Nonlinear Gravitational Dynamics Research Articles

Preheating in full general relativity

We investigate the importance of local gravity during preheating, the non-linear dynamics that may be responsible for starting the process of reheating the universe after inflation. We introduce three numerical methods that study a simple preheating scenario while relaxing gravitational assumptions, culminating in studying the process in full numerical relativity. We confirm that perturbation theory is no longer valid when one considers modes whose wavelengths are comparable to the size of the horizon at the end of inflation; however, this breakdown does not necessarily lead to a breakdown of the preheating process in non-linear gravity. For the specific model we test we find no evidence for the creation of primordial black holes from the instabilities in this model. Finally, we remark on the opportunity for future numerical study of non-linear gravitational dynamics in the early universe.

Semiclassical path to cosmic large-scale structure

We chart a path toward solving for the nonlinear gravitational dynamics of cold dark matter by relying on a semiclassical description using the propagator. The evolution of the propagator is given by a Schr\"odinger equation, where the small parameter $\hbar$ acts as a softening scale that regulates singularities at shell-crossing. The leading-order propagator, called free propagator, is the semiclassical equivalent of the Zel'dovich approximation (ZA), that describes inertial particle motion along straight trajectories. At next-to-leading order, we solve for the propagator perturbatively and obtain, in the classical limit the displacement field from second-order Lagrangian perturbation theory (LPT). The associated velocity naturally includes an additional term that would be considered as third order in LPT. We show that this term is actually needed to preserve the underlying Hamiltonian structure, and ignoring it could lead to the spurious excitation of vorticity in certain implementations of second-order LPT. We show that for sufficiently small $\hbar$ the corresponding propagator solutions closely resemble LPT, with the additions that spurious vorticity is avoided and the dynamics at shell-crossing is regularised. Our analytical results possess a symplectic structure that allows us to advance numerical schemes for the large-scale structure. For times shortly after shell-crossing, we explore the generation of vorticity, which in our method does not involve any explicit multi-stream averaging, but instead arises naturally as a conserved topological charge.

Lagrangian theory for cosmic structure formation with vorticity: Newtonian and post-Friedmann approximations

We study the nonlinear gravitational dynamics of a universe filled with a pressureless fluid and a cosmological constant $\Lambda$ in the context of Newtonian gravity, and in the relativistic post-Friedmann approach proposed in paper I [I. Milillo et al., Phys. Rev. D 92, 023519 (2015).]. The post-Friedmann approximation scheme is based on the $1/c$ expansion of the space-time metric and the energy-momentum tensor, and includes nonlinear Newtonian cosmology. Here we establish the nonlinear post-Friedmann framework in the Lagrangian-coordinates approach for structure formation. For this we first identify a Lagrangian gauge which is suitable for incorporating nonzero vorticity. We analyze our results in two limits: at the leading order we recover the fully nonlinear Newtonian cosmological equations in the Lagrangian formulation, and we provide a space-time metric consistent from the perspective of general relativity. We then linearize our expressions and recover the relativistic results at first order in cosmological perturbation theory. Therefore, the introduced approximation scheme provides a unified treatment for the two leading-order regimes, from the small scales described by Newtonian gravity to the large linear scale, where first-order relativistic cosmological perturbation theory gives a very good description of structure formation.

On separate universes

The separate universe conjecture states that in General Relativity a density perturbation behaves locally (i.e. on scales much smaller than the wavelength of the mode) as a separate universe with different background density and curvature. We prove this conjecture for a spherical compensated tophat density perturbation of arbitrary amplitude and radius in ΛCDM. We then use Conformal Fermi Coordinates to generalize this result to scalar perturbations of arbitrary configuration and scale in a general cosmology with a mixture of fluids, but to linear order in perturbations. In this case, the separate universe conjecture holds for the isotropic part of the perturbations. The anisotropic part on the other hand is exactly captured by a tidal field in the Newtonian form. We show that the separate universe picture is restricted to scales larger than the sound horizons of all fluid components. We then derive an expression for the locally measured matter bispectrum induced by a long-wavelength mode of arbitrary wavelength, a new result which in standard perturbation theory is equivalent to a relativistic second-order calculation. We show that nonlinear gravitational dynamics does not generate observable contributions that scale like local-type non-Gaussianity flocNL, and hence does not contribute to a scale-dependent galaxy bias Δ b ∝ k−2 on large scales; rather, the locally measurable long-short mode coupling assumes a form essentially identical to subhorizon perturbation theory results, once the long-mode density perturbation is replaced by the synchronous-comoving gauge density perturbation. Apparent flocNL-type contributions arise through projection effects on photon propagation, which depend on the specific large-scale structure tracer and observable considered, and are in principle distinguishable from the local mode coupling induced by gravity. We conclude that any observation of flocNL beyond these projection effects signals a departure from standard single-clock inflation.

Coupled oscillator model for nonlinear gravitational perturbations

Motivated by the gravity/fluid correspondence, we introduce a new method for characterizing nonlinear gravitational interactions. Namely we map the nonlinear perturbative form of the Einstein equation to the equations of motion of a collection of nonlinearly-coupled harmonic oscillators. These oscillators correspond to the quasinormal or normal modes of the background spacetime. We demonstrate the mechanics and the utility of this formalism within the context of perturbed asymptotically anti-de Sitter black brane spacetimes. We confirm in this case that the boundary fluid dynamics are equivalent to those of the hydrodynamic quasinormal modes of the bulk spacetime. We expect this formalism to remain valid in more general spacetimes, including those without a fluid dual. In other words, although borne out of the gravity/fluid correspondence, the formalism is fully independent and it has a much wider range of applicability. In particular, as this formalism inspires an especially transparent physical intuition, we expect its introduction to simplify the often highly technical analytical exploration of nonlinear gravitational dynamics.

General relativistic corrections and non-Gaussianity in large-scale structure

General relativistic cosmology cannot be reduced to linear relativistic perturbations superposed on an isotropic and homogeneous (Friedmann–Robertson–Walker) background, even though such a simple scheme has been successfully applied to analyse a large variety of phenomena (such as cosmic microwave background primary anisotropies, matter clustering on large scales, weak gravitational lensing, etc). The general idea of going beyond this simple paradigm is what characterizes most of the efforts made in recent years: the study of second and higher-order cosmological perturbations including all general relativistic contributions—also in connection with primordial non-Gaussianities—the idea of defining large-scale structure observables directly from a general relativistic perspective, the various attempts to go beyond the Newtonian approximation in the study of nonlinear gravitational dynamics, by using e.g., post-Newtonian treatments, are all examples of this general trend. Here we summarize some of these directions of investigation, with the aim of emphasizing future prospects in this area of cosmology, both from a theoretical and observational point of view.

Cosmological dynamics: from the Eulerian to the Lagrangian frame. Part I. Newtonian approximation

We analyse the non-linear gravitational dynamics of a pressure-less fluid in the Newtonian limit of General Relativityin both the Eulerian and Lagrangian pictures. Starting from the Newtonian metric in the Poisson gauge, we transform to the synchronousand comoving gauge and obtain the Lagrangian metric within the Newtonian approximation. Our approach is fully non-perturbative,which implies that if our quantities are expanded according to the rules of standard perturbation theory,all terms are exactly recovered at any order in perturbation theory, only provided they are Newtonian.We explicitly show this result up to second order and in both gauges. Our transformation clarifies the meaning of the change of spatialand time coordinates from the Eulerian to the Lagrangian frame in the Newtonian approximation.

The dynamics of brane-world cosmological models

Brane-world cosmology is motivated by recent developments in string/M-theory and offers a new perspective on the hierarchy problem. In the brane-world scenario, our Universe is a four-dimensional subspace or brane embedded in a higher-dimensional bulk spacetime. Ordinary matter fields are confined to the brane while the gravitational field can also propagate in the bulk, and it is not necessary for the extra dimensions to be small, or even compact, leading to modifications of Einstein's theory of general relativity at high energies. In particular, the Randall–Sundrum-type models are relatively simple phenomenological models that capture some of the essential features of the dimensional reduction of eleven-dimensional supergravity introduced by Hořava and Witten. These curved (or warped) models are self-consistent and simple and allow for an investigation of the essential nonlinear gravitational dynamics. The governing field equations induced on the brane differ from the general relativistic equations in that there are nonlocal effects from the free gravitational field in the bulk, transmitted via the projection of the bulk Weyl tensor, and the local quadratic energy-momentum corrections, which are significant in the high-energy regime close to the initial singularity. In this review, we investigate the dynamics of the five-dimensional warped Randall–Sundrum brane worlds and their generalizations, with particular emphasis on whether the currently observed high degree of homogeneity and isotropy can be explained. In particular, we discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models containing both a perfect fluid and a scalar field close to the initial singularity. Using dynamical systems techniques, it is found that, for models with a physically relevant equation of state, an isotropic singularity is a past-attractor in all orthogonal spatially homogeneous models (including Bianchi type IX models). In addition, we describe the dynamics in a class of inhomogeneous brane-world models, and show that these models also have an isotropic initial singularity. These results provide support for the conjecture that typically the initial cosmological singularity is isotropic in brane-world cosmology. Consequently, we argue that, unlike the situation in general relativity, brane-world cosmological models may offer a plausible solution to the initial conditions problem in cosmology. PACS Nos.: 98.89.Cq/Jk, 04.20–q

Nonlinear dynamics of the large-scale structure in the universe

This paper briefly reviews the nonlinear gravitational dynamics in an expanding universe. The gravitational instability is believed to play a key role in the formation of the large-scale structure of the universe, amplifying primordial small-amplitude random density fluctuations of Gaussian type and then, at the nonlinear stage, transforming them into thin dense pancakes, filaments, and compact clumps of matter. Although there are evidences that the distribution of galaxies and the distribution of the underlying mass density are not quite the same, it is widely accepted that they become similar when smoothed with a window of a sufficiently large size. At present the formation of galaxies themselves is very poorly understood and is not discussed here. We discuss scales in the range roughly from 1 Mpc to 100 Mpc (≈ 3 × 10 24 - 3 × 10 26 cm). The lower limit is at least 50 times greater than the size of the luminous part of a large galaxy, and the upper limit is at least 30 times smaller than the size of the observed universe. The problem of the formation of the large-scale structure is one of the key problems in modern cosmology and has too many aspects to mention in a short paper. Instead I review only the nonlinear evolution of density inhomogeneities on scales in question, which occured relatively recently. The paper discusses mainly analytical and semianalytical methods used in examining the nonlinear gravitational dynamics of the large-scale structure, though many hints were found in N-body simulations which have played a very important role in confronting theoretical models with observations.