorem in false vacuum decay with gravity

The so-called negative mode problem in the path integral approach to the false vacuum decay with the effect of gravity has been an unsolved problem. Several years ago, we proposed a conjecture which is to be proved in order to give a consistent solution to the negative mode problem. We called it the “no-negative mode conjecture”. In the present paper, we give a proof of this conjecture for rather general models. Recently, we also proposed the “no-supercritical supercurvature mode conjecture” that claims the absence of supercritical supercurvature modes in the one-bubble open inflation model. In the same paper, we clarified the equivalence between the “no-negative mode conjecture” and the “no-supercritical supercurvature mode conjecture”. Hence, the latter is also proved at the same time when the former is proved.

Spectrum of cosmological perturbations in the one-bubble open universe

The spectrum of cosmological perturbations in the context of the one-bubble open inflation model is discussed, taking into account fluctuations of the metric. We find that, quite generically, thin wall single field models have no supercurvature modes. However, single field models with supercurvature modes do exist. In these models the density parameter Ω becomes a random variable taking a range of values inside of each bubble. We also show that the model dependence of the continuous spectrum for both scalar and tensor-type perturbations is small as long as the kinetic energy density of the background field does not dominate the total energy density. We conclude that the spectrum of the density perturbation predicted in the single-field model of the one-bubble open inflation is rather robust. We also consider the spectrum of scalar and tensor perturbations in a model of the Hawking-Turok type, without a false vacuum.

No supercritical supercurvature mode conjecture in one-bubble open inflation

In the path integral approach to false vacuum decay with the effect of gravity, there is an unsolved problem, called the negative mode problem. We show that the appearance of a supercritical supercurvature mode in the one-bubble open inflation scenario is equivalent to the existence of a negative mode around the Euclidean bounce solution. Supercritical supercurvature modes are those whose mode functions diverge exponentially for large spatial radius on the time constant hypersurface of the open universe. Then we propose a conjecture that there should be ``no supercritical supercurvature mode''. For a class of models that contains a wide variety of tunneling potentials, this conjecture is shown to be correct.

Where do all the supercurvature modes go?

In the hyperbolic slicing of de Sitter space appropriate for open universe models, a curvature scale is present and supercurvature fluctuations are possible. In some cases, the expansion of a scalar field in the Bunch-Davies vacuum includes supercurvature modes, as shown by Sasaki, Tanaka and Yamamoto. We express the normalizable vacuum supercurvature modes for a massless scalar field in terms of the basis modes for the spatially-flat slicing of de Sitter space.

Fluctuations in the Hawking-Turok model

Scalar and gravity wave subcurvature fluctuations are calculated for a background approximating the Hawking and Turok open universe model. The gravity wave cosmic microwave background contribution is finite and it appears that a normalizable scalar supercurvature mode is possible in some regions of parameter space.

Open hybrid inflation

We propose an open hybrid inflation scenario that produces an open universe with a ``tilted'' $n>1$ spectrum of metric perturbations. The model contains a symmetry-breaking field that tunnels to its true vacuum, producing a single bubble inside which hybrid inflation drives the universe to almost flatness. In order to obtain density perturbations with $n>1$ we analyze a recently proposed new version of hybrid inflation scenario called tilted hybrid inflation. In this scenario, unlike in the previously known versions of hybrid inflation, a considerable tilt of the spectrum can be obtained without fine tuning. The stage of inflation in this model is rather short, which allows us to obtain an inflationary universe with $\ensuremath{\Omega}<1$ in a more natural way. We study the separate contribution of scalar perturbations coming from the continuum subcurvature modes, the discrete supercurvature mode, and the bubble wall mode to the angular power spectrum of temperature fluctuations in open inflation. We derive bounds on the parameters of the model so that the predicted spectrum is compatible with the observed anisotropy of the microwave background.

Complete power spectrum for an induced gravity open inflation model

We study the phenomenological constraints on a recently proposed model of open inflation in the context of induced gravity. The main interest of this model is the relatively small number of parameters, which may be constrained by many different types of observation. We evaluate the complete spectrum of density perturbations, which contains continuum sub-curvature modes, a discrete super curvature mode, and a mode associated with fluctuations in the bubble wall. From these, we compute the angular power spectrum of temperature fluctuations in the microwave background, and derive bounds on the parameters of the model so that the predicted spectrum is compatible with the observed anisotropy of the microwave background and with large-scale structure observations. We analyze the matter era and the approach of the model to general relativity. The model passes all existing constraints.

Normalization of modes in an open universe

We discuss the appropriate normalization of modes required to generate a homogeneous random field in an open Friedmann-Robertson-Walker universe. We consider scalar random fields and certain tensor random fields that can be obtained by covariantly differentiating a scalar. Modes of interest fall into three categories: the familiar sub-curvature modes, the more recently discussed super-curvature modes, and a set of discrete modes with positive eigenvalues which can be used to generate homogeneous tensor random fields even though the underlying scalar field is not homogeneous. A particular example of the last case which has been discussed in the literature is the bubble wall fluctuation in open inflationary universes.

Open inflationary universes in the induced gravity theory

The induced gravity theory is a variant of Jordan--Brans--Dicke theory where the `dilaton' field possesses a potential. It has the unusual feature that in the presence of a false vacuum there is a {\em stable} static solution with the dilaton field displaced from the minimum of its potential, giving perfect de Sitter expansion. We demonstrate how this solution can be used to implement the open inflationary universe scenario. The necessary second phase of inflation after false vacuum decay by bubble nucleation is driven by the dilaton rolling from the static point to the minimum of its potential. Because the static solution is stable whilst the false vacuum persists, the required evolution occurs for a wide range of initial conditions. As the exterior of the bubble is perfect de Sitter space, there is no problem with fields rolling outside the bubble, as in one of the related models considered by Linde and Mezhlumian, and the expansion rates before and after tunnelling may be similar which prevents problematic high-amplitude super-curvature modes from being generated. Once normalized to the microwave background anisotropies seen by the COBE satellite, the viable models form a one-parameter family for each possible $\Omega_0$.

The Spectrum of Gravitational Wave Perturbations in the One-Bubble Open Inflationary Universe

We give the initial spectrum of quantized gravitational waves in the context of the one-bubble open inflationary universe scenario. In determining the quantum state after the bubble nucleation we adopt the prescription to require the analyticity of positive frequency functions in half of the Euclidean extension of the background O(3,1)-symmetric spacetime. We find the spectrum is well behaved at the infrared limit and there appears no supercurvature mode. In the thin wall approximation, the explicit form of the spectrum of gravitational wave perturbations is calculated.

Quantum fluctuations and CMB anisotropies in one-bubble open inflation models

We first develop a method to calculate a complete set of mode functions that describe the quantum fluctuations generated in one-bubble open inflation models. We consider two classes of models. One is a single scalar field model proposed by Bucher, Goldhaber, and Turok and by us as an example of the open inflation scenario, and the other is a two-field model such as the supernatural inflation proposed by Linde and Mezhlumian. In both cases we assume the difference in the vacuum energy density between inside and outside the bubble is negligible. There are two kinds of mode functions. One kind has the usual continuous spectrum and the other has a discrete spectrum with characteristic wavelengths exceeding the spatial curvature scale. The latter can be further divided into two classes in terms of its origin. One is called the de Sitter supercurvature mode, which arises due to the global spacetime structure of de Sitter space, and the other is due to fluctuations of the bubble wall. We calculate the spectrum of quantum fluctuations in these models and evaluate the resulting large angular scale CMB anisotropies. We find there are ranges of model parameters that are consistent with observed CMB anisotropies.