Cosmological perturbations

The scalar and Einstein field equations are considered in the general symmetric comoving system. The self-gravitational interaction of the scalar field is expressed by identifying the energy momentum tensor with the one canonically associated to the scalar field. It is shown that the scalar field can depend only on the radial and time variables and that the energy momentum tensor is implicitly conserved. Two classes of solutions are then studied. A first class, that contains the static case previously considered and the complete time-dependent case, is determined by a priori assuming the unknown functions to depend on one only variable. This class is meaningful essentially for massless field. A second class of solutions refers to the a priori assumption that the scalar field alone depends on r or on t. In both cases some comments are developed for the massive scalar field and for a large class of massless fields we are able to determine the solutions.

Low energy effective field theories motivated by string theory will likely contain several scalar moduli fields which will be relevant to early Universe cosmology. Some of these fields are expected to couple with non-standard kinetic terms to gravity. In this paper, we study the splitting into adiabatic and isocurvature perturbations for a model with two scalar fields, one of which has a non-standard kinetic term in the Einstein-frame action. Such actions can arise, e.g., in the Pre-Big-Bang and Ekpyrotic scenarios. The presence of a non-standard kinetic term induces a new coupling between adiabatic and isocurvature perturbations which is non-vanishing when the potential for the matter fields is nonzero. This coupling is un-suppressed in the long wavelength limit and thus can lead to an important transfer of power from the entropy to the adiabatic mode on super-Hubble scales. We apply the formalism to the case of a previously found exact solution with an exponential potential and study the resulting mixing of adiabatic and isocurvature fluctuations in this example. We also discuss the possible relevance of the extra coupling in the perturbation equations for the process of generating an adiabatic component of the fluctuations spectrum from isocurvature perturbations without considering a later decay of the isocurvature component.

To explain the accelerated expansion of late universe, the 1/R correction to Einstein gravity is usually considered, where R is the Ricci scalar. This correction term is generally believed to be negligible in the early universe. However, if the 1/R term is inflaton-dependent, it will dramatically change the story of inflation. The entropy perturbation will naturally appear and drive the evolution of curvature perturbation outside the Hubble horizon. In a large class of models, the entropy perturbation can be made nearly scale-invariant. In Einstein gravity the single-field inflation with a quartic potential has been ruled out by recent observations, but it revives when the 1/R term is turned on. The evolution of non-Gaussianities on large scale are also studied and applied to inflation with 1/R correction. In some specific models, a large non-Gaussianity can be naturally generated outside the horizon.

We study the background dynamics and primordial perturbation in $\alpha$-attractor-type double inflation. The model is composed of two minimally coupled scalar fields, where each of the fields has a potential of the $\alpha$-attractor type. We find that the background trajectory in this double inflation model is given by two almost straight lines connected with a turn in the field space. With this simple background, we can classify the property of the perturbations generated in this double inflation model just depending on whether the turn occurs before the horizon exit for the mode corresponding to the scale of interest or not. In the former case, the resultant primordial curvature perturbation coincides with the one obtained by the single-field $\alpha$-attractor. On the other hand, in the latter case, it is affected by the multifield effects, like the mixing with the isocurvature perturbation and the change of the Hubble expansion rate at the horizon exit. We obtain the approximated analytic solutions for the background, with which we can calculate the primordial curvature perturbation by the $\delta N$ formalism analytically, even when the multifield effects are significant. We also impose observational constraints on the model parameters and initial values of the fields in this double inflation model based on the Planck result.

This paper describes CosmoLattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoLattice incorporates a series of features that makes it very versatile and powerful: i) it is written in C++ fully exploiting the object oriented programming paradigm, with a modular structure and a clear separation between the physics and the technical details, ii) it is MPI-based and uses a discrete Fourier transform parallelized in multiple spatial dimensions, which makes it specially appropriate for probing scenarios with well-separated scales, running very high resolution simulations, or simply very long ones, iii) it introduces its own symbolic language, defining field variables and operations over them, so that one can introduce differential equations and operators in a manner as close as possible to the continuum, iv) it includes a library of numerical algorithms, ranging from O(δt2) to O(δt10) methods, suitable for simulating global and gauge theories in an expanding grid, including the case of ‘self-consistent’ expansion sourced by the fields themselves. Relevant observables are provided for each algorithm (e.g. energy densities, field spectra, lattice snapshots) and we note that, remarkably, all our algorithms for gauge theories (Abelian or non-Abelian) always respect the Gauss constraint to machine precision. Program summaryProgram Title::CosmoLatticeCPC Library link to program files:https://doi.org/10.17632/44vr5xssc6.1Developer's repository link:http://github.com/cosmolattice/cosmolatticeLicensing provisions: MITProgramming language: C++, MPINature of problem: The phenomenology of high energy physics in the early universe is typically characterized by non-linear dynamics, which cannot be captured accurately with analytical techniques. In order to fully understand the non-linearities developed in a given scenario, one needs to carry out lattice simulations. A number of public packages for lattice simulations have appeared over the years, but most of them are only capable of simulating scalar fields. However, realistic models of particle physics do contain other kind of field species, such as (Abelian or non-Abelian) gauge fields, whose non-linear dynamics can also play a relevant role in the early universe. Tensor modes representing gravitational waves are also naturally expected in many scenarios.Solution method:CosmoLattice represents a modern code for lattice simulations of scalar-gauge field theories in an expanding universe. It allows for the simulation of the evolution of interacting (singlet) scalar fields, charged scalar fields under U(1) and/or SU(2) gauge groups, and the corresponding associated Abelian and/or non-Abelian gauge fields. From version 1.1 onward, CosmoLattice also allows to simulate the production of gravitational waves. Simulations can be done either in a flat space-time background, or in a homogeneous and isotropic (spatially flat) expanding FLRW background. CosmoLattice provides symplectic integrators, with accuracy ranging from O(δt2) up to O(δt10), to simulate the non-linear dynamics of the appropriate fields in comoving three-dimensional lattices. The code is parallelized with MPI, and uses a discrete Fourier Transform parallelized in multiple spatial dimensions, which makes it a very powerful code for probing physical problems with well-separated scales. Moreover, the code has been designed as a ‘platform’ to implement any system of dynamical equations suitable for discretization on a lattice.

The equations of relativistic, perfect-fluid hydrodynamics are cast in Eulerian form using six scalar fields, each of which has an equation of evolution. These equations determine the motion of the fluid through the equation Uʋ=µ-1 (o,ʋ + αβ,ʋ + ƟS,ʋ). Einstein's equations and the velocity-potential hydrodynamical equations follow from a variational principle whose action is I = (R + 16π p) (-g)1/2 d4x, where R is the scalar curvature of spacetime and p is the pressure of the fluid. These equations are also cast into Hamiltonian form, with Hamiltonian density –T00 (-goo)-1/2. The second variation of the action is used as the Lagrangian governing the evolution of small perturbations of differentially rotating stellar models. In Newtonian gravity this leads to linear dynamical stability criteria already known. In general relativity it leads to a new sufficient condition for the stability of such models against arbitrary perturbations. By introducing three scalar fields defined by ρ ᵴ = ∇ λ + ∇ x(x i + ∇ xɣ i ) (where ᵴ is the vector displacement of the perturbed fluid element, ρ is the mass-density, and i , is an arbitrary vector), the Newtonian stability criteria are greatly simplified for the purpose of practical applications. The relativistic stability criterion is not yet in a form that permits practical calculations, but ways to place it in such a form are discussed.

Current data are in good agreement with the predictions of single field inflation. However, the hemispherical asymmetry, seen in the cosmic microwave background data, may hint at a potential problem. Generalizing to multi-field models may provide one possible explanation. A useful way of modeling perturbations in multi-field inflation is to investigate the projection of the perturbation along and perpendicular to the background fields' trajectory. These correspond to the adiabatic and isocurvature perturbations. However, it is important to note that in general there are no corresponding adiabatic and isocurvature fields. The purpose of this article is to highlight the distinction between a field redefinition and a perturbation projection. We provide a detailed derivation of the evolution of the isocurvature perturbation to show that no assumption of an adiabatic or isocurvature field is needed. We also show how this evolution equation is consistent with the field covariant evolution equations for the adiabatic perturbation in the flat field space limit.

We study adiabatic ~curvature! and entropy ~isocurvature! perturbations produced during a period of cosmological inflation that is driven by multiple scalar fields with an arbitrary interaction potential. A local rotation in field space is performed to separate out the adiabatic and entropy modes. The resulting field equations show explicitly how on large scales entropy perturbations can source adiabatic perturbations if the background solution follows a curved trajectory in field space, and how adiabatic perturbations cannot source entropy perturbations in the long-wavelength limit. It is the effective mass of the entropy field that determines the amplitude of entropy perturbations during inflation. We present two applications of the equations. First, we show why one in general expects the adiabatic and entropy perturbations to be correlated at the end of inflation, and calculate the cross correlation in the context of a double inflation model with two non-interacting fields. Second, we consider two-field preheating after inflation, examining conditions under which entropy perturbations can alter the large-scale curvature perturbation and showing how our new formalism has advantages in numerical stability when the background solution follows a non-trivial trajectory in field space.

This is the third in a series of papers [J. Math. Phys. 40, 3978 (1999); 40, 3995 (1999)], the overall objective of which is the demonstration that a set of 26 gauge-invariant variables, denoted collectively by D and referred to as the complete set of basic variables, can be used to describe the equivalence classes of perturbations in a Bianchi type I or type V universe filled with a nonbarotropic perfect fluid. The object here is the derivation of a full system of propagation and constraint equations for these basic variables. We show that the constraint equations, which involve only the spatial derivatives of D, are preserved in time along the unperturbed fluid flow lines, i.e., that the time derivative of each constraint equation is identically satisfied as a consequence of the other equations that hold. Let us put things another way. What we prove is the statement that if the constraints in our system are satisfied at one time and the evolution equations are satisfied at all times, then the constraints are satisfied at all times. A further important point is simply this. When the linearized field equations of Einstein’s gravity theory are re-expressed in a manifestly gauge-invariant form, an open set of equations is obtained for D since there are too many unknowns. Thus this set must be suitably closed by means of accurate “closure” relations. In order to find them, we observe that the definition of basic gauge-invariant variables gives rise to additional geometrical identities from which an exact method of closure can be determined. Our formalism turns out to be especially appropriate for handling the linearized perturbations in a Bianchi type V universe model where the standard approaches conceptually break down.

We study adiabatic and isocurvature perturbation spectra produced by a period of cosmological inflation driven by two scalar fields. We show that there exists a model-independent consistency condition for all two-field models of slow-roll inflation, despite allowing for model-dependent linear processing of curvature and isocurvature perturbations during and after inflation on super-horizon scales. The scale dependence of all spectra are determined solely in terms of slow-roll parameters during inflation and the dimensionless cross-correlation between curvature and isocurvature perturbations. We present additional model-dependent consistency relations that may be derived in specific two-field models, such as the curvaton scenario.

Evolution equation for the expectation value of a scalar field in spatially flat RW universes

The generalized f(R) gravity with coupling in five-dimensional (5D) spacetime is studied on the basis of both the 4D generalized f(R) gravity with coupling and the pure f(R) gravity without coupling in 5D spacetime. Specifically, by assuming a hypersurface-orthogonal space-like Killing vector field in 5D spacetime, the generalization of the 4D generalized f(R) gravity to 5D can be realized, which can give the reduced 4-metric coupled with two scalar fields. In particular, we discuss a special class of models, i.e., f1(R) = f2(R) = αRm (m ≠ 1), and choose B(Lm) = Lm = −ρ in the homogeneous and isotropic cosmology with the 4D Friedmann—Robertson—Walker metric. The numerical analysis shows that the parameter m can be constrained by means of the current observations for the deceleration parameter, which implies that this generalized f(R) model with coupling in 5D spacetime can account for the present accelerated expansion of the universe.

We study the cosmological sector of the Lorentzian Barrett-Crane (BC) model coupled to a free massless scalar field in its Group Field Theory (GFT) formulation, corresponding to the mean-field hydrodynamics obtained from coherent condensate states. The relational evolution of the condensate with respect to the scalar field yields effective dynamics of homogeneous and isotropic cosmologies, similar to those previously obtained in SU(2)-based EPRL-like models. Also in this manifestly Lorentzian setting, in which only continuous SL(2,ℂ)-representations are used, we obtain generalized Friedmann equations that generically exhibit a quantum bounce, and can reproduce all of the features of the cosmological dynamics of EPRL-like models. This lends support to the expectation that the EPRL-like and BC models may lie in the same continuum universality class, and that the quantum gravity mechanism producing effective bouncing scenarios may not depend directly on the discretization of geometric observables.

Using the 3rd quantization formalism we study the quantum entanglement of universes created in pairs within the framework of standard homogeneous and isotropic cosmology. In particular, we investigate entanglement quantities (entropy, temperature) around maxima, minima and inflection points of the classical evolution. The novelty from previous works is that we show how the entanglement changes in an extended minisuperspace parameterised by the scale factor and additionally, by the massless scalar field. We study the entanglement quantities for the universes which classically exhibit Big-Bang and other than Big-Bang (exotic) singularities such as Big-Brake, Big-Freeze, Big-Separation, and Little-Rip. While taking into account the scalar field, we find that the entanglement entropy is finite at the Big-Bang singularity and diverges at maxima or minima of expansion. As for the exotic singularity models we find that the entanglement entropy or the temperature in all the critical points and singularities is either finite or infinite, but it never vanishes. This shows that each of the universes of a pair is entangled to a degree parametrized by the entanglement quantities which measure the quantumness of the system. Apart from the von Neumann entanglement entropy, we also check the behaviour of the the Tsallis and the Renyi entanglement entropies, and find that they behave similarly as the meters of the quantumness. Finally, we find that the best-fit relation between the entanglement entropy and the Hubble parameter (which classically marks special points of the universe evolution) is of the logarithmic shape, and not polynomial, as one could initially expect.

Exact general solutions for spatially flat isotropic and homogeneous cosmologies with a nonminimally coupled ghost scalar field that has polynomial potentials of the fourth degree are obtained in the framework of the Einstein–Cartan theory (ECT) and general relativity (GR). The special values of coupling constant \(\xi \) and restrictions on \(\xi \) are found for the above solutions. Some effects of torsion and scalar field potential are elucidated. It is shown that solutions can describe the bouncing models with the late-time accelerated expansion. It is demonstrated that some models admit the unified scenario for dark matter and dark energy: (i) both singular models in ECT with a de Sitter-like asymptotic and with the power-law \((t^{4/3})\) asymptotic at late times, (ii) singular and nonsingular models in GR with a de Sitter-like asymptotic at late times.